Logics for Analyzing Games

First published Mon Mar 4, 2019; substantive revision Mon Dec 19, 2022

In light of logic’s historical roots in dialogue andargumentation, games and logic are a natural fit. Argumentation is agame-like activity that involves taking turns, saying the right thingsat the right time, and, in competitive settings, has clear pay-offs interms of winning and losing. Pursuing this connection, specializedlogic games have already been used in the middle ages as a tool forlogic training (Hamblin 1970). Themodern area augmented this picture with formal dialogue games asfoundation for logic, relating winning strategies in argumentation tocogent proofs (Kamlah 1973 [1984]).Today, connections between logic and game theory span across a greatnumber of different strands, involving the interface with game theory,but also linguistics, computer science, and further fields.

Themes from the extensive and growing area surrounding logic and gamesoccur in various entries of this Encyclopedia, in particular onuses of games in logic,epistemic foundations of game theory,formal approaches to social procedures,logics for analyzing powers of agents, andgame semantics for programming and process languages. These entries differ in their emphasis, which may be on logic, gametheory, or foundations of computer science. The present entry isconcerned with logics for analyzing games, broadly speaking. It makesreference to other perspectives in this Encyclopedia whererelevant.

1. Overview

The present entry provides a comprehensive survey of logics foranalyzing games, arranged under a number of unifying themes andperspectives. Also, occasional connections are made with other strandsat the interface of logic and games covered elsewhere in thisEncyclopedia. This overview section is a brieftourd’horizon for topics that will return in more detail lateron.

1.1 Logic of Games

Specific games stand for significant recurrent patterns of socialinteraction. In a perspective called ‘logic of games’,notions and results from logic are used to analyze the structure ofvarious games. In fact, much classic reasoning about games involvesnotions that are familiar from logic.

Example Game solution reasoning.

Consider the following game tree for two playersA,E,with turns marked, and with pay-offs written with the value forA first. Alternatively, these values can be interpreted asencoding players’ qualitative (or ordinal) preferences betweenoutcomes.

This is a game tree diagram illustrating what is described in the paragraph following the figure. The extended description (link in figure caption) will describe the tree.

Figure 1.ⓘ

Here is how players might reason. At her turn,E faces astandard decision problem, with two available actions and the outcomeof actionleft better for her than that ofright. Soshe will choose left. Knowing this,A expects that his choosingright will give him outcome 0, while goingleftgives him outcome 1, so he chooses left. As a result, both players areworse off than they would have been, had they playedright/right. The reasoning in this scenario, inshort, leads to an outcome that is not Pareto-optimal.

The example raises the question just why players should act this way,and whether, say, a more cooperative behavior could also be justified.An answer obviously depends on the players’ information andstyle of reasoning. Here it becomes of interest to probe the structureof the example. Looking more closely, many notions are involved in theabove scenario: actions and their results, players knowledge about thestructure of the game, their preferences about its results, but alsohow they believe the game will proceed. There are even counterfactualconditionals in the background, such asA’s explaininghis choice afterwards by saying that “if I had playedright,E would have playedleft”. Thesenotions, moreover, are entangled in subtle ways. For instance,A does not chooseleft because it dominatesright in the standard sense of always being better for him,but rather becauseleft dominatesright according tohis beliefs. How these beliefs are formed, in turn, depends on manyother features of the game, including the nature of the players.

In short, even a very simple game like the one discussed bringstogether large parts of the agenda of philosophical logic in one veryconcrete setting. This entry will zoom in on the aspects mentionedhere, withSection 3 dedicated to players’ preferences and beliefs, whileSection 4 addresses reasoning styles and the dynamics of attitudes as the gameproceeds.

The analysis is structured by a few broad distinctions. Intuitively,games involve several phases that involve logic in different ways:deliberation prior to the game, as many game-theoretic solutionconcepts are in fact deliberation procedures that create initialexpectations about how a game will go on. Observation and beliefrevision during game play, including reactions to deviations fromprior expectations. And finally, post-game analysis, say, to settlewhat can be learnt from a defeat, or to engage in spin aboutone’s performance. Moreover all this can be considered in twomodes, assuming either a first-person participant or a third-personobserver view of games and play.

1.2 Logic and Game Theory

In many of the above topics, logics meets game theory. One suchinterface area isepistemic game theory where game play and solution concepts are analyzed and justified inlight of various assumptions about players and their epistemic states,such as common knowledge or common belief in rationality. Epistemicgame theory may be viewed as a joint offspring of logic and gametheory, a form of progeny which constitutes a reliable sign of successof an interdisciplinary contact.

There are also other viable logical perspectives. In particular, onecan look at game theory the way mathematical logicians look at anybranch of mathematics. Following the style of the famous ErlangenProgram, one can discuss the structures studied in that field and lookfor structural invariance relations and matching logical languages.Game theory is rich in structure, as it has several different naturalnotions of invariance. The tree format of extensive games offers adetailed view of what happens step by step as players make theirmoves, whereas the matrix format of strategic form games offers ahigh-level view that centers on outcomes. Yet other formats, to bediscussed below, focus on players’ control over the variousoutcomes. All these different levels of game structure come with theirown logical systems, as will be detailed inSection 2. Moreover, these different logics do not just provide isolatedsnapshots: they can be related in a systematic manner.

In this way, the usual logical techniques can be brought to bear. Forinstance, formal languages can express basic properties of games,while model-checking techniques can determine efficiently whetherthese hold in given concrete games (cf. Clarke,Grumberg, & Peled 1999).

Example Winning strategies.

Consider the following game tree, with move relations for bothplayers, and propositional letters \(\win_{i}\) marking winningpositions for playeri.

A game tree with extended description in the link in the caption below

Figure 2.ⓘ

Clearly, playerE has a winning strategy against playerA, i.e., a recipe that guarantees her to win, no matter whatA does. This is expressed by a modal formula capturing exactlythe right dynamics:

\[[\move_A]\langle \move_E\rangle \win_E\]

Here \([\move_A]\) is the universal modality “for all moves byplayerA”, and \(\langle \move_E\rangle\) is the existentialmodality “for some move by playerE”. This two-stepmodality- or quantifier-based response pattern is typical forstrategic powers of players in arbitrary games, as it captures theessence of sequential interaction. Crucially, logical laws can acquiregame-theoretic import. For instance, the law of Excluded Middleapplied to the above formula yields:

\[{[\move_A]\langle \move_E\rangle \win_E} \lor {\neg[\move_A]\langle \move_E\rangle \win_E}\]

or in a logically equivalent formulation:

\[ {[\move_A]\langle \move_E\rangle \win_E} \lor {\langle \move_A\rangle [\move_E]\neg \win_E}\]

In two-step games like the above, where exactly one player wins (i.e.,\(\win_A\leftrightarrow\neg\win_E\)), the latter formula expressesthat either playerE or playerA has a winning strategy.More generally, this disjunctive assertion is a special case ofZermelo’s theorem, stating that every finite full informationgame is determined.

Having established the connection to logical languages, furthermodel-theoretic themes can be applied fruitfully to games. Languagebased reasoning allows, for instance, to examine the preservation ofproperties between different games, based on the exact syntactic shapeof their definition. Besides, logical syntax also supports logicalproof theory. Hence, the latter’s rich pool of proof calculi mayhelp to analyze basic results in game theory. This entry illustratesmajor recurring patterns of reasoning about interaction that come tolight in this way.

Game theory also has a further natural level of representation,suppressing details of local moves and choices. The most familiarformat for this are games instrategic form. In the simplestcase of only two players, these correspond to a two-dimensionalmatrix, with rows standing for some player’s strategies, andcolumns for the other’s. Individual cells of such matrix hencecorrespond to the different possiblestrategy profiles of thegame. Typically, all cells are labeled with information about theoutcomes resulting from playing the corresponding strategiesagainst one another. This labeling specifies players’ attitudesto outcome in terms of pay-offs, more abstract utilities, or ordinalmarkers for players’ preferences orders among outcomes.

Strategic form games, too, can model significant social scenarios.Here is an illustration from the philosophical literature on theevolution of social behavior.

Example The following game in matrix form is theStag Hunt of Skyrms (2003),going back to ideas of David Hume. It serves as a metaphor for thesocial contract.

		E
		H	S
A	H	1,1	1,0
A	S	0,1	2,2

Each agent must decide between pursuing their own little project,hunting a hare, or joining in a larger collective endeavor, huntingstag. The former gives a moderate but guaranteed income, no matterwhat others do. The collective endeavor, on the other hand, can onlysucceed if all contribute, in which case everybody receives a highprofit. If, however, some do not join, all contributions are lost andno contributor receives anything. In the corresponding strategic formgame, all players have to decide on what to do in parallel, withoutknowing the actions of others.

The Stag Hunt game has two pure strategy Nash equilibria:everycontributes, andnobody contributes. Which of thesestable outcomes ensues will crucially depend on the players’reasoning, their expectations about each other, and perhaps evenfurther information stemming, for instance, from pre-gamecommunication.

Clearly, analyzing strategic games involves agentive information,reasoning and expectations. All these aspects have tight connectionsto logic. Viewing outcomes as possible worlds, three relevantrelations emerge between these. Within the matrix above, relating allcells in the same row fixes a unique choice already made by the rowplayerA, while leavingE’s move completely open.In short, each horizontal row lists all possible choices of the columnplayerE whichA has to take into account. Thecorresponding modality may hence be said to describeA’sknowledge about the outcomes of the game given his choice. Stillassuming the row player’s perspective, relating cells verticallyrather than horizontally corresponds toA’s freedom ofchoice among his available strategies. Of course, one could alsoassume playerE’s perspective instead, viewing thehorizontal direction asE’s freedom of movement, whilethe vertical directions captures her epistemic uncertainty.

Thus, a bimodal logic arises for matrix games with laws such as

\[{K_{E}K_{A}\varphi} \leftrightarrow {K_{A}K_{E}\varphi},\]

capturing the grid structure of matrix games. For more than twoplayers, this logic gains some additional options and subtleties to bediscussed inSection 2.6.

The crucial third relation is that of player’s preferences amongoutcomes. These, again, have matching modalities, now taken frompreference logic (Hansson 2001). Withthe help of some auxiliary devices, the three modalities can definethe central game-theoretic notion of a Nash equilibrium (Harrenstein2004; van der Hoek & Pauly2007).

Logics for matrix games differ from those for extensive games, asgrids behave quite differently from trees in terms of complexity. Yet,both fall under the same general methodology. Towards a commonunderstanding, one might view the logic of matrix games as capturingthe basic laws of parallel, rather than sequential action.

1.3 Computation and Agency

Philosophical logic and mathematical logic are not the onlyilluminating perspectives on games. A third relevant viewpoint is thatof computational logic. In modern computation, the paradigm is nolonger single Turing machines but interacting systems of multipleprocessors. These processors may cooperate, but they might alsocompete for resources. In general, hence, it is useful to studymultiple agents engaging in computation, be it within human,artificial or mixed societies. Though doing so, games become a naturalmodel for computation, too. In fact, games are rich multi-agentsystems where agents process information, communicate, and engage inactions, all driven by their respective preferences and goals. In theconverse direction, computer science themes such as complexity andalgorithmics have entered game theory, resulting in the area of computational game theory (Nisan et al.2007). For a richer survey of computational logics of agencyand games, see van der Hoek and Pauly(2007) and Shoham and Leyton-Brown(2008). The present entry contains occasional links tocomputation. These are especially prominent for reasoning abouttemporally extended games and their strategies (Sections4.2,4.4) and in the context of gamification (Section 6), where games are explored as a novel semantics for classical logicalsystems.

1.4 Games in Logic

Finally, recall the start of this section, but with reverseperspective: instead of asking what logic can do for games, ask whatgames can do for logic. Argumentation and dialogue are basic notionsfor logic. Both can be studied using techniques and results from gametheory (Lorenzen & Lorenz 1978; Hamblin1970). In this perspective, logical validity of consequencerests on there being a winning strategy for a Proponent claiming theconclusion against an Opponent granting the premises in a game wheremoves are regulated by the logical constants. Many games have founduses in modern logic since the 1950s, withEhrenfeucht-Fraïssé games for model comparison being aparadigmatic example. Besides these, also semantic verification ormodel construction can be cast asnatural logic games.

This raises an intricate issue within in the philosophy of logic,concerning the nature of logic and in particular that of logicalconstants. A ‘weak thesis’ would hold that gamesconstitute a natural technique for analyzing logical notions, as wellas a didactic tool for teaching logic that appeals directly to vividintuitions. Parts of the literature, however, also defend a‘strong thesis’, suggesting that the primary semantics ofcertain logical systems may be procedural and game-theoretic, ratherthan denotational in a standard sense. This perspective, sometimescalled ‘logic as games’, occurs in some attractivesemantics for first-order languages (Hintikka& Sandu 1997), as well as ingame semantics for programming languages.

The theme of logic as games will appear only briefly in the presententry, which is mainly directed toward logics of games.Section 6 will discuss which questions arise from joining both perspectives onthe interface of logic and games.

As it happens, the logic-as-games perspective is of broader relevance.Logic games were originally designed for particular tasks insidelogic. Yet, taken to reality, they can help analyze or streamlineactual lines of argumentation. As such they may be compared todesigned parlor games that challenge reasoning skills. A game likeClue involves an intriguing mix of logical deduction, newinformation from drawing cards or public observation of moves, butalso private communication acts by players (vanDitmarsch 2000). Other parlor games, such asNine MenMorris (Gasser 1996) are graphgames (Grädel, Thomas, & Wilke2002) with added chance moves that serve to diminish the riskof finding a repeatable simple strategy on the fixed board. Thelogical study of playable designed games for bounded agents, and thedesign of new such games, is a natural sequel to this entry (cf.van Benthem & Liu 2019).

1.5 Probability

Game theory may be understood as generalized interactive decisiontheory. A major vehicle for the latter, just as for standard decisiontheory, is probability theory. Within games, probability can assumemany roles. It may, for instance, express players’ degrees ofbelief quantitatively, but it can also enrich the space of actionswith mixed strategies, thereby laying the ground for generalequilibrium results. Probability can even play a role in the verydefinition of certain important games, especially in evolutionary gametheory (Osborne & Rubinstein 1994).In this entry, probability is only mentioned in passing.Section 5, however, maps some combinations of logic and probability that aresuggested by the study of games.

1.6 Zooming In

Games have a natural interface with logic in all its varieties,including mathematical, philosophical, and computational logic. In onedirection of contact, logic can provide new abstract notionsunderneath game theory. Conversely, game-theoretic notions can alsoserve to enrich logical analysis. The present entry mainlyconcentrates on the first of these directions, the use of logic foranalyzing games. It does so mostly from a semantic perspective, thedominant paradigm so far in the area. Though proof-theoreticapproaches will be mentioned occasionally. The sections to followelaborate on this theme along several dimensions. Specificperspectives include logics for game structures (Section 2), logical analysis of the nature of players (Section 3) and of the process of game play (Section 4). Additional spotlights are put on the relationship between logic andprobability in the context of games (Section 5) and the endeavor of Gamification (Section 6). Each section forms a free-standing exposition, which results in someunavoidable, and perhaps useful, overlap. Throughout the exposition,some familiarity is assumed with the basic concepts of logic and gametheory. In particular, notions of game theory left unexplained herecan be found in the correspondingentry and in Leyton-Brown and Shoham(2008).

2. Game Structure and Game Logics

This first spotlight section focuses on game structures in a narrowsense.Game forms leave aside agents and notions typical forthese, such as preferences or information. Players, as well as thetemporal progression of play will be added in later sections. Even so,there is already a good deal of structure in game forms to be studiedby logical techniques.

2.1 Levels of Representation

The starting point of any logical analysis is to fix its perspectiveon games. This section will review several major candidates for doingso, starting with the two most prominent perspectives. The first ofthese makes the temporal structure of a game explicit, representing itas a tree in the standard mathematical sense.

Example A two-player extensive form game.

This is a game tree diagram illustrating the example. The extended description (link in figure caption) will describe the tree.

Figure 3.ⓘ

A game inextensive form is a tree where each non-terminalnode orstate specifies which player is to move next, whileedges correspond to the players’ possible moves. The leaves ofthe tree, finally, denote the possible outcomesO of game play.There are many possible variations on these stipulations for statesand moves, but they do not affect the essentials of a logicalanalysis.

The second major perspective on games emphasizes the players’available strategies. Suppressing all information about temporalstructure, a game instrategic form yields the matrixpictures known from game theory. In its classic interpretation, a gamein strategic form represents a set of players that each select acomplete strategy for the entire game without knowledge of the otherplayers’ choices. Each strategy profile, i.e., combination ofone strategy per player, then induces an outcome \(O_i\). Themotivation for this structure might seem complex on first sight. Yet,it can also be viewed as something quite simple: a one-step game withparallel rather than sequential moves, which is the simplest case ofsimultaneous action.

Example A two-player strategic form game.

		E
		a	b
A	c	\(O_1\)	\(O_2\)
A	d	\(O_3\)	\(O_4\)

Extensive and strategic forms differ in their focus. The formeremphasize the sequential temporal structure of a game, while thelatter highlights strategy choice prior to play. One can freely switchbetween both when appropriate to the purpose at hand. Besides thesetwo, there are other natural dimensions, highlighting players’powers for influencing outcomes (cf.Section 2.5) or players’ information about the game (cf.Section 3.6).

Remark While all examples so far concerned two-playergames, no such restriction is needed. Both extensive games andstrategic form games work for any number of players, althoughoccasional subtleties may occur. A few will be mentioned below.Moreover, with more players, possible coalitions enter the picture, atopic that will not be treated in this entry. Finally, selectedaspects of agency may sometimes enter through the back door. Manyscenarios in real life contain external chance events outside of anyplayer’s control, such as a roll of a die, weather conditions,or technical malfunctions. Such factors can usually be incorporatedinto a logical analysis by admitting Nature as additional player.

2.2 Invariance Relations Between Games

With different ways of representing a game at hand, there is a naturalfollow up question concerning equivalence. Given two game structures,when are they representations of the same underlying game? The answeris that it very much depends on what aspects one is interested in.

Example The same game, or not?

This has two game tree diagrams illustrating the example. The extended description (link in figure caption) will describe the tree.

Figure 4.ⓘ

Consider the two game forms above. If one cares about exact sequencesof moves or the choices players have along the way, these games aredifferent. The game to the left hasA move first, whileE begins in the game on the right. In the game on the right,A may face a choice betweenp andq. This cannothappen on the left.

Caring about exact moves as done here constitutes a fine-grainedperspective on games. There are others. When focusing onplayers’ powers for bringing about certain outcomes, forinstance, the analysis changes. In the game on the left,A hasa strategy (playingleft) that ensures the game to end up inan outcome satisfyingp, and one (playingright) thatrestricts possible outcomes to those satisfying \(q \lor r\). Withthis second strategy, the further choice which of \(q or r\) getsrealized is left up to playerE. Also the second player,E, has two strategies in the game on the left, one (playingleft) ensuring the outcome to satisfy \(p\lor q\), the other(playingright) guaranteeing that the outcome satisfies \(p\lor r\).

Performing the same calculations for the game on the right, virtuallythe same player powers emerge. More precisely,A’suniform strategiesleft-left andright-right yieldp and \(q\lor r\) respectively, exactly the same powers as inthe left game. The two remaining strategiesleft-right andright-left yield \(p\lor q\) and \(p\lor r\), both of whichare mere weakenings ofA’s power to achievep.Thus, at the level of players’ powers, the above two game formsshould be considered the same.

As this example illustrates, there are several legitimate ways ofcomparing games. When taking a fine-grained focus on the internalstructure of a game, a natural candidate is the notion of abisimulation (cf. Blackburn, de Rijke,& Venema 2001). A bisimulation \(Z\subseteq G_1\times G_2\)relates states of two game forms \(G_1\) and \(G_2\) subject to fourconditions: Statesm andn may only be related when(i) the same player is to move inm andn,(ii)m andn do not differ in any of theirbasic local properties, while (iiia) whenever there is anavailable move of typea in \(G_1\) leading to a state \(m'\),there is a matching available move of typea in \(G_2\) leadingto a state \(n'\) with \(m'Zn'\), and vice versa (iiib)whenever there is a move in \(G_2\) that leads to a state \(n'\),there there is a move of the same type in \(G_1\) leading to a state\(m'\) with \(m'Zn'\).

Example A bisimulation between games.

Figure 5.ⓘ

This particular notion of bisimulation is not the only invariance thatmakes sense for games. A more coarse-grained perspective, forinstance, might not distinguish moves by their particular actiontypes, but merely by which player is to perform them. A correspondingbisimulation can be defined by omitting references to particularaction types in conditions (iiia) and (iiib)above.

Further notions of bisimulation take an even coarser perspective onthe games’ move structure, for instance by allowing to contractzones where the same player moves several times in a row. Finally,dropping all information about players and their choices, games can becompared by the sequences of moves they admit. This purelyobservational notion, known astrace equivalence incomputation, may, however, be less relevant in the context of games.An alternative approach to coarsening focuses on the players’powers to control outcomes, cf.Section 2.5 and van Benthem, Bezhanishvili and Enqvist(2019a).

While most notions of invariance discussed so far related to extensiveform games, a similar style of analysis applies to games in strategicform. Van Benthem, Pacuit, and Roy(2011) define modal bisimulations that connect outcome statesof different matrices, and apply bisimulation’s back-and-forthconditions to the relevant relations of players’ choice,freedom, and preference.

This may be a good point to stress once more that the present sectionis concerned with game forms only, omitting any player related aspectsuch as preferences between outcomes. When these are added,identifying appropriate notions of invariance becomes morechallenging, as will be discussed inSection 3 below.

2.3 Languages Matching Invariance Relations

The choice of invariance relations mirrors which structure is deemedrelevant within a given perspective on games. A central tool forbringing out such relevant aspects is the existence of a logicallanguage matching some invariance relation. In general, the morefine-grained the invariance perspective, the more distinctions amatching language should be able to make.

For a start, if one is interested in the properties a player can bringabout through moves, a good choice of language is based on modalities\(\langle \move_i\rangle \varphi\), expressing that at least one ofi’s available moves leads to a next stage satisfying\(\varphi\). The following illustrates how this language works in agiven extensive form game.

Example Modal game language.

Figure 6.ⓘ

The modal formula \([\move_A]\langle \move_E\rangle \win_E\), true atrootr, expresses thatE has a strategy that ensures hera win in two steps: whateverA does,E can react in sucha way that she ends up in a node where she wins. In a morefine-grained perspective, the modal language could add expressions\([a],[b]\ldots\) for specific move types \(a,b,\ldots\). In thislanguage, the coarse-grained modality \(\langle \move_i\rangle\varphi\) is definable by the disjunction \(\bigvee_{a\text{ is a movefor }i}\langle a \rangle\varphi\), making the new language arefinement of the old.

In this way, general results of modal logic apply to games. Forinstance, take pointed models such as game trees with an indicator forthe current moment. Whenever two such pointed models \(\G,m\) and\(\G',m'\) are bisimilar in the first sense defined above, theequivalence \(\G, m\vDash\varphi\) iff \(\G', m'\vDash\varphi\) holdsfor all formulas \(\varphi\) in a sufficiently rich modal languagewith modalities \([a]\) for each move label. Thus, one can switchbetween syntactic, language based perspectives and semantic invariancerelations, depending on what is convenient for a given perspective ongames. Entirely similar points hold for bisimulations and modallanguages for power perspectives, or for strategic form games.

Finally, modal languages do not have exclusive rights. If still morefine-grained perspectives are needed, more expressive first-order orhigher-order languages become serious contenders for describinggames.

2.4 Modal Logic of Extensive Games

A language for games facilitates both, defining properties of gamesand reasoning about them. An example are winning strategies forplayers in a two-step extensive game as just discussed. Moregenerally, for any finite extensive game, there are formulas\(\varphi_j\) for each agentj that are true iffj has awinning strategy:

\[\varphi_j:=[\move_i]\langle \move_j\rangle[\move_i]\ldots \win_j\]

where the number of operators in the formula corresponds to the depthof the tree.

Thus, logical laws governing reasoning with such formulas acquiregame-theoretic content. For instance, the negation of the statementthat one player,A, has a winning strategy is provablyequivalent to saying that the other player,E, has a winningstrategy, at least in those cases whereA wins if and only ifE does not:

\[\begin{aligned}\neg \varphi_A &= \neg\langle \move_A\rangle [\move_E]\langle \move_E\rangle\ldots \win_A \\& \leftrightarrow [\move_A]\langle \move_E\rangle[\move_E]\ldots \neg \win_A\\& \leftrightarrow [\move_A]\langle \move_E\rangle[\move_E]\ldots \win_E=\varphi_E\end{aligned}\]

Hence, the logical law of excluded middle in its modal guisecorresponds to Zermelo’s theorem, stating determinacy for finitegames.

Yet, there are limitations to such characterizations of game-theoreticproperties in terms of logical laws. Formulas stating whether someplayer has a winning strategy change from model to model, as thenumber of modal operators depends on the size of the game tree. Infact, there is no uniform formula in the basic modal languageexpressing that playeri can win in an arbitrary finiteextensive form game. Such a formula can only be found in the modalμ-calculus (Venema 2008), where thestatement thati has winning strategy can be expressed with thefixed-point formula

\[\mu p. \, (\win_{i} \lor\, (\turn_i\land \langle i\rangle p)\lor\bigwedge_{j\neq i}(\turn_j\land\, [j]p)\]

The more general point here is that the recursive nature ofgame-theoretic equilibria and solution concepts reflects naturally inlogics with fixed-point operators for induction and recursion.

In this setting, known results about modal logic acquire a newsignificance. In the realm of finite models, for instance, having thesame modal formulas true at two states isequivalent to therebeing a bisimulation connecting those two states (cf.Blackburn, de Rijke, & Venema 2001).Hence, whenever two finite games satisfy the same modal propositionsin their respective roots they are equivalent in the sense ofbisimulation. For infinite models, such results are less direct. Afull equivalence between bisimulation and satisfying the sameformulas, for instance, only holds for an extended modal language withinfinite conjunctions and disjunctions. Other relevant results includethe existence of modal formulas that define given pointed models up tobisimulation. Such formulas sometimes exist in the basic modallanguage, sometimes in the μ-calculus, and always in the infinitarymodal language. Applied to concrete gamesG, these modaldefinitions can be viewed as complete descriptions of all propertiesofG at the relevant level of invariance.

Finally, modal logic has many complete proof systems for capturing thevalid consequences on various classes of models (Blackburn,de Rijke, & Venema 2001).These calculi of reasoning also apply to games, where they can captureaspects of specialized game-theoretic argumentation. Proof-theoreticperspectives are not the focus of this entry, but a number of strandswill be mentioned where appropriate.

2.5 Modal Neighborhood Logic for Powers

Besides extensive form games, standard modal logic is also suitablefor the power perspective on game structure. Sometimes, one ignoresthe internal mechanisms of a game altogether, merely viewing it as ablack box social mechanisms where players control outcomes to acertain extent. In this perspective, a player canforce theoutcome of the game to be in some setX if she commands astrategy that ensures the game to end up in an outcome ofX, nomatter what the other players do (van der Hoek& Pauly 2007). Similarly, a player can force that someproposition \(\varphi\) holds if she has the power to enforce that thegame ends in a \(\varphi\) state. The collection of all sets ofoutcomes an agent can force are often called herforcingpowers. In classical game theory, these forcing powers sometimesgo by the name ofeffectivity functions (Peleg1997), which are often also studied forcoalitions of players (see Pauly 2001; Goranko,Jamroga, & Turrini 2013; and the entry onlogics for analyzing power in normal form games).

Example Powers in extensive games.

Figure 7.ⓘ

Notably, forcing powers are not closed under conjunction. In the gameabove, agentA can forcep andq individuallywithout being able to force \(p \land q\). In modal logic terms,forcing powers give rise to aneighborhood logic (Pacuit2017), where the neighborhoodfunctions list the set of outcomes players can enforce from a givenstate. Reasoning about forcing powers can then employ a logicallanguage with forcing modalities \(\{i\}\) for each player:

\(\{i\}\varphi\): agenti can force the outcome of the game tosatisfy \(\varphi\).

These modalities can be interpreted over the extended game forms withneighborhood functions described above. On the semantic side, ageneralization of the above neighborhood models support a generalizednotion ofpower bisimulation, see vanBenthem, Pacuit, and Roy (2011).

The modal logic of powers allows to reason about games at a globallevel of description. The modal logic of neighborhood models validatesthe standard modal monotonicity principle

\[\{i\}\varphi\rightarrow\{i\}(\varphi\lor\psi),\]

as follows already from the truth definition of forcing modalities.However, as forcing powers are not closed under intersection, theaggregation law fails:

\[(\{i\}\varphi\land\{i\}\psi)\not\rightarrow\{i\}(\varphi\land\psi).\]

Instead, the logic contains new valid principles relating forcingmodalities for different players. For instance, ifi can forcethe truth of \(\varphi\), then no other playerj can force itsfalsity. Thus,

\[\{i\}\varphi\rightarrow\neg\{j\}\neg\varphi\]

is a valid principle of ‘consistency of powers’ in thelogic of forcing powers. The converse of this principle for games withtwo players \(i, j\)

\[\neg\{j\}\neg\varphi \rightarrow \{i\}\varphi\]

expresses the notion of determinacy from the last section. Thisformula is not generally valid, but is an axiom for the special classof determined games.

Finally, there also is an alternative, more algebraic perspective onpowers, assuming an earlier-mentioned perspective of logic games. Thetwo games depicted in the core example ofSection 2.2 may be seen as evaluation games for propositional formulas

\[p \land (q \lor r) \quad \text{and} \quad (p \land q) \lor (p \land r).\]

Their equivalencequa powers, described earlier, then matchesthe standard propositional law of distribution. This algebraicperspective will return inSection 2.9.

More recent views of forcing and powers re-interpret the setsXof outcomes employed in the above definitions as referring to bothplayers: one player restricts the total set of outcomes, while theother players can achieve all outcomes within that set. This balancebetween constraint and freedom significantly impacts the correspondingnotions of game equivalence, as well as the modal languages used (vanBenthem, Bezhanishvili, & Enqvist2019b).

2.6 Modal Logic for Strategic Games

In the strategic perspective on games, players select actionssimultaneously, without having learned about their opponents’choices of actions. This requires an additional level of analysis.Besides the various possible moves, an adequate representation mustalso track players’ uncertainty about how their opponents mightact.

In terms of matching logical languages, this suggest amulti-modal approach, with \([\approx_i]\) ranging overi’s possible choices, and \([\equiv_i]\) representing heruncertainty about the opponents, see vanBenthem, Pacuit, and Roy (2011). Moreover, when consideringgames rather than game forms, this picture needs to be enriched with athird feature, viz. preference modalities \([\preceq_i]\), seeSection 3.

Games in strategic form can be viewed naturally as models for a modallanguage of choice and uncertainty, where eachstatemconsists of a strategy profiles, i.e., a sequence \((m_1,m_2\ldots)\)listing each player’s choice of action. For convenience, thepreference modality has been included:

\(\G,m\vDash[\approx_i]\varphi\)	Given the opponents’ actions, \(\varphi\) holds whateveri does.
\(\G,m\vDash[\equiv_i]\varphi\)	Giveni’s choice, \(\varphi\) holds whatever theopponents do.
\(\G,m\vDash[\preceq_i]\varphi\)	\(\varphi\) holds in all states at least as good as the currentone.

This multi-modal language can express a variety of statements aboutstrategic form games, such as:

\({\langle\approx_i\rangle\langle\equiv_i\rangle} \varphi\)	\(\varphi\) is a possible outcome of the game
\({[\approx_i][\equiv_i]}\varphi\)	all outcomes of the game satisfy \(\varphi\)
\([\preceq_i]\langle\preceq_i\rangle\varphi\)	some optimal states for playeri satisfy \(\varphi\)

In the case of two players, one agent’s choices corresponds tothe other’s uncertainty and vice versa. This shows in thevalidity of principles such as

\[[\equiv_i]\varphi\leftrightarrow[\approx_j]\varphi\]

More generally, the logic of matrix games includes the \(\mathsf{S}5\)axioms for both \([\approx_i]\) and \([\equiv_i]\), but also thecommutation law

\[[\approx_i] [\equiv_i]\varphi\leftrightarrow[\equiv_i][\approx_i]\varphi\]

expressing the grid-like structure of matrix games. This logic bearssome resemblance to STIT-type logics of actions (Herzig& Lorini 2010). Technically, agrid structure in models allows for encoding of undecidablecomputational problems (Blackburn, de Rijke,& Venema 2001), rendering it an open problem whetherexpressive modal logics of game matrices are decidable.

The step from two to more players, often routine in epistemic logics,can be delicate in the logic of matrix games. Accessibility relationsof type \([\approx_i]\), interpreted as identity of profiles exceptfor the \(i^{}\)-coordinate, yield a product logic akin to thethree-variable fragment of first-order logic which is known to beundecidable (Bezhanishvili 2006).However, with only relations of identity at thei-coordinate,i.e., \([\equiv_i]\), the logic remains decidable (Venema1998; Van De Putte, Tamminga, & Duijf2017; Lomuscio, van der Meyden, & Ryan 2000).

2.7 Strategies as Logical Objects

There is further structure in extensive games than just single moves.In game trees, a player’sstrategy specifies what to doat each turn, whether this turn will ever be reached or not. Anincreasing body of work examines such strategies and their underlyingformats, see van Benthem, Ghosh, and Verbrugge(2015) for an overview of various logical frameworks forreasoning about strategies.

In one concrete perspective, a strategy is akin to a program thatinstructs the agent on how to navigate a game tree. Hence, a naturallogic of strategies uses the language ofpropositional dynamic logic of programs PDL, an approach that will return later. As programs are in generalnon-deterministic, such logics let a strategy recommend one or moreactions the agent should take at each turn. In this perspective,strategies resemble plans that might remain partial.

In a program format, strategies start with basic actions, representingindividual moves in a game tree. From there, complex programs \(\pi\)can be created using operations including sequential compositions\(\pi_{1} \,;\pi_{2}\) (\(\pi_{1}\) is to be performed followed by\(\pi_{2}\)), or choice \(\pi_{1}\, \cup_{i} \pi_{2}\) (agentiis to pick between actions \(\pi_{1}\) and \(\pi_{2}\) ). Moreover, atest operation \(?\varphi\) for checking whether \(\varphi\) holds,enables strategies to react to properties of states oropponents’ past actions. Finally, to describe continuousexecution of a strategy along a game tree, it makes sense to have anoperation \(\pi^{*}\) of program iteration, stating that \(\pi\) beexecuted arbitrarily often.

The language of PDL then has modal operators \([\pi]\) for everyprogram \(\pi\) that can be defined from the basic actions and theoperations just described. A simple such strategy advises playeri to doa whenever it is her turn. The following formulastates that this strategy ensures that \(\varphi\) holdsthroughout:

\[[((?\turn_i \mathbin{;} a)\, \cup \, (?\turn_j \mathbin{;} \move_j))^{*}]\varphi\]

Program definitions for strategies given here are closely related tothe use of finite automata for defining strategies in computer scienceand game theory (Osborne & Rubinstein 1994;Grädel, Thomas, & Wilke 2002; Ramanujam & Simon2008).

2.8 Simultaneous Moves and Imperfect Information

In the extensive form games ofSection 2.1, players move in sequence and can base their decisions on fullinformation of what has happened so far. The other extreme were gamesin strategic form, where agents move in parallel or, in theinterpretation of strategy selection, have no means of picking upinformation during actual play. There are ample scenarios in betweenthese extremes. Public good games with optional retribution againstnon-cooperators (Andrighetto et al.2013), for instance, combine moments where some or all playersmake simultaneous moves with information collection along the way.Such parallel action can be mimicked in sequential games by limitingthe information available to players at various states of the game.The resulting games of imperfect information will be discussed inSection 3, alongside other sources of imperfect information.

Further well-known logical approaches to parallel action employ STITlogic (Horty & Belnap 1995; Broersen2009), and temporal logics such as ATL (Alur,Henzinger, & Kupferman 2002) or itsepistemic variant ATEL (van der Hoek &Wooldridge 2003).

2.9 Game Algebra and Dynamic Logic of Computation

So far, games were treated as monolithic entities that agents reasonabout in their entirety. This can be at odds with how real life agentsconceptualize games. To facilitate reasoning, games are often brokenup into smaller tasks that are easier to handle separately. A chessplayer, for instance, may know how to solve different end games.Rather than reasoning about every possible situation until its end,she will evaluate different options in mid-play by considering whichof these end games they will, most likely, lead up to. In thisperspective, complex games are constructed out of simpler games thatmay profit from separate analysis. Games then form an algebra withoperations that construct complex games from simpler ones. This styleof thinking is reinforced when games are viewed as scenarios forinteractive computation, where again algebraic methods are used widely(Bergstra, Ponse, & Smolka2001).

Here is an illustration of this approach. For simplicity, consideronly two players,A andE, the latter of which startsthe game. One influential game algebra has the following operations,cf. Parikh (1985).

\(G\,\cup G'\)	AgentE has the choice between playingG and\(G'\), i.e., represented by a choice node with two outcomesGand \(G'\)
\(G\,;G'\)	G is played first, followed by \(G'\)
\((\cdot)^d\)	The roles of the playersA andE areinterchanged
\(?\varphi\)	Test game whether some property \(\varphi\) holds.

For instance, take a chess player in mid-game reasoning. Forsimplicity, restrict the possible end games to \(G_{F_1}\) and\(G_{F_2}\). The player can then conceptualize mid-play as a game\(G_{mid}\) with end nodes labeled by propositions \(p_1\) or \(p_2\),describing which of the two end games follows. The full remainingchess tree is then given by

\[G_{\textit{complete}}=G_{\textit{mid}};((?p_1;G_{F_1})\cup (?p_2;G_{F_2}))\]

Equational axiomatizations for this game algebra can be found in Goranko(2003) and Venema(2003). However, following the analogyof propositional dynamic logic for an algebra of programs, there alsois adynamic game logic for this algebra of games, (Parikh1985). It adds a modality\(\{G\}\varphi\) for each gameG, with \(\{G\}\varphi\)expressing that in gameG, the first playerE has astrategy to force the truth of \(\varphi\). For the case ofnon-determined games, the language will be extended further to includeseparate modalities \(\{G, i\}\varphi\), one for each playeri.Dynamic game logic shows in a perspicuous manner how strategicabilities for complex games supervene on abilities in simpler games.This is done by means of reduction laws such as

\[\{G;G'\}\varphi\leftrightarrow \{G\}\{G'\}\varphi, \quad \{G\cup G'\}\varphi\leftrightarrow\{G\}\varphi\lor\{G'\}\varphi\]

For a complete list of reduction laws, as well as open problems inthis dynamic game logic see Pauly (2001), vanBenthem (2014). For other styles of game algebra, includingalso forms of parallel composition, cf. Abramsky(1997).

It should be said that imperfect information challenges this approachto game algebra. For instance, one may have to decompose a larger gameinto smaller subgames where agents need not know which of thesesubgames they are in. Game algebras with imperfect information havebeen studied in the context of Boolean Games (Harrensteinet al. 2001). A recentpower-based game algebra with operations encoding imperfectinformation, showing some analogies with IF logic (Mann,Sandu, & Sevenster 2011) can befound in van Benthem, Bezhanishvili, andEnqvist (2019b).

2.10 Special Topics

Coalitions and Networks Nothing has been said so farabout social or structural relations between players: they moveindividually and in interaction with all other players. However, inmany games, groups of players can team up to jointly pursue goals,possibly in competition with other groups. Coalitions are a natural,but non-trivial extension of the logical frameworks introduced here,as strategic abilities of groups may exceed those of all memberscombined, see Peleg (1997), Vande Putte & Klein (2021, 2022), and theentry oncoalition powers in games. In other studies of social phenomena, the set of players is equippedwith an additional network structure. An agents’ outcome orbehavior will then depend upon what network neighbors do (Baltaget al. 2019; Christoff 2016). Lastly,games on networks are closely related to information flows in socialnetworks, as studied in depth by Liu, Seligman,and Girard (2014) and Seligman andThompson (2015) from a logical perspective.

Tracking This section contains a wide variety ofperspectives on games. These differ in their invariance relations andtheir matching languages, offering different foci such as outcomes,powers, or the detailed temporal evolution of games. Even furtherperspectives will no doubt keep emerging. This diversity may seemoverwhelming, making the field rather scattered. But here, anotherrole of logic shows, by not just proliferating systems, but also asconnecting them. Various logical translations exist between thelanguages and levels involved. Often, reasoning about games in a logicfor some level can be mirrored precisely under translation into thelogic of another level. Moreover, these translations can often keeptrack of changes in games under actions of information updates, atopic to be taken up in Section 3. Tracking of this kind is definedand studied in general logical terms in vanBenthem (2016) and Cinà(2017), and in category-theoretic terms in Ye(2022).

Infinite games So far, games were tacitly assumedfinite in length. This assumption is innocuous for many real lifescenarios, yet there are notable exceptions. A prominent example aresafety games, where one of the players, the guard, has to ensure asystem to never leave a certain state, while the opponent attempts todeviate. Many technical tools for finite games also work for infinitegames. There is, however, a number of conceptual and logicaldiscontinuities. Since infinite games have no last moments, forinstance, outcomes must be attached to complete histories of gameplay, rather than leaves of a tree. Reasoning about games thenrequires temporal modalities for a given history, but also modalitiesranging over all open future histories. For analyzing powers, then,temporal versions of forcing modalities are needed. With thesemodifications, a logical style of analysis still applies. Forinstance, it is well-known that determinacy fails for infinite games(Jech 2003). However, what holds for allgames is a law of ‘weak determinacy’ stating that, ifi has no strategy to force a set of histories satisfying\(\varphi\), her opponentj can ensure thati will neverobtain such a \(\varphi\)-strategy in the future. The differencebetween standard determinacy and weak determinacy is captured by thefollowing two formulas, that are entirely in line with thissection’s style of analysis:\(\{i\}\varphi\,\lor\,\{j\}\neg\varphi\) (determinacy) versus\(\{i\}\varphi\lor\{j\}G\neg\{i\}\varphi\) (weak determinacy), whereG is the temporal modality of ‘always in the future onthe current history’. A more radical use of infinite games isfound in coalgebra (Abramsky 1997; Jacobs 2016), where infiniteprocesses are modeled that can only be observed from the outside, butnot built up finitely from ground elements.

3. The Nature of Players

Game forms may be seen as spaces where players can operate. A game,however, is not fully determined by its game form alone. Rather, theplayers involved may import additional features relevant for gameplay. Players can, for instance, be limited in their powers ofobservation, either by aspects of the game structure or throughcognitive limitations. The most striking added feature, however, isthat players have preferences. Agents not only observe the world oract in it. While these describe mere kinematics of a game, agents alsoevaluate current state and various possible futures. Being driven bypreferences, it is such evaluations that are the moving force behindplayer’s choices. Preference, hence, take a prominentexplanatory role for true game dynamics.

This section places its focus on the preferential and epistemicdimensions of players. Such factors are essential to notions ofrationality where information, action, and preference are oftenentangled. In game theory, a harmony between these is often sought innotions of equilibrium for strategy profiles.

3.1 Preference and Equilibria

Game trees and game matrices specify the moves available to players atdifferent moments in time. They also indicate all possible outcomes,either as cells in a matrix, or as leaf nodes in an extensive game.However, to study what playersshould orwill do ina game, a further component is needed: players’ preferences.Such preferences need not only reflect material pay-offs or otherfeatures of outcome states. Rather, they may also relate to theprocess of play itself, and which moves lead to a certain outcome.Moreover, preferences may contain irreducibly subjective elements.Even when assuming the same role in a game, different players maydisagree about the relative desirability of certain outcomes (Fehr& Schmidt 1999).

Within a static, outcome-oriented perspective on games, a majoremphasis is on equilibria: strategy combinations where all players dothe best they can in light of their preferences and theopponents’ strategies. A further, dynamic perspective focuses onhow such equilibria relate to the individual players’ stepwiselocal reasoning on how to act in light of their beliefs and desires.This perspective is taken up inSection 4.

3.2 Preference Logics for Games

For reasoning about preferences, it must first be specified what it isthat agents’ preferences apply to. The orthodox account letspreferences exclusively range over possible outcomes (Osborne& Rubinstein 1994). However, agrowing trend in the logical literature assumes agents to rather careabout the truth value of general propositions than can describe boththe progression or outcome of a game. While not equivalent, the twoperspectives are compatible. Both will be discussed in thissection.

In the classical picture, playeri’s preferences on agame tree are represented by a preference relation \(\prec_i\) rangingover the set of outcomes. Such a relation is usually assumedtransitive and reflexive, but need not be total.

Example A game tree with preferences.

Figure 8.ⓘ

Just as with the earlier modal logics for game forms, a relativelysimple logical formalism can already express relevant aspects ofagency in games. It offers a low-complexity language for stating basicfeatures of action and information, without going into details of theunderlying quantitative mechanisms. More precisely, games withpreferences naturally support a logic with modal operators\([\preceq_i]\) interpreted as:

\([\preceq_i]\varphi\): \(\varphi\) holds in all states at least as good as the currentone to agenti.

Logics of this type can express various properties relevant to games.They can, for instance, say that all states better than the currentone are \(\varphi\) states, making moving towards \(\varphi\) states anecessary condition for maximizing utility. They can also express,that all best states are \(\varphi\) states, with the formula

\[\langle\preceq_i\rangle[\preceq_i]\varphi\]

For more on modal preference logics, see Hansson (1990, 2001),Girard(2008) and van der Torre(1997).

Modal preference logic has further extensions with natural connectionsto games. In a refined perspective, for instance, preferences mayderive from reasons, say, criteria or goals various agents want toachieve. This gives rise to a duality between preference relationsamong outcome states and priority orders over formulas, describing theagents’ goals. Dynamic accounts, finally, track how preferencecan change under various input events. For more on both of theseissues, see Liu (2011).

However, modal preference logics, construed either way, are not yetrich enough to express one of the essential notions of game theory.Further extensions are needed to deal withbest responses,expressing that the current move of a player is the best she can do inlight of her opponent’s actions.

Best response moves are the main ingredient for game equilibria.Formally, aNash equilibrium is a strategy profile, fixing aunique choice for each player, where nobody can improve byunilaterally changing strategy when all others maintain theirs. Thereare several ways of defining this property in extended modalpreference languages. One possibility is to simply introduce a newatom \(b_i\), stating that the current world is the best playeri could have achieved in light of the opponents’ actions.In this language, Nash equilibrium are characterized by

\[\bigwedge_{i\in \text{Players}}b_i.\]

More explicit definitions exist, building on the strict preferencemodalities of van Benthem, Girard, and Roy(2009). Yet, perhaps the simplest illuminating approach uses anintersection modality from hybrid logic (Arecesand ten Cate 2007) combining theagent’s preference relation with her uncertainty between theopponent’s action to characterize best responses and Nashequilibria (cf.Section 2.6):

\[\bigwedge_{i\in \text{Players}}[\prec_i\cap\equiv_i]\bot\]

Expressing Nash equilibrium has served as a benchmark for logics ofstrategic games (van der Hoek & Pauly2007). Yet there are other desiderata, often connected toanalyzing standard game-theoretic solution concepts for games. Theseare usually designed to find Nash equilibria or at least narrow downthe strategy profiles to those compatible with certain requirements ofrationality. Well-known methods of this kind are Backward Inductionfor extensive games and Iterated Removal of Strictly DominatedStrategies for strategic form games (Osborne& Rubinstein 1994). These will be discussed now, as theyraise intriguing further logical issues.

3.3 Backward Induction in Extensive Form Games

Here is a high-level description of Backward Induction. In extensivegame forms, the aim is to introduce a new preference-based relation\(\best_i\), denoting that some move is the best a player can do atsome given state. Thus, \(\best_i\) is a subset of playeri’s total move relation, to be defined in a suitablemanner.

For final moves, standard decision theory suggests that a choice isbest for the active player if no other move leads to a better outcome.When extending the analysis to earlier positions of the game, thingsdepend crucially on players’ expectations about theiropponents’ future behavior. Several possible policies exist,depending on the types of player involved. A widespread assumption inepistemic game theory is common belief in rationality, i.e., that allplayers involved are rational, believe their opponents to be rational,believe their opponents to believe that opponents are rational, and soon. In line with this assumption, the following algorithm extends the\(\best_i\) relation recursively to non-terminal nodes:

Whenever playeri is to move at states, possiblechoices are assessed by comparing what would happen if, after thatmove, everybody followed their \(\best\) relation. A possible move ats is included ini’s \(\best\) relation if thebest outcome of this move followed by repeated \(\best\) moves by allplayers is at least as good as every other movei could make ats, followed by \(\best\) moves by all players.

Here is how this bottom-up procedure works in practice.

Example Backward Induction.

Figure 9.ⓘ

This procedure is a qualitative version of classic game theory’sBackward Induction, which is based on utility values rather thanpreference relations (Leyton-Brown & Shoham2008).

Backward Induction and the resulting \(\best\) relation is a primeexample for the complex entanglement of preferences, information andaction. A key modal axiom governing this relation was identified byvan Benthem, van Otterloo, and Roy(2006). \(\best^*\) denotes here the transitive closure of theunion of all \(\best_i\) relations.

\[(\turn_i\land\langle \best\rangle[ \best^*](\operatorname{end}\rightarrow p))\rightarrow[\move_i]\langle \best^*\rangle(\operatorname{end}\land\langle \preceq_i\rangle p)\]

Describing the limit of a dynamic process with a static property, thisequivalence exemplifies a family ofcharacterization theoremsthat play a crucial role within the logical analysis of games. Otherdynamic perspectives can be analyzed in a similar logical style (Liu2011).

3.4 Iterated Removal of Dominated Strategies

Iterative reasoning strategies akin to Backward Induction also existfor games in strategic form. Rather than defining a new unarymove-predicate \(\best\), however, these procedures work byeliminating suboptimal actions. An actiona is labeledsuboptimal ordominated if there is some other availableaction,b, that guarantees a better result thana, nomatter what the opponents do. In this case a rational player shoulddropa from her space of admissible acts, as she would neverplay it.

Just as Backward Induction, dominance reasoning has an iterativeflavor. Assuming common belief of rationality, players can expecttheir opponents to also drop dominated actions from consideration.Doing so reduces the game and might render further moves dominated, asis illustrated in the following example. Within the left-to-righttemporal progression, moves are greyed out as they get discarded.Players’ preferences are represented with numerical values with1 the best and 4 the worst.

	c	d
a	1,1	4,3
b	2,2	3,4

	c	d
a	1,1	4,3
b	2,2	3,4

	c	d
a	1,1	4,3
b	2,2	3,4

The fact that further strategies may become dominated suggests torepeat the procedure, turning removal of dominated strategies into aniterated process. When games are finite, this process is guaranteed toconverge in finite time. Iterated removal of dominated strategies onbinary preference relations is a qualitative variant of the versionemployed in classical game theory, where cardinal utility values areassumed (Leyton-Brown & Shoham2008).

A closely related process is iterated removal ofweaklydominated strategies, where some movea is deleted ifthere exists ab that outperformsa on some of theopponents’ moves, while being at least as good on the remainingones. Unlike its strict counterpart, iterated removal of weaklydominated strategies suffers from a number of technical and conceptualintricacies, such as order dependence of iterated deletion (Samuelson1992; Pacuit & Roy 2011).

3.5 Goals

Generalizing the concept of winning or losing, agents can be assignedgoals they pursue in a game. Restricting to a single goal per agentretains a binary perspective: A goal is reached or not. Goals,however, allow for additional flexibility. Besides pure competition asin win-lose games, these can also express pure coordination games,with everybody pursuing the same goal, or mixed motive games withpartial overlap between different players’ goals.

The concept of goal functions is particularly prominent in the logicalframework ofBoolean games (Harrenstein 2004). There, each agentis given control over some atomic propositions, permitting her tofreely decide on their truth value. Goals are then formulated aspropositional formulas over the set of all players’ atoms.Crucially, a player’s goal formula might hence involve atomsthat are not under her control. In iterated extensive Boolean games,goal formulas might also refer to properties of histories of playdefined in temporal logics (Gutierrez,Harrenstein, & Wooldridge 2015)

3.6 Knowledge, Belief, and Limits of Information

There are various types of information players can possess or lackabout a game. First and foremost, players can be uncertain about thetypes of opponents they face: their preferences, their reasoning aboutthe game and how they expect the game to unfold. Second, agents’uncertainty can extend to the game itself. Players, of course’won’t know their opponents choices in a simultaneous move game.Besides, agents might also have limited information about past movesand events. Such uncertainty can arise from the game structureeschewing certain observations, but also from failing to record pastinformation properly. In yet more extreme cases, agents might even beunsure about the moves available to their opponents.

Given the various limitations to their knowledge, players mayentertain beliefs to structure their uncertainty. Such beliefs may,naturally, change over time, as players communicate or observe thegame unfold. The importance of beliefs in the logical analysis ofgames has been emphasized in Stalnaker(1998), who was the first to highlight the role of beliefrevision in analyzing reasoning about game solution.

3.6.1 Uncertainty about moves

In one sense of uncertainty, even highly idealized agents might havebut limited information of what has happened so far. In certain cases,the game’s structure may limit some players’ observationalpowers of their opponents’ moves. In other instances, agents maysuffer under cognitive limitations restricting their perspective onthe game. Or, sometimes, agents might simply fail to record some ofthe moves made by themselves or others.

Withinextensive games with imperfect information, all suchcases are represented by indistinguishability relations\(-\,-\,-\,-_A\) between states \(m,m'\), expressing that agentA cannot distinguish between being atm and \(m'\).Notably, this does not preclude the player from learning later on inthe game whether he has been atm or \(m'\).

Example A game with imperfect information.

Figure 10.ⓘ

While allowing agents to lack information of various kinds, the aboveanalysis makes one structural assumption about players: they alwaysknow which moves are available to them at a given node. In extensivegames with imperfect information, this translates to the requirementthat whenever two states are indistinguishable to some agent, theycoincide on the set of her possible actions.

The move from perfect to imperfect information has major implicationsfor strategic reasoning. In the game depicted above, playerAcannot distinguish between being atm and \(m'\). When at theformer, she may, for all she knows, be at \(m'\) instead. HenceA’s decision needs to account for both possibilities; shecannot base her choice on any property that holds at only of thoselocations. In particular,A has no available strategy thatguarantees her ending up in a \(\win_A\) node. SinceE cannotensure a win either, no player has a winning strategy. This is acentral difference with finite perfect information games, where it isguaranteed that one of the players has a winning strategy, cf.Section 2.4.

3.6.2 The logic of imperfect information

Reasoning about imperfect information requires to extended languagesfor extensive form games with epistemic modalities. For each playeri, modality \(K_i\varphi\) representsi’sknowledge. The usual semantics of epistemic logic relates this touncertainty, as encoded by the players’s indistinguishabilityrelation:

\(\mathcal{M},m\vDash K_i\varphi\quad\) all states \(m' \text{ with }m\,-\,-\,-\,-_i\,m'\) satisfy \(\mathcal{M},m'\vDash\varphi\)

This language is best illustrated with the above game tree. To thisend, interpret the tree as the classic children’s game where oneplayer,A, has to guess in which hand her opponent,E,hides some little token. OnceE has hidden the token in, say,her right hand (move \(h_R\)), the guessing player has a winning movede re: she should pickright (\(p_R\)). However, asthe token was placed in secret, she may not know that pickingright is a winning move: the player has no winning strategyde dicto. This is expressed by:

\[\mathcal{M},m\vDash [p_R]\win_A\land\neg K_A[p_R] \win_A.\]

In a game theoretic setting, thede re vs.de dictodistinction has been studied by Horty andPacuit (2017) and van Benthem(2001).

In light of these considerations, many logics for defining strategiesinvolve epistemic elements. It seems reasonable to demand that agentscannot base their choice of strategy on everything that has happenedbefore, but only on what they know, i.e., their current information(Pacuit, Parikh, & Cogan 2006). Theresultinguniform strategies (Maubert2014), can be defined by the knowledge programs of Fagin,Halpern et al. (1997). Furtherrestrictions are possible, for instance granting agents limited memorythat only reaches back a fixed number of moves (Gutierrez,Harrenstein, & Wooldridge2015).

The epistemic action language can express many further phenomena inimperfect information games. The following game is anillustration.

This is a game tree diagram. The extended description (link in figure caption) will describe the tree.

Figure 11.ⓘ

Once playerE arrives at noden, she cannot discern thatactual situation from node \(n'\). However,E must havepossessed information earlier on that distinguishesn from\(n'\): to arrive atn, she must have playeda as herfirst choice, while \(n'\) can only be reached after playingb.Thus,E can only be uncertain between these two nodes if sheforgot about her own previous actions.

The epistemic action language can distinguish between such scenarioswith memory loss and those without. The property ofPerfectRecall states that players retain full memory of all moves theyobserved. This can be expressed by the following axiom scheme (Halpern& Vardi 1986; Bonanno 2004)

\[K_i[a]\varphi\rightarrow[a]K_i\varphi.\]

Also the converse of this scheme admits a natural interpretation:

\[[a]K_i\varphi\rightarrow K_i[a]\varphi.\]

ThisNo Miracles property expresses that players can onlylearn by observing moves, not by any other methods extraneous to thegame.

Of course, logic does not presuppose that all players have perfectmemory, or that they cannot pick up any information outside theprogression of play. Epistemic action language can equally well beemployed to analyze more general scenarios where the above axioms donot hold. Especially within dynamic-epistemic versions, epistemiclogics can produce modified versions that cover many more cases thanthose described here (van Benthem 2014).Moreover, further modalities from epistemic logic make sense, inparticular, those for common or distributed knowledge in groups ofplayers (Fagin, Halpern et al. 1995; Meyer& van der Hoek 1995).

An epistemic component with operators \(K_i\) fits with many logicalperspectives on games. In particular, epistemic extensions are ascompatible with coarse logics such as the earlier-mentioned\([\move_i]\)-setting with a single move-modality per player, as withfine logics where each individual action type is represent by adistinct modality \([a]\). In fact, inSection 2.6, epistemic operators were used for analyzing games in strategic form,where modalities naturally relate to uncertainty about the otherplayers’ strategies.

3.6.3 Uncertainty about options and preferences

In more general settings, uncertainty does not stop at theopponents’ informational states. In international relations oreconomic bargaining, also the players’ motivations andpreferences are not fully known to all parties involved. Within thecorresponding extended form games, players may be uncertain abouttheir opponents’ preferences and strategic options, whether theycan afford a certain move or whether they actually possess theinformation they threatened to reveal. Obviously, uncertainty aboutpreferences or available options will impact reasoning about theequilibria of a game. Strategic players might even try to exploit suchuncertainties, for instance by pretending to have options they do notpossess.

In a first pass, this type of uncertainty can be expressed byintroducing nature as hypothetical player, with a first move thatdetermines the preferences and available options for all players. Asimple example is the game depicted below. At the start,A isuncertain whetherE can reply toA’s movefby playinge. Likewise, she lacks information on whetherE prefers \(O_3\) over \(O_4\), or vice versa.

This is a game tree diagram for the previous paragraph. The extended description (link in figure caption) will describe the tree.

Figure 12.ⓘ

From a logical point of view, this miraculous initial move by Natureis not needed. Standard epistemic models can represent the abovescenario, and many more complex ones, by means of theindistinguishability relations introduced above. Technically, thisrequires to move beyond standard imperfect information trees toso-calledepistemic forests (vanBenthem, Gerbrandy, Hoshi, & Pacuit 2009), sets of treeslinked by epistemic relations. In particular, the above game treetransforms into

This has two game tree diagrams. The extended description (link in figure caption) will describe the tree.

Figure 13.ⓘ

The epistemic action language for trees works just as well onepistemic forests. However, in suitably expressive languages, thelogic of forests is weaker than that of trees, as the set ofvalidities on the class ofn-player trees is a strict supersetof the validities onn-player forests.

3.6.4 Imperfect information and beliefs

Further enrichments of the logical framework add semantic structure tothe agents’ uncertainty. When unable to determine the exactsituation, players may classify options with respect to plausibility.To this end, epistemic models have been equipped with plausibilityorderings \(\geq_i\) for playersi (Boutilier1994; Stalnaker 1968; Baltag & Smets2008).

In the preceding example, plausibility order might work asfollows:

Figure 14.ⓘ

This richer structure is reflected by introducing new modalities foragents’ beliefs, determined by the most plausible states:

\(\mathcal{M},w\vDash B_i^{\phantom{\psi}}\varphi\quad\) \(\varphi\)holds in all \(\geq_i\)-maximal states ini’s epistemicrange.

Conditional belief, important to players’ planning within agame, can be interpreted in the same style:

\(\mathcal{M},w\vDash B_i^\psi\varphi\quad\) \(\varphi\) holds in all\(\geq_i\)-maximal \(\psi\)-states ini’s epistemicrange.

These clauses are intended to work in finite as well as in infinitesettings. However, in the latter case minor modifications might beneeded akin to those inconditional logic. These have been proposed in various alternatives. Notably, thisenriched epistemic-doxastic logic allows for further, less standardinterpretations beyond those illustrated so far. Examples are‘strong belief’, expressing that all relevant\(\varphi\)-states are more plausible than all relevant\(\neg\varphi\) states, or ‘safe belief’ saying that\(\varphi\) holds at all states that are at least as plausible as thecurrent one. See van Benthem and Smets(2015) for an overview of plausibility semantics and itsconnections to conditional logic, belief revision theory,dynamic-epistemic logic, and a wide range of philosophical andtechnical issues.

3.7 Higher Order Uncertainty and Type Spaces

In various scenarios, agents reason not only about theopponents’ preferences or admissible moves, but also about theirbeliefs about the game and others’ behavior therein. In fact,such higher-order reasoning can have a major impact on game play. Aprime example is the Backward Induction procedure ofSection 3.3, where the construction of a best move relation crucially relied oncommon knowledge of rationality. More generally, agent’s bestmoves frequently depend upon what they expect others to do. Thisphenomenon is especially prominent for simultaneous move games, whereit occurs in both coordinative scenarios (Skyrms2003; Lewis 2002) as well ascompetitive ones (Hotelling 1929). Moredetails can be found in the entry onepistemic game theory.

Arbitrary first and higher order levels of knowledge and belief can berepresented with the above relational models, the standard tool inepistemic and doxastic logic. For information in extensive form games, theepistemic-doxastic perspective on states can be combined with\(\move\)-relations in exactly the way described before. The resultareepistemic-doxastic trees orforests that canrepresent most types of knowledge or belief players might have aboutthe game, including its exact shape, previous moves, opponents’preferences or opponents’ first- and higher-order beliefs on anyof these matters.

Outside of logic, higher-order information has also been modeled inclassical game theory. Quantitative frameworks represent informationas probability distributions over a given event space. In thissetting, higher-order information corresponds to probabilitydistributions over probability distributions of the right kind. Morespecifically, \(n^{\textrm{th}}\) order information corresponds to aprobability distribution over the space of \((n-1)^{\textrm{th}}\)order beliefs. As shown by Harsanyi(1967–1968), the limit of specifying higher and higherlevels of information can be represented as a type space, where eachagents’ type is a probability distribution over states of natureand the other players’ types. In an abstract sense to bediscussed below, these types correspond to states in standard modelsof modal logic.

In addition to standard modal models, logic also has a straightforwardanalogue to probabilistic type spaces:logical type spaces.In a formal framework first introduced by Fagin,Geanakoplos et al. (1999), ann-type is a sequence \(\mathfrak{f}_n=\langlef_0,f_1\ldots,f_{n}\rangle\) where \(f_0\) specifies the state ofnature, i.e., a valuation recording which atomic propositions are trueor false, and \(f_1\) lists for all players the states of nature theyconsider possible. \(f_m\) for \(m\geq 0\) then specifies for allplayers which \((m-1)\)-types, i.e., sequences \(\langleg_0,\ldots,g_{m-1}\rangle\) they consider possible. In this way, ann-type fixes the player’s higher-order beliefs up toleveln. These types are, of course, subject to coherenceconditions: the agents’k-types for differentkmust fit together. For instance, whenever some agent considers ak-type \(\mathfrak{f}_k\) possible, she must also consider anyinitial segment \(\mathfrak{f}_{k'}\) for \(k'<k\) possible.Conversely, any \(k'\) type the agent considers must be the initialsegment ofsomek type the agent holds possible. Typespaces offer a semantics for the epistemic language: Inductively, fora formula \(\varphi\) of modal depthm, \(K_i\varphi\) is trueat a type \(\mathfrak{f}_n\) if all \(g_{m}\in f_{m+1}(i)\) satisfy\(\varphi\) or if \(m>n\).

Alternatively, the set ofn-types allows for a naturalinterpretation as relational models with accessibility relationsdefined by

\(\langle f_0,\ldots f_n\rangle R_i\langle g_0,\ldotsg_n\rangle\quad\) For all \(m\leq n\) holds \(g_{m-1}\in f_m(i)\)

Interpreting the set ofn-types as a relational model yields asecond way of evaluating the epistemic language on logical typespaces. For formulas of modal depth less thann the twointerpretations coincide. Hence, up to finite depths, type spaces andtheir associated relational models are two perspectives on the sameinformational situation.

To fix all of the agents’ beliefs, the analysis moves to types\(\mathfrak{f}=\langle f_0,f_1,\ldots\rangle\), containing some\(f_n\) for every natural numbern. In this extended frameworkthe situation becomes more complicated. The space of all such types isuniversal in the following sense: every relational model can be mappedin a truth-preserving manner to the space of all types by sending eachstate to a full description of the agents’ corresponding first-and higher-order informational attitudes. This map, however, isusually not a modal bisimulation. In fact, the process of typeconstruction could be continued indefinitely, yielding a transfinitehierarchy of mutually non-bisimilar type spaces (Heifetz& Samet 1998). Such transfinitetypes can become relevant when the epistemic language is enriched withmodalities for common group knowledge, in which case a fulldescription of all expressible attitudes involves infinite hierarchiesof higher-order information (Fagin, Geanakoploset al. 1999). A recent logical study of type spaces includingtheir probabilistic structure can be found in Bjorndahland Halpern (2017).

The tight connection between type spaces and relational models iscompatible with additional assumptions that might be imposed on theplayers’ mental states. Fagin,Geanakoplos et al. (1999) characterizes when type spaces giverise to \(S5\) models, while Galeazzi &Lorini (2016) do the same for multi-agentKD45belief.

While relational models and logical type spaces represent exactly thesame information, their main differences is in perspective. Relationalmodels take a third person bird’s eye view on possible worlds.Their starting point is a set of worlds rich enough to contain allstates considered possible by the relevant agents, together withaccessibility relations modeling players’ information. Fromthere, agents’ first-order beliefs at the various worlds can beread off and, subsequently, also all higher levels of information.Logical type spaces, by contrast, assume a first personperspective. They take a full description of first and higher-orderbeliefs as primitive and treat indistinguishability as a derivedrelation.

Finally, it should be noted that type spaces assume a staticperspective on games. No provisions are taken for representing movesor strategies explicitly, nor for incorporating updates of knowledgeand belief that occur as a game in extensive form unfolds, cf. thediscussion inSection 4. Thus, there is some distance between type spaces and the earlierepistemic-doxastic forest models for extensive games. As a first steptowards filling this gap, it has been shown how type spaces canaccommodate product updates from dynamic epistemic logic (Klein& Pacuit 2014).

3.8 Reasoning, Bounded Agency, and Player Types

Besides variations in preferences and beliefs, a third crucial aspectof players is their styles of information processing, decision making,and reasoning. Real cognitive agents are bounded in their informationprocessing, as both their memory and reasoning capacities are limited.In particular, players may not be able to represent the entire gamethey are in, nor reason until the end of the game. This phenomenon ofshort sight has been studied in Grossiand Turrini (2012) and Turrini(2016). Moreover, in real-life iterated social interaction,payoffs are generated along the game, and may not be clear beforehand,(Axelrod & Hamilton 1981). In suchcontexts, the best strategy in terms of short-term payoffs need not beoptimal in the long run, but bounded agents may miss this longerhorizon, (Klein, Marx, & Scheller2021).

Logical literature on bounded agency is too broad to be surveyed here.For a few research lines relevant to games, see Faginand Halpern (1987) and Heifetz, Meier, andSchipper (2006) on epistemic logics with awareness, Artemov(2008) onjustification logics, van Benthem and Pacuit (2011) onevidence logics, and Hansson (1998) and Lorini(2018) on doxastic logics with computationally tractable beliefbases.

In the game theoretic literature, bounded agents have often beenrepresented as finite state machines (Gutierrez,Harrenstein, & Wooldridge 2015;Binmore & Samuelson 1992). Limitations on reasoningcapacities or memory size then translate into bounds on the machinesize. The resulting hierarchy allows for a fine grained analysis ofinformation processing, reasoning, and thus bounded players ofdifferent types. This perspective fits well with the logical study ofagency in computer science, (Grädel,Thomas, & Wilke 2002; Wooldridge 2009).

In an integrated perspective, preferences, beliefs and reasoningstyles can all be subsumed under the game-theoretic notion of aplayer type. For reasoning about the future course of a game,players will hence often entertain beliefs about each others’types. A simple example is Backward Induction, where players assumeall opponents fully rational throughout. In more complex settings,individual actors may attempt to rationalize various moves observedand derive predictions about their opponents’ future behavior bytaking a broader range of options into account. Such players may startby assuming the counter-player to be a simple machine, and only moveup to more complex views when required by evidence. In particular,there is no reason to assume uniformity of players or views. Within agiven scenario, a diversity of player types might be present (Liu2009; Liu & Wang 2013; Paul & Ramanujam2011; Ghosh & Verbrugge 2018; Bergwerff et al. 2014). Forsome game-theoretic proposals concerning most frequently occurringplayer types, see Camerer (2003).

3.9 Thin and Thick Models for Players

This section has outlined a variety of ways for incorporating playersand agency into game forms. These come in a hierarchy of richness,ranging from annotated game trees to epistemic forests, type spaces,or yet more abstract models of games.

At the thinner end, the focus is on structural aspects of the game,incorporating players’ preferences, but not necessarily theirbeliefs. Such frameworks are typically just rich enough to representequilibria or backward induction paths, and to reason about these inlogics of action and preference. Thin models leave much informationabout players’ knowledge, beliefs, or theirmodusoperandi unspecified, and put less emphasis on the actualdynamics of game play.

At the thicker end, models for games have lush worlds encodingplayers’ preferences, information, beliefs, and perhaps eventheir complete types, including memory and reasoning capacities.Typical models of this kind are found in Stalnaker(1998) and Halpern(2001). When applied to extensivegames rather than strategic-form games, thick models can anticipateanything that can happen in one large temporal universe, allowing toderive a full prediction about how play will proceed.

The distinction between thick and thin logical models seems folklorein applied logic. In fact, it occurred already in theearlier-mentioned choice between local logics with single stepmodalities versus temporal logics built over a complete universe ofhistories. Yet, there does not seem to be a unique best perspective.Rather, the choice between thick and thin models often depends on theexact goals pursued. A central consideration in this trade off iswhere the dynamics of stepwise play should be located: Thick modelspre-encoded such dynamics, whereas thin models allow for an externaldynamic logics for updates (Baltag, Smets, andZvesper 2009). The next section will highlight a number of waysfor complementing a thin perspective with dynamic information on gameplay through representations of actions and updates.

3.10 Special Topics

Deontic reasoning Preference is closely related toobligation and permission. This shows in particular on the formalside, where preference logic in both static and dynamic variants (Hansson1990; van Benthem, Grossi, & Liu2014) has clear analogies withdeontic logic. Moreover, deontic and game-theoretic perspectives have given rise tomany fruitful connections. In one direction, deontic notions may beseen as high-level descriptions of optimal actions given theinformation and obligations of agents (Kooi& Tamminga 2008; Anglberger, Gratzl, & Roy 2015).Conversely, an agent’s goals and intentions may be derived from theirobligations (Klein & Marra 2020).More generally, game solution procedures can enrich accounts ofdeontic notions (Başkent, Loohuis andParikh 2012; Horty 2018). The connection between games anddeontic reasoning may also highlight new topics, for instance on therelation between individual and collective preferences (Duijf,Tamminga & Van De Putte 2021).Lastly, in artificial intelligence, deontic perspectives arise intracking the behavior of a distributed system in relation to itsgoals, (Ågotnes & Wooldridge2010).

Mathematical foundations Incorporating players raisesnew questions about game equivalence. When agency matters, adequatenotions of equivalence cannot stop at preserving properties of theunderlying game form. Rather,player-dependent equivalenceswill also require preservation of players’ beliefs, preferencesor reasoning types. Incorporating such additional parameters makes itharder for two games to be equivalent, as new space for variationcomes into play. On the other hand, agent limitations might alsocreate new simpler game equivalences that can be studied by the toolsof this entry.

4. Analyzing Play

The term ‘game theory’ suggests that everything ofinterest is captured in the format of a game with its moves andoutcomes. The present entry reassembles this perspective, consideringadditional structure. A first extension was offered inSection 3, treating the nature of players as a topic in its own right. Thissections puts a spotlight on a second topic, game play in a widersense.

Many themes in the literature on logic and games fall into threephases connected to play. Certain activities can already be conductedbefore the actual game. Examples are assessing the opponentor forming a plan. Most relevant choices and decisions, however, takeplaceduring the game — at least unless one thinks of playersas automata blindly following preset strategies. Lastly, alsoafter a game significant activities occur. These involvelearning about opponent types, identifying crucial mistakes made, orrationalizing the moves taken. In what follows, examples will bepresented of each phase.

4.1 Game Solution and Pregame Deliberation

When viewing games as static structures, rationality can be defined interms of coherence between players’beliefs, preferences and choices or intentions (Elster 1988). However, rationalityalso describes a quality of behavior, related to how players act orwhat they take advantage of when deliberating about a game. Therelation between both perspectives can be made concrete wheninterpreting game solution procedures as styles of pregamedeliberation. To follow is a dynamic analysis of Backward Induction(cf.Section 3.3) that differs conceptually fromcharacterization theorems in terms of static properties such as common knowledge or commonbelief. In the dynamic analysis, these group properties are notassumed as preconditions. Rather, they are produced through the logicof deliberation.

4.1.1 Backward Induction via public announcement

The Backward Induction algorithm is usually presented in aquantitative setting, where each outcome is associated with utilityvalues for all players. (Leyton-Brown andShoham 2008). However, the same algorithm also works in aqualitative setting, with attitudes expressed by a preference relationbetween outcomes.

Backward Induction Backward Induction computesoptimal moves for players. More specifically, at each choice node ofan extensive form game, one or more of the available moves is labeledas optimal. For each player this set of optimal moves often forms astrategy in the usual game-theoretic sense, i.e., a function selectinga unique action to take at each of her choice nodes. Yet, there aredegenerate cases where backward induction merely creates a relationalstrategy, restricting the available moves, while still leaving somechoices to the player.

The principle driving the Backward Induction algorithm is that noplayer should ever select a move that is dominated by another moveavailable at the same moment. Dominance here works in a recursivemanner. Movea dominates moveb if the correspondingplayer prefers each final outcome reachable froma by followingBackward Induction moves to every outcome reachable fromb byBackward Induction moves.

Public announcement of rationality In oneperspective, Backward Induction can be understood as a process ofprior-to-play deliberation, executed by players whose minds proceed inharmony. Deliberation steps are repeated public announcements(\(!\rat\)) of rationality-at-nodes:

\(\rat \): no player arrived at the current node via a strictly dominatedmove

Dominance here is a relation between the outcomes available after acertain move has been made. In one interpretation, some moveadominates another moveb if every outcome that remainsobtainable aftera is preferred to any outcome reachable afterab move. However, there is a dynamic twist: crucially, thegame tree considered, and hence the outcomes available, changes duringthe deliberation procedure.

The semantics ofannouncement updates works by trimming models. \(!\varphi\) transforms a modelMinto a sub-model \(M_{|\varphi}\) consisting of all those points inM that satisfy \(\varphi\), while deleting all \(\neg\varphi\)nodes. Relations on \(M_{|\varphi}\) are those inherited fromM. Crucially, deletion may change the truth value of formulas:after announcing \(!\varphi\), some nodes in \(M_{|\varphi}\) maysatisfy \(\neg\varphi\). In particular, withM the set ofpoints in a game tree, the set of available histories may keepshrinking as successive announcements are made. Hence, repeatedannouncements of \(!\rat \) make sense. In finite games, this processalways reaches a limit, a smallest subgame where no move is dominatedby another.

Example Solving games through iterated assertions ofrationality.

Consider the following game, already introduced inSection 1.1. Iterated announcements of \(!\rat \) removes nodes that can only bereached by dominated moves as long as this can be done. The trace ofthis procedure is:

This has three game tree diagrams illustrating the example. The extended description (link in figure caption) will describe the tree.

Figure 15.ⓘ

Here, the Backward Induction solution emerges step by step. Stage 1 ofthe procedure rules out the leaf labeled with (\(2,2\)) as the onlypoint where \(\rat \) fails. Stage 2 then rules outE’schoice node as new node where \(\rat\) fails. In the resulting gametree, \(\rat \) holds throughout.

More generally, let \((!\varphi, M)^\#\) be the limit (i.e., the firstfixed point) ofM under repeatedly announcing \(\varphi\) aslong as it still true. In any game tree, the fixed-point \((!\rat ,M)^\#\) has \(\rat\) true throughout. Its nodes contain the actualplay computed by the Backward Induction algorithm (vanBenthem 2014).

Limit behavior Rationality is‘self-fulfilling’ in the limit: if players commit to it indeliberation for long enough, they prune away all irrational movesand,a fortiori, all moves incompatible with common belief ofrationality. The final outcome is a model with rational play at everypoint, a form of common knowledge of rationality. However, iteratedannouncements can also yield a different type of limit behavior:self-refutation. A prime example for this is the classic MuddyChildren puzzle (Gierasimczuk & Szymanik2011) where repeatedly communicating ignorance leads toknowledge in the end. Also within game theory, a number of situationsexist where (credible) announcements of future irrationality can leavesome player better off than the Backward Induction solution (Leyton-Brown& Shoham 2008). Limitbehavior of announcements in infinite games is studied in Baltag,Bezhanishvili & Fernández-Duque(2022), linking up with Cantor’s derivative in classicaltopology.

4.1.2 Backward Induction via belief revision

Iterated belief revision. A different perspective ofpre-game deliberation is couched in terms of belief instead ofknowledge. The driving force here is rationality-in-belief:

\(\rat^*\): players never chose any move dominated by another in light oftheir beliefs how play will proceed from then on.

In this setting, the game tree itself remains invariant duringdeliberation: no histories are removed or ruled out. What may changeinstead is the relative plausibility of occurrence players assign toend nodes or, in infinite games, histories.

In plausibility semantics (briefly introduced inSection 3.6), an agent believes propositions that hold true in all thoseepistemically accessible worlds that are maximal in her plausibilityorder. The corresponding dynamics of deliberation does not proceed bypoint deletion, but bysoft updates modifying theagents’ plausibility ordering. For Backward Induction, a‘radical upgrade’ \(\Uparrow\varphi\) suffices, that movesall \(\varphi\)-worlds above all \(\neg\varphi\)-worlds states whilemaintaining the ordering within these two sets (Baltag& Smets 2008).

Here is how this mechanism works in a game setting. Start with all endnodes equiplausible for all players. Since upgrades proceed by publicannouncement, all players will share the same beliefs throughout. Inthe procedure to follow, a movex isdominated inbelief by a movey of the same choice node if, in theacting agent’s plausibility ordering, the most plausible endnodes reachable aftery are all better for her than each mostplausible end node compatible with movex. Now perform radicalupgrades of type

\(\Uparrow\rat^*\): Ify is dominated in belief byx, make all end nodesfollowingx more plausible than those aftery.

Example Backward Induction, soft version.

Here are the stages for the new procedure in the preceding example,where the letters \(x, y, z\) stand for end nodes or histories of thegame:

Figure 16.ⓘ

In the top node of the leftmost tree, going right is not dominated inbeliefs for playerA by going left. So, \(\rat ^*\) onlyaffectsE’s turn, and radical upgrade with \(\Uparrow\rat^*\) makes \((0, 3)\) more plausible than \((2,2)\) . After thischange, going right has become dominated in beliefs in the top node,and a new upgrade takes place, makingA’s going left mostplausible.

Iterated upgrade with \(\rat ^*\) always stabilizes to a fixedplausibility order, which is the same for all players. Identifyingeach history of a game with its end node allows for a belief analysisof Backward Induction (van Benthem &Gheerbrant 2010). On finite trees, the histories emerging whenall players resort to their Backward Induction strategies exactlycorrespond to the most plausible end nodes created by iterated radicalupgrade with rationality-in-belief. An alternative dynamic-epistemiccharacterization of Backward Induction, using similar ideas in adifferent mix, can be found in Baltag, Smets,and Zvesper (2009).

Stabilization cannot be taken for granted. For other assertions\(\varphi\), iterated upgrades \(\Uparrow\varphi\) can lead tooscillating or divergent plausibility orders. However, this divergenceis limited. While cycles can occur for conditional beliefs, everytruthful iterated sequence of radical upgrades eventually stabilizesall propositional beliefs (Baltag & Smets2009).

Fixed-point logic In a more technical perspective,Backward Induction strategies can be defined as largest subrelation ofthe total \(\move\) relation that has at least one successor at eachnon-terminal node, while satisfying a confluence property betweenaction and preference:

\(CF(s)\): \(\forall x\forall y(( \turn_i(x)\land xsy)\rightarrow\forall z(x\, \move\, z\rightarrow\) \(\exists u\exists v(\sfend(u)\land\sfend(v)\land y s^* v\land z s^* u\land u\leq_i v)))\)

This fact is the basis for proving that Backward Induction isdefinable in First Order Fix point logicLFP(FFO) (vanBenthem & Gheerbrant 2010). Resultsin this line of research connect game solution and game-theoreticequilibria with fixed-point logics of computation. In simple settings,such as that of Zermelo’s Theorem mentioned earlier, modalfixed-point logics akin to μ-calculus suffice.

4.1.3 Iterated removal of strictly dominated strategies

Further game solution concepts can be analyzed with logics of iteratedupdate. In particular, iterated updates are not restricted toextensive form games, but can also provide insights for games instrategic form. A paradigmatic algorithm is Iterated Removal ofStrictly Dominated Strategies \((SD^\infty)\). In this setting, astrategy is considered dominated if there exists another strategy thatyields a strictly higher payoff against any of the opponent’sactions.

Example Iterated removal of strictly dominatedstrategies \((SD^\infty)\).

Consider the following matrix. As usual, pairs listA’sutility first,E’s second.

		E
		a	b	c
A	d	2, 3	2, 2	1, 1
	e	0, 2	4, 1	1, 0
	f	0, 1	1, 4	2, 0

First remove the right-hand column, i.e.,E’s actionc which is dominated by either ofa andb. Withc being removed,A’s actionf has becomestrictly dominated. After its removal,E’s actionb becomes strictly dominated, and after that,A’sactione. At the end of the process, iterated removal leavesnothing but the state \((d,a)\), the game’s unique Nashequilibrium. In general, the resulting game matrix after all removalsis guaranteed to contain all Nash equilibria of the original game, butit may also contain further strategy combinations.

In this setting, the formal dynamic apparatus involves assertionsappropriate to the matrix games ofSection 2. In fact, various different types of rationality can be defined inlogics for matrix games. Here is an illustration for two-player games,involving announcements of ‘Weak Rationality’:

WR: Each player thinks that, compared with each of her availablealternative actions, her current move might be at least as good forher.

This statement is the negation, for each player, of her current actionbeing strongly dominated. Naturally, this property can be expressedformally with suitable epistemic action modalities. Yet, even as itstands, it is clear that Weak Rationality can be announced to pruneaway strategy profiles, and that in a iterated manner. The strategicgame will change every time weak rationality is announced, initiatinga stepwise process that resembles the earlier iterative announcementsof rationality for Backward Induction. As observed there, limits ofpublic announcements are always reached eventually, as models can onlyget smaller. For announcing Weak Rationality, these limits match theoutcome of \(SD^\infty\) precisely (van Benthem2007).

A similar style of analysis can be extended to other notions ofrationality. For instance, taking \(B_i\) to stand for ‘thecurrent action of playeri is best for her against all actionsof the opponent’, the following formula may be dubbed strongrationalitySR

\[\langle E\rangle B_E\land \langle A\rangle B_A\]

Briefly, the formula expresses that both players have a reasonablehope of doing well.Strong Rationality in this sense isrelated to the rationalizability program for game solution (Pearce1964; de Bruin 2005), where actionsare discarded if a better response exists under all circumstances.Strong Rationality, too, drives a game solution method.

Example Updates with iterated announcements of strongrationality (SR).

Consider a slight variation on the previous example. Below is thesequence of updates for iterated announcements of strong rationality\((\textit{SR}^\omega)\)

2, 3	2, 2	1, 1
0, 2	4, 1	1, 0
1, 1	3, 4	2, 0

2, 3	2, 2
0, 2	4, 1
1, 1	3, 4

2, 3	2, 2
0, 2	4, 1

2, 3
0, 2

2, 3

Each box may be viewed as an epistemic game model, as explainedearlier. Again, every step of announcement increases players’knowledge, until a fixed-point is reached, constituting an equilibriumwhere each player knows as much as they can.

Strong rationality is a more demanding condition than weakrationality. WhileSR impliesWR, there can be movesthat satisfy weak but not strong rationality. This shows in thefollowing difference between the current and the previous example. Inthe present matrix, announcing Weak Rationality stops after the firststep elimination of actionc. The reason is that, in the secondmatrix, the row player’s bottom move is not strictly dominatedby any other action, so this row remains after re-announcingWR. However, under no possible circumstance is the rowplayer’s bottom action best for her. This contradicts strongrationality and hence that row is eliminated by the nextSRannouncement. More generally, the game matrix resulting from\((\textit{SR})^\infty\) is a sub-matrix of that produced by\((\textit{WR})^\infty\). Notably, this is not entirely obvious, asthe two update sequences may produce different epistemic modelssatisfying different formulas. A proof can be found in vanBenthem (2014).

Just as with Backward Induction, there are connections betweeniterated announcements and fixed-point logics. The set of strategyprofiles that survive iterated announcements of strong rationality canbe defined in modal μ-calculi (Kozen 1983;Venema 2008). If announcements are generalized to arbitraryformulas, so-called deflationary fixed-point logics are needed forstudying limit behavior (Ebbinghaus & Flum1995; Dawar, Grädel, & Kreutzer 2004).

Further game solution concepts have been analyzed in a similar dynamicupdate style. The iterated regret minimization of Halpernand Pass (2012), for instance, hasbeen captured in terms of iterated announcements (Cui& Luo 2013).

It should be noted, that there are also more deductive takes onsolution concepts, where successive inferences assume the role of thesemantic updates above. A systematic proof-theoretic perspective ongame-theoretic reasoning toward solution can be found in deBruin (2005). Lastly, alternative analysesof Strong and Weak Rationality as well as other game solutionconcepts, in an abstract rewriting format of computational logic canbe found in Apt (2005).

Digression: comparing across representation levelsDifferent iterative announcement procedures lead to different analysesof games. Moreover when comparing various procedures across differentframeworks, surprises may occur. For an illustration, take BackwardInduction. Its dynamic analysis produced a new ‘best move’relation or plausibility order on an extensive form game. Theresulting strategy profiles may differ from a Nash equilibriumanalysis of the associated strategic form game:

Example Backward Induction and Nash equilibria.

Consider the following game.E has no preferences between anyoutcomes, butA does, as marked by the utility values.

Figure 17.ⓘ

In the earlier BI analysis, neither move forA dominates theother in beliefs, so no move is eliminated. Now consider the twopossible strategy profiles of each player and compute Nashequilibria:

(Left, right) is not a Nash equilibrium, sinceA woulddo better by playingRight, but(Left, left) is.

This illustrates differences in logical perspective on strategic andextensive form games. Primitive elements of the former, strategies,are complex objects in the latter’s game tree that cannot beidentified completely at the level of individual nodes alone. Arelated point of connecting perspectives in classic game theory gaverise to the concept of subgame perfect Nash equilibria (Selten1975).

Further scenarios Casting game solution in terms ofdeliberation renders it an internal mental process: normally opponentsdo not sit down together to discuss their game play in advance, butreason about the opponents’ possible actions and considerations.However, the deliberative techniques introduced above also apply toreal conversational scenarios. An example related to game theory isthe topic of disagreement, first introduced in an epistemic setting byAumann’s 1976seminal agreeing to disagree result.Dégremont and Roy (2012)investigate this topic with techniques of dynamic logic, building onclassical results from Geanakoplos andPolemarchakis (1982). In this framework, any dialogue whereagents keep stating whether or not they believe some formula\(\varphi\) leads to agreement in the limit model, where updates nolonger have any effect. Briefly said, agents cannot disagree forever,at least when starting with different hard information, while sharinga well-founded plausibility order.

4.2 Information Flow, Knowledge, and Belief During Play

Game play is a dynamic process, where players repeatedly obtain newinformation about other players. Certain aspects of informationcollection are hard-wired into the game’s structure, such asobserving moves, or, in settings of imperfect observation, changingfrom one information state to another. Other updates may beextraneous, such as signals about the type of opponent one is dealingwith. As of now, there is no general logical theory encompassing allthese phenomena. Yet, instructive samples exist. The first topic toaddress concerns the players’ knowledge, the second theirbelief.

4.2.1 Epistemic update and imperfect information

In one perspective, games annotated with imperfect information cellscan be interpreted as recording a process of actual play. However, animperfect information tree does not suffice to fully specify the traceof a real game. This raises the question on how to tease out what hasreally happened. One style of analysis involves techniques fromdynamic-epistemic logic. In this approach, players are assumed to have perfect recall, they donot forget anything they once knew, while also satisfying No Miracles:observation of actual game play is their only source of information,(cf.Section 3.6).

In a first approximation, every move triggers a public announcement,informing all players what just happened. Many games, however, includepartially observable moves, where some players merely learn that anact has been performed, but not necessarily which. In this case,information processing requiresproduct updates fromdynamic-epistemic logic (cf. Baltag & Moss2004), allowing for an appropriate mixture of knowledge anduncertainty.

Example Decorating a game tree by updates.

The left-hand side of the following diagram displays the game’sbare action structure, without any information on observability.However, when moving, players can distinguish their own actions, butnot all moves of their opponents. Their precise observational powersare described byevent models for the individual moves (cf.van Ditmarsch, van der Hoek, & Kooi2007).

This is a game tree diagram and an Event Model illustrating the example. The extended description (link in figure caption) will describe the tree.

Figure 18.ⓘ

The observational structure on possible moves is encoded by relationsbetween the corresponding nodes, as described for games of imperfectinformation (Section 3.6). Here are the successive updates that create the uncertainty links inthe tree:

This three diagrams representing the updates. The extended description (link in figure caption) will describe the tree.

Figure 19.ⓘ

The resulting annotated tree is the following imperfect informationgame:

Figure 20.ⓘ

A similar analysis applies to infinite trees as well as to epistemicforests (cf.Section 3.6). More generally, any imperfect information structure can arise frominformation updates, provided players satisfy perfect recall and nomiracles, and moves in the game have logically definable preconditionsgoverning their availability. Precise formulations and proofs can befound in van Benthem, Gerbrandy, Hoshi, andPacuit (2009). A generalization to game play without assumptionof synchronicity is provided in Dégremont,Löwe, and Witzel(2011).

No miracles and perfect recall are typical assumptions for most typesof agents in game theory. However, certain scenarios requiremodifications, (cf. Osborne & Rubinstein1994 on the ‘drunken driver’ scenario). Moreover,if players are represented as finite automata, (cf.Section 3.8), perfect recall fails, and quite different patterns of uncertaintybecome possible. Characterizations results for memory-free andmemory-bounded players can be found in Liu(2011).

In addition to observational restrictions built into the game setup,product updates can also model extraneous communication or otherinformation flow parallel to actual play. Some such scenarios will belisted under Further Directions below.

4.2.2 Belief revision and Forward Induction

Certain types of information may be judged inconclusive or not fullyreliable. While unsuitable for advancing knowledge, such informationmay prompt agents to alter some of their beliefs. Such inconclusiveevidence often concerns expectations concerning opponents’player types. Leaving aside the possibility of mere mistakes, allmoves can be assumed to result from intentional, strategicconsiderations. By interpreting opponents’ past moves, agentsmay hence infer about their beliefs, preferences, risk attitudes orreasoning types. Naturally, most such observations are not fullyconclusive. The corresponding updates hence cannot delete anyalternatives. Rather, they merely change the agent’splausibility ordering \(\leq_i\) among different options. Formally,this can be handled with the plausibility updates introduced forBackward Induction. However, the interpretation differs. Here, theseupdates do not represent steps in pregame deliberation, but resultfrom actual moves during the game. Besides the radical upgradeintroduced above, a number of further updating policies reflectingdifferent attitudes to the information acquired are defined in Baltagand Smets (2008). Theepistemic-plausibility patterns that can arise from systematicplausibility update in games have been identified in Dégremont(2010), using twocounterparts of the earlier perfect recall and no miracles properties:‘Plausibility Revelation’ and ‘PlausibilityPropagation’.

These results refer to but one aspect of belief in games. There areothers. A further type of belief describes agents prior attitudes tothe game, generated by past experience or deliberation. Another refersto the agents’ beliefs about where they are located in the gametree, based on previous observations during play. To keep thesenotions separate, one might distinguish between more local‘beliefs’ during play and future-oriented‘expectations’ about the game’s progression.Plausibility orders created by Backward Induction, for instance,describe expectations about future game play. These are not based onobservations already made in the present game, and significantly, failto satisfy the properties of plausibility revelation and propagation.Here is a particular strand of logical belief and its revision that isof independent game-theoretic interest.

Forward Induction Suppose some player has deviatedfrom her Backward Induction strategy as computed in pre-gamedeliberation. What are others to make of this? Answers offered in theliterature range from interpreting the deviation as an error withoutany future implications (Aumann 1995) totreating it as significant in various ways (Bicchieri1993). In the latter vein, thedeviation could be a signal for cooperation (believable or not), asign of limited resources, or it could reveal other relevantinformation about the player’s type.

More explicitly, the situation has the following aspects. At any stageof a game, players have several types of information, including theirprior expectations of how the game would proceed and the perhapssurprising observations made along the way. If the game is to continuefurther, as in the state marked below, agents need to integrate bothinto expectations about the future course of the game.

This is a game tree diagram illustrating the previous paragraph. The extended description (link in figure caption) will describe the tree.

Figure 21.ⓘ

Rationalizing There is no unique best way ofintegrating information of the various kinds. Yet, a natural option isto maintain the assumption of opponents’ rationality, taken inthe earlier sense. Assuming preferences to be common knowledge,observed moves hence provide new information about theopponent’s belief. More specifically, these beliefs have twocomponents: expectations about what other players will do, andintentions about their own future actions. The driving principle willthen be

Rationalizing By playing amove, a rational player communicates that this move is not strictlydominated-in-beliefs for her.

Clearly, rationalizing can only be maintained as long as the playerdoes not choose a move that is strictly dominated under allcircumstances. In that case, one must ascend a ladder of furtherhypotheses about the opponent, including the possibility of her makingmistakes.

Reasoning policies of the above type are calledForwardInduction. Battigalli and Siniscalchi (2002) and Brandenburger(2007), analyze Forward Induction in extensive form games based on itsknown tight connection with Iterated Removal of Weakly DominatedStrategies in strategic form games. The following example involvingexplicit reasoning is from Perea(2012).

Example A Forward Induction scenario.

Figure 22.ⓘ

In the matrix game, no move dominates any other. HenceE shouldconsider all outcomes possible. In this case, goingleft issafer for her than goingright, and henceA shouldplayLeft at the start. However, ifE rationalizes,and observesA goingRight, she has extra informationavailable at her choice node. Following the rationality assumption,A expects to do better than 3, which is only possible if heintends to playUp in the matrix game. Now this tellsE to proceed to the matrix and play theleft columntherein.E’s results in a better payoff than the 2 of heroriginal safe option.

From a logical perspective, a study of Forward Induction requiresepistemic-doxastic models with ternary world-dependent plausibilityrelations, combined with the public announcement updates orplausibility upgrade described above (Section 4.1.2; see. also van Benthem 2014). No definitive logical analysis ofForward Induction has been published so far.

4.2.3 Post-game rationalization

Comparatively little attention has been paid in the literature to whatplayers do after a game. Yet, these follow up-activities are oftencrucial, for instance to establish general lessons learned that may bevaluable for future game play. Such interpretations are especiallyprominent in small or isolated groups, where the same opponent mightbe encountered again in the future.

Preference change after a game At a simple level,post-game activity can consist in setting, or altering, the secondinput parameter of rational choice beside belief: the players’preferences. Several folklore results relate to this option. Forinstance, when playing against a given strategy of another player withknown preferences, any strategy can be rationalized by choosingsuitable preferences among outcomes. Liu(2011) discusses several preference based rationalizationalgorithms using dynamic logics of preference change.

Preference change can also occur during a game. Players may receivenew information about the game’s end states and theirproperties. They may also follow a command or a suggestion from anauthority, establishing a preference or reversing an earlier one.Relatedly, players may change their external goals pursued in thegame, or they may adjust their preferences for more internal reasons,as in the phenomenon of ‘sour grapes’ (Elster1983).

4.2.4 Play in a long-term temporal perspective

The main focus of this section is the local dynamics of what happensbefore, during and after a single game. There is also a broaderperspective of time, in which all these activities are embedded in anextended temporal universe, large enough to include all possibletrajectories of the game, finite or just as well infinite. Inevolutionary game theory, in particular, infinite games typically arise from iterated play offinite games, with Lewis’signaling games (2002) as prominentexample in philosophy.

Assuming an extended infinite temporal perspective raises additionalquestions about players’ strategic foresight and adaptation(Christoff 2016). Various long-termperspectives differ drastically in this respect, ranging from theminimal rationality assumptions typical of evolutionary game theory tohigh reasoning complexities of agents anticipating the long termeffects of their choices.

Temporal logics While infinite game forms werebriefly alluded to inSection 2.10, infinite play focuses on agency over time. A host of temporal logicsfor this end have been put forward, including interpreted systems(Fagin, Halpern et al. 1995),epistemic-temporal logic (Parikh &Ramanujam 2003), STIT (Belnap &Perloff 1988; Horty & Belnap 1995), ATL (Alur,Henzinger, & Kupferman 2002; van der Hoek& Wooldridge 2003), and others. Many of these systemscombine multiple modalities. Consequentially, their complexity can bevery high (undecidable, non-axiomatizable, or even non-arithmetical),as Halpern & Vardi (1986) show in apioneering study for the case of combining time and knowledge.Surveying this area is beyond the scope of this entry, cf. the entryontemporal logics. A unifying view of connections between the various paradigms ispresented in van Benthem and Pacuit(2006).

This is a tree diagram. The extended description (link in figure caption) will describe the tree.

Figure 23.ⓘ

Just as with finite games, players’ preferences must bespecified for studying equilibria in potentially infinite games. Inlack of outcome nodes or final moments to attach preferences to, theseare naturally thought of in terms of players’ goals, expressedas properties the game’s histories should satisfy. Such goalscan be local propositional facts true at some particular moment. Butgoals can also concern global properties of histories such as avoidingor reaching the same position some specified number of times, or moreabstractly, achieving safety or fairness in some appropriate sense.All such properties can be specified in temporal logics. For the caseof Linear Temporal Logic (LTL), the ‘Booleangames’ of Gutierrez, Harrenstein, andWooldridge (2015) have developed the temporal goal basedapproach in depth. Notably, this framework validates a logical versionof the ‘folk theorem’ for iterated games, cf. Osborne& Rubinstein (1994): Under naturalconditions on goals, iterated games can have novel equilibria notsupervenient on the base game’s Nash equilibria. Furthersignificant uses of temporal logics connect game theory with beliefrevision theory (Battigalli & Bonanno 1999;Perea 2012; Stalnaker 1998).

Evolutionary game theory and dynamical systems Aprominent application of iterated games occurs in evolutionary gametheory (Maynard Smith 1982; Hofbauer &Sigmund 1998; Gintis 2000), a framework that has manyapplications in biology, formal sociology, but also linguistics andphilosophy (Lewis 2002; Skyrms 2010; Alexander2007; Clark 2012).

Little work has been done so far on the logical analysis ofevolutionary games along the dimensions of this entry. In fact, thereare striking conceptual differences between evolutionary games and thestyle of analysis pursued here that might be dubbed ‘highrationality’-oriented. Rather than incorporating intentional,strategic actors, evolutionary games work by temporal progression of adynamical system which is driven by individuals’ fitness valuesderived from game-like encounters with others. Within such systems,behavior is not driven by belief updates or complex strategicconsiderations. Rather, players typically display ‘lowrationality, following certain hard-wired strategies. Much of theevolutionary system’s dynamics is then driven by changes in thepopulation’s composition of strategy types.

Even so, evolutionary game theory does invite connections to logic.The evolutionary success of simple strategies like Tit-for-Tat (Axelrod& Hamilton 1981) raises thequestion of just when complex logically based high-rationalitystrategies can be replaced by equally efficient alternatives simpleenough to be played by automata or similar models of bounded agents,(Grädel, Thomas, & Wilke 2002).At a higher level of abstraction, there also is an incipient line ofresearch into the connection between logic and dynamical systems, astandard tool for analyzing evolutionary games. This strand includes abimodal topological logic of time (Kremer &Mints 2007), fixed-point logics of oscillation (vanBenthem 2015), and a systematic linkagebetween dynamic-epistemic update logics and dynamical systems overmetric spaces (Klein & Rendsvig2019; 2020). At a much concreter level, one important species ofevolutionary games are signaling games (Lewis2002; Cho & Kreps 1987; Osborne & Rubinstein 1994; Skyrms2010: van Rooy 2004), where agents send and receive signalsabout the state of the world. Signaling games match up naturally withthe earlier dynamics logic of information flow during play.

4.3 Conclusion: Theory of Play

Topics discussed in this section are less standard in the literaturethan those of the sections before. In an orthodox reading, various ofthe aspects addressed would not be considered part of game theoryproper. The extended agenda followed here has been embraced by vanBenthem, Pacuit, and Roy (2011) as alarger program for logic, going under the heading ‘Theory ofPlay’. The underlying line of reasoning is that games do notfully determine their outcomes, as they allow for various styles ofplay. Hence, it might be the process of play itself, includingplayers’ types and how they change over time, that might thebest focus for understanding interaction, rather than mere games orgame forms alone. Similar lines of argument can be found in thefoundations of computation where it has been proposed that theessential topic of study should bebehavior (Abramsky2008).

4.4 Further Directions

Belief revision and learning theory Belief revisionin repeated games bears natural resemblance to limit learning offormal learning theory (Kelly 1996). Baltag,Gierasimczuk, and Smets (2011) analyzelearning in terms of initial epistemic-doxastic models over whichfinite histories of signals trigger the learner to revise beliefs,represented as changes in epistemic accessibility or plausibilityorder. It turns out that both iterated public announcement anditerated radical upgrade as discussed above are universal learningmethods, though only the latter maintains this property in thepresence of (finitely many) errors in the input stream.

Goal dynamics and intentions While preferences andgoals have so far been assumed fixed and universally known, this is byno means necessary. van Otterloo (2005)presents a dynamic logic of strategic powers, where information aboutplayers’ intentions and preferences can be announced duringplay. Roy (2008) uses announcements ofintentions to obtain simplified solution procedures for strategicgames. More concrete scenarios of extraneous information flow arefound in Parikh, Taşdemı̇r, andWitzel (2013), where agents manipulate the knowledge of othersduring play.

Game change In many real life scenarios, players donot know the full game tree they are playing. Even if they did, itmight change during play. Or, at least, players may attempt to changethe game. A concrete example is provided by the game tree inSection 4.1.1. There, the inefficient Backward induction outcome \((1, 0)\) could beavoided byE promising not to go left. When made binding (forinstance through imposing a fine) this announcement eliminateshistories and, consequentially, a new backward Induction outcome of\((2,2)\) results. Hence, both players can be made better off byrestricting the freedom of one. Game theory has sophisticated analysesof such scenarios, including an analysis of ‘cheap talk’(Osborne & Rubinstein 1994), askingwhen such announcements are credible. On the logical side, thissuggests an analysis of signaling games (vanRooy 2004). We are not aware of any logical work done in thisdirection.

Real games The discrepancy between specification of agame and the realities of play is especially striking in real gameplay, either of the ‘natural kind’ in common parlor games(van Ditmarsch & Kooi 2015), or ofthe artificial kind found in the laboratory experiments ofexperimental game theory (Camerer 2003).Little work has been done by logicians in this realm, though there isa broad tradition of computational analysis of games (Schaeffer& van der Herik 2002; Kurzen2011). Any adequate logical analysis would clearly need toincorporate the considerations on bounded agency discussed inSection 3.

Mathematical foundations Logic of play as discussedhere raises issues of how to interfaces local with global dynamics.This shows in particular with logical limit behavior, whereobservations and assertions are made repeatedly. Limit models ofpublic announcement, as described earlier, can be‘self-fulfilling’ or ’self-refuting’. In thefirst case, the property asserted becomes common knowledge among allagents, whereas in the second it eventually becomes false at theactual world. With soft update on plausibility models, a third optionarises, namely, infinite oscillation, or even divergence (Baltag& Smets 2009). To date, there isno general logical theory of these phenomena, but see vanBenthem (2011) on the use of fixed-pointlogics for limit models, Miller and Moss(2005) on the high complexity of public announcement logic withfinitely iterated announcements, and Klein andRendsvig (2017) on limit behavior of product updates.

The topic of limit behavior also raises the issue of how local dynamiclogics of agency relate to the global temporal logics discussed inSection 4.2.4. Towards clarifying the connection, vanBenthem, Gerbrandy, Hoshi, and Pacuit (2009), show howdynamic-epistemic logics can be seen as decidable fragments of moreexpressive temporal logics. Baltag, Smets, andZvesper (2009) discuss the related theme of how dynamicrepresentations can decrease complexity by shifting information fromthe temporal universe to dynamic events.

5. Interfacing Logic and Probability in Games

Probabilities are central to game theory, where they serve twoprominent roles. First, they structure players’ uncertaintyabout various aspects of the game, including the state of nature, thetype of opponent faced, and past, present, and future moves of otherplayers. Second, ever since the origins of Game Theory (vonNeumann & Morgenstern 1944),probabilistic randomization has served to expand the agents’space of possible actions. While the interpretation of such mixedstrategies has been the subject of extensive debate (Sugden1991), it is uncontroversial thatrandomized moves add significant depth to the analysis of games. Infact, the concept of mixed strategies is vital for a number of seminalresults in classic game theory including the existence of Nashequilibria in finite games of imperfect information.

Probability and its logic is a major topic in both mathematics andphilosophy, as discussed in the entries oninterpretations of probability, andlogic and probability. The current section merely outlines a few key connections betweenprobability and the logical analysis of games. The followingpresentation assumes all state spaces to be finite, ignoring importanttechnical and conceptual issues around the transition from finite toinfinite state spaces.

5.1 Probabilities and Beliefs

Probabilistic methods are widely employed to represent beliefs ofagents within and outside of games, witnessBayesian epistemology. Typically, a probabilistic belief model consists of two components: aspace of possible states that the agent’s beliefs range over,plus a quantitative probability function denoting how probable theagent judges different propositions or states to be. In qualitativelogical models, on the other hand, agentive belief is represented invarying degree of detail. The coarsest approach only distinguishesstates the agent considers possible from those she rules out. This isthe perspective of standard epistemic-doxastic logics, such asmulti-agent S5 and KD45 discussed inSection 3.6. More fine-grained perspectives are employed in the plausibilitymodels of Boutilier (1994) and Baltagand Smets (2008), where the range ofepistemic options is structured further by plausibility orderingsencoding which options agents take to be less or more likely. See alsoSections3.6.4 and4.2.

Both probabilistic and plausibilistic perspectives can express thatsome alternative is more likely than another. However, there are alsoconceptual differences between the two frameworks. Probabilisticmodels can aggregate, allowing their logic to express, for instance,whether many low-probability events combined can outweigh even thehighest-probability worlds. Aggregated probabilities play a key role,for instance, in calculations of expected utility. Yet no such thingcan be expressed in plausibility semantics. For another strikingdifference, plausibility models lead to a notion of belief that isclosed under conjunction. This conjunction closure typically fails forprobabilistic accounts of belief. See however Leitgeb(2017) for a sophisticated bridgebetween both types of modeling.

5.2 From Logic to Probability and Back

On a received view, logical and probabilistic frameworks emphasizedifferent aspects of belief. Logic emphasizes coherence propertiesbetween the propositions believed, such as closure under logicalimplications or conjunction. Probabilistic reasoning, on the otherhand, stresses graded information and attitudes towards uncertainevents such as lotteries. Even so, there is a variety of approachesattempting to unify the two types of reasoning by constructing bridgesbetween the frameworks.

From qualitative to quantitative probability Earlyattempts in this direction go back to Finetti(1970 [1974]), striving for a purely qualitative axiomatizationof probability theory. A step further towards logical reasoning arevarious theories of qualitative probability (Kyburg1994), often based on the assumptionthat classical quantitative notions of probability are too demandingfor real-life agents. In this line of research agents need only reasonwith partial, comparative probability assessments, rather than havingfully specified probabilities for all events. Logical frameworks forqualitative probability include Segerberg(1971), Fagin, Halpern, and Megiddo (1990), and Delgrandeand Renne (2015). While differing indetail, all logics in this line have in common that they allow forexpressions of the form \(\varphi\preceq\psi\), indicating that\(\psi\) is judged at least as probable at \(\varphi\). Recentframeworks add various additional refinements to this language.Similarly, Heifetz and Mongin (2001)expand the axiomatic analysis of probabilistic beliefs to higher orderreasoning, working on probabilistic type space akin to thoseintroduced inSection 3.7.

Yet, the concept of qualitative probability is sometimes consideredflawed: a complete set of probabilistic-logical principlesguaranteeing that every complete description in the logical languagecorresponds to a unique probability measure turns out to require acomplex calculus, involving an opaque infinite rule (Kraft,Pratt, & Seidenberg 1959; Scott1964). A new angle has been proposed in Harrison-Trainor,Holliday, and Icard (2016) andDing, Holliday, and Icard (2021) through axiomatizing alow-complexity qualitative probabilistic logic that emerges fromrelaxing the above unique correspondence requirement to merelyrequiring compatibility with all probability measures of a certainfamily.

However, none of the frameworks described here are specific for games.In fact, it remains to be seen whether qualitative probabilisticlogics, old or recent, can be used for a qualitative analysis ofgame-theoretic solution concepts.

From quantitative to qualitative probability Whilethe preceding line of research aims at recovering quantitativeprobability from qualitative notions, a converse project shows howubiquitous qualitative patterns might arise naturally within aquantitative probabilistic setting. Building on what is sometimesdubbed the Lockean Thesis,threshold approaches connectprobability to logic by stipulating that some \(\varphi\) is to bebelieved simpliciter if the probability of \(\varphi\) is above someappropriate numerical thresholdt. For most choices ofthresholdt, such translations do not square well with standardlogical desiderata, as beliefs will in general not be closed underconjunction. However, in recent work Leitgeb(2017) and Lin and Kelly (2012)have identified conditions under which one can do better. Building onideas of Skyrms (1977), Leitgebidentifies strong, context-dependent‘robustness conditions’ on thresholds that guarantee thedefined belief operator to satisfy the KD45 axioms after all. Linand Kelly, on the other hand, work withnon-uniform thresholds for transitioning between logical andprobabilistic notions of belief, allowing to derive coherence betweendifferent forms of belief dynamics on the two sides. For a recentstudy of the mathematical foundations and limits of these approaches,as well as the conditional logics they generate see Mierzewski(2020).

5.3 Updating and Tracking Probabilities

In the dynamic perspective of game play, every move constitutes a newpiece of information agents have to take into account. Besides,players may also change their beliefs about the game upondeliberation, through communication or any other signals they receive,be they reliable or not ( cf.Section 4). All such dynamic events raise the question of how new information isto be incorporated into the agents’ beliefs and whenprobabilistic updates have corresponding logical revisions or viceversa.

If the new information is of the hard type, accepted as irrevocablytrue by all agents, the probabilistic counterpart of logical publicannouncements is Bayesian conditioning. Both notions track each otherat a semantic level, meaning that their outputs amount to the samething. Computing beliefs after a public announcement means recomputingin the submodel consisting of all states where the informationreceived was true. This is exactly the same mechanism as forrecalculating probabilities in Bayesian update. Moreover, forreasoning about updates, both approaches require conditional notions:conditional belief and conditional probability respectively. Thequantitative notion of conditional belief relates to the logicalnotion of conditional belief \(B(\varphi|\psi)\), with the slightcaveat that the latter also allows for epistemic or doxastic operatorsinside either argument. More refined logical notions of conditionalbelief arise in the earlier-mentioned plausibility semantics ofSection 4.1.2, (Baltag & Smets 2008).

Given the co-existence of qualitative and quantitative perspectives,it makes sense to ask whether one cantrack the other. In astatic sense, tracking asks whether different notions of belief can betranslated into each other by omitting or transforming some of thesemantic details involved. A dynamic interpretation expands on this byasking whether updating, either in the hard or soft varieties ofSection 4, is compatible with these translations in a commutative diagram:Information update in a new perspective after translation should yieldsame result as first performing a matching update in the oldperspective and then translating (van Benthem2016). In games, the topic of tracking may refer not just toinformation update, but also to solution concepts or moves in gameplay described at the various levels considered inSection 2.

The existence of tracking maps depends on the exact type of updateconsidered. By now there is a wide variety of updating policies onplausibility models (van Benthem & Smets2015), not all of which have obvious probabilisticcounterparts. Likewise, for well-known varieties of probabilisticupdate, such as Jeffrey update, where the probability of selectedpropositions can be reset at will, plausibility counterparts are noteasy to find, though the attempt of vanBenthem, Gerbrandy, and Kooi (2009) modifies dynamic-epistemiclogic to allow for Jeffrey update and other generalized probabilisticpolicies.

5.4 Specializing to Games

Virtually all aspects of game theory provide contacts between logicaland probabilistic perspectives. Clearly, this is true for thedifferent representations of knowledge, beliefs and their dynamicsjust discussed. Other contacts occur at the level of game forms, cf.Section 2, where probabilities enrich the space of strategies. The resultingmixed moves require players to expand their preferences to mixedoutcomes, cf.Section 3. Finally, at the level of reasoning about game play, cf.Section 4, probabilistic beliefs play a role in solution techniques such asdominance or expected utility based reasoning.

Available actions and mixed strategies Probabilisticmixtures of pure strategies are prominent in game theory, as they cansecure outcomes and payoffs that no pure strategy alone couldguarantee.

Example Matching Pennies.

Consider the well-known game of matching pennies in matrix form:

		Bob
		x	y
Ann	a	1,−1	−1, 1
Ann	b	−1,1	1,−1

For Ann, a mixed strategy of playinga exactly half of the timeguarantees an expected outcome of 0, no matter what Bob does. No purestrategy could have achieved this.

From a logical point of view, mixed strategies can be conceptualizedas new primitive actions in the earlier logics of games ( cf.Section 2). Yet, this treatment immediately renders the set of available actionsinfinite. A cautiously refined logical language, extending logicalapproaches to qualitative probability, can allow for expressions suchas an agent playing actiona with probability at leastq, (Delgrande and Renne2015).

A more challenging general question is how to relate classicprobabilistic approaches with their fixed-point results and ensuingequilibrium existence theorems, with the logical fixed-pointapproaches mentioned at several places in this entry. The latteroperate with step-by-step ordinal iterations, as opposed to thegradual, approximative procedures that underlie the Brouwer orKakutani fixed-point theorems relevant for classical game theory. Arelated question is just how much logic is needed to reproduceprobabilistic existence theorems within a qualitative framework.

Adding players’ preferences Once preferencesare added, mixed strategies trigger additional intricacies. A strategyprofile where some players pursue mixed strategies does not produce aunique outcome, but a weighted combination of outcomes. Thus,permitting mixed strategies requires lifting preference relations toprobabilistic mixtures of outcomes or strategy profiles. Incorporatingsuch mixtures may implicitly depart from the standard, purelyqualitative perspective on outcomes (Ramsey1931; Savage 1954).

Example Extended preference comparison.

The following two games are equivalent in terms of qualitative (i.e.,ordinal) preferences between outcomes for both players. However, theydiffer in preferences between mixed outcomes, with

\[0.5(b,x)+0.5(b,y)\succeq_A (a,x)\]

holding in the game to the left, but not in that to the right.

	x	y
a	1,0	2,1
b	0,1	4,0

	x	y
a	1,0	2,1
b	−10,1	4,0

Going this way poses some logical challenges. For example, consider apreference relation over probabilistic mixtures of outcomes, where\(m^t(a,b)\) stands for obtaininga with a probability oft, andb otherwise. This setting is in the scope ofvon Neumann and Morgenstern’s1944 well-known ‘continuityaxiom’ that is characterized by an implicit infinitedisjunction:

\[ {a \preceq b\preceq c} \Rightarrow {\exists t \in(0,1):\ m^t(a,c)\preceq b \preceq m^{1-t}(a,c)} \]

This seems well beyond the expressive power of standard probabilisticlogics.

Solution concepts and game play Probabilization alsoimpact the process of game play and its reasoning dynamics, forinstance by changing the earlier-mentioned calculus of weak and strongdominance (Section 3.4). Consider the following game, cf. de Bruin(2005):

		B
		x	y
A	a	0, 5	5, 0
	b	5, 0	0, 5
	c	1, 1	1, 1

None ofA’s strategies are dominated in terms of purestrategies. However, in terms of expected outcomes,c isdominated by an equal mixture ofa andb. Thus, solutionprocedures analyzed earlier such as iterated removal of weakly orstrongly dominated strategies may provide different and incompatibleoutcomes, depending on whether mixed strategies are considered or not.No satisfactory logical analysis of the earlier kind seems to existfor this setting.

5.5 Further Challenges for a Logical Approach

Further challenges to the interplay of logic and probabilisticreasoning abound. By way of conclusion, here is a dimension that seemshard to capture in purely qualitative logical terms. A characteristicfeature of game- and decision-theoretic reasoning is that beliefs andpreferences areentangled in various ways (Liu2011). For instance, the crucial notionof expected utility entangles probability, representing beliefs, withutilities, standing for preferences. Players faced with probabilisticuncertainty about the opponent’s present and future actions areoften advised to maximize expected utility (vonNeumann & Morgenstern 1944; Savage 1954). Hence, even ifone has found qualitative counterparts for probabilistic belief andcardinal utility separately, entanglement poses the additionaldifficulty of merging these two qualitative analyses in a way thatmatches what the quantitative side achieves easily by forming somearithmetical combination of both components.

6. Gamification

The main topic of this entry is a logical approach to game theory,bringing classical notions and methods from logic to bear upon games.This project is sometimes called ‘logic of games’. Therealso is a converse direction of ‘logic as games’, wheregame theoretic concepts are employed to elucidate basic notions oflogic. This section presents a brief discussion on this direction as anatural counterpoint to the main lines of the entry. For an extensivesurvey see the entries onlogic and games andgames, abstraction and completeness.

6.1 Logic Games

Many notions in logic have been analyzed in game-theoretic terms

Evaluation games There are well-known two-playergames for evaluating a first-order formula \(\varphi\) within a givenlogical model. These games are played between Verifier and Falsifier,who can both test atomic assertions, and specify the value ofvariables from a given domain (Hintikka1973). The schedule of the game is determined from thesyntactic structure of the formula \(\varphi\). Disjunctions andexistential quantifiers require choices of the Verifier, conjunctionsand universal quantifiers of the Falsifier, and negations trigger arole switch between the two players. The result is the following matchbetween winning strategies and the ordinary semantic notion oftruth:

Formula \(\varphi\) is true in modelM under assignments iff the Verifier has a winning strategy in the associatedgame \(game(M, s, \varphi)\).

Correspondingly, Falsifier has a winning strategy if the formula\(\varphi\) is false in the model. Evaluation games turn out to be anextremely flexible tool. By suitably modulating rules and winningconventions, adequate evaluation games can be found for most logicalsystems. Doing so, however, can be a highly non-trivial task, aswitnessed by the intricate infinite ‘parity games’corresponding to fixed-point logics such as the modal μ-calculus(Venema 2008). For the present purpose,it should be noted that this style of analysis ties the very logicaloperations, conjunctions, disjunctions, modal operators, to naturalmoves in a game. Similarly, the notion of truth is linked to thefundamental game-theoretic notion of a strategy in an extensive formgame (cf.Section 2): a complex, structured object which may here be understood as a reasonor an explanation for the truth or falsity of the formula.

The links between both perspectives are so close, that validprinciples of logic come to express game-theoretic facts. Forinstance, after a little analysis, the law of excluded middle impliesthat always either Verifier or Falsifier has a winning strategy, cf.Section 2.4. In other words, logical evaluation games for classical logic aredetermined in the game-theoretic sense. In fact, This property extendsto most games for non-classical logics.

Further logic games Logic games exist for many otherpurposes. Ehrenfeucht-Fraïssé games serve model comparison(Ehrenfeucht 1961; Ebbinghaus & Flum1995), Lorenzen games perform proof analysis (Kamlah1973 [1984]) and tableau games executemodel construction (Hodges 1985). Ineach case, strategies in the game match important logical notions. InLorenzen dialogue games, for instance, winning strategies for theProponent of a claim correspond to proofs of that claim from premisesgranted by the Opponent, whereas winning strategies for the Opponentare constructions of counter-models. Thus, proofs and models, twoquite distinct notions in logic, co-exist within a single game.

There exists an alternative, game-theoretic way of interpreting theseconnective results. Suppose the game under study is fixed, andassociated with some sort of ‘game board’ representingmajor features of the game’s general state (think of Chess,though more abstract game boards may occur). Then the aboveequivalences suggest that winning strategies, i.e., a typicalgame-theoretic notion defined in terms of the complete extensive gametree, is equivalent to a simpler ‘invariant’ that can bedefined entirely in terms of some game board associated with thetree’s nodes. Identifying useful such invariants is a well-knownart in the analysis of concrete games. In terms of a main theme ofthis entry, invariants can live at different levels of representationassociated with a given class of games.

Game semantics One can view logic games as meredidactic devices analyzing logical notions that were alreadywell-understood. Or, in other terms, as offering a concrete way ofteaching logic that draws on game-theoretic intuitions. However, logicgames have more to offer. First of all, new logics are suggested bypursuing natural variations in winning conventions, moves, orscheduling within existing logic games. Moreover, viewing logicaloperations as game constructors suggests a new, refined view onlogical constants. Conjunction, for instance, now splits naturallyinto a sequential and a parallel version. Similar examples ofparallelism also exist in logics of computation. Moreover, associatingquantifiers with object picking, as in evaluation games, turnsquantifiers into special types of atomic games that connect to thefollowing formula by an abstract operation of game composition. Thegeneral logic of this abstract composition operation combined withpropositional operations of choice and switch has been showndecidable, providing a new decidable core logic inside first-orderlogic whose existence had not been suspected (vanBenthem 2014). Games, hence, can offer afresh perspective on existing logical systems.

A major source of independent, game-theoretic perspectives on logic isthe game semantics of computational logics (Abramsky1997). In this setting, the statusof logic games may change. Rather than being a mere pedagogical orexploratory device, to some, these games are considered the truemeaning of logical constants. In this setting, infinite games areoften the paradigm, standing for ongoing processes, rather than finiteterminating games. This shows in the importance of infinite paritygames for the modal μ-calculus, Venema(2008), as well as in the earlier-mentioned co-algebraicperspective on games. Baltag, van Benthem &Westerståhl (forthcoming) discuss the outside-inperspective of co-algebraic game semantics as a viable generalparadigm for linguistic meaning representing a significant shift instandard discussions of compositionality.

6.2 Other Logic-Related Games

The distinction between game logics and logic games is not alwayssharp. Recent literature has seen a number of games whose design isconnected to logic, yet they are not meant to analyze logical notionsper se. Many of these games are played on graphs, as underlying gameboards where players can perform different moves. Graph games arewidely used in the computational literature for analyzing logical andalgorithmic tasks ( Flum, Grädel &Wilke 2007). However, in this section, we merely highlight oneparticular kind of graph game for its fruitful connections with logicwhich suggests a number of unusual issues.

Example Sabotage Games.

Sabotage games were proposed to analyze algorithmic tasks in adversecircumstances. Consider the below network between some Europeancities:

This is a diagram of four cities in a ring around the fifth. The extended description (link in figure caption) will describe the tree.

Figure 24.ⓘ

It is easy to travel either way between Amsterdam and the German townof Saarbruecken. Now, let a malevolent Demon start cancelingconnections in the network. At every stage, let the Demon take out onelink, while the Traveler can afterwards follow one of the remaininglinks. This turns a one-agent planning problem into a two-playersabotage game. Zermelo-style reasoning shows that, from Saarbruecken,a German Traveler still has a winning strategy, while in Amsterdam,the Demon has the winning strategy against the Dutch Traveler, byfirst cutting a link close to Saarbruecken. The symmetry of theoriginal search problem is broken.

The sabotage game has been applied to a variety of scenarios,including learning (Gierasimczuk, Kurzen, &Velázquez-Quesada 2009), and communication networks(Aucher, van Benthem, & Grossi2018). On finite graphs, the game is clearly determined, withthe computational complexity of identifying who has a winning strategybeing Pspace-complete (Löding & Rohde2003).

The existence of this winning strategy can be expressed by afirst-order formula. More specifically, winning conditions can bedefined in a bimodal logic that combines a standard modality fortravel steps with a new modality for one-step arrow deletion,interpreted in models \(M = (W, R, V)\):

\(M, s \vDash [{-}]\varphi\quad\) For each edge \((u, v)\) in \(R:\,\, (W, R {-} \{(u. v)\}, V) \vDash \varphi\)

This logic fits the sabotage game closely. On top, it is a naturalfragment of the first-order language of graphs. Surprisingly, thislogic is undecidable (Löding & Rohde2003), making it one of the simplest examples of an undecidablemodal logic over arbitrary models.

Further graph games in a similar spirit exist, including the poisongame of Duchet and Meyniel (1993), wherethe Demon poisons nodes, rather than deleting edges. Extensive studiesof modal logics for changing graphs, and μ-calculi for defininggeneric solutions to graph games are given in Areces,Figueira et al. (2011); Areces, Fervari, andHoffmann (2015); and Aucher, van Benthem & Grossi (2018). Aclassification of graph games including the effects of complex goalformulas and imperfect information is found in vanBenthem and Liu (2019).

One perspective on such logics for reasoning about model change is thesemantic games approach ofSection 6.1. In standard evaluation games, the initial model does not change.Modalities for model change, however, require a process of formulaevaluation where the model of evaluation changes as, say, witnessesfor quantifiers are not replaced (unlike in standard semantics forfirst-order logic), or moves change facts by causing damage toaccessibility relations. In other contexts, similar modalities arejustified by physical measurements that change the phenomenon underinvestigation, (Hintikka 2002; Renardel 2001;Ågotnes & Wáng 2017), while another naturalexample are the dynamic-epistemic logics for information updatementioned in the discussion of game play (Baltag & Renne 2016). Such generalized form of semanticsare of independent logical interest.

Example Knowledge Games.

New logical games also arise naturally within the dynamics ofinformation, knowledge or belief as triggered by the process of gameplay (cf.Section 4). In particular, information update suggests conversation games betweenparticipants with similar or different goals. These games may becooperative, with players aiming to pool their information, therebyturning distributed knowledge into common knowledge (Meyer& van der Hoek 1995). But they canalso be competitive, say, when players strive to be the first to knowwhether some relevant proposition holds. Mixtures between both modesalso occur, for instance with some players aiming to communicate afact that outsiders should not learn about (vanDitmarsch 2003).

A concrete example are the ‘announcement games’ of Ågotnesand van Ditmarsch (2011).Players speak simultaneously and only once, while pursuing goal thatare specified as epistemic formulas. Speaking is modeled by publicannouncement, and players preferences are binary. They prefer finalmodels where their goal formula holds over those where it isfalse.

These games are conducted under imperfect information, as players maynot know the true state of their epistemic model. Accordingly, therelevant strategies need to be uniform. Players must say the samething in all states they cannot distinguish. In general, then, manysolution concepts produce mixed strategy outcomes. In fact, it can beshown that there exists simple announcement games without any uniqueequilibrium in pure strategies. However, there is a relevant role forlogic to play. Suppose that the goal statements are all‘universal’, i.e., constructed from literals by applyingonly conjunction, disjunction, knowledge operators, and dynamicmodalities with universal announcements. Truth of such formulas ispreserved when transitioning from a model to a submodel. Consequently,epistemic uncertainty becomes less harmful and knowledge games withuniversal goals have equilibria in pure strategies. Recently,knowledge games have been expanded further to include both questionsand answers as separate actions of issue change and information change(Ågotnes, van Benthem et al.2012).

Example Boolean games.

A third example of game design in between game logics and logic gamesare the Boolean games of Harrenstein et al.(2001) and Gutierrez, Harrenstein, andWooldridge (2015) that have been mentioned several timesalready. Each player is handed control over a subset of thepropositional variables, and can pick truth values for these at will.Using goals specified in temporal logics, these games can model alarge number of relevant scenarios of agency. By now, a growing bodyof work addresses various aspects of Boolean games including theircomputational characteristics (both single-shot and iterated), theirgame-theoretic properties and equilibria (Gutierrez,Harrenstein, & Wooldridge2015), and their connections with games played on socialnetworks (Seligman & Thompson 2015).Further discussion can be found in the entry oncoalitional powers.

6.3 Special Topics

Back and forth between game logics and logic gamesThe topic of this section suggests cycles between the two perspectiveson logic and games. Given a logical system, one can design logicalgames for it, which can then again be studied using some appropriategame logic. Conversely, given a game, one can introduce a logic fordescribing it, and then introduce evaluation games for that logic, andso on. Sometimes these cycles reach fixed-points, where, say, theevaluation game for a formula describing some game is isomorphic tothat game itself. But sometimes, the cycling continues. Fordiscussion, see Rebuschi (2006) and van Benthem(2014).

Imperfect information Logic games naturally supportimperfect information, where players do not have complete access towhat their opponents do. Epistemic variations can have far-reachingconsequences for the corresponding logics. A particularly prominentframework among this lines is theindependence friendly logic of Hintikka and Sandu (1989), see alsoHintikka and Sandu (1997) and Mann, Sandu, andSevenster (2011).

Argumentation games and graph games Another strand ofgame analysis with a connection to logic is the study of argumentationnetworks (Dung 1995; Caminada & Gabbay2009), with uses in AI and philosophy (Grossi2013; Shi 2018)

Computational logic The material in this section isclosely connected to games in computational logic, which serve toanalyze expressive power of languages. For relevant results andconnections with automata theory, see Grädel,Thomas, and Wilke (2002) andvan Benthem (2014).

Gaming and mechanism design Game design is awell-known aspect in the area of gaming (Rouse2000). Likewise, mechanism design is an established topic ingame theory (Nisan & Ronen 2001; Osborne& Rubinstein 1994). For connections between between logic,game design and planning, see (Löwe,Pacuit, & Witzel 2011; Löwe 2008).

7. Discussion and Further Directions

This entry presents an overview of current work at the interface oflogic and games. The topics surveyed fall in a number of strandsincluding current logical analysis of games in the broadest sense,contacts between logic and classical game theory, connections withprobability and with computation, and, lastly, the game-theoreticcontent of logic itself.

All this produced a perhaps bewildering variety of logical systems atthe interface with games. Yet, this entry also shows a certain unityin approach, since the same sort of (modal) logics turn out suitableto deal with quite different aspects of game structure and game play.Moreover, when we step back a little, broader perspectives across thespecific systems surveyed here arise.

One noteworthy phenomenon is the existence of two different styles oflogical analysis. Some logical systems ‘zoom in’ onparticular aspects of an activity or reasoning practice, providingmore detail than what is usually found in standard mathematical orphilosophical analysis. Other logical systems do the opposite, and‘zoom out’ to general patterns that may not have beenvisible at the more detailed level of the original practice.

Most of the logical systems in this entry are of the fine-grained‘zooming-in’ variety. Even so, coarse-grained logics ofthe ‘zooming-out’ variety are interesting, too, as theymay highlight laws or general patterns in social behavior that liebeyond the details attended to by games and game theory. One exampleis the current interest in logics for the abstract notion ofdependence (Väänänen 2007;Baltag 2016). Many forms of dependence and independencepermeate social life. Baltag and van Benthem(2021) present a simple modal base logic of functionaldependence and independence which fits this perspective, suggesting ananalysis of strategies in extensive games as dynamic devices thatcreate dependencies among players. Chen, Shi & Wang (2022, OtherInternet Resources) then combine this framework, applied to strategicgames, with the modal preference logic discussed above to find acommon logical structure underlying both competitive and cooperativegames. Another example of the logical search for generality in gamesis the proof-theoretic analysis of Hu andKaneko (2012) of the general postulates for social interactionin Johansen (1982).

The interface area of logic and games still isin statunascendi. Correspondingly, there are obvious gaps and desiderataon the logic side, which are reflected in the material in thisentry.

In particular, one fundamental theme are syntactic perspectives ongame-theoretic reasoning. Samples of a proof-theoretic style ofanalysis for play can be found in de Bruin(2005). More concretely, Zvesper(2010) analyzes classical results in epistemic game theory,(cf. Tan & Werlang 1988; Aumann1999), in terms of abstract modalities for belief andoptimality, showing how a few simple proof rules from modalμ-calculus can capture the essence of famous results in epistemicgame theory. Proof-theoretic aspects of logic have so far beenovershadowed by semantic analyses, although this situation is changingslowly (Artemov 2014; Kaneko 2002; Kaneko &Suzuki 2003). Model-based reasoning provides abstract semanticperspectives on games that can aid conceptual clarification, and thediscovery of general laws. But it might turn out to be proof theorythat governs the concrete reasoning used in the semantics, and thatmay be able to guide the context of justification in establishinggeneral facts about games.

A further contact with logic that has been ignored in this entry isthe rich interface between games and descriptive set theory (Woodin2010; Kanamori 2003).

It should be stressed once more that logic is not the only formaldiscipline that throws light on games. Quantitative probability entersthe study of games in many ways, both in classical and in evolutionarygame theory. The interface of logic and games may well profit from themany old and new contacts between logic and probability (Leitgeb2017; Lin & Kelly 2012;Harrison-Trainor, Holliday, & Icard 2016).

Another link that remained underrepresented in this entry arecomputational aspects. The study of games, play and players hasnatural connections with computation and agency in computer scienceand AI (Grädel, Thomas, & Wilke 2002;Abramsky 2008; Halpern 2013; Perea 2012; Brandenburger 2014).The proper perspective on what has been presented here may well turnout to be a triangle of interfaces between logic, games, andcomputation.

As for still broader connections, we have not done justice to alllinks between logic, games and philosophy, of which more are found inStalnaker (1996, 1999). The same is truefor links to linguistics and psychology (Clark2012). In this language-oriented connection, one should alsomention the work of Bjorndahl, Halpern, andPass (2017) on the natural language used in specifying gamesand reasoning about them, thus making game analysis moredescription-dependent.

Finally, the main thrust of this entry is theoretical andfoundational. However, there also is a more practical aspect to logicand games. Logic plays a role in cognitive psychology and experimentalgame theory, if only to identify testable hypotheses related to Theoryof Mind or strategic reasoning (Ghosh,Meijering, & Verbrugge 2014; Ghosh & Verbrugge 2018; Bicchieri1993; Fagin, Halpern et al. 1995). Lastly, some work at theinterface of logic and games suggests outreach to the world of actualparlor games (van Ditmarsch & Kooi 2015;van Benthem & Liu 2019).

All in all, the claim of this entry is a modest one. Logic and gamesform a natural combination, that may reveal interesting things whenpursued explicitly. Even so, too much logic may import too much of aformal apparatus, which may end up strangling the games perspective:logical systems are infinite machineries that can easily overwhelm aconcrete game of interest. In short, the contact has to be managedwith care.

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