Stat155 Game Theory Lecture 15: Evolutionary Game Theory Outline

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Stat155 Game Theory Lecture 15: Evolutionary game theory Nash equilibria Criticisms of Nash equilibria Example: hawks and doves Peter Bartlett Evolutionarily stable strategies

October 18, 2016

1 / 21 2 / 21 Multiplayer general-sum games Multiplayer general-sum games

Notation Definition A k-person general-sum game is specified by k utility functions, A sequence x = (x∗,..., x∗) ∆S ∆S (called a strategy profile) 1 k ∈ 1 × · · · × k uj : S1 S2 Sk R. is a Nash equilibrium for utility functions u1,..., uk if, for each player × × · · · × → Player j can choose strategies s S . j 1,..., k , j ∈ j ∈ { } Simultaneously, each player chooses a strategy. max uj (xj , x∗ j ) = uj (xj∗, x∗ j ). xj ∆S − − Player j receives payoff uj (s1,..., sk ). ∈ j

More generally, Player j can choose mixed strategies xj ∆S , and ∈ j If the players play these mixed strategies xj∗, nobody has an incentive receives payoff uj (x) = Es1 x1,...,sk xk uj (s1,..., sk ), where ∼ ∼ to unilaterally deviate: each player’s mixed strategy is a best response x = (x1,..., xk ). to the other players’ mixed strategies.

3 / 21 4 / 21 What’s wrong with Nash equilibria? Outline

Will all players know everyone’s utilities? Maximizing expected utility does not (explicitly) model risk aversion. Will players maximize utility and completely ignore the impact on other players’ utilities? Nash equilibria How can the players find a Nash equilibrium? Criticisms of Nash equilibria How can the players agree on a Nash equilibrium to play? Example: hawks and doves Will players actually randomize? Evolutionarily stable strategies

Alternative equilibrium concepts Correlated equilibrium Evolutionary stability Equilibria in perturbed games.

5 / 21 6 / 21 Hawks and Doves Chicken

(Karlin and Peres, 2016)

vocabulary.com 7 / 21 8 / 21 Hawks and Doves Hawks and Doves

Expected utility Payoff Fix v = 2, c = 2. H D H (-1,-1) (2,0) D (0,2) (1,1)

If your opponent plays Pr(H) = x, your expected utility is:

2 3x for H − 1 x for D −

Let’s play it, against a random (previous) opponent. 9 / 21 10 / 21 Outline Evolutionarily stable strategies

There is a population of individuals. Nash equilibria There is a game played between pairs of individuals. Criticisms of Nash equilibria Each individual has a pure strategy encoded in its genes. Example: hawks and doves The two players are randomly chosen individuals. Evolutionarily stable strategies A higher payoff gives higher reproductive success. This can push the population towards stable mixed strategies.

11 / 21 12 / 21 John Maynard Smith and George Price Hawks and Doves

Expected utility

© 1973 Nature Publishing Group

Nature

13 / 21 14 / 21 Evolutionarily stable strategies Evolutionarily stable strategies

Suppose that x is invaded by a small population of mutants z: x is replaced by (1 )x + z. − Will the mix x survive?

Consider a two-player game with payoff matrices A, B. x’s utility: x>A (z + (1 )x) = x>Az + (1 )x>Ax Suppose that it is symmetric (A = B>). − − Consider a mixed strategy x. z’s utility: z>A (z + (1 )x) = z>Az + (1 )z>Ax − − Think of x as the proportion of each pure strategy in the population.

Definition Mixed strategy x ∆ is an evolutionarily stable strategy (ESS) if, for any ∈ n pure strategy z, 1 z Ax x Ax (x, x) is a Nash equilibrium. > ≤ > ←− 2 If z>Ax = x>Ax then z>Az < x>Az.

15 / 21 16 / 21 Hawks and Doves Hawks and Doves

Payoff Fix v = 2, c = 2.

H D x = (1/2, 1/2) is a Nash equilibrium. H (-1,-1) (2,0) Is it an ESS? D (0,2) (1,1) Consider a mutant pure strategy z. We need

1 z>Ax x>Ax? True, because( x, x) is a Nash equilibrium. Expected utility ≤ 2 If z>Ax = x>Ax then z>Az < x>Az? For z = (1, 0) (that is, H), z Az = 1 < 1/2 = x Az. > − − > For z = (0, 1) (that is, D), z>Az = 1 < 3/2 = x>Az. So x is an ESS.

17 / 21 18 / 21 Rock-Paper-Scissors Uta Stansburiana: Rock-paper-scissors in nature

Payoff R P S R 0 -1 1 P 1 0 -1 S -1 1 0

x = (1/3, 1/3, 1/3) is a Nash equilibrium. Is it an ESS? Consider a mutant pure strategy z. We need (Karlin and Peres, 2016) 1 z>Ax x>Ax? True, because( x, x) is a Nash equilibrium. ≤ Orange-throat (aggressive, large harems) 2 If z>Ax = x>Ax then z>Az < x>Az? defeats blue-throat (less aggressive, small harems) But for any pure strategy z, z Ax = 0 = z Az. > > defeats yellow-striped (submissive, look like females) So x is not an ESS. defeats orange-throat. Cycles can occur, with the population shifting between strategies. Six-year cycle of shifting population proportions. 19 / 21 20 / 21 Outline

Nash equilibria Criticisms of Nash equilibria Example: hawks and doves Evolutionarily stable strategies

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