Evolution’s strategies

Genetic algorithms and game theory models

Steven Hamblin and Peter L. Hurd Department of Psychology, University of Alberta

What I’ll be discussing…

1.  Extensive form games and alternatives to ESS.

2.  Solving game theory models using genetic algorithms.

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.

Payoff matrix (normal form) Extensive form game - usually better for biological games.

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.

ES set

! A set of strategies that would, individually, be ESSs except that they all invade each other. (Thomas, 1985; Cressman, 1992)

The ESS formalism is not enough for games with realistic complexity.

Solving them another way…

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).

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!

0 500

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

As the mutation rate goes higher, it becomes harder and harder (or impossible) for the GA to find the ESS.

0 500

The ESS goes extinct very quickly. Pink/Red - A previously unknown ES set solution to the e85 game.

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.

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.

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.