Evolution and the Theory of Games
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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.
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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.
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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.
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.
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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.
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Neither of us has any special claim as against the other. Out positions are entirely symmetric. 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. Consequently, this application of evolutionary game theory does not provide a neutral analysis of the norm in question, but merely acts as a vehicle for advancing particular values, namely those smuggled in the premises.
Bibliography Ackley, David and Michael Littman Langton, ed. Reading, MA: Addison-Wesley, pp. Adachi, N. Alexander, J. McKenzie Alexander, Jason and Brian Skyrms Axelrod, R. The Evolution of Cooperation. New York: Basic Books. Axelrod, Robert M. Banerjee, Abhijit V. Barrett, Jeffrey A. Bergin, J. Bicchieri, Cristina Binmore, Ken and Samuelson, Larry Blume, L.
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