ABS 2006
description
Transcript of ABS 2006
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Evolution’s strategies
Genetic algorithms and game theory models
Steven Hamblin and Peter L. Hurd Department of Psychology, University of Alberta
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What I’ll be discussing…
1. Extensive form games and alternatives to ESS.
2. Solving game theory models using genetic algorithms.
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ESS - Evolutionarily stable strategy
! An uninvadable strategy: if every member of a population plays that one strategy, then no mutant can invade. (Maynard Smith, 1982)
! An ESS is a mathematical description of a population equilibrium.
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Payoff matrix (normal form) Extensive form game - usually better for biological games.
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A common problem with more complex games is strategies that are not pervasive.
Here, it never pays for player 1 to choose the last branch, so player 2’s choice at that branch is moot.
Pervasive - all information sets reached with non-zero probability.
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ES set
! A set of strategies that would, individually, be ESSs except that they all invade each other. (Thomas, 1985; Cressman, 1992)
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The ESS formalism is not enough for games with realistic complexity.
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Solving them another way…
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An alternative tool: Genetic algorithms.
! Algorithms that simulate evolution to solve optimization problems.
! Heuristic search as opposed to analytical solutions.
! Scales more effectively to larger games (greater biological realism).
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The e85 model (Enquist, 1985)
324 pure strategies with a pervasive ESS.
If we add another state variable or signal, we can end up with over ten million strategies!
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Graph shows strategy evolution over time.
Strategies split into two halves: when ego strong and when ego weak.
18 colours for each half: 18 * 18 = 324 total
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As the mutation rate goes higher, it becomes harder and harder (or impossible) for the GA to find the ESS.
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The ESS goes extinct very quickly. Pink/Red - A previously unknown ES set solution to the e85 game.
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Results
! A set of strategies whose end move is always “attack”, is a previously unknown ES set solution to the e85 game.
! The ES set has much greater attractive power than the ESS.
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Take home message… ! ESS is useful intuitively, but
limited practically.
! Most games with temporal sequence / underlying state / etc., won’t have an ESS.
! Even more useful solution tools (e.g. ES sets) are too complicated to calculate for larger, more realistic games.
! Genetic algorithms are a sensible choice to solve complex game theory models.
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Acknowledgements
! Pete Hurd, for … well, just about everything.
! Eldridge Adams, for valuable discussion on the inability of GAs to find the e85 ESS.
! The members of the Hurd lab for feedback and advice.
! Brandy Williams, for design input.