Evolutionarily Stable Strategy (ESS)

From Theory

Notes for CS 8803 - Game Theory and Computer Science. Spring 2008

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Evolutionarily stable strategy, ESS; (or also evolutionary stable strategy) was formalized by Maynard Smith. This is a strategy which, if adopted by a population of players, cannot be invaded by any mutant strategy that is initially rare.

A good motivating example is the Hawk - Dove Game , also sometimes called Chicken.

	Hawk	Dove
Hawk	-10, -10	30, 0
Dove	0, 30	10, 10
Hawk-Dove game

This is a symmetric 2 Player Game.

u₁(a₁,a₂) = u₂(a₂,a₁)), where A₁ = A₂, Δ₁ = Δ₂.

Contents 1 Hawk-Dove Game 2 Theorem 1 3 Stability Condition 4 Theorem 2 4.1 Proof 5 Existence

Hawk-Dove Game

Review: From previous lectures we found that in nature there are 2/3 Hawks and 1/3 Doves.

This can be found by solving this game from Player 2's perspective given that Player 1 is a Hawk with probability p and a Dove with probability(1-p).

The following sets of equations are developed:

Player 2 is a Hawk, expectation is: -10p + 30(1-p)= 30-40p

Player 2 is a Dove, expectation is: 0p + 10(1-p)= 10-10p

Solving simultaneously, Player 2 is a Hawk p = 2/3 and Player 1 is a Dove 1/3 through the symmetric nature of the game.

Theorem 1

Every symmetric finite game has a symmetric Nash Equilibrium. The proof will be a later homework exercise.

Definition. A mixed strategy σ is an ESS if for all either:

(1)

(2) and

This implies that (σ,σ) is a NE...no other choice is better.

Stability Condition

If you move away from the equilibrium you'll want to move back.

Restated as a Theorem 2:

Theorem 2

Suppose σis an ESS of a symmetric 2 player game.

Then (σ,σ) is a NE and for all such that is the Best Response.

Let β be the mutant fraction in a population and (1 − β) be the fraction of healthy population.

for all

=

Proof

(σ,σ) is a NE by definition.

Regrouping above Theorem,

=

So, though the population may drift slightly from NE it will eventually move back.

Existence

Although, ESS doesn't always exist.

See the following two games for examples:

Dog vs Dog'

Who gets $10?

	Dog	Dog'
Dog	0, 0	0, 0
Dog'	0, 0	0, 0
Dog vs Dog'

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The production of this material was supported in part by NSF award SES-0734780.