Evolutionary game theory originated as an application of themathematical theory of games to biological contexts, arising from therealization that frequency dependent fitness introduces a strategicaspect to evolution. Recently, however, evolutionary game theory hasbecome of increased interest to economists, sociologists, andanthropologists--and social scientists in general--as well asphilosophers. The interest among social scientists in a theory withexplicit biological roots derives from three facts. First, the‘evolution’ treated by evolutionary game theory need notbe biological evolution. ‘Evolution’ may, in this context,often be understood ascultural evolution, where this refersto changes in beliefs and norms over time. Second, the rationalityassumptions underlying evolutionary game theory are, in many cases,more appropriate for the modelling of social systems than thoseassumptions underlying the traditional theory of games. Third,evolutionary game theory, as an explicitly dynamic theory, provides animportant element missing from the traditional theory. In the prefacetoEvolution and the Theory of Games, Maynard Smith notesthat “[p]aradoxically, it has turned out that game theory ismore readily applied to biology than to the field of economicbehaviour for which it was originally designed.” It is perhapsdoubly paradoxical, then, that the subsequent development ofevolutionary game theory has produced a theory which holdsgreat promise for social scientists, and is as readily applied to thefield of economic behaviour as that for which it was originallydesigned.

• 1. Historical Development
• 2. Two Approaches to Evolutionary Game Theory
  • 2.1 Definitions of evolutionary stability
  • 2.2 Dynamic concepts of evolutionary stability
• 3. Dynamics, Stability, and Rational Outcomes
• 4. Why Evolutionary Game Theory?
  • 4.1 The equilibrium selection problem
  • 4.2 The problem of hyperrational agents
  • 4.3 The lack of a dynamical theory in the traditional theory of games
• 5. Applications of Evolutionary Game Theory
  • 5.1 A sense of fairness
  • 5.2 The emergence of language.
• 6. Philosophical Problems of Evolutionary Game Theory
  • 6.1 The meaning of fitness in cultural evolutionary interpretations
  • 6.2 The explanatory irrelevance of evolutionary game theory
  • 6.3 The value-ladenness of evolutionary game theoretic explanations
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. Historical Development

Evolutionary game theory was first developed by R. A. Fisher [seeThe Genetic Theory of Natural Selection (1930)] in hisattempt to explain the approximate equality of the sex ratio inmammals. The puzzle Fisher faced was this: why is it that the sexratio is approximately equal in many species where the majority ofmales never mate? (See, for example, the Northern elephant sealMirounga angustirostris.) In these species, the non-matingmales would seem to be excess baggage carried around by the rest ofthe population, having no real use. Fisher realized that if we measureindividual fitness in terms of the expected number ofgrandchildren, then individual fitness depends on thedistribution of males and females in the population. When there is agreater number of females in the population, males have a higherindividual fitness; when there are more males in the population,females have a higher individual fitness. Fisher pointed out that, insuch a situation, the evolutionary dynamics lead to the sex ratiobecoming fixed at equal numbers of males and females. The fact thatindividual fitness depends upon the relative frequency of males andfemales in the population introduces a strategic element toevolution.

Fisher’s argument can be understood game theoretically, but hedid not state it in those terms. In 1961, R. C. Lewontin made thefirst explicit application ofgame theory to evolutionary biology in “Evolution and the Theory ofGames” (not to be confused with the Maynard Smith work of thesame name). In 1972, Maynard Smith first introduced the concept of anevolutionarily stable strategy (hereafter ESS) in the chapter“Game Theory and the Evolution of Fighting.” However, itwas the publication of “The Logic of Animal Conflict,” byMaynard Smith and Price in 1973 that introduced the concept of an ESSinto widespread circulation. In 1982, Maynard Smith’s seminaltextEvolution and the Theory of Games appeared, followedshortly thereafter by Robert Axelrod’s famous workTheEvolution of Cooperation in 1984. Since then, there has been averitable explosion of interest by economists and social scientists inevolutionary game theory (see the bibliography below).

Initially, it was thought that evolutionary game theory might providean inroad into solving theequilibrium selection problem oftraditional game theory. Although the fundamental solution concept oftraditional game theory, theNash equilibrium, had thedesirable property of always existing for any game with a finitenumber of players and strategies, provided that mixed strategies wereallowed, it had several deficencies. A Nash equilibrium was notguaranteed to be unique (sometimes even with uncountable many Nashequilibria existing), did not always seem to correspond to areasonable outcome (see Hargreaves Heap and Varoufakis, 2004), andoccasionally conflicted with people’s intuitions as to what ought tocount as arational outcome. In contrast, it could be shownthat a completely mixed evolutionarily stable strategy was unique,that there were at most only a finite number of evolutionarily stablestrategies, and that several intuitive definitions of evolutionarilystability were equivalent to the original definition of Maynard Smithand Price.

It was soon realised that evolutionary game theory itself had problemsstructurally similar to that of the equilibrium selection problem.Several competing definitions of evolutionary stability were putforward, each of which had certain intuitive merit. In addition, asthe connection between thestatic anddynamicapproaches to evolutionary game theory were explored in detail, it wasfound that there was, at best, an imperfect fit between staticconcepts of evolutionary stability and that of dynamic stability.Furthermore, dynamical models of evolutionary game theory led tooutcomes which were expresselyirrational from the point ofview of traditional game theory, such as the preservation ofstrictly dominated strategies.

2. Two Approaches to Evolutionary Game Theory

There are two approaches to evolutionary game theory. The firstapproach derives from the work of Maynard Smith and Price and employsthe concept of an evolutionarily stable strategy as the principal toolof analysis. The second approach constructs an explicit model of theprocess by which the frequency of strategies change in the populationand studies properties of the evolutionary dynamics within thatmodel.

The first approach can thus be thought of as providing a staticconceptual analysis of evolutionary stability. “Static”because, although definitions of evolutionary stability are given, thedefinitions advanced do not typically refer to the underlying processby which behaviours (or strategies) change in the population. Thesecond approach, in contrast, does not attempt to define a notion ofevolutionary stability: once a model of the population dynamics hasbeen specified, all of the standard stability concepts used in theanalysis of dynamical systems can be brought to bear.

2.1 Definitions of evolutionary stability

In game theory, the main solution concept is theNashequilibrium. A Nash equilibrium is a profile of strategies (thatis, an assignment of strategies to each player) which is a mutual bestresponse, meaning that no player has any incentive to deviate fromtheir chosen strategy.

To see why the traditional game theoretic solution concept of a Nashequilibrium is too weak to capture the notion of evolutionarystability, consider the game of figure 1. There are two Nashequilibria in pure strategies: \( (S_1, S_1) \) and \( (S_2, S_2) \).Since a Nash equilibrium is a set of mutual best responses, no playercanimprove their payoff by adopting a different strategy,but a Nash equilibrium allows for the possibility that a player whodeviates from their equilibrium strategy receives thesamepayoff. This is the case for the \((S_2, S_2)\) equilibrium. And thatis why a Nash equilibrium does not suffice for evolutionary stability:it allows for the possibility of drift away from the equilibrium,eventually leading to the replacement of the incumbent strategy.

To see this, suppose that a population of individuals all followed thestrategy \(S_2\). If a mutant appeared who played the strategy\(S_1\), the payoff of the \(S_1\)-mutant would be the same as therest of population, and hence there would be no selection pressureagainst the mutant. If a second mutant appeared, the payoff earned byan \(S_1-S_1\) interaction would yield a fitness to both greater thanthe average fitness of the population. This would allow the \(S_1\)mutant to spread and eventually take over the rest of thepopulation.

Astrict Nash equilibrium is one where any unilateraldeviation from a player’s equilibrium strategy leaves that playerworse off. Although a strict Nash equilibrium does intuitively captureone sense of evolutionary stability (it can be thought of as a kind of“local optimum”), it can also be shown that a strict Nash equilibriumis too strong to capture the idea of evolutionary stability, ingeneral.

To see this, consider the Hawk-Dove game, analyzed by Maynard Smithand Price in their 1973 paper “The Logic of AnimalConflict.” In this game, two individuals compete for a resourceof a fixed value \(V\). (In biological contexts, the value \(V\) ofthe resource corresponds to an increase in the Darwinian fitness ofthe individual who obtains the resource; in a cultural context, thevalue \(V\) of the resource would need to be given an alternateinterpretation more appropriate to the specific model at hand.) Eachindividual follows exactly one of two strategies described below:

Now assume the following:

1. Whenever two players both initiate aggressive behaviour, conflicteventually results and both are equally likely to be injured.
2. The conflict reduces the individual fitness of the injured partyby some constant value \(C\).
3. When a Hawk meets a Dove, the Dove immediately retreats and theHawk obtains the resource.
4. When two Doves meet the resource is shared equally betweenthem.

Given this, the fitness payoffs for the Hawk-Dove game are summarizedaccording to the following matrix:

The Hawk-Dove game has no Nash equilibria in pure strategies andexactly one Nash equilibrium in mixed strategies. The mixed strategyNash equilibrium has both individuals playing Hawk with probability\(\frac{V}{C}\) and Dove with probability \(1-\frac{V}{C}\). Denotethis strategy by \(\sigma\). According to the fundamental theorem ofmixed-strategy Nash equilibria (see Gintis, 2009), it is the case that\[ \pi(\text{Hawk} \mid \sigma) = \pi(\text{Dove} \mid \sigma) =\pi(\sigma \mid \sigma), \] where “\(\pi( x \mid y)\)” denotes thepayoff obtained when playing strategy \(x\) against someone using thestrategy \(y\). From this, it follows that forany othermixed strategy \(\mu\) it is the case that \(\pi( \mu \mid \sigma ) =\pi(\sigma \mid \sigma)\), and so the Nash equilibrium is notstrict. Yet a population where everyone follows the strategy\(\sigma\) is still nevertheless able to resist invasion, for it canbe shown that \(\pi(\sigma \mid \mu) > \pi(\mu \mid \mu)\). Thatis, the incumbent strategy \(\sigma\) receives a higher payoff whenplayed against any mutant strategy \(\mu\), than the mutant strategyreceives when played against itself.

These considerations lead Maynard Smith (1982) to propose thefollowing definition:^[1]

Definition. A strategy \(\sigma\) is anevolutionarily stable strategy (ESS) if and only if for allother strategies \(\mu \neq \sigma\) it is the case thateither \(\pi(\sigma \mid \sigma) \gt \pi(\mu \mid \sigma)\)or that \(\pi(\sigma \mid \sigma) = \pi(\mu \mid \sigma)\)and \(\pi(\sigma \mid \mu) \gt \pi(\mu \mid \mu)\).

Or alternatively:

Definition. A strategy \(\sigma\) is anevolutionarily stable strategy (ESS) if and only if, for allother strategies \(\mu \neq \sigma\),

1. \(\pi(\sigma \mid \sigma) \geq \pi(\mu\mid\sigma)\)
2. If \(\pi(\sigma \mid \sigma) = \pi(\mu \mid \sigma)\), then \(\pi(\sigma \mid \mu) \gt \pi( \mu \mid \mu) \).

The second definition, while trivially logically equivalent to thefirst, has the advantage of making it clear that every evolutionarilystable strategy is also a Nash equilibrium. The first definition, fromMaynard Smith, has the advantage of making it clear that every strictNash equilibrium is also evolutionarily stable.

From this, we see that an evolutionarily stable strategy is a Nashequilibrium with an additional second-order stability criterion. It isa proper strengthening of the Nash equilibrium concept, becausewhereas every game with finitely many players and a finite number, ofstrategies has at least one Nash equilibrium (if mixed strategies areallowed), not every game has an evolutionarily stable strategy. It iseasily shown that the game of Rock-Scissors-Paper, shown in figure 3,does not have an evolutionarily stable strategy. (The only potentialcandidate for being evolutionarily stable is the Nash equilibriummixed strategy \(\sigma\) which assigns equal probability to all threepure strategies. But since \( \pi( \sigma \mid \sigma) = \pi(\text{Rock} \mid \sigma) \) and \(\pi(\sigma \mid \text{Rock}) = \pi(\text{Rock} \mid \text{Rock})\), it is not evolutionarily stable.)

Despite the fact that evolutionarily stable strategies do not alwaysexist, one advantage of the Maynard Smith and Price definition of anevolutionarily stable strategy is that it can be shown to beequivalent to two other concepts of evolutionary stability which weredefined. Since these three definitions of evolutionary stability arenot obviously the same, the fact that they turn out to be logicallyequivalent is interesting. We briefly discuss these other conceptsbelow, before stating the equivalence result.

To begin, recall how in the discussion above regarding the game offigure 1, we appealed informally to the idea of a mutant attempting toinvade a population, all of which follow a single incumbent strategy.To make this idea precise, we need to introduce some notation: let\(\mu, \sigma\) be two strategies and let \(0 \lt \epsilon \lt 1\).Then \(\epsilon \mu + (1-\epsilon) \sigma\) denotes the strategy whichplays \(\mu\) with probability \(\epsilon\) and plays \(\sigma\) withprobability \(1-\epsilon\). However, we may also interpret this asrepresenting the strategy used by aplayer chosen at randomfrom a large population in which the majority \((1-\epsilon)\) followthe strategy \(\sigma\) and a minority (\(\epsilon\)) follow thestrategy \(\mu\) which is attempting to invade.

Definition. A strategy \(\sigma\) hasauniform invasion barrier if there exists an \(\bar\epsilon\gt 0\) such that for all \(\mu \neq \sigma\) and \(\epsilon \in(0,\bar\epsilon)\), \[ \pi( \sigma \mid \epsilon \mu +(1-\epsilon)\sigma) \gt \pi( \mu \mid \epsilon \mu + (1-\epsilon)\sigma). \]

A uniform invasion barrier is a natural concept of evolutionarystability. It says that, when a population all follows the samestrategy \(\sigma\), except for a small proportion of mutants, all ofwhom follow asingle strategy \(\mu\), the incumbent strategy\(\sigma\) has a strictly higher expected fitness in the mixedpopulation than the invading strategy \(\mu\).^[2] Hence there would be selection against the invading strategy and, ina sufficiently large population with a sufficiently small number ofmutants, \(\sigma\) would be evolutionarily stable.

A second concept of evolutionary stability draws upon the intuitionthat a stable strategy should rule out the possibility of drift. Thatis, after all, what created problems for the second Nash equilibria inpure strategies for the game of figure 1. One way of characterisingthis is as follows.

Definition. A strategy \(\sigma\) is said to belocally superior if there exists a neighbourhood \(U\) around\(\sigma\) such that for all strategies \(\mu \in U\), where \(\mu\neq\sigma\), it is the case that \(\pi( \sigma \mid \mu) > \pi (\mu\mid \mu)\).

One can then prove the following:

Theorem (Hofbauer et al., 1979) The following areequivalent:

1. \(\sigma\) is an evolutionarily stable strategy.
2. \(\sigma\) has a uniform invasion barrier.
3. \(\sigma\) is locally superior.

In the years following the original work of Maynard Smith and Price,alternate analytic solution concepts have been proposed forevolutionary game theory. One such alternative is the idea of anevolutionarily stable set (see Thomas 1984, 1985a,b).Consider, by way of motivation, the game shown in figure 4. In thatgame, there are no evolutionarily stable strategies, since both\(S_1\) and \(S_2\) receive the same payoff when played against eachother. However, any population containing amix of \(S_1\)and \(S_2\) is, in a sense, stable: although drift will certainlyoccur regarding the exact proportion of \(S_1\) and \(S_2\),regardless of the specific mix the population will still tend to driveout any \(S_3\) or \(S_4\) mutants due to the differences in expectedfitness.

Other analytic solution concepts exist, as well. Swinkels (1992)introduced the idea of anequilibrium evolutionarily stableset, providing a further refinement of the idea of anevolutionarily stable set. (Every evolutionarily stable set containedsome equilibrium evolutionarily stable set, but not every equilibriumevolutionarily stable set was an evolutionarily stable set.) Hence wesee that the search for static solution concepts which suffice forcapturing the idea of evolutionary stability encounters a problemstructurally similar to that of the equilibrium selection problem intraditional game theory: there are multiple competing concepts ofevolutionary stability, all of which have some intuitive claim toplausibility.

2.2 Dynamic concepts of evolutionary stability

As an example of the second approach, consider the well-knownPrisoner’s Dilemma. In this game, individuals choose one of twostrategies, typically called “Cooperate” and“Defect.” Here is the general form of the payoff matrixfor the prisoner’s dilemma:

In figure 5, the payoffs are assumed to satisfy the ordering \(T \gt R\gt P \gt S\) and \(\frac{T + S}2 \lt R\). The latter requirement,although often omitted in discussions of the Prisoner’s Dilemma,ensures that in the context of an indefinitely repeated game there isno net overall advantage in the players alternating betweenCooperate-Defect and Defect-Cooperate.

How will a population of individuals that plays the Prisoner’sDilemma evolve over time? We cannot answer that question withoutintroducing a few assumptions concerning the nature of the population.First, assume that the population is quite large and that theprobability of interacting with a Cooperator or Defector equals theproportion of the population following that strategy. This allows usto represent the state of the population by simply keeping track ofwhat proportion follow the strategies Cooperate and Defect. Let\(p_C\) and \(p_D\) denote these proportions. Denote the expectedfitness of cooperators and defectors by \(W_C\) and \(W_D\),respectively, and let \(\overline{W}\) denote the average fitness ofthe entire population. (These quantities may vary over time; thetime-dependency has been suppressed for clarity of notation.) Giventhese assumptions, the values of \(W_C, W_D\), and \(\overline{W}\)can be expressed in terms of the population proportions and payoffvalues as follows, where \(F_0\) stands for the base fitness level ofan individual prior to any interaction:

Second, assume that the proportion of the population following thestrategies Cooperate and Defect in the next generation is related tothe proportion of the population following the strategies Cooperateand Defect in the current generation according to the followingrule:

The justification for these transition rules is as follows: if \(W_C\lt \overline{W}\), then the expected fitness of Cooperate is lowerthan the average fitness of the population. That means it is moreadvantageous to Defect than Cooperate, and so we would expect someproportion of the population to switch. The rate at which individualsswitch is proportional to how much worse Cooperate does than thepopulation average. (We are being deliberately ambiguous in terms ofwhether we are thinking in terms of biological or cultural evolution,but at this level of abstraction it makes little difference.) Since\(\frac{W_C}{\overline{W}} \lt 1\), it follows that \(p_c' \lt p_c\),as would be expected.

We can rewrite these expressions in the following form:

If we assume that the change in the strategy frequency from onegeneration to the next are small, these difference equations may beapproximated by the differential equations:

These equations were offered by Taylor and Jonker (1978) and Zeeman(1979) to provide continuous dynamics for evolutionary game theory andare known as thereplicator dynamics.

Since it is the case that, for any value of \(p_c\) and \(p_d\), \(W_C \lt \overline{W}\), future population states will always featurefewer cooperators than before. This is represented in the diagram offigure 6.

Figure 6: The Replicator Dynamical Modelof the Prisoner’s Dilemma

This diagram is interpreted as follows: the leftmost point representsthe state of the population where everyone defects, the rightmostpoint represents the state where everyone cooperates, and intermediatepoints represent states containing a mix of both cooperators anddefectors. (One maps states of the population onto points in thediagram by mapping the state when \(N\)% of the population defectsonto the point of the line \(N\)% of the way to the leftmost point.)Arrows on the line indicate the evolutionary trajectory followed bythe population over time. The open circle at the rightmost pointrepresents the fact that the state where everybody cooperates is anunstable equilibrium, in that if a small portion (any amount\(\epsilon > 0\)) of the population deviates from the strategyCooperate, then the evolutionary dynamics will lead the populationaway from the all-Cooperate state. The solid circle at the leftmostpoint indicates that the state where everybody Defects is a stableequilibrium, in the sense that if some portion of the populationdeviates from the strategy Defect, then the evolutionary dynamics willdrive the population back to the all-Defect state.

Although the replicator dynamics were the first dynamics to be used inevolutionary game theory, many alternative dynamics have since beenexplored. In what follows, we shall talk about evolutionary dynamicsentirely from the perspective ofcultural evolution, whichmeans nothing more than change in belief (e.g., strategy) overtime.

In his comprehensive workPopulation Games and Evolutionary Dynamics,^[3] William Sandholm provides a useful framework which allows us torelate the particular learning rules used by individuals to theevolutionary dynamics at the population-level of description. This canbe thought of as providing the “microfoundations” of evolutionary gametheory, analogous to how the study of individual decision makingprovides the microfoundations of macroeconomics.

We begin with a sketch of the framework for modelling learning rules,and then give several examples of individual learning rules and theevolutionary dynamics which result at the population-level. Forsimplicity, all of the mathematical details are suppressed in thefollowing discussion; for elaboration, see Sandholm (2010).(Sandholm’s framework, in general, allows for games featuring multiplenonoverlapping populations. One of the simplifying assumptions madehere is that there is only one population.) Assume that we have asymmetric game \(G\) with \(n\) strategies \(S_1,\dots, S_n\). (Asymmetric game is one where the payoff for playing a particularstrategy depends only on the strategies used by the other players, noton who is playing what strategy.) In addition, assume that the onlypossible information which could be taken into account by individualsis the following: (1) the current state of the population, representedas a distribution over the pure strategies of the game, and (2) theexpected payoffs of each strategy, given the current state of thepopulation.

An individual learning rule (or, to use Sandholm’s terminology, arevision protocol) can be represented as a function whichtakes these two pieces of information as arguments, mapping them to amatrix of theconditional switch rates between strategies.That is, the \(ij^{th}\)-entry of the matrix contains the rate atwhich followers of strategy \(S_i\) will switch to the strategy\(S_j\). It is called theconditional switch rate because the\(S_i\to S_j\) switch rate will typically depend on — that is,be conditional on — both the state of the population and thevector of expected payoffs. Note that the function may not actuallyuse all of the information contained in its arguments: some learningrules may be more sophisticated than others.

From this, the population-level evolutionary dynamics can be derivedelegantly: the instantaneous rate of change for the proportion of thepopulation following strategy \(S_i\) simply equals the total rate atwhich players followingother strategies switchto\(S_i\), minus the total rate at whichcurrent followers of\(S_i\) switch tosome other strategy. Substituting anindividual learning rule into the following equation schema, and thensolving the resulting system of equations, gives the population-leveldynamics.

This general framework allows one to investigatethe relationship between particular learning rules, at the individuallevel, and the evolutionary dynamics, at the population level. Hereare three examples.

The Replicator Dynamics. Suppose each player selectssomeone else from the population at random (with all individualsequally likely to be selected), and compares their payoff in the lastround of play with the payoff earned by the person selected. If theperson selected received a higher payoff, then the player adopts thestrategy used by the person selected with a probability proportionalto the payoff difference. Schlag (1998) showed that this learning ruleyields the replicator dynamics.

The Brown-Nash-von Neumann dynamics. One keyassumption made by the learning rule which yields the replicatordynamics is that imitation is a reliable guide to future payoffs. Thiscan be problematic for two reasons. First, the fact that a strategyhas an expected payoff greater than the average payoff of thepopulation may simply be indicative of peculiarities of the currentpopulation composition, and not of any particular strategic merit thestrategy possesses. Such a learning rule may end up with much of thepopulation shifting to adopt a strategy, only for its transientfitness benefits to disappear. Second, if a strategy isnotpresent in the population at all, it has no chance of beingadopted by imitation.

As an alternative, one might consider a learning rule where the rateplayers switch to strategy \(S_i\) only depends on whether theexpected payoff of \(S_i\) exceeds the average payoff of thepopulation at the current time. Notice that such a learning ruleattributes a higher degree of rationality to the individual playersthan the learning rule which generates the replicator dynamics. Why?This learning rule requires people to know the entire set of possiblestrategies, as well as the associated payoff matrix, so that they candetermine if a strategy presently absent from the population would beworth adopting. When this learning rule is plugged into the aboveschema, one obtains the Brown-Nash-von Neumann (BNN) dynamic (seeBrown and von Neumann, 1950). Unlike the replicator dynamics, the BNNdynamic can introducenew strategies into the populationwhich are not represented. When the population is started in thestates where everyone plays Rock, Paper or Scissors, the BNN dynamicwill eventually end up in a state where all strategies arerepresented.

The Smith Dynamics. One unusual feature of thelearning rule which generates the BNN dynamic is that it compares theexpected payoff of alternative possible strategies with the averagepayoff of the population. One might wonder why performing better thanthe population average is a sensible point of comparison, since theaverage payoff of the population is often not actually achievable byanyparticular strategy available to the players. Instead,consider the learning rule which compares the expected payoff of one’scurrent strategy, in the present population state, with theexpected payoff of other possible strategies, in the presentpopulation state, but where only those alternative strategies whichhave a higher expected payoff have a nonzero probability of beingadopted. When plugged into the Sandholm framework above, this learningrule generates an evolutionary dynamic first studied by Smith (1984)and is thus known as theSmith dynamic.

3. Dynamics, Stability, and Rational Outcomes

Given the number of different types of evolutionary dynamics, as seenin section 2.2, and the number of different concepts of evolutionarystability, as seen in section 2.1, a first question to ask is whatrelationships exist between the two? A second question to ask is whatrelationships exist between the various families of evolutionarydynamics and what one might consider to be the “rational” outcome of agame? Answers to these questions turn out to be more subtle andcomplex than one might first anticipate.

One complication, at the outset, is that an ESS is astrategy, possibly mixed, which satisfies certain properties.In contrast, all of the evolutionary dynamics described above modelpopulations where individuals employ only pure strategies. How, then,are we to relate the two concepts?

One natural suggestion is to interpret the probabilities which appearin an evolutionarily stable strategy aspopulationfrequencies. When the probabilities are understood in this way,one speaks of anevolutionarily stable state, in order tostress the difference in interpretation. One can then ask under whatconditions a particular evolutionary dynamic will converge to anevolutionarily stable state.

In the case of the replicator dynamics, it is immediately apparentthat the replicator dynamics need not converge to an evolutionarilystable state. This is because, as noted previously, the replicatordynamics cannot introduce strategies into the population if they areinitially absent. Hence, if an evolutionarily stable state requirescertain pure strategies to be present, and those pure strategies donot appear in the initial population state, then the replicatordynamics will not converge to the evolutionarily stable state.

This can be seen in a particularly stark form in figure 6, above. Inthe case of the Prisoner’s Dilemma, if the population begins in thestate whereeveryone cooperates, the replicator dynamics willremain in that state forever, because the strategy of Defect cannot beintroduced. This shows that, under the replicator dynamics, there canbe instances where evenstrictly dominated strategies willpersist.

That said, it can also be seen in figure 6 that whenever there is anonzero proportion of Defectors present in the population, that theywill increase in number and eventually drive Cooperate to extinction.(In the limit, because another property of the replicator dynamics isthat no strategy which appears can ever go extinct in a finite amountof time.) This motivates the following result:

Theorem (Akin, 1980) Let \(G\) be a symmetric,two-player game, and let \(\vec p(0)\) be an initial population statein which all pure strategies are represented (that is, appear with afrequency greater than zero). Then, under the replicator dynamicsbeginning at the initial state \(\vec p(0)\), all strictly dominatedstrategies will disappear in the limit.

What the above theorem shows is that, although there are cases wherestrictly dominated strategies may persist under the replicatordynamics, such cases are rare. As long as all strategies are initiallypresent, no matter to how small an extent, the replicator dynamicseliminates strictly dominated strategies.

However, the same does not hold forweakly dominatedstrategies. A strategy \(A\) is said to weakly dominate the strategy\(B\) if \(A\) does at least as well as \(B\) against all possiblecompetitors, and there is at least one case where \(A\) does strictlybetter. When this happens, the strategy \(B\) is known as a weaklydominated strategy. Weakly dominated strategiescan canappear in a Nash equilibrium, as figure 7 shows below. When bothplayers adopt \(S_2\), it is in neither player’s interest to switchbecause they continue to receive a payoff of 100 — and so wehave a Nash equilibrium when both players adopt \(S_2\). Yet it isalso the case that \(S_1\) weakly dominates \(S_2\).

It can be shown (see Weibull, 1995) that a weakly dominated strategycannever be an ESS. In the case of the game shown in figure7, this is surprising because the equilibrium generated by the weaklydominated strategy is Pareto optimal and has a much higher expectedpayoff than any other Nash equilibrium. However, it is also the case(see Skyrms, 1996) that the replicator dynamics need not eliminateweakly dominated strategies. In fact, in chapter 2 ofEvolution ofthe Social Contract, Brian Skyrms shows that there are some gamesin which the replicator dynamics almost always yields outcomescontaining a weakly dominated strategy! This shows that there can beconsiderable disagreement between the evolutionary outcomes of thereplicator dynamics and what the static approach identifies as anevolutionarily stable strategy.

The potential disagreement between the outcomes of an evolutionarydynamic and what ordinary game theory would consider to be a“rational” outcome of play is not just limited to the replicatordynamics. For example, consider the BNN dynamic and the Smith dynamic,described in section 2.2. In both cases, the underlying learning rulewhich generates those dynamics has some intuitive plausibility. Inparticular, each of those learning rules can be seen as employingslightly more rational approaches to the problem of strategy revisionthan the imitative learning rule which generated the replicatordynamics. Yet Hofbauer and Sandholm (2011) show that both the BNNdynamic and the Smith dynamic are not guaranteed to eliminate strictlydominated strategies!

Consider the game of figure 8 below. This is known as the game of“Rock-Paper-Scissors with a feeble twin”. In this game, the Twinstrategy is identical to Paper with the exception that all of itspayoffs are decreased uniformly by some small amount \(\varepsilon> 0\) (hence, the “feeble twin”). This means that the Twin strategyis strictly dominated by Paper, since there are absolutely noinstances in which it is rationally preferable to play Twin instead ofPaper. Yet, under the Smith dynamic, there are a nontrivial number ofinitial conditions which end up trapped in cycles where the Twinstrategy is played by a nontrivial portion of the population.

The connection between ESSs and stable states under an evolutionarydynamical model is weakened further if we do not model the dynamicsusing a continuous population model. For example, suppose we use alocal interaction model in which each individual plays theprisoner’s dilemma with his or her neighbors. Nowak and May(1992, 1993), using a spatial model in which local interactions occurbetween individuals occupying neighboring nodes on a square lattice,show that stable population states for the prisoner’s dilemmadepend upon the specific form of the payoff matrix.^[4] (What is interesting about this finding is that, in all cases, itremains true that Defect strictly dominates Cooperate, so thefundamental underlying strategic problem described by the game has notchanged.)

When the payoff matrix for the population has the values \(T = 2.8\),\(R = 1.1\), \(P = 0.1\), and \(S = 0\), the evolutionary dynamics ofthe local interaction model agree with those of the replicatordynamics, and lead to a state where each individual follows thestrategy Defect—which is, as noted before, the onlyevolutionarily stable strategy in the prisoner’s dilemma. Thefigure below illustrates how rapidly one such population converges toa state where everyone defects.

Generation 1	Generation 2	Generation 3
Generation 4	Generation 5	Generation 6

Figure 9: Prisoner’s Dilemma: AllDefect
[View a movie of this model]

However, when the payoff matrix has values of \(T = 1.2, R = 1.1, P =0\).1, and \(S = 0\), the evolutionary dynamics carry the populationto a stable cycle oscillating between two states. In this cyclecooperators and defectors coexist, with some regions containing“blinkers” oscillating between defectors and cooperators(as seen in generation 19 and 20).

Generation 1	Generation 2
Generation 19	Generation 20

Figure 10: Prisoner’s Dilemma:Cooperate
[View a movie of this model]

Notice that with these particular settings of payoff values, theevolutionary dynamics of the local interaction model differsignificantly from those of the replicator dynamics. Under thesepayoffs, the stable states have no corresponding analogue in eitherthe replicator dynamics nor in the analysis of evolutionarily stablestrategies.

A phenomenon of greater interest occurs when we choose payoff valuesof \(T = 1.61, R = 1.01, P = 0\).01, and \(S = 0\). Here, the dynamicsof local interaction lead to a world constantly in flux: under thesevalues regions occupied predominantly by Cooperators may besuccessfully invaded by Defectors, and regions occupied predominantlyby Defectors may be successfully invaded by Cooperators. In thismodel, there is no “stable strategy” in the traditionaldynamical sense.^[5]

Generation 1	Generation 3	Generation 5
Generation 7	Generation 9	Generation 11
Generation 13	Generation 15

Figure 11: Prisoner’s Dilemma:Chaotic
[view a movie of this model]

These results demonstrate that, although there are cases where boththe static and dynamic approaches to evolutionary game theory agreeabout the expected outcome of an evolutionary game, there are enoughdifferences in the outcomes of the two modes of analysis to justifythe development of each program independently.

4. Why Evolutionary Game Theory?

4.1 The equilibrium selection problem

The concept of a Nash equilibrium (see the entry ongame theory) has been the most used solution concept in game theory since itsintroduction by John Nash (1950). A selection of strategies by a groupof agents is said to be in a Nash equilibrium if each agent’sstrategy is a best-response to the strategies chosen by the otherplayers. By best-response, we mean that no individual can improve herpayoff by switching strategies unless at least one other individualswitches strategies as well. This need not mean that the payoffs toeach individual are optimal in a Nash equilibrium: indeed, one of thedisturbing facts of the prisoner’s dilemma is that the only Nashequilbrium of the game—when both agents defect—is suboptimal.^[6]

Yet a difficulty arises with the use of Nash equilibrium as a solutionconcept for games: if we restrict players to using pure strategies,not every game has a Nash equilbrium. The game “MatchingPennies” illustrates this problem.

While it is true that every noncooperative game in which players mayuse mixed strategies has a Nash equilibrium, some have questioned thesignificance of this for real agents. If it seems appropriate torequire rational agents to adopt only pure strategies (perhaps becausethe cost of implementing a mixed strategy runs too high), then thegame theorist must admit that certain games lack solutions.

A more significant problem with invoking the Nash equilibrium as theappropriate solution concept arises because some games havemultiple Nash equilibria (see the section onSolution Concepts and Equilibria, in the entry on game theory). When there are several different Nashequilibria, how is a rational agent to decide which of the severalequilibria is the “right one” to settle upon?^[7] Attempts to resolve this problem have produced a number of possiblerefinements to the concept of a Nash equilibrium, each refinementhaving some intuitive purchase. Unfortunately, so many refinements ofthe notion of a Nash equilibrium have been developed that, in manygames which have multiple Nash equilibria, each equilibrium could bejustified by some refinement present in the literature. The problemhas thus shifted from choosing among multiple Nash equilibria tochoosing among the various refinements.

Samuelson (1997), in his workEvolutionary Games and EquilibriumSelection) expressed hope that further development ofevolutionary game theory could be of service in addressing theequilibrium selection problem. At present, this hope does not seem tohave been realised. As section 2.1 showed, there are multiplecompeting concepts of evolutionary stability in play. Furthermore, assection 3 showed, there is an imperfect agreement between what isevolutionary stable, in the dynamic setting, and what is evolutionarystable, in the static setting.

4.2 The problem of hyperrational agents

The traditional theory of games imposes a very highrationality requirement upon agents. This requirement originates inthe development of the theory of utility which provides gametheory’s underpinnings (see Luce and Raiffa, 1957, for anintroduction). For example, in order to be able to assign a cardinalutility function to individual agents, one typically assumes that eachagent has a well-defined, consistent set of preferences over the setof “lotteries” over the outcomes which may result fromindividual choice. Since the number of different lotteries overoutcomes is uncountably infinite, this requires each agent to have awell-defined, consistent set of uncountably infinitely manypreferences.

Numerous results from experimental economics have shown that thesestrong rationality assumptions do not describe the behavior of realhuman subjects. Humans are rarely (if ever) the hyperrational agentsdescribed by traditional game theory. For example, it is not uncommonfor people, in experimental situations, to indicate that they prefer\(A\) to \(B, B\) to \(C\), and \(C\) to \(A\). These “failuresof the transitivity of preference” would not occur if people hada well-defined consistent set of preferences. Furthermore, experimentswith a class of games known as a “beauty pageant” show,quite dramatically, the failure of common knowledge assumptionstypically invoked to solve games.^[8] Since evolutionary game theory successfully explains the predominanceof certain behaviors of insects and animals, where strong rationalityassumptions clearly fail, this suggests that rationality is not ascentral to game theoretic analyses as previously thought. The hope,then, is that evolutionary game theory may meet with greater successin describing and predicting the choices of human subjects, since itis better equipped to handle the appropriate weaker rationalityassumptions. Indeed, one of the great strengths of the frameworkintroduced by Sandholm (2010) is that it provides a general method forlinking the learning rules used by individuals, at the micro level,with the dynamics describing changes in the population, at the macrolevel.

4.3 The lack of a dynamical theory in the traditional theory of games

At the end of the first chapter ofTheory of Games and EconomicBehavior, von Neumann and Morgenstern write:

We repeat most emphatically that our theory is thoroughly static. Adynamic theory would unquestionably be more complete and thereforepreferable. But there is ample evidence from other branches of sciencethat it is futile to try to build one as long as the static side isnot thoroughly understood. (Von Neumann and Morgenstern, 1953, p. 44)

The theory of evolution is a dynamical theory, and the second approachto evolutionary game theory sketched above explicitly models thedynamics present in interactions among individuals in the population.Since the traditional theory of games lacks an explicit treatment ofthe dynamics of rational deliberation, evolutionary game theory can beseen, in part, as filling an important lacuna of traditional gametheory.

One may seek to capture some of the dynamics of the decision-makingprocess in traditional game theory by modeling the game in itsextensive form, rather than its normal form. However, for most gamesof reasonable complexity (and hence interest), the extensive form ofthe game quickly becomes unmanageable. Moreover, even in the extensiveform of a game, traditional game theory represents anindividual’s strategy as a specification of what choice thatindividual would make at each information set in the game. A selectionof strategy, then, corresponds to a selection, prior to game play, ofwhat that individual will do at any possible stage of the game. Thisrepresentation of strategy selection clearly presupposes hyperrationalplayers and fails to represent the process by which one playerobserves his opponent’s behavior, learns from theseobservations, and makes the best move in response to what he haslearned (as one might expect, for there is no need to model learningin hyperrational individuals). The inability to model the dynamicalelement of game play in traditional game theory, and the extent towhich evolutionary game theory naturally incorporates dynamicalconsiderations, reveals an important virtue of evolutionary gametheory.

5. Applications of Evolutionary Game Theory

Evolutionary game theory has been used to explain a number of aspectsof human behavior. A small sampling of topics which have been analysedfrom the evolutionary perspective include:altruism(Fletcher and Zwick, 2007; Gintiset al., 2003;Sánchez and Cuesta, 2005; Trivers, 1971),behavior inpublic goods game (Clemens and Riechmann, 2006; Hauert, 2006;Hauertet al., 2002, 2006; Huberman and Glance, 1995),empathy (Page and Nowak, 2002; Fishman, 2006),human culture (Enquist and Ghirlanda, 2007; Enquistet al., 2008),moral behaviour (Alexander,2007; Boehm, 1982; Harms and Skyrms, 2008; Skyrms 1996, 2004),private property (Gintis, 2007),signalingsystems and other proto-linguistic behaviour (Barrett, 2007;Hausken and Hirshleirfer, 2008; Hurd, 1995; Jäger, 2008; Nowaket al., 1999; Pawlowitsch, 2007, 2008; Skyrms, 2010; Zollman,2005),social learning (Kameda and Nakanishi, 2003;Nakahashi, 2007; Rogers, 1988; Wakano and Aoki, 2006; Wakanoetal., 2004), andsocial norms (Axelrod, 1986;Bicchieri, 2006; Binmore and Samuelson, 1994; Chalubet al.,2006; Kendalet al., 2006; Ostrum, 2000).

The following subsections provide a brief illustration of the use ofevolutionary game theoretic models to explain two areas of humanbehavior. The first concerns the tendency of people to share equallyin perfectly symmetric situations. The second shows how populations ofpre-linguistic individuals may coordinate on the use of a simplesignaling system even though they lack the ability to communicate.These two models have been pointed to as preliminary explanations ofour sense of fairness and language, respectively. They were selectedfor inclusion here for three reasons: (1) the relative simplicity ofthe model, (2) the apparent success at explaining the phenomenon inquestion, and (3) the importance of the phenomenon to beexplained.

5.1 A sense of fairness

One natural game to use for investigating the evolution of fairness isdivide-the-cake (this is the simplest version of the Nashbargaining game). In chapter 1 ofEvolution of the SocialContract, Skyrms presents the problem as follows:

Here we start with a very simple problem; we are to divide a chocolatecake between us. Neither of us has any special claim as against theother. Out positions are entirely symmetric. The cake is a windfallfor us, and it is up to us to divide it. But if we cannot agree how toshare it, the cake will spoil and we will get nothing. (Skyrms, 1996,pp. 3–4)

More formally, suppose that two individuals are presented with aresource of size \(C\) by a third party. Astrategy for aplayer, in this game, consists of an amount of cake that he wouldlike. The set of possible strategies for a player is thus any amountbetween 0 and \(C\). If the sum of strategies for each player is lessthan or equal to \(C\), each player receives the amount he asked for.However, if the sum of strategies exceeds \(C\), no player receivesanything. Figure 13 illustrates the feasible set for this game.

Figure 13: The feasible set for the gameof Divide-the-Cake. In this figure, the cake is of size \(C=10\) butall strategies between 0 and 10 inclusive are permitted for eitherplayer (including fractional demands).

We have a clear intuition that the “obvious” strategy foreach player to select isC/2; the philosophical problem liesin explainingwhy agents would choose that strategy ratherthan some other one. Even in the perfectly symmetric situation,answering this question is more difficult than it first appears. Tosee this, first notice that there are an infinite number of Nashequilibria for this game. If player 1 asks for \(p\) of the cake,where \(0 \le p \le C\), and player 2 asks for \(C - p\), then thisstrategy profile is a Nash equilibrium for any value of \(p \in[0,C]\). (Each player’s strategy is a best response given whatthe other has chosen, in the sense that neither player can increaseher payoff by changing her strategy.) Thus the equal split is only oneof infinitely many Nash equilibria.

One might propose that both players should choose that strategy whichmaximizes their expected payoff on the assumption they are uncertainas to whether they will be assigned the role of Player 1 or Player 2.This proposal, Skyrms notes, is essentially that of Harsanyi (1953).The problem with this is that if players only care about theirexpected payoff, and they think that it is equally likely that theywill be assigned the role of Player 1 or Player 2, then this, too,fails to select uniquely the equal split. Consider the strategyprofile \(\langle p, C - p\rangle\) which assigns Player 1 \(p\)slices and Player 2 \(C - p\) slices. If a player thinks it is equallylikely that he will be assigned the role of Player 1 or Player 2, thenhis expected utility is \(\frac{1}{2} p + \frac{1}{2}(C - p) =\frac{C}{2}\), for all values \(p \in[0, C]\).

Now consider the following evolutionary model: suppose we have apopulation of individuals who pair up and repeatedly play the game ofdivide-the-cake, modifying their strategies over time in a way whichis described by the replicator dynamics. For convenience, let usassume that the cake is divided into 10 equally sized slices and thateach player’s strategy conforms to one of the following 11possible types: Demand 0 slices, Demand 1 slice, … , Demand 10slices. For the replicator dynamics, the state of the population isrepresented by a vector \(\langle p_0, p_1 , \ldots ,p_{10}\rangle\)where each \(p_i\) denotes the frequency of the strategy “Demand\(i\) slices” in the population.

The replicator dynamics allows us to model how the distribution ofstrategies in the population changes over time, beginning from aparticular initial condition. Figure 14 below shows two evolutionaryoutcomes under the continuous replicator dynamics. Notice thatalthough fair division can evolve, as in Figure 14(a), it is not theonly evolutionary stable outcome, as Figure 14(b) illustrates.

(a) The evolution of fair division.

(b) The evolution of an unequal division rule.

Figure 14: Two evolutionary outcomesunder the continuous replicator dynamics for the game ofdivide-the-cake. Of the eleven strategies present, only three arecolour-coded so as to be identifiable in the plot, as noted in thelegend.

Recall that the task at hand was to explain why we think the“obvious” strategy choice in a perfectly symmetricresource allocation problem is for both players to ask for half of theresource. What the above shows is that, in a population of boundedlyrational agents who modify their behaviours in a manner described bythe replicator dynamics, fair division is one, although not the only,evolutionary outcome. The tendency of fair division to emerge,assuming that any initial condition is equally likely, can be measuredby determining the size of thebasin of attraction of the state where everyone in the population uses the strategyDemand 5 slices. Skyrms (1996) measures the size of the basin ofattraction of fair division usingMonte Carlo methods, finding that fair division evolves roughly 62% of the time.

However, it is important to realise that the replicator dynamicsassumes any pairwise interaction between individuals is equallylikely. In reality, quite often interactions between individuals arecorrelated to some extent. Correlated interaction can occuras a result of spatial location (as shown above for the case of thespatial prisoner’s dilemma), the structuring effect of socialrelations, or ingroup/outgroup membership effects, to list a fewcauses.

When correlation is introduced, the frequency with which fair divisionemerges changes drastically. The amount of correlation in the model isrepresented by thecorrelation coefficient \(\varepsilon\),which can range between 0 and 1. When \(\varepsilon=0\), there is nocorrelation at all and the likelihood of pairwise interactions isdetermined simply by the proportion of agents in the populationfollowing a particular strategy. When \(\varepsilon=1\), correlationis perfect and agents following a particular strategy only interactwith their own kind. Intermediate levels of correlation introduce sometendency for agents to interact with their own kind, where thetendency increases with the value of \(\varepsilon\). Figure 15illustrates how the basin of attraction of All Demand 5 changes as thecorrelation coefficient \(\varepsilon\) increases from 0 to 0.2.^[9] Once the amount of correlation present in the interactions reaches\(\varepsilon = 0.2\), fair division is virtually an evolutionarycertainty. Note that this does not depend on there only being threestrategies present: allowing for some correlation between interactionsincreases the probability of fair division evolving even if theinitial conditions contain individuals using any of the elevenpossible strategies.

(a) \(\varepsilon = 0\)	(b) \(\varepsilon = 0\).1
(c) \(\varepsilon = 0.2\)

Figure 15: Three diagrams showing how,as the amount of correlation among interactions increases, fairdivision is more likely to evolve. In figures 15(a) and 15(b), thereis an unstable fixed point in the interior of space where all threestrategies are present in the population. (This is the point where theevolutionary trajectories appear to intersect.) This fixed point iswhat is known as asaddle point in dynamical systems theory:the smallest perturbation will cause the population to evolve awayfrom that point to one of the other two attractors.

What, then, can we conclude from this model regarding the evolution offair division? It all depends, of course, on how accurately thereplicator dynamics models the primary evolutionary forces (culturalor biological) acting on human populations. Although the replicatordynamics are a “simple” mathematical model, it doessuffice for modelling both a type of biological evolution (see Taylorand Jonker, 1978) and a type of cultural evolution (see Börgersand Sarin, 1996; Weibull, 1995). As Skyrms (1996) notes:

In a finite population, in a finite time, where there is some randomelement in evolution, some reasonable amount of divisibility of thegood and some correlation, we can say that it is likely that somethingclose to share and share alike should evolve in dividing-the-cakesituations. This is, perhaps, a beginning of an explanation of theorigin of our concept of justice.

This claim, of course, has not gone without comment. For a selectionof some discussion see, in particular, D’Arms (1996, 2000);D’Armset al., 1998; Danielson (1998); Bicchieri(1999); Kitcher (1999); Gintis (2000); Harms (2000); Krebs (2000);Alexander and Skyrms (1999); and Alexander (2000, 2007).

5.2 The emergence of language.

In his seminal workConvention, David Lewis developed theidea of sender-receiver games. Such games have been used to explainhow language, and semantic content, can emerge in a community whichoriginally did not possess any language whatsoever.^[10] His original definition is as follows (with portions of extraneouscommentary deleted for concision and points enumerated for clarity andlater reference):

Atwo-sided signaling problem is a situation \(S\) involvingan agent called thecommunicator and one or more other agentscalled theaudience, such that it is true that, and it iscommon knowledge for the communicator and the audience that:

1. Exactly one of several alternative states of affairs \(s_1 ,\ldots ,s_m\) holds. The communicator, but not the audience, is in agood position to tell which one it is.
2. Each member of the audience can do any one of several alternativeactions \(r_1 , \ldots ,r_m\) calledresponses. Everyoneinvolved wants the audience’s responses to depend in a certainway upon the state of affairs that holds. There is a certainone-to-one function \(F\) from \(\{s_i\}\) onto \(\{r_j\}\) such thateveryone prefers that each member of the audience do \(F(s_i)\) oncondition that \(s_i\) holds, for each \(s_i\).
3. The communicator can do any one of several alternative actions\(\sigma_1 , \ldots ,\sigma_n (n \ge m)\) calledsignals. Theaudience is in a good position to tell which one he does. No oneinvolved has any preference regarding these actions which is strongenough to outweigh his preference for the dependence \(F\) ofaudience’s responses upon states of affairs. […]
4. Acommunicator’s contingency plan is any possibleway in which the communicator’s signal may depend upon the stateof affairs that he observes to hold. It is a function \(Fc\) from\(\{s_i\}\) into \(\{\sigma_k \}\). […]
5. Similarly, anaudience’s contingency plan is anypossible way in which the response of a member of the audience maydepend upon the signal he observes the communicator to give. It is aone-to-one function \(Fa\) from part of \(\{\sigma_k\}\) into\(\{r_j\}\). […]

Whenever \(Fc\) and \(Fa\) combine […] to give the preferreddependence of the audience’s response upon the state of affairs,we call \(\langle Fc, Fa\rangle\) asignaling system. (Lewis,1969, pp. 130–132)

Since the publication ofConvention, it is more common torefer to the communicator as thesender and the members ofthe audience asreceivers. The basic idea behindsender-receiver games is the following: Nature selects which state ofthe world obtains. The person in the role of Sender observes thisstate of the world (correctly identifying it), and sends a signal tothe person in the role of Receiver. The Receiver, upon receipt of thissignal, performs a response. If what the Receiver does is the correctresponse, given the state of the world, then both players receive apayoff of 1; if the Receiver performed an incorrect response, thenboth players receive a payoff of 0. Notice that, in this simplifiedmodel, no chance of error exists at any stage. The Sender alwaysobserves the true state of the world and always sends the signal heintended to send. Likewise, the Receiver always receives the signalsent by the Sender (i.e., the channel is not noisy), and the Receiveralways performs the response he intended to.

Whereas Lewis allowed the “audience” to consist of morethan one person, it is more common to consider sender-receiver gamesplayed between two people, so that there is only a single receiver(or, in Lewisian terms, a single member of the audience).^[11] For simplicity, in the following we will consider a two-player,sender-receiver game with two states of the world \(\{S_1, S_2\}\),two signals \(\{\sigma_1, \sigma_2\}\), and two responses \(\{r_1,r_2\}\). (We shall see later why larger sender-receiver games areincreasingly difficult to analyse.)

Notice that, in point (2) of his definition of sender-receiver games,Lewis requires two things: that there be a unique best response to thestate of the world (this is what requiring \(F\) to be one-to-oneamounts to) and that everyone in the audience agrees that this is thecase. Since we are considering the case where there is only a singleresponder, the second requirement is otiose. For the case of twostates of the world and two responses, there are only two ways ofassigning responses to states of the world which satisfy Lewis’srequirement. These are as follows (where \(X \Rightarrow Y\) denotes“in state of the world \(X\), the best response is to do\(Y\)”):

1. \(S_1 \Rightarrow r_1, S_2 \Rightarrow r_2\).
2. \(S_1 \Rightarrow r_2, S_2 \Rightarrow r_1\).

It makes no real difference for the model which one of these wechoose, so pick the intuitive one: in state of the world \(S_i\), thebest response is \(r_i\) (i.e., function 1).

Astrategy for the sender (what Lewis called a“communicator’s contingency plan”) consists of afunction specifying what signal he sends given the state of the world.It is, as Lewis notes, a function from the set of states of the worldinto the set of signals. This means that it is possible thata sender may send thesame signal in two different states ofthe world. Such a strategy makes no sense, from a rational point ofview, because the receiver would not get enough information to be ableto identify the correct response for the state of the world. However,we do not exclude these strategies from consideration because they arelogically possible strategies.

How many sender strategies are there? Because we allow for thepossibility of the same signal to be sent for multiple states of theworld, there are two choices for which signal to send given state\(S_1\) and two choices for which signal to send given state \(S_2\).This means there are four possible sender strategies. These strategiesare as follows (where \(\mathrm{`}X \rightarrow Y\text{'}\) means thatwhen the state of the world is \(X\) the sender will send signal\(Y)\):

Sender 1: \(S_1 \rightarrow \sigma_1, S_2\rightarrow \sigma_1\).
Sender 2: \(S_1 \rightarrow \sigma_1, S_2\rightarrow \sigma_2\).
Sender 3: \(S_1 \rightarrow \sigma_2, S_2\rightarrow \sigma_1\).
Sender 4: \(S_1 \rightarrow \sigma_2, S_2\rightarrow \sigma_2\).

What is a strategy for a receiver? Here, it proves useful to deviatefrom Lewis’s original definition of the “audience’scontingency plan”. Instead, let us take a receiver’sstrategy to be a function from the set of signals into the set ofresponses. As in the case of the sender, we allow the receiver toperform the same response for more than one signal. By symmetry, thismeans there are \(\mathbf{4}\) possible receiver strategies. Thesereceiver strategies are:

Receiver 1: \(\sigma_1 \rightarrow r_1, \sigma_2\rightarrow r_1\).
Receiver 2: \(\sigma_1 \rightarrow r_1, \sigma_2\rightarrow r_2\).
Receiver 3: \(\sigma_1 \rightarrow r_2, \sigma_2\rightarrow r_1\).
Receiver 4: \(\sigma_1 \rightarrow r_2, \sigma_2\rightarrow r_2\).

If the roles of Sender and Receiver are permanently assigned toindividuals — as Lewis envisaged — then there are only twosignaling systems: \(\langle\)Sender 2, Receiver 2\(\rangle\) and\(\langle\)Sender 3, Receiver 3\(\rangle\). All other possiblecombinations of strategies result in the players failing tocoordinate. The coordination failure occurs because the Sender andReceiver only pair the appropriate action with the state of the worldin one instance, as with \(\langle\)Sender 1, Receiver 1\(\rangle\),or not at all, as with \(\langle\)Sender 2, Receiver 3\(\rangle\).

What if the roles of Sender and Receiver are not permanently assignedto individuals? That is, what if nature flips a coin and assigns oneplayer to the role of Sender and the other player to the role ofReceiver, and then has them play the game? In this case, aplayer’s strategy needs to specify what he will do when assignedthe role of Sender, as well as what he will do when assigned the roleof Receiver. Since there are four possible strategies to use as Senderand four possible strategies to use as Receiver, this means that thereare a total of \(\mathbf{16}\) possible strategies for thesender-receiver game when roles are not permanently assigned toindividuals. Here, a player’s strategy consists of an orderedpair (Sender \(X\), Receiver \(Y)\), where \(X, Y \in \{1, 2, 3,4\}\).

It makes a difference whether one considers the roles of Sender andReceiver to be permanently assigned or not. If the roles are assignedat random, there are four signaling systems amongst two players^[12]:

1. Player 1: (Sender 2, Receiver 2),Player2: (Sender 2, Receiver 2)
2. Player 1: (Sender 3, Receiver 3),Player2: (Sender 3, Receiver 3)
3. Player 1: (Sender 2, Receiver 3),Player2: (Sender 3, Receiver 2)
4. Player 1: (Sender 3, Receiver 2),Player2: (Sender 2, Receiver 3)

Signaling systems 3 and 4 are curious. System 3 is a case where, forexample, I speak in French but listen in German, and you speak Germanbut listen in French. (System 4 swaps French and German for both youand me.) Notice that in systems 3 and 4 the players are able tocorrectly coordinate the response with the state of the worldregardless of who gets assigned the role of Sender orReceiver.

The problem, of course, with signaling systems 3 and 4 is that neitherPlayer 1 nor Player 2 would do well when pitted against a clone ofhimself. They are cases where the signaling system would not work in apopulation of players who are pairwise randomly assigned to play thesender-receiver game. In fact, it is straightforward to show that thestrategies (Sender 2, Receiver 2) and (Sender 3, Receiver 3) are theonly evolutionarily stable strategies (see Skyrms 1996,89–90).

As a first approach to the dynamics of sender-receiver games, let usrestrict attention to the four strategies (Sender 1, Receiver 1),(Sender 2, Receiver 2), (Sender 3, Receiver 3), and (Sender 4,Receiver 4). Figure 16 illustrates the state space under thecontinuous replicator dynamics for the sender-receiver game consistingof two states of the world, two signals, and two responses, whereplayers are restricted to using one of the previous four strategies.One can see that evolution leads the population in almost all cases^[13] to converge to one of the two signaling systems.^[14]

Figure 16: The evolution of signalingsystems.

Figure 17 illustrates the outcome of one run of the replicatordynamics (for a single population model) where all sixteen possiblestrategies are represented. We see that eventually the population, forthis particular set of initial conditions, converges to one of thepure Lewisian signalling systems identified above.

Figure 17: The evolution of a signallingsystem under the replicator dynamics.

When the number of states of the world, the number of signals, and thenumber of actions increase from 2, the situation rapidly becomes muchmore complex. If there are \(N\) states of the world, \(N\) signals,and \(N\) actions, the total number of possible strategies equals\(N^{2N}\). For \(N=2\), this means there are 16 possible strategies,as we have seen. For \(N=3\), there are 729 possible strategies, and asignalling problem where \(N=4\) has 65,536 possible strategies. Giventhis, one might think that it would prove difficult for evolution tosettle upon an optimal signalling system.

Such an intuition is correct. Hofbauer and Hutteger (2008) show that,quite often, the replicator dynamics will converge to a suboptimaloutcome in signalling games. In these suboptimal outcomes, apooling orpartial pooling equilibrium will emerge.A pooling equilibrium occurs when the Sender uses the same signalregardless of the state of the world. A partial pooling equilibriumoccurs when the Sender is capable of differentiating between somestates of the world but not others. As an example of a partial poolingequilibrium, consider the following strategies for the case where\(N=3\): Suppose that the Sender sends signal 1 in state of the world1, and signal 2 in states of the world 2 and 3. Furthermore, supposethat the Receiver performs action 1 upon receipt of signal 1, andaction 2 upon receipt of signals 2 and 3. If all states of the worldare equiprobable, this is a partial pooling equilibrium. Given thatthe Sender does not differentiate states of the world 2 and 3, theReceiver cannot improve his payoffs by responding differently tosignal 2. Given the particular response behaviour of the Receiver, theSender cannot improve her payoffs by attempting to differentiatestates of the world 2 and 3.

6. Philosophical Problems of Evolutionary Game Theory

6.1 The meaning of fitness in cultural evolutionary interpretations

As noted previously, evolutionary game theoretic models may often begiven both a biological and a cultural evolutionary interpretation. Inthe biological interpretation, the numeric quantities which play arole analogous to “utility” in traditional game theorycorrespond to the fitness (typically Darwinian fitness) of individuals.^[15] How does one interpret “fitness” in the culturalevolutionary interpretation?

In many cases, fitness in cultural evolutionary interpretations ofevolutionary game theoretic models directly measures some objectivequantity of which it can be safely assumed that (1) individuals alwayswant more rather than less and (2) interpersonal comparisons aremeaningful. Depending on the particular problem modeled, money, slicesof cake, or amount of land would be appropriate cultural evolutionaryinterpretations of fitness. Requiring that fitness in culturalevolutionary game theoretic models conform to this interpretativeconstraint severely limits the kinds of problems that one can address.A more useful cultural evolutionary framework would provide a moregeneral theory which did not require that individual fitness be alinear (or strictly increasing) function of the amount of some realquantity, like amount of food.

In traditional game theory, a strategy’s fitness was measured bythe expected utility it had for the individual in question. Yetevolutionary game theory seeks to describe individuals of limitedrationality (commonly known as “boundedly rational”individuals), and the utility theory employed in traditional gametheory assumes highly rational individuals. Consequently, the utilitytheory used in traditional game theory cannot simply be carried overto evolutionary game theory. One must develop an alternate theory ofutility/fitness, one compatible with the bounded rationality ofindividuals, that is sufficient to define a utility measure adequatefor the application of evolutionary game theory to culturalevolution.

6.2 The explanatory irrelevance of evolutionary game theory

Another question facing evolutionary game theoretic explanations ofsocial phenomena concerns the kind of explanation it seeks to give.Depending on the type of explanation it seeks to provide, areevolutionary game theoretic explanations of social phenomenairrelevant or mere vehicles for the promulgation of pre-existingvalues and biases? To understand this question, recognize that onemust ask whether evolutionary game theoretic explanations target theetiology of the phenomenon in question, the persistence of thephenomenon, or various aspects of the normativity attached to thephenomenon. The latter two questions seem deeply connected, forpopulation members typically enforce social behaviors and rules havingnormative force by sanctions placed on those failing to comply withthe relevant norm; and the presence of sanctions, if suitably strong,explains the persistence of the norm. The question regarding aphenomenon’s etiology, on the other hand, can be consideredindependent of the latter questions.

If one wishes to explain how some currently existing social phenomenoncame to be, it is unclear why approaching it from the point of view ofevolutionary game theory would be particularily illuminating. Theetiology of any phenomenon is a unique historical event and, as such,can only be discovered empirically, relying on the work ofsociologists, anthropologists, archaeologists, and the like. Althoughan evolutionary game theoretic model may exclude certain historicalsequences as possible histories (since one may be able to show thatthe cultural evolutionary dynamics preclude one sequence fromgenerating the phenomenon in question), it seems unlikely that anevolutionary game theoretic model would indicate a unique historicalsequence suffices to bring about the phenomenon. An empirical inquirywould then still need to be conducted to rule out the extraneoushistorical sequences admitted by the model, which raises the questionof what, if anything, was gained by the construction of anevolutionary game theoretic model in the intermediate stage. Moreover,even if an evolutionary game theoretic model indicated that a singlehistorical sequence was capable of producing a given socialphenomenon, there remains the important question of why we ought totake this result seriously. One may point out that since nearly anyresult can be produced by a model by suitable adjusting of thedynamics and initial conditions, all that the evolutionary gametheorist has done is provide one such model. Additional work needs tobe done to show that the underlying assumptions of the model (both thecultural evolutionary dynamics and the initial conditions) areempirically supported. Again, one may wonder what has been gained bythe evolutionary model—would it not have been just as easy todetermine the cultural dynamics and initial conditions beforehand,constructing the model afterwards? If so, it would seem that thecontributions made by evolutionary game theory in this context simplyare a proper part of the parent social science—sociology,anthropology, economics, and so on. If so, then there is nothingparticular about evolutionary game theory employed in theexplanation, and this means that, contrary to appearances,evolutionary game theory is really irrelevant to the givenexplanation.

If evolutionary game theoretic models do not explain the etiology of asocial phenomenon, presumably they explain the persistence of thephenomenon or the normativity attached to it. Yet we rarely need anevolutionary game theoretic model to identify a particular socialphenomenon as stable or persistent as that can be done by observationof present conditions and examination of the historical records; hencethe charge of irrelevancy is raised again. Moreover, most of theevolutionary game theoretic models developed to date have provided thecrudest approximations of the real cultural dynamics driving thesocial phenomenon in question. One may well wonder why, in thesecases, we should take seriously the stability analysis given by themodel; answering this question would require one engage in anempirical study as previously discussed, ultimately leading to thecharge of irrelevance again.

It is sometimes argued that evolutionary game theoretic models answer“how possibly” questions. That is, an evolutionary gametheoretic model shows how some phenomenon could possibly be generatedby an underlying dynamical process of interacting, boundedly rationalagents. Although this is certainly the case, one might wonder whetherthis subtly shifts the explanatory target. Answering a “howpossibly” question is most interesting when we do not knowwhether something is possible at all. The challenge faced by someevolutionary game theoretic accounts of social phenomena is that theyanswer a “how possibly” question regarding something whichwe already knew was possible, because the phenomenon actually exists.What we would like to know is how the answer to the “howpossibly” question connects to the actual real-world processesgenerating the phenomenon. This suggests that evolutionary gametheoretic explanations of social phenomena are, even in the bestcases,incomplete.

6.3 The value-ladenness of evolutionary game theoretic explanations

If one seeks to use an evolutionary game theoretic model to explainthe normativity attached to a social rule, one must explain how suchan approach avoids committing the so-called “naturalisticfallacy” of inferring an ought-statement from a conjunction of is-statements.^[16] Assuming that the explanation does not commit such a fallacy, oneargument charges that it must then be the case that the evolutionarygame theoretic explanation merely repackages certain key value claimstacitly assumed in the construction of the model. After all, since anyargument whose conclusion is a normative statement must have at leastone normative statement in the premises, any evolutionary gametheoretic argument purporting to show how certain norms acquirenormative force must contain—at least implicitly—anormative statement in the premises. Consequently, this application ofevolutionary game theory does not provide a neutral analysis of thenorm in question, but merely acts as a vehicle for advancingparticular values, namely those smuggled in the premises.

This criticism seems less serious than the charge of irrelevancy.Cultural evolutionary game theoretic explanations of norms need not“smuggle in” normative claims in order to draw normativeconclusions. The theory already contains, in its core, a propersubtheory having normative content—namely a theory of rationalchoice in which boundedly rational agents act in order to maximize, asbest as they can, their own self-interest. One may challenge thesuitability of this as a foundation for the normative content ofcertain claims, but this is a different criticism from the abovecharge. Although cultural evolutionary game theoretic models do act asvehicles for promulgating certain values, they wear those minimalvalue commitments on their sleeve. Evolutionary explanations of socialnorms have the virtue of making their value commitments explicit andalso of showing how other normative commitments (such as fair divisionin certain bargaining situations, or cooperation in theprisoner’s dilemma) may be derived from the principled action ofboundedly rational, self-interested agents.

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Other Internet Resources

• Evolving Artificial Moral Ecologies, (with interactive simulators), by Peter Danielson (U. British Columbia) and William Harms (Bowling Green State).
• Brookings Center on Social and Economic Dynamics.
• Complexity of Cooperation, website on Robert Axelrod's book.

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