evolutionarily stable strategy

Definition

A mixed strategy $x^*$ in a two-player symmetric game is an evolutionarily stable strategy (ESS) if for every mixed strategy $x$ that differs from $x^*$ there exists $\epsilon_0 = \epsilon_0(x) > 0$ such that, for all $\epsilon \in (0,\epsilon_0)$,

where $u_1$ denotes the utility function for player 1

Application to dove-hawk population game

• see dove-hawk game#Equal payoffs

Theorem (symmetric Nash equilibrium)

If $x^*$ is an evolutionarily stable strategy in a two-player symmetric game, then $(x^*, x^*)$ is a symmetric Nash equilibrium in the game.

Theorem (condition for ESS)

A strategy $x^*$ is evolutionarily stable if and only if for each $x \neq x^*$ only one of the following two conditions obtains:

• $u_1(x, x^*) < u_1(x^*, x^*)$, or
  • (if a mutation deviates from $x^*$, it will lose in its encounters with the normal population)
• $u_1(x, x^*) < u_1(x^*, x^*)$ and $u_1(x, x) < u_1(x^*, x)$
  • (if the payoff a mutation receives from encountering a normal individual is equal to that received by a normal individual encountering a normal individual, mutation will receive a smaller payoff when it encounters the same mutation than a normal individual would receive when encountering the mutation)

Corollary (strict symmetric equilibrium)

In a symmetric game, if $(x^*, x^*)$ is a strict symmetric equilibrium then $x^*$ is an evolutionarily stable equilibrium.

(recall strict symmetric equilibrium means no player can change strategy without reducing payoff)

Notes

• Vincent et al. 2011 generalize ESS to evolutionary games governed by a system of ODEs

References

1. M. Maschler, E. Solan, and Shmuel Zamir, Game Theory, Cambridge University Press, 2013, p. 190.
2. Smith, J. M., & Price, G. R. (1973). The logic of animal conflict. Nature, 246(5427), 15-18. https://doi.org/10.1038/246015a0
3. https://wiki.mbalib.com/wiki/进化稳定策略
4. https://knowledgehive.github.io/Game-Theory/lecture 12.html
5. https://plato.stanford.edu/entries/game-evolutionary/
6. Vincent, Thomas L., Tania L.S. Vincent, and Yosef Cohen. 2011. “Darwinian Dynamics and Evolutionary Game Theory.” Journal of Biological Dynamics 5 (3): 215–26. https://doi.org/10.1080/17513758.2010.526306.
7. https://vknight.org/Year_3_game_theory_course/Content/Chapter_11_Population_Games_and_Evolutionary_stable_strategies/