Evolutionary Game Theory: A Multidisciplinary Approach, Study notes of Computer Science

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Evolutionary Game Theory

Patrick Roos

Department of Computer Science Universities of Maryland

CMSC 828N, Game Theory - Spring 2009

Evolutionary Game Theory (EGT)

Origin in Biology

• (^) Ronald A. Fisher
  • (^) The Genetic Theory of Natural Selection 1930
  • (^) Why equal sex ratio?
  • (^) Frequency Dependent Individual Fitness
• (^) Richard C. Lewontin
  • (^) Evolution and the Theory of Games 1961
  • (^) Explicitly: game theory → evolutionary biology
• (^) Taylor and Jonker (1978) and Zeeman (1979)
  • Replicator Equations as evolutionary dynamic in EGT
• John Maynard Smith
  • "The Logic of Animal Conflict" - Nature 1973 (G. R. Price)
  • Evolution and the Theory of Games 1982
  • Evolutionarily Stable Strategy (ESS)

Evolutionary Game Theory

Maynard Smith’s shifts from Classical GT

• Strategy
  • (^) Species have strategy sets (not players)
  • (^) Individuals inherit strategy - possibly mutated
• (^) Equilibrium
  • (^) Evolutionarily Stable Strategy (ESS) in place of NE
  • (^) Population using strategy A cannot be invaded by a small group using strategy B
• (^) Player Interactions
  • (^) Repeated, random pairings of agents in population

Hawk Dove Game

(Chicken Game if C > G)

• "The Logic of Animal Conflict"
• (^) Population of birds fighting over food
• Hawk: escalate battle
• (^) Dove: retreat if opponent escalates

Payoff Matrix

H D

H (G − C)/ 2 G

D 0 G/ 2

G = Payoff from food, C = Cost of injury

Evolutionary Stability

A strategy is evolutionarily stable if no other strategy can invade it under the influence of natural selection. We say a strategy μ can invade a population of σ if Fμ ≥ Fσ.

Strategy σ is an evolutionary stable strategy if, for all strategies μ 6 = σ, Wσσ ≥ Wμσ

and if Wσσ = Wμσ, Wσμ > Wμμ

Hawk Dove Game Stability

• (^) S = {Hawk, Dove}
• (^) Is a strategy evolutionarily stable if G ≥ C?
  • (^) WHH ≥ WDH?

Payoff Matrix

H D

H (G − C)/ 2 G

D 0 G/ 2

Payoff Matrix

H D

H 1 3

D 0 1. 5

Hawk Dove Game Stability

• (^) S = {Hawk, Dove}
• (^) Is a strategy evolutionarily stable if G < C?
  • (^) WHH ≥ WDH?
  • (^) WDD ≥ WHD?

Payoff Matrix

H D

H (G − C)/ 2 G

D 0 G/ 2

Payoff Matrix

H D

H − 1 2

D 0 1

Hawk Dove Game Stability

• S = {Hawk, Dove}
• Is a strategy evolutionarily stable if G < C?
  • WHH ≥ WDH?
  • WDD ≥ WHD?
• (^) Neither is evolutionarily stable
• So, what happens in a pop of H and D?

Payoff Matrix

H D

H (G − C)/ 2 G

D 0 G/ 2

Payoff Matrix

H D

H − 1 2

D 0 1

Replicator Equation

If we express evolutionary success as the difference between the fitness of a replicator (player or strategy in evolutionary game theory) and the average fitness in the population, we obtain the ODE: x^ ˙i = xi [ Fi ( x ) − θ( x )],

where x is a vector holding the proportions of all player types in the population, xi is the proportion of player type i in the population, x ˙i is the rate of change, Fi ( x ) is the average fitness of a player of type i (depending on the population make-up x ), and θ( x ) is the average fitness in the population.

Hawk Dove Game Stability

• S = {Hawk, Dove}
• (^) Is a strategy evolutionarily stable if G < C?
  • (^) WHH ≥ WDH?
  • (^) WDD ≥ WHD?
• Neither is evolutionarily stable
• (^) So, what happens in a pop of H and D?

Payoff Matrix

H D

H 0 7

D 2 6

Hawk Dove Under Replicator

Replicator Equation

Rest Points

If we have an n × n matrix U , such that Fi ( x ) = ( Ux ) i , then the replicator equation

x^ ˙i = xi [ Fi ( x ) − θ( x )],

takes the form x^ ˙i = xi [( Ux ) i − x · Ux ],

the rest points of which are the solutions of

( Ux ) 1 = ... = ( Ux ) n

Evolutionary Stability

Implication

Theorem If x ∗ is an evolutionarily stable strategy, then x ∗ is an evolutionary equilibrium of the replicator dynamic. Moreover, if x ∗ uses all strategies with positive probability, then σ is a globally stable fixed point.

Evolutionary equilibrium = asymptotically stable fixed point

Replicator Dynamic

Implications

Theorem Under replicator dynamic,

• If x ∗ is a Nash equilibrium of the evolutionary game, x ∗ is a fixed (rest) point of the replicator dynamic.
• If x ∗ is an evolutionary equilibrium of the replicator dynamic, then it is a Nash equilibrium.