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Ingame theory, acooperative orcoalitional game is agame with groups ofplayers who form binding "coalitions" with external enforcement of cooperative behavior (e.g. throughcontract law). This is different fromnon-cooperative games in which there is either no possibility to forge alliances or all agreements need to beself-enforcing (e.g. throughcredible threats).^[1]

Cooperative games are analysed by focusing on coalitions that can be formed, and the joint actions that groups can take and the resulting collective payoffs.^[2][3]

A cooperative game is given by specifying a value for every coalition. Formally, the coalitional game consists of a finite set of players${\displaystyle N}$, called thegrand coalition, and acharacteristic function${\displaystyle v:2^N\to \mathbb{R}}$^[4] from the set of all possible coalitions of players to a set of payments that satisfies${\displaystyle v(\mathrm{\varnothing })=0}$. The function describes how much collective payoff a set of players can gain by forming a coalition.

Cooperative game theory is a branch of game theory that deals with the study of games where players can form coalitions, cooperate with one another, and make binding agreements. The theory offers mathematical methods for analysing scenarios in which two or more players are required to make choices that will affect other players wellbeing.^[5]

• Common interests: In cooperative games, players share a common interest in achieving a specific goal or outcome. The players must identify and agree on a common interest to establish the foundation and reasoning for cooperation. Once the players have a clear understanding of their shared interest, they can work together to achieve it.^{[citation needed]}
• Necessary information exchange: Cooperation requires communication and information exchange among the players. Players must share information about their preferences, resources, and constraints to identify opportunities for mutual gain. By sharing information, players can better understand each other's goals and work towards achieving them together.^{[citation needed]}
• Voluntariness, equality, and mutual benefit: In cooperative games, players voluntarily come together to form coalitions and make agreements. The players must be equal partners in the coalition, and any agreements must be mutually beneficial. Cooperation is only sustainable if all parties feel they are receiving a fair share of the benefits.^{[citation needed]}
• Compulsory contract: In cooperative games, agreements between players are binding and mandatory. Once the players have agreed to a particular course of action, they have an obligation to follow through. The players must trust each other to keep their commitments, and there must be mechanisms in place to enforce the agreements. By making agreements binding and mandatory, players can ensure that they will achieve their shared goal.^{[citation needed]}

Let${\displaystyle S⊊N}$ be a non-empty coalition of players. Thesubgame${\displaystyle v_S:2^S\to \mathbb{R}}$ on${\displaystyle S}$ is naturally defined as

${\displaystyle v_S(T)=v(T),\mathrm{\forall }T\subseteq S.}$

In other words, we simply restrict our attention to coalitions contained in${\displaystyle S}$. Subgames are useful because they allow us to applysolution concepts defined for the grand coalition on smaller coalitions.

Characteristic functions are often assumed to besuperadditive (Owen 1995, p. 213). This means that the value of a union ofdisjoint coalitions is no less than the sum of the coalitions' separate values:

${\displaystyle v(S\cup T)\ge v(S)+v(T)}$ whenever${\displaystyle S,T\subseteq N}$ satisfy${\displaystyle S\cap T=\mathrm{\varnothing }}$.

Larger coalitions gain more:

${\displaystyle S\subseteq T\Rightarrow v(S)\le v(T)}$.

This follows fromsuperadditivity. i.e. if payoffs are normalized so singleton coalitions have zero value.

A coalitional gamev is consideredsimple if payoffs are either 1 or 0, i.e. coalitions are either "winning" or "losing".^[6]

Equivalently, asimple game can be defined as a collectionW of coalitions, where the members ofW are calledwinning coalitions, and the otherslosing coalitions.It is sometimes assumed that a simple game is nonempty or that it does not contain an empty set. However, in other areas of mathematics, simple games are also calledhypergraphs orBoolean functions (logic functions).

• A simple gameW ismonotonic if any coalition containing a winning coalition is also winning, that is, if${\displaystyle S\in W}$ and${\displaystyle S\subseteq T}$ imply${\displaystyle T\in W}$.
• A simple gameW isproper if the complement (opposition) of any winning coalition is losing, that is, if${\displaystyle S\in W}$ implies${\displaystyle N\setminus S\notin W}$.
• A simple gameW isstrong if the complement of any losing coalition is winning, that is, if${\displaystyle S\notin W}$ implies${\displaystyle N\setminus S\in W}$.
  • If a simple gameW is proper and strong, then a coalition is winning if and only if its complement is losing, that is,${\displaystyle S\in W}$ iff${\displaystyle N\setminus S\notin W}$. (Ifv is a coalitional simple game that is proper and strong,${\displaystyle v(S)=1-v(N\setminus S)}$ for anyS.)
• Aveto player (vetoer) in a simple game is a player that belongs to all winning coalitions. Supposing there is a veto player, any coalition not containing a veto player is losing. A simple gameW isweak (collegial) if it has a veto player, that is, if the intersection${\displaystyle \bigcap W:=\bigcap _{S\in W}S}$ of all winning coalitions is nonempty.
  • Adictator in a simple game is a veto player such that any coalition containing this player is winning. The dictator does not belong to any losing coalition. (Dictator games in experimental economics are unrelated to this.)
• Acarrier of a simple gameW is a set${\displaystyle T\subseteq N}$ such that for any coalitionS, we have${\displaystyle S\in W}$ iff${\displaystyle S\cap T\in W}$. When a simple game has a carrier, any player not belonging to it is ignored. A simple game is sometimes calledfinite if it has a finite carrier (even ifN is infinite).
• TheNakamura number of a simple game is the minimal number ofwinning coalitions with empty intersection. According to Nakamura's theorem, the number measures the degree of rationality; it is an indicator of the extent to which an aggregation rule can yield well-defined choices.

A few relations among the above axioms have widely been recognized, such as the following(e.g., Peleg, 2002, Section 2.1^[7]):

• If a simple game is weak, it is proper.
• A simple game is dictatorial if and only if it is strong and weak.

More generally, a complete investigation of the relation among the four conventional axioms(monotonicity, properness, strongness, and non-weakness), finiteness, and algorithmic computability^[8]has been made (Kumabe and Mihara, 2011^[9]),whose results are summarized in the Table "Existence of Simple Games" below.

The restrictions that various axioms for simple games impose on theirNakamura number were also studied extensively.^[11]In particular, a computable simple game without a veto player has a Nakamura number greater than 3 only if it is aproper andnon-strong game.

LetG be a strategic (non-cooperative) game. Then, assuming that coalitions have the ability to enforce coordinated behaviour, there are several cooperative games associated withG. These games are often referred to asrepresentations of G. The two standard representations are:^[12]

• The α-effective game associates with each coalition the sum of gains its members can 'guarantee' by joining forces. By 'guaranteeing', it is meant that the value is the max-min, e.g. the maximal value of the minimum taken over the opposition's strategies.
• The β-effective game associates with each coalition the sum of gains its members can 'strategically guarantee' by joining forces. By 'strategically guaranteeing', it is meant that the value is the min-max, e.g. the minimal value of the maximum taken over the opposition's strategies.

The main assumption in cooperative game theory is that the grand coalition${\displaystyle N}$ will form.^[13] The challenge is then to allocate the payoff${\displaystyle v(N)}$ among the players in some way. (This assumption is not restrictive, because even if players split off and form smaller coalitions, we can apply solution concepts to the subgames defined by whatever coalitions actually form.) Asolution concept is a vector${\displaystyle x\in {\mathbb{R}}^N}$ (or a set of vectors) that represents the allocation to each player. Researchers have proposed different solution concepts based on different notions of fairness. Some properties to look for in a solution concept include:

• Efficiency: The payoff vector exactly splits the total value:${\displaystyle \sum _{i\in N}{x}_i=v(N)}$.
• Individual rationality: No player receives less than what he could get on his own:${\displaystyle x_i\ge v(\{i\}),\mathrm{\forall }i\in N}$.
• Existence: The solution concept exists for any game${\displaystyle v}$.
• Uniqueness: The solution concept is unique for any game${\displaystyle v}$.
• Marginality: The payoff of a player depends only on the marginal contribution of this player, i.e., if these marginal contributions are the same in two different games, then the payoff is the same:${\displaystyle v(S\cup \{i\})=w(S\cup \{i\}),\mathrm{\forall }S\subseteq N\setminus \{i\}}$ implies that${\displaystyle x_i}$ is the same in${\displaystyle v}$ and in${\displaystyle w}$.
• Monotonicity: The payoff of a player increases if the marginal contribution of this player increase:${\displaystyle v(S\cup \{i\})\le w(S\cup \{i\}),\mathrm{\forall }S\subseteq N\setminus \{i\}}$ implies that${\displaystyle x_i}$ is weakly greater in${\displaystyle w}$ than in${\displaystyle v}$.
• Computational ease: The solution concept can be calculated efficiently (i.e. in polynomial time with respect to the number of players${\displaystyle |N|}$.)
• Symmetry: The solution concept${\displaystyle x}$ allocates equal payments${\displaystyle x_i=x_j}$ to symmetric players${\displaystyle i}$,${\displaystyle j}$. Two players${\displaystyle i}$,${\displaystyle j}$ aresymmetric if${\displaystyle v(S\cup \{i\})=v(S\cup \{j\}),\mathrm{\forall }S\subseteq N\setminus \{i,j\}}$; that is, we can exchange one player for the other in any coalition that contains only one of the players and not change the payoff.
• Additivity: The allocation to a player in a sum of two games is the sum of the allocations to the player in each individual game. Mathematically, if${\displaystyle v}$ and${\displaystyle \omega }$ are games, the game${\displaystyle (v+\omega )}$ simply assigns to any coalition the sum of the payoffs the coalition would get in the two individual games. An additive solution concept assigns to every player in${\displaystyle (v+\omega )}$ the sum of what he would receive in${\displaystyle v}$ and${\displaystyle \omega }$.
• Zero Allocation to Null Players: The allocation to a null player is zero. Anull player${\displaystyle i}$ satisfies${\displaystyle v(S\cup \{i\})=v(S),\mathrm{\forall }S\subseteq N\setminus \{i\}}$. In economic terms, a null player's marginal value to any coalition that does not contain him is zero.

An efficient payoff vector is called apre-imputation, and an individually rational pre-imputation is called animputation. Most solution concepts are imputations.

The stable set of a game (also known as thevon Neumann-Morgenstern solution (von Neumann & Morgenstern 1944)) was the first solution proposed for games with more than 2 players. Let${\displaystyle v}$ be a game and let${\displaystyle x}$,${\displaystyle y}$ be twoimputations of${\displaystyle v}$. Then${\displaystyle x}$dominates${\displaystyle y}$ if some coalition${\displaystyle S\ne \mathrm{\varnothing }}$ satisfies${\displaystyle x_i>y_i,\mathrm{\forall }i\in S}$ and${\displaystyle \sum _{i\in S}{x}_i\le v(S)}$. In other words, players in${\displaystyle S}$ prefer the payoffs from${\displaystyle x}$ to those from${\displaystyle y}$, and they can threaten to leave the grand coalition if${\displaystyle y}$ is used because the payoff they obtain on their own is at least as large as the allocation they receive under${\displaystyle x}$.

Astable set is a set ofimputations that satisfies two properties:

• Internal stability: No payoff vector in the stable set is dominated by another vector in the set.
• External stability: All payoff vectors outside the set are dominated by at least one vector in the set.

Von Neumann and Morgenstern saw the stable set as the collection of acceptable behaviours in a society: None is clearly preferred to any other, but for each unacceptable behaviour there is a preferred alternative. The definition is very general allowing the concept to be used in a wide variety of game formats.^{[citation needed]}

• A stable set may or may not exist (Lucas 1969), and if it exists it is typically not unique (Lucas 1992). Stable sets are usually difficult to find. This and other difficulties have led to the development of many other solution concepts.
• A positive fraction of cooperative games have unique stable sets consisting of thecore (Owen 1995, p. 240).
• A positive fraction of cooperative games have stable sets which discriminate${\displaystyle n-2}$ players. In such sets at least${\displaystyle n-3}$ of the discriminated players are excluded (Owen 1995, p. 240).

Let${\displaystyle v}$ be a game. Thecore of${\displaystyle v}$ is the set of payoff vectors

${\displaystyle C(v)=\left\{x\in {\mathbb{R}}^N:\sum _{i\in N}{x}_i=v(N);\sum _{i\in S}{x}_i\ge v(S),\mathrm{\forall }S\subseteq N\right\}.}$

In words, the core is the set ofimputations under which no coalition has a value greater than the sum of its members' payoffs. Therefore, no coalition has incentive to leave the grand coalition and receive a larger payoff.

• Thecore of a game may be empty (see theBondareva–Shapley theorem). Games with non-empty cores are calledbalanced.
• If it is non-empty, the core does not necessarily contain a unique vector.
• Thecore is contained in any stable set, and if the core is stable it is the unique stable set; see (Driessen 1988) for a proof.

For simple games, there is another notion of the core, when each player is assumed to have preferences on a set${\displaystyle X}$ of alternatives.Aprofile is a list${\displaystyle p=({\succ }_i^p{)}_{i\in N}}$ of individual preferences${\displaystyle {\succ }_i^p}$ on${\displaystyle X}$.Here${\displaystyle x{\succ }_i^{p}y}$ means that individual${\displaystyle i}$ prefers alternative${\displaystyle x}$to${\displaystyle y}$ at profile${\displaystyle p}$.Given a simple game${\displaystyle v}$ and a profile${\displaystyle p}$, adominance relation${\displaystyle {\succ }_v^p}$ is definedon${\displaystyle X}$ by${\displaystyle x{\succ }_v^{p}y}$ if and only if there is a winning coalition${\displaystyle S}$(i.e.,${\displaystyle v(S)=1}$) satisfying${\displaystyle x{\succ }_i^{p}y}$ for all${\displaystyle i\in S}$.Thecore${\displaystyle C(v,p)}$ of the simple game${\displaystyle v}$ with respect to the profile${\displaystyle p}$ of preferencesis the set of alternatives undominated by${\displaystyle {\succ }_v^p}$(the set of maximal elements of${\displaystyle X}$ with respect to${\displaystyle {\succ }_v^p}$):

${\displaystyle x\in C(v,p)}$ if and only if there is no${\displaystyle y\in X}$ such that${\displaystyle y{\succ }_v^{p}x}$.

TheNakamura number of a simple game is the minimal number of winning coalitions with empty intersection.Nakamura's theorem states that the core${\displaystyle C(v,p)}$ is nonempty for all profiles${\displaystyle p}$ ofacyclic (alternatively,transitive) preferencesif and only if${\displaystyle X}$ is finiteand the cardinal number (the number of elements) of${\displaystyle X}$ is less than the Nakamura number of${\displaystyle v}$.A variant by Kumabe and Mihara states that the core${\displaystyle C(v,p)}$ is nonempty for all profiles${\displaystyle p}$ of preferences that have amaximal elementif and only if the cardinal number of${\displaystyle X}$ is less than the Nakamura number of${\displaystyle v}$. (SeeNakamura number for details.)

Because thecore may be empty, a generalization was introduced in (Shapley & Shubik 1966). Thestrong${\displaystyle \epsilon }$-core for some number${\displaystyle \epsilon \in \mathbb{R}}$ is the set of payoff vectors

${\displaystyle C_{\epsilon }(v)=\left\{x\in {\mathbb{R}}^N:\sum _{i\in N}{x}_i=v(N);\sum _{i\in S}{x}_i\ge v(S)-\epsilon ,\mathrm{\forall }S\subseteq N\right\}.}$

In economic terms, the strong${\displaystyle \epsilon }$-core is the set of pre-imputations where no coalition can improve its payoff by leaving the grand coalition, if it must pay a penalty of${\displaystyle \epsilon }$ for leaving.${\displaystyle \epsilon }$ may be negative, in which case it represents a bonus for leaving the grand coalition. Clearly, regardless of whether thecore is empty, the strong${\displaystyle \epsilon }$-core will be non-empty for a large enough value of${\displaystyle \epsilon }$ and empty for a small enough (possibly negative) value of${\displaystyle \epsilon }$. Following this line of reasoning, theleast-core, introduced in (Maschler, Peleg & Shapley 1979), is the intersection of all non-empty strong${\displaystyle \epsilon }$-cores. It can also be viewed as the strong${\displaystyle \epsilon }$-core for the smallest value of${\displaystyle \epsilon }$ that makes the set non-empty (Bilbao 2000).

TheShapley value is the unique payoff vector that is efficient, symmetric, and satisfies monotonicity.^[14] It was introduced byLloyd Shapley (Shapley 1953) who showed that it is the unique payoff vector that is efficient, symmetric, additive, and assigns zero payoffs to dummy players. The Shapley value of asuperadditive game is individually rational, but this is not true in general. (Driessen 1988)

Let${\displaystyle v:2^N\to \mathbb{R}}$ be a game, and let${\displaystyle x\in {\mathbb{R}}^N}$ be an efficient payoff vector. Themaximum surplus of playeri over playerj with respect tox is

${\displaystyle s_{ij}^v(x)=max\left\{v(S)-\sum _{k\in S}{x}_k:S\subseteq N\setminus \{j\},i\in S\right\},}$

the maximal amount playeri can gain without the cooperation of playerj by withdrawing from the grand coalitionN under payoff vectorx, assuming that the other players ini's withdrawing coalition are satisfied with their payoffs underx. The maximum surplus is a way to measure one player's bargaining power over another. Thekernel of${\displaystyle v}$ is the set ofimputationsx that satisfy

• ${\displaystyle (s_{ij}^v(x)-s_{ji}^v(x))\times (x_j-v(j))\le 0}$, and
• ${\displaystyle (s_{ji}^v(x)-s_{ij}^v(x))\times (x_i-v(i))\le 0}$

for every pair of playersi andj. Intuitively, playeri has more bargaining power than playerj with respect toimputationx if${\displaystyle s_{ij}^v(x)>s_{ji}^v(x)}$, but playerj is immune to playeri's threats if${\displaystyle x_j=v(j)}$, because he can obtain this payoff on his own. The kernel contains allimputations where no player has this bargaining power over another. This solution concept was first introduced in (Davis & Maschler 1965).

TheHarsanyi dividend (named afterJohn Harsanyi, who used it to generalize theShapley value in 1963^[15]) identifies the surplus that is created by a coalition of players in a cooperative game. To specify this surplus, the worth of this coalition is corrected by subtracting the surplus that was already created by subcoalitions. To this end, the dividend${\displaystyle d_v(S)}$ of coalition${\displaystyle S}$ in game${\displaystyle v}$ is recursively determined by

An explicit formula for the dividend is given by$d_v(S)=\sum _{T\subseteq S}(-1{)}^{|S\setminus T|}v(T)$. The function${\displaystyle d_v:2^N\to \mathbb{R}}$ is also known as theMöbius inverse of${\displaystyle v:2^N\to \mathbb{R}}$.^[16] Indeed, we can recover${\displaystyle v}$ from${\displaystyle d_v}$ by help of the formula$v(S)=d_v(S)+\sum _{T⊊S}{d}_v(T)$.

Harsanyi dividends are useful for analyzing both games and solution concepts, e.g. theShapley value is obtained by distributing the dividend of each coalition among its members, i.e., the Shapley value${\displaystyle {\varphi }_i(v)}$ of player${\displaystyle i}$ in game${\displaystyle v}$ is given by summing up a player's share of the dividends of all coalitions that she belongs to,${\varphi }_i(v)=\sum _{S\subset N:i\in S}{d}_v(S)/|S|$.

Let${\displaystyle v:2^N\to \mathbb{R}}$ be a game, and let${\displaystyle x\in {\mathbb{R}}^N}$ be a payoff vector. Theexcess of${\displaystyle x}$ for a coalition${\displaystyle S\subseteq N}$ is the quantity${\displaystyle v(S)-\sum _{i\in S}{x}_i}$; that is, the gain that players in coalition${\displaystyle S}$ can obtain if they withdraw from the grand coalition${\displaystyle N}$ under payoff${\displaystyle x}$ and instead take the payoff${\displaystyle v(S)}$. Thenucleolus of${\displaystyle v}$ is theimputation for which the vector of excesses of all coalitions (a vector in${\displaystyle {\mathbb{R}}^{2^N}}$) is smallest in theleximin order. The nucleolus was introduced in (Schmeidler 1969).

(Maschler, Peleg & Shapley 1979) gave a more intuitive description: Starting with the least-core, record the coalitions for which the right-hand side of the inequality in the definition of${\displaystyle C_{\epsilon }(v)}$ cannot be further reduced without making the set empty. Continue decreasing the right-hand side for the remaining coalitions, until it cannot be reduced without making the set empty. Record the new set of coalitions for which the inequalities hold at equality; continue decreasing the right-hand side of remaining coalitions and repeat this process as many times as necessary until all coalitions have been recorded. The resulting payoff vector is the nucleolus.

• Although the definition does not explicitly state it, the nucleolus is always unique. (See Section II.7 of (Driessen 1988) for a proof.)
• If the core is non-empty, the nucleolus is in the core.
• The nucleolus is always in the kernel, and since the kernel is contained in the bargaining set, it is always in the bargaining set (see (Driessen 1988) for details.)

Introduced byShapley in (Shapley 1971), convex cooperative games capture the intuitive property some games have of "snowballing". Specifically, a game isconvex if its characteristic function${\displaystyle v}$ issupermodular:

${\displaystyle v(S\cup T)+v(S\cap T)\ge v(S)+v(T),\mathrm{\forall }S,T\subseteq N.}$

It can be shown (see, e.g., Section V.1 of (Driessen 1988)) that thesupermodularity of${\displaystyle v}$ is equivalent to

${\displaystyle v(S\cup \{i\})-v(S)\le v(T\cup \{i\})-v(T),\mathrm{\forall }S\subseteq T\subseteq N\setminus \{i\},\mathrm{\forall }i\in N;}$

that is, "the incentives for joining a coalition increase as the coalition grows" (Shapley 1971), leading to the aforementioned snowball effect. For cost games, the inequalities are reversed, so that we say the cost game isconvex if the characteristic function issubmodular.

Convex cooperative games have many nice properties:

• Supermodularity trivially impliessuperadditivity.
• Convex games aretotally balanced: Thecore of a convex game is non-empty, and since any subgame of a convex game is convex, thecore of any subgame is also non-empty.
• A convex game has a unique stable set that coincides with itscore.
• TheShapley value of a convex game is the center of gravity of itscore.
• Anextreme point (vertex) of thecore can be found in polynomial time using thegreedy algorithm: Let${\displaystyle \pi :N\to N}$ be apermutation of the players, and let${\displaystyle S_i=\{j\in N:\pi (j)\le i\}}$ be the set of players ordered${\displaystyle 1}$ through${\displaystyle i}$ in${\displaystyle \pi }$, for any${\displaystyle i=0,\dots ,n}$, with${\displaystyle S_0=\mathrm{\varnothing }}$. Then the payoff${\displaystyle x\in {\mathbb{R}}^N}$ defined by${\displaystyle x_i=v(S_{\pi (i)})-v(S_{\pi (i)-1}),\mathrm{\forall }i\in N}$ is a vertex of thecore of${\displaystyle v}$. Any vertex of thecore can be constructed in this way by choosing an appropriatepermutation${\displaystyle \pi }$.

Submodular andsupermodular set functions are also studied incombinatorial optimization. Many of the results in (Shapley 1971) have analogues in (Edmonds 1970), wheresubmodular functions were first presented as generalizations ofmatroids. In this context, thecore of a convex cost game is called thebase polyhedron, because its elements generalize base properties ofmatroids.

However, the optimization community generally considerssubmodular functions to be the discrete analogues of convex functions (Lovász 1983), because the minimization of both types of functions is computationally tractable. Unfortunately, this conflicts directly withShapley's original definition ofsupermodular functions as "convex".

Corporate strategic decisions can develop and create value through cooperative game theory.^[17] This means that cooperative game theory can become the strategic theory of the firm, and different CGT solutions can simulate different institutions.

• Consensus decision-making
• Coordination game
• Intra-household bargaining
• Hedonic game
• Linear production game
• Minimum-cost spanning tree game - a class of cooperative games.

• Bilbao, Jesús Mario (2000),Cooperative Games on Combinatorial Structures, Kluwer Academic Publishers,ISBN 9781461543930

• Davis, M.;Maschler, M. (1965), "The kernel of a cooperative game",Naval Research Logistics Quarterly,12 (3):223–259,doi:10.1002/nav.3800120303

• Driessen, Theo (1988),Cooperative Games, Solutions and Applications, Kluwer Academic Publishers,ISBN 9789401577878

• Edmonds, Jack (1970), "Submodular functions, matroids and certain polyhedra", in Guy, R.; Hanani, H.; Sauer, N.; Schönheim, J. (eds.),Combinatorial Structures and Their Applications, New York: Gordon and Breach, pp. 69–87

• Lovász, László (1983), "Submodular functions and convexity", in Bachem, A.;Grötschel, M.; Korte, B. (eds.),Mathematical Programming—The State of the Art, Berlin: Springer, pp. 235–257

• Leyton-Brown, Kevin; Shoham, Yoav (2008),Essentials of Game Theory: A Concise, Multidisciplinary Introduction, San Rafael, CA: Morgan & Claypool Publishers,ISBN 978-1-59829-593-1. An 88-page mathematical introduction; see Chapter 8.Free online(subscription required)Archived 2000-08-15 at theWayback Machine at many universities.

• Lucas, William F. (1969), "The Proof That a Game May Not Have a Solution",Transactions of the American Mathematical Society,136:219–229,doi:10.2307/1994798,JSTOR 1994798.

• Lucas, William F. (1992), "Von Neumann-Morgenstern Stable Sets", inAumann, Robert J.;Hart, Sergiu (eds.),Handbook of Game Theory, Volume I, Amsterdam:Elsevier, pp. 543–590

• Luce, R.D. andRaiffa, H. (1957)Games and Decisions: An Introduction and Critical Survey, Wiley & Sons. (see Chapter 8).
• Maschler, M.; Peleg, B.;Shapley, Lloyd S. (1979), "Geometric properties of the kernel, nucleolus, and related solution concepts",Mathematics of Operations Research,4 (4):303–338,doi:10.1287/moor.4.4.303

• Osborne, M.J. andRubinstein, A. (1994)A Course in Game Theory, MIT Press (see Chapters 13,14,15)
• Moulin, Herve (1988),Axioms of Cooperative Decision Making (1st ed.), Cambridge:Cambridge University Press,ISBN 978-0-521-42458-5

• Owen, Guillermo (1995),Game Theory (3rd ed.), San Diego:Academic Press,ISBN 978-0-12-531151-9

• Schmeidler, D. (1969), "The nucleolus of a characteristic function game",SIAM Journal on Applied Mathematics,17 (6):1163–1170,doi:10.1137/0117107.

• Shapley, Lloyd S. (1953), "A value for${\displaystyle n}$-person games", in Kuhn, H.; Tucker, A.W. (eds.),Contributions to the Theory of Games II, Princeton, New Jersey: Princeton University Press, pp. 307–317

• Shapley, Lloyd S. (18 March 1952),A Value for N-Person Games, Santa Monica, California: The RAND Corporation

• Shapley, Lloyd S. (1971), "Cores of convex games",International Journal of Game Theory,1 (1):11–26,doi:10.1007/BF01753431,S2CID 123385556

• Shapley, Lloyd S.; Shubik, M. (1966), "Quasi-cores in a monetary economy with non-convex preferences",Econometrica,34 (4):805–827,doi:10.2307/1910101,JSTOR 1910101

• Shoham, Yoav; Leyton-Brown, Kevin (2009),Multiagent Systems: Algorithmic, Game-Theoretic, and Logical Foundations, New York:Cambridge University Press,ISBN 978-0-521-89943-7. A comprehensive reference from a computational perspective; see Chapter 12.Downloadable free online.

• von Neumann, John;Morgenstern, Oskar (1944),"Theory of Games and Economic Behavior",Nature,157 (3981), Princeton:Princeton University Press: 172,Bibcode:1946Natur.157..172R,doi:10.1038/157172a0,S2CID 29754824

• Yeung, David W.K. and Leon A. Petrosyan. Cooperative Stochastic Differential Games (Springer Series in Operations Research and Financial Engineering), Springer, 2006. Softcover-ISBN 978-1441920942.
• Yeung, David W.K. and Leon A. Petrosyan. Subgame Consistent Economic Optimization: An Advanced Cooperative Dynamic Game Analysis (Static & Dynamic Game Theory: Foundations & Applications), Birkhäuser Boston; 2012.ISBN 978-0817682613

• "Cooperative game",Encyclopedia of Mathematics,EMS Press, 2001 [1994]

• Cooperative games
• Game theory
• Mathematical and quantitative methods (economics)

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