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lecture6c - CMSC 498T Game Theory 6c Evolutionary Games...

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Nau: Game Theory 1 Updated 11/7/11 CMSC 498T, Game Theory 6c. Evolutionary Games Dana Nau University of Maryland
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Nau: Game Theory 2 Updated 11/7/11 Evolutionary Games Evolutionary game : a repeated stochastic game whose structure is intended to model certain aspects of evolutionary environments Øඏ Consists of a number of stages or generations In each stage, there is a set of k agents ( k is the total population size ) Øඏ Usually k is very large For mathematical analysis, k is often assumed to be infinite The agents interact in some kind of game-theoretic scenario Øඏ Different agents have different strategies (ways of choosing actions) Øඏ Each agent gets a numeric payoff that’s a stochastic function of the strategy profile (the strategies of all the agents) The payoffs are used in deciding what the set of agents will be at the next stage Agents at stage 1 Agents at stage 2 Agents at stage 3
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Nau: Game Theory 3 Updated 11/7/11 Evolutionary Games Consider the set of all agents that use strategy s Øඏ In a biological setting, s may represent a type of animal Øඏ In a cultural setting, s may represent a learned behavior Over time, the proportion of agents using s may grow or shrink depending on how well s performs Øඏ How this happens depends on the population dynamic (next slide) The measure of s ’s reproductive success is s ’s frequency (the proportion of the population that uses s ) after some long period of time Agents with strategy s 1 Agents with strategy s 2 stage 1 stage 2 stage 3
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Nau: Game Theory 4 Updated 11/7/11 Population Dynamics The population dynamic (or reproduction dynamic ) is the mechanism for deciding Øඏ which strategies will disappear Øඏ which strategies will reproduce Øඏ how many progeny they’ll have Many different possible reproduction dynamics
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Nau: Game Theory 5 Updated 11/7/11 Population Dynamics The population dynamic (or reproduction dynamic ) is the mechanism for deciding Øඏ which strategies will disappear Øඏ which strategies will reproduce Øඏ how many progeny they’ll have Many different possible reproduction dynamics Øඏ I’ll briefly discuss two of them No, not these two
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Nau: Game Theory 6 Updated 11/7/11 Replicator Dynamic In the replicator dynamic, Øඏ The change in frequency of each strategy s i is proportionate to the average performance of all the agents that use s i Øඏ Let f i ( j ) = the frequency of s i at stage j r i ( j ) = average payoff for those agents R = average payoff for all agents Øඏ At stage j +1, the frequency of s i will be Does well at reflecting growth of animal populations (where strategy ó༿ type of animal) Also has some nice mathematical properties Øඏ e.g., the definition of evolutionary stability in Chapter 3 f i ( j + 1) = f i ( j ) r i ( j ) R
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Nau: Game Theory 7 Updated 11/7/11 Imitate-the-Better Dynamic Model of learning by imitating others At stage j , let A j = {all agents at stage j } To build A j +1 , do the following steps k times:
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