Evolutionary Game Theory and Frequency Dependent Selection

Transcription

1 Evolutionary Game Theory and Frequency Dependent Selection 1

2 Game Theory A major component of economics Given a set of rules, predict how would rational beings behave in an interaction Who is in the game? What do they know? What are their options? What are their motives? What are the payoffs?

3 Biological Context How would evolved beings behave in an interaction Replicator dynamics is added to the model Just a way to calculate the fitness of a strategy Should they fight for a resource? Should they help some other individual? Should they continue investing resources in hopes of a payout? 3

4 Contextual Example: Hawk-Dove Game Members of the population want a resource with value V Hawk and Dove are names of two possible strategies Hawk: Always fight to get everything 0.5 chance of success Fighting costs energy (v) Dove: Share or run Opponent Player Hawk Hawk V C Dove 0 Dove V V 4

5 Calculating Payoffs Maximize the payoff of the average encounter h: Frequency of hawks (1-h): Frequency of doves F h, F d : Relative fitness values F h =h ( V C )+(1 h)v F d =h(0)+(1 h)( 1 Opponent Hawk Dove ) Player Hawk V C Dove 0 V V 5

6 Equilibrium Frequencies Equilize relative fitnesses and solve for h F h =h ( V C )+(1 h)v =F d =h(0)+(1 h)( 1 ) h= V C Yes this is a stable equilibrium since no strategy is inherently better than the other 6

7 Nash Equilibrium No individual player can benefit from changing tactics. Tragedy of Commons: Why we can't have nice things (e.g. world peace) If V>C: Hawks can invade a dove population and take over Doves can't survive in a hawk population Opponent Player Hawk Hawk V C Dove 0 Dove V V 7

8 A man who doesn't spend time with his family can never be a real man

9 Relatives in Hawk-Dove Hawk against a family member is bad Hawk against brother V C + V C 4 Dove against brother V + V 4 Net loss for the hawk 3C 4 Does it worth remembering? p familiarencounter ( 3C 4 M ) (1 p familiarencounter ) M Hamilton will come in a moment. Not really needed here since there is no cost to behaving altruistically in this game This is a toy example, I hoped to expalin this in words. M is the cost of remembering who your brother is per encounter. It represents the energy cost of having high cognitive power Hence in a large dispersed population that you are unlikely to meet your family members there is only the cost 9

10 Some considerations It is easy to think of alternative strategies But they have their own costs 10

11 Prisoner's dilemma Opponent Player Cooperate Defect Cooperate b-c -c Defect b 0 In isolation defectors always invade F coop = p coop (b c)+(1 p coop )( c)= p coop b c F defect = p coop (b)+(1 p coop )(0)= p coop b Iteration promotes cooperation ( tit for tat ) Cognitive abilitiy Population size Note that even with iteration, defection would be a stable strategy. But as long as a wide variety of strategies are present, those who are forgiving have an advantage as shown in Axelrod's tournaments. 11

12 Hamilton's Rule rb>c r: relatedness Opponent Player Cooperate Defect Cooperate b-c -c Defect b 0 B: benefit to the family member C: cost to individual What happens when we add relatives? Direct alturism towards kin F coop = p coop (b c+r(b c))+(1 p coop )( c+br)= p(b rc) c+br F defect = p coop (b+rc)+(1 p coop )(0+0 r)= p(b cr) b (b rc) c+br> p (b cr) br>c 1

13 Limitations of game theory High rationality requirement Games might lack a solution Modeling becomes unmanagable with complex problems Hard to predict all possible strategies and their mechanisms for evolving Payouts are not clear 13

14 In the wilds Connection of behavior to genes is not trivial Artificial selection for aggressive behavior in drosophila (Dierick and Greenspan 006) Look at gene expression to find differentially expressed genes in brain Not sequencing but still something... Induce mutations in mice to see what happens. (Brown et al. 1997) Fosb gene was found to be related to nurturing behavior Sequencing would give a direct gene to phenotype link while differential expression is not that helpfull. Just something I wanted to mention 14

15 Conclusion Behavioral characteristics of replicators are under evolutionary pressure Depending on the population structure this can lead to altruistic strategies and cooperation Though theoretically useful, it is often hard to characterize the exact parameters that effect fitness and phenotype. 15

16 Side Note Darwinbots: A nice little game to play with behavioral selection Code your bots, have them replicate and mutate Set costs for having more genes This is a nice concept. Physically identical bots fight over resources and reproduce based on their programming and random mutations are introduced to the code with every replication. It is even possible to create cooperating bots solely from an empty script with random mutations 16

17 References How to Find Hamilton s Rule. The Cultural Niche. N.p., n.d. Web. 17 Nov Alexander, J. McKenzie. Evolutionary Game Theory. The Stanford Encyclopedia of Philosophy. Ed. Edward N. Zalta. Fall 009. N.p., 009. Stanford Encyclopedia of Philosophy. Web. 17 Nov Brown, Jennifer R et al. A Defect in Nurturing in Mice Lacking the Immediate Early Gene fosb. Cell 86. (1996): ScienceDirect. Web. 17 Nov Dierick, Herman A., and Ralph J. Greenspan. Molecular Analysis of Flies Selected for Aggressive Behavior. Nature Genetics 38.9 (006): Web. 17 Nov