Routledge, London Google Scholar. Hodgson DH Consequences of utilitarianism. Clarendon Press, Oxford Google Scholar. Hollis M Trust within reason. Hurley SL Natural reasons. Regan D Utilitarianism and cooperation. Sugden R Thinking as a team: toward an explanation of nonselfish behavior. Sugden R Team preferences. Sugden R The logic of team reasoning. Sugden R Fellow-feeling.

Tuomela R The importance of US: a philosophical study of basic social notions. Personalised recommendations. Cite article How to cite? ENW EndNote. Furthermore, let us denote the average fitness of cooperators and defectors by W C and W D , respectively, and let W denote the average fitness of the entire population. The values of W C , W D , and W can be expressed in terms of the population proportions and payoff values as follows:.

Second, let us assume that the proportion of the population following the strategies Cooperate and Defect in the next generation is related to the proportion of the population following the strategies Cooperate and Defect in the current generation according to the rule:. These equations were offered by Taylor and Jonker and Zeeman to provide continuous dynamics for evolutionary game theory and are known as the replicator dynamics.

The replicator dynamics may be used to model a population of individuals playing the Prisoner's Dilemma. For the Prisoner's Dilemma, the expected fitness of Cooperating and Defecting are:. We interpret this diagram as follows: the leftmost point represents the state of the population where everyone defects, the rightmost point represents the state where everyone cooperates, and intermediate points represent states where some proportion of the population defects and the remainder cooperates.

Arrows on the line represent the evolutionary trajectory followed by the population over time. The open circle at the rightmost point indicates that the state where everybody cooperates is an unstable equilibrium, in the sense that if a small portion of the population deviates from the strategy Cooperate, then the evolutionary dynamics will drive the population away from that equilibrium.

The solid circle at the leftmost point indicates that the state where everybody Defects is a stable equilibrium, in the sense that if a small portion of the population deviates from the strategy Defect, then the evolutionary dynamics will drive the population back to the original equilibrium state. At this point, one may see little difference between the two approaches to evolutionary game theory.

Since this state is the only stable equilibrium under the replicator dynamics, the two notions fit together quite neatly: the only stable equilibrium under the replicator dynamics occurs when everyone in the population follows the only ESS. In general, though, the relationship between ESSs and stable states of the replicator dynamics is more complex than this example suggests. Taylor and Jonker , as well as Zeeman , establish conditions under which one may infer the existence of a stable state under the replicator dynamics given an evolutionarily stable strategy. Roughly, if only two pure strategies exist, then given a possibly mixed evolutionarily stable strategy, the corresponding state of the population is a stable state under the replicator dynamics.

If the evolutionarily stable strategy is a mixed strategy S , the corresponding state of the population is the state in which the proportion of the population following the first strategy equals the probability assigned to the first strategy by S , and the remainder follow the second strategy.

### See a Problem?

However, this can fail to be true if more than two pure strategies exist. The connection between ESSs and stable states under an evolutionary dynamical model is weakened further if we do not model the dynamics by the replicator dynamics. For example, suppose we use a local interaction model in which each individual plays the prisoner's dilemma with his or her neighbors. Nowak and May , , using a spatial model in which local interactions occur between individuals occupying neighboring nodes on a square lattice, show that stable population states for the prisoner's dilemma depend upon the specific form of the payoff matrix.

The figure below illustrates how rapidly one such population converges to a state where everyone defects. Notice that with these particular settings of payoff values, the evolutionary dynamics of the local interaction model differ significantly from those of the replicator dynamics. Under these payoffs, the stable states have no corresponding analogue in either the replicator dynamics nor in the analysis of evolutionarily stable strategies.

Here, the dynamics of local interaction lead to a world constantly in flux: under these values regions occupied predominantly by Cooperators may be successfully invaded by Defectors, and regions occupied predominantly by Defectors may be successfully invaded by Cooperators.

These models demonstrate that, although numerous cases exist in which both approaches to evolutionary game theory arrive at the same conclusion regarding which strategies one would expect to find present in a population, there are enough differences in the outcomes of the two modes of analysis to justify the development of each program.

Yet a difficulty arises with the use of Nash equilibrium as a solution concept for games: if we restrict players to using pure strategies, not every game has a Nash equilbrium. While it is true that every noncooperative game in which players may use mixed strategies has a Nash equilibrium, some have questioned the significance of this for real agents. If it seems appropriate to require rational agents to adopt only pure strategies perhaps because the cost of implementing a mixed strategy runs too high , then the game theorist must admit that certain games lack solutions.

A more significant problem with invoking the Nash equilibrium as the appropriate solution concept arises because games exist which have multiple Nash equilibria see the section on Solution Concepts and Equilibria , in the entry on game theory. Unfortunately, so many refinements of the notion of a Nash equilibrium have been developed that, in many games which have multiple Nash equilibria, each equilibrium could be justified by some refinement present in the literature.

The problem has thus shifted from choosing among multiple Nash equilibria to choosing among the various refinements. Some see Samuelson , Evolutionary Games and Equilibrium Selection hope that further development of evolutionary game theory can be of service in addressing this issue. Numerous results from experimental economics have shown that these strong rationality assumptions do not describe the behavior of real human subjects.

Humans are rarely if ever the hyperrational agents described by traditional game theory. For example, it is not uncommon for people, in experimental situations, to indicate that they prefer A to B , B to C , and C to A. The hope, then, is that evolutionary game theory may meet with greater success in describing and predicting the choices of human subjects, since it is better equipped to handle the appropriate weaker rationality assumptions.

The theory of evolution is a dynamical theory, and the second approach to evolutionary game theory sketched above explicitly models the dynamics present in interactions among individuals in the population. Since the traditional theory of games lacks an explicit treatment of the dynamics of rational deliberation, evolutionary game theory can be seen, in part, as filling an important lacuna of traditional game theory. One may seek to capture some of the dynamics of the decision-making process in traditional game theory by modeling the game in its extensive form, rather than its normal form.

However, for most games of reasonable complexity and hence interest , the extensive form of the game quickly becomes unmanageable. Moreover, even in the extensive form of a game, traditional game theory represents an individual's strategy as a specification of what choice that individual would make at each information set in the game. A selection of strategy, then, corresponds to a selection, prior to game play, of what that individual will do at any possible stage of the game. This representation of strategy selection clearly presupposes hyperrational players and fails to represent the process by which one player observes his opponent's behavior, learns from these observations, and makes the best move in response to what he has learned as one might expect, for there is no need to model learning in hyperrational individuals.

The inability to model the dynamical element of game play in traditional game theory, and the extent to which evolutionary game theory naturally incorporates dynamical considerations, reveals an important virtue of evolutionary game theory. Evolutionary game theory has been used to explain a number of aspects of human behavior. A small sampling of topics which have been analysed from the evolutionary perspective include: altruism Fletcher and Zwick, ; Gintis et al.

The following subsections provide a brief illustration of the use of evolutionary game theoretic models to explain two areas of human behavior. The first concerns the tendency of people to share equally in perfectly symmetric situations. The second shows how populations of pre-linguistic individuals may coordinate on the use of a simple signaling system even though they lack the ability to communicate. These two models have been pointed to as preliminary explanations of our sense of fairness and language, respectively.

They were selected for inclusion here primarily because of the relative simplicity of the model and apparent success at explaining the phenomenon in question. One natural game to use for investigating the evolution of fairness is divide-the-cake this is the simplest version of the Nash bargaining game. In chapter 1 of Evolution of the Social Contract , Skyrms presents the problem as follows:.

More formally, suppose that two individuals are presented with a resource of size C by a third party. A strategy for a player, in this game, consists of an amount of cake that he would like. The set of possible strategies for a player is thus any amount between 0 and C. If the sum of strategies for each player is less than or equal to C , each player receives the amount he asked for.

However, if the sum of strategies exceeds C , no player receives anything. Figure 8 illustrates the feasible set for this game. Even in the perfectly symmetric situation, answering this question is more difficult than it first appears. To see this, first notice that there are an infinite number of Nash equilibria for this game. Each player's strategy is a best response given what the other has chosen, in the sense that neither player can increase her payoff by changing her strategy.

However, the equal split is only one of infinitely many Nash equilibria. One might propose that both players should choose that strategy which maximizes their expected payoff on the assumption they are uncertain as to whether they will be assigned the role of Player 1 or Player 2.

This proposal, Skyrms notes, is essentially that of Harsanyi The problem with this is that if players only care about their expected payoff, and they think that it is equally likely that they will be assigned the role of Player 1 or Player 2, then this, too, fails to select uniquely the equal split. Now consider the following evolutionary model: suppose we have a population of individuals who pair up and repeatedly play the game of divide-the-cake, modifying their strategies over time in a way which is described by the replicator dynamics.

For convenience, let us assume that the cake is divided into 10 equally sized slices and that each player's strategy conforms to one of the following 11 possible types: Demand 0 slices, Demand 1 slice, … , Demand 10 slices. The replicator dynamics allows us to model how the distribution of strategies in the population changes over time, beginning from a particular initial condition. Figure 9 below shows two evolutionary outcomes under the continuous replicator dynamics. Notice that although fair division can evolve, as in Figure 9 a , it is not the only evolutionary outcome, as Figure 9 b illustrates.

What the above shows is that, in a population of boundedly rational agents who modify their behaviours in a manner described by the 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 measured by determining the size of the basin of attraction of the state where everyone in the population uses the strategy Demand 5 slices.

However, it is important to realise that the replicator dynamics assumes any pairwise interaction between individuals is equally likely. In reality, quite often interactions between individuals are correlated to some extent. When correlation is introduced, the frequency with which fair division emerges changes drastically.

Note that this does not depend on there only being three strategies present: allowing for some correlation between interactions increases the probability of fair division evolving even if the initial conditions contain individuals using any of the eleven possible strategies.

What, then, can we conclude from this model regarding the evolution of fair division? It all depends, of course, on how accurately the replicator dynamics models the primary evolutionary forces cultural or biological acting on human populations. As Skyrms notes:. This claim, of course, has not gone without comment. For a selection of some discussion see, in particular, D'Arms , ; D'Arms et al.

In his seminal work Convention , David Lewis developed the idea of sender-receiver games. Such games have been used to explain how language, and semantic content, can emerge in a community which originally did not possess any language whatsoever. Lewis, , pp. Since the publication of Convention , it is more common to refer to the communicator as the sender and the members of the audience as receivers.

The basic idea behind sender-receiver games is the following: Nature selects which state of the world obtains. The person in the role of Sender observes this state of the world correctly identifying it , and sends a signal to the person in the role of Receiver. The Receiver, upon receipt of this signal, performs a response. If what the Receiver does is the correct response, given the state of the world, then both players receive a payoff of 1; if the Receiver performed an incorrect response, then both players receive a payoff of 0.

Notice that, in this simplified model, no chance of error exists at any stage. The Sender always observes the true state of the world and always sends the signal he intended to send.

Likewise, the Receiver always receives the signal sent by the Sender i. We shall see later why larger sender-receiver games are increasingly 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 the state of the world this is what requiring F to be one-to-one amounts to and that everyone in the audience agrees that this is the case. Since we are considering the case where there is only a single responder, the second requirement is otiose. For the case of two states of the world and two responses, there are only two ways of assigning responses to states of the world which satisfy Lewis's requirement.

It makes no real difference for the model which one of these we choose, so pick the intuitive one: in state of the world S i , the best response is r i i. It is, as Lewis notes, a function from the set of states of the world into the set of signals. This means that it is possible that a sender may send the same signal in two different states of the world. Such a strategy makes no sense, from a rational point of view, because the receiver would not get enough information to be able to identify the correct response for the state of the world.

However, we do not exclude these strategies from consideration because they are logically possible strategies. How many sender strategies are there? Because we allow for the possibility of the same signal to be sent for multiple states of the world, 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. What is a strategy for a receiver? Instead, let us take a receiver's strategy to be a function from the set of signals into the set of responses.

As in the case of the sender, we allow the receiver to perform the same response for more than one signal. By symmetry, this means there are 4 possible receiver strategies. These receiver strategies are:. All other possible combinations of strategies result in the players failing to coordinate. What if the roles of Sender and Receiver are not permanently assigned to individuals? That is, what if nature flips a coin and assigns one player to the role of Sender and the other player to the role of Receiver, and then has them play the game? In this case, a player's strategy needs to specify what he will do when assigned the role of Sender, as well as what he will do when assigned the role of Receiver.

Since there are four possible strategies to use as Sender and four possible strategies to use as Receiver, this means that there are a total of 16 possible strategies for the sender-receiver game when roles are not permanently assigned to individuals. It makes a difference whether one considers the roles of Sender and Receiver to be permanently assigned or not. If the roles are assigned at random, there are four signaling systems amongst two players [ 10 ] :.

Signaling systems 3 and 4 are curious. System 3 is a case where, for example, I speak in French but listen in German, and you speak German but listen in French. System 4 swaps French and German for both you and me. Notice that in systems 3 and 4 the players are able to correctly coordinate the response with the state of the world regardless of who gets assigned the role of Sender or Receiver.

## Henry Brighton

The problem, of course, with signaling systems 3 and 4 is that neither Player 1 nor Player 2 would do well when pitted against a clone of himself. They are cases where the signaling system would not work in a population of players who are pairwise randomly assigned to play the sender-receiver game. In fact, it is straightforward to show that the strategies Sender 2, Receiver 2 and Sender 3, Receiver 3 are the only evolutionarily stable strategies see Skyrms , 89— As a first approach to the dynamics of sender-receiver games, let us restrict attention to the four strategies Sender 1, Receiver 1 , Sender 2, Receiver 2 , Sender 3, Receiver 3 , and Sender 4, Receiver 4.

Figure 11 illustrates the state space under the continuous replicator dynamics for the sender-receiver game consisting of two states of the world, two signals, and two responses, where players are restricted to using one of the previous four strategies. One can see that evolution leads the population in almost all cases [ 11 ] to converge to one of the two signaling systems. Figure 12 illustrates the outcome of one run of the replicator dynamics for a single population model where all sixteen possible strategies are represented.

We see that eventually the population, for this particular set of initial conditions, converges to one of the pure Lewisian signalling systems identified above. When the number of states of the world, the number of signals, and the number of actions increase from 2, the situation rapidly becomes much more complex. If there are N states of the world, N signals, and N actions, the total number of possible strategies equals N 2N.

Given this, one might think that it would prove difficult for evolution to settle upon an optimal signalling system. Such an intuition is correct. Hofbauer and Hutteger show that, quite often, the replicator dynamics will converge to a suboptimal outcome in signalling games. In these suboptimal outcomes, a pooling or partial pooling equilibrium will emerge.

A pooling equilibrium occurs when the Sender uses the same signal regardless of the state of the world. A partial pooling equilibrium occurs when the Sender is capable of differentiating between some states of the world but not others. Furthermore, suppose that the Receiver performs action 1 upon receipt of signal 1, and action 2 upon receipt of signals 2 and 3. If all states of the world are equiprobable, this is a partial pooling equilibrium.

Given that the Sender does not differentiate states of the world 2 and 3, the Receiver cannot improve his payoffs by responding differently to signal 2. Given the particular response behaviour of the Receiver, the Sender cannot improve her payoffs by attempting to differentiate states of the world 2 and 3.

As noted previously, evolutionary game theoretic models may often be given both a biological and a cultural evolutionary interpretation. In many cases, fitness in cultural evolutionary interpretations of evolutionary game theoretic models directly measures some objective quantity of which it can be safely assumed that 1 individuals always want more rather than less and 2 interpersonal comparisons are meaningful.

Depending on the particular problem modeled, money, slices of cake, or amount of land would be appropriate cultural evolutionary interpretations of fitness. Requiring that fitness in cultural evolutionary game theoretic models conform to this interpretative constraint severely limits the kinds of problems that one can address. A more useful cultural evolutionary framework would provide a more general theory which did not require that individual fitness be a linear or strictly increasing function of the amount of some real quantity, like amount of food.

In traditional game theory, a strategy's fitness was measured by the expected utility it had for the individual in question. Consequently, the utility theory used in traditional game theory cannot simply be carried over to evolutionary game theory. Another question facing evolutionary game theoretic explanations of social phenomena concerns the kind of explanation it seeks to give. Depending on the type of explanation it seeks to provide, are evolutionary game theoretic explanations of social phenomena irrelevant or mere vehicles for the promulgation of pre-existing values and biases?

To understand this question, recognize that one must ask whether evolutionary game theoretic explanations target the etiology of the phenomenon in question, the persistence of the phenomenon, or various aspects of the normativity attached to the phenomenon. The latter two questions seem deeply connected, for population members typically enforce social behaviors and rules having normative force by sanctions placed on those failing to comply with the relevant norm; and the presence of sanctions, if suitably strong, explains the persistence of the norm.

The question regarding a phenomenon's etiology, on the other hand, can be considered independent of the latter questions. If one wishes to explain how some currently existing social phenomenon came to be, it is unclear why approaching it from the point of view of evolutionary game theory would be particularily illuminating. The etiology of any phenomenon is a unique historical event and, as such, can only be discovered empirically, relying on the work of sociologists, anthropologists, archaeologists, and the like.

Although an evolutionary game theoretic model may exclude certain historical sequences as possible histories since one may be able to show that the cultural evolutionary dynamics preclude one sequence from generating the phenomenon in question , it seems unlikely that an evolutionary game theoretic model would indicate a unique historical sequence suffices to bring about the phenomenon. An empirical inquiry would then still need to be conducted to rule out the extraneous historical sequences admitted by the model, which raises the question of what, if anything, was gained by the construction of an evolutionary game theoretic model in the intermediate stage.

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Moreover, even if an evolutionary game theoretic model indicated that a single historical sequence was capable of producing a given social phenomenon, there remains the important question of why we ought to take this result seriously. One may point out that since nearly any result can be produced by a model by suitable adjusting of the dynamics and initial conditions, all that the evolutionary game theorist has done is provide one such model.

Additional work needs to be done to show that the underlying assumptions of the model both the cultural evolutionary dynamics and the initial conditions are empirically supported. Again, one may wonder what has been gained by the evolutionary model--would it not have been just as easy to determine the cultural dynamics and initial conditions beforehand, constructing the model afterwards? If so, it would seem that the contributions made by evolutionary game theory in this context simply are a proper part of the parent social science--sociology, anthropology, economics, and so on.

If so, then there is nothing particular about evolutionary game theory employed in the explanation, and this means that, contrary to appearances, evolutionary game theory is really irrelevant to the given explanation. If evolutionary game theoretic models do not explain the etiology of a social phenomenon, presumably they explain the persistence of the phenomenon or the normativity attached to it. Yet we rarely need an evolutionary game theoretic model to identify a particular social phenomenon as stable or persistent as that can be done by observation of present conditions and examination of the historical records; hence the charge of irrelevancy is raised again.

Moreover, most of the evolutionary game theoretic models developed to date have provided the crudest approximations of the real cultural dynamics driving the social phenomenon in question. One may well wonder why, in these cases, we should take seriously the stability analysis given by the model; answering this question would require one engage in an empirical study as previously discussed, ultimately leading to the charge of irrelevance again.

This criticism seems less serious than the charge of irrelevancy. The theory already contains, in its core, a proper subtheory having normative content--namely a theory of rational choice in which boundedly rational agents act in order to maximize, as best as they can, their own self-interest.

One may challenge the suitability of this as a foundation for the normative content of certain claims, but this is a different criticism from the above charge. Although cultural evolutionary game theoretic models do act as vehicles for promulgating certain values, they wear those minimal value commitments on their sleeve. Evolutionary explanations of social norms have the virtue of making their value commitments explicit and also of showing how other normative commitments such as fair division in certain bargaining situations, or cooperation in the prisoner's dilemma may be derived from the principled action of boundedly rational, self-interested agents.

Historical Development 2. Two Approaches to Evolutionary Game Theory 2. Why Evolutionary Game Theory? Applications of Evolutionary Game Theory 4. Philosophical Problems of Evolutionary Game Theory 5. Historical Development Evolutionary game theory was first developed by R. Each individual follows exactly one of two strategies described below: Hawk Initiate aggressive behaviour, not stopping until injured or until one's opponent backs down.

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Dove Retreat immediately if one's opponent initiates aggressive behaviour. Payoffs listed as row, column. This means that and Since the strategy frequencies for Defect and Cooperate in the next generation are given by and respectively, we see that over time the proportion of the population choosing the strategy Cooperate eventually becomes extinct.

Figure 3 illustrates one way of representing the replicator dynamical model of the prisoner's dilemma, known as a state-space diagram. Figure 3: The Replicator Dynamical Model of the Prisoner's Dilemma We interpret this diagram as follows: the leftmost point represents the state of the population where everyone defects, the rightmost point represents the state where everyone cooperates, and intermediate points represent states where some proportion of the population defects and the remainder cooperates.

Generation 1 Generation 2 Generation 19 Generation 20 Figure 5: Prisoner's Dilemma: Cooperate [View a movie of this model] Notice that with these particular settings of payoff values, the evolutionary dynamics of the local interaction model differ significantly from those of the replicator dynamics. Although evolutionary game theory has provided numerous insights to particular evolutionary questions, a growing number of social scientists have become interested in evolutionary game theory in hopes that it will provide tools for addressing a number of deficiencies in the traditional theory of games, three of which are discussed below.

A selection of strategies by a group of agents is said to be in a Nash equilibrium if each agent's strategy is a best-response to the strategies chosen by the other players. By best-response, we mean that no individual can improve her payoff by switching strategies unless at least one other individual switches strategies as well. This need not mean that the payoffs to each individual are optimal in a Nash equilibrium: indeed, one of the disturbing facts of the prisoner's dilemma is that the only Nash equilbrium of the game--when both agents defect--is suboptimal.

Heads Tails Heads 0,1 1,0 Tails 1,0 0,1 Figure 7: Payoff matrix for the game of Matching Pennies Row wins if the two coins do not match, whereas Column wins if the two coins match. This requirement originates in the development of the theory of utility which provides game theory's underpinnings see Luce for an introduction.

Since the number of different lotteries over outcomes is uncountably infinite, this requires each agent to have a well-defined, consistent set of uncountably infinitely many preferences. A dynamic theory would unquestionably be more complete and therefore preferable. But there is ample evidence from other branches of science that it is futile to try to build one as long as the static side is not thoroughly understood.

Von Neumann and Morgenstern, , p. Applications of Evolutionary Game Theory Evolutionary game theory has been used to explain a number of aspects of human behavior.

In chapter 1 of Evolution of the Social Contract , Skyrms presents the problem as follows: Here we start with a very simple problem; we are to divide a chocolate cake between us. Neither of us has any special claim as against the other. Out positions are entirely symmetric.

## Evolutionary Game Theory – Systems Innovation

The cake is a windfall for us, and it is up to us to divide it. But if we cannot agree how to share it, the cake will spoil and we will get nothing. Skyrms, , pp. Figure 8: The feasible set for the game of Divide-the-Cake. Figure 9: Two evolutionary outcomes under the continuous replicator dynamics for the game of divide-the-cake. Of the eleven strategies present, only three are colour-coded so as to be identifiable in the plot see the legend.

As Skyrms notes: In a finite population, in a finite time, where there is some random element in evolution, some reasonable amount of divisibility of the good and some correlation, we can say that it is likely that something close to share and share alike should evolve in dividing-the-cake situations. This is, perhaps, a beginning of an explanation of the origin of our concept of justice. The communicator, but not the audience, is in a good position to tell which one it is.

Each member of the audience can do any one of several alternative actions r 1 , …, r m called responses. Everyone involved wants the audience's responses to depend in a certain way upon the state of affairs that holds. The audience is in a good position to tell which one he does. No one involved has any preference regarding these actions which is strong enough to outweigh his preference for the dependence F of audience's responses upon states of affairs.

Figure The evolution of a signalling system under the replicator dynamics. Philosophical Problems of Evolutionary Game Theory The growing interest among social scientists and philosophers in evolutionary game theory has raised several philosophical questions, primarily stemming from its application to human subjects. After all, since any argument whose conclusion is a normative statement must have at least one normative statement in the premises, any evolutionary game theoretic argument purporting to show how certain norms acquire normative force must contain--at least implicitly--a normative statement in the premises.