Game Theory: lecture of April 16, 2019: The replicator equation

Game Theory: lecture of April 16, 2019: The replicator equation

Chiara Mocenni

Course of Game Theory

Main idea behind Evolutionary game theory

Evolutionary game theory

Infinite population of equal individuals (players or replicators) Each individual exhibits a certain phenotype (= it is

preprogrammed to play a certain strategy).

An individual asexually reproduces himself by replication.

Offspring will inherit the strategy of parent.

Nature favors the fittest

Two replicators are randomly drawn from the population.

Only one of them can reproduce himself.

They play a game in which strategies are the phenotypes.

Fitness of a replicator is evaluated like for game theory!

The fittest is able to reproduce himself. The share of population with that phenotype will increase.

Natural selection!

Replicator equation

The replicator equation describes the evolution of the pure strategies distribution over the whole population:

˙

x _i = x _i (p _i (x ) − φ(x ))

x _i is the share of population with strategy i 0 ≤ x _i ≤ 1, P

i x _i = 1 ∀t ≥ 0

p _i (x ) = [Bx ] _i is the payoff obtained by a player with strategy i φ(x ) = x ^T Bx is the average payoff obtained by a player B is the payoff matrix

Weibull, J.W., 1995. Evolutionary Game Theory. MIT Press

Hofbauer J., Sigmund K., 1998. Evolutionary games and population dynamics, Cambridge University Press

Replicator equation - 2 strategy games

A A A A

B A dominates B B dominates B Bistability B Coexistence

Stable Unstable

Payoff matrices Rest points

NE

NE NE NE

NE

Some "A", some "B"

All "B"

All "A"

NE

Marginal stability and existence of infinite equilibria are also

possible, e.g with payoff matrices [a b; a b].

Steady states of the RE and Nash equilibria of the game

The steady states (st.st.) of RE are states x = (x ₁ , . . . , x _N ) such that ˙ x i = 0, ∀i = 1, . . . , N.

All Nash equilibria are steady states of RE

Pure NE Pure NE are such that x _i = 0 or x _i = 1. If x _i = 0 then ˙ x i = 0. If x i = 1 then [Bx ] i = x ^T Bx , then ˙ x i = 0. We notice that all pure strategies are st. st. of RE.

Mixed NE Mixed NE are such that [Bx ] i = [Bx ] j , ∀i , j . then [Bx ] _i = x ^T Bx = M, ∀i , then ˙ x _i = 0, ∀i . Thus, mixed NE are st.st of RE.

Pure/Mixed NE Pure/Mixed NE are such that x _h = 0 for some h and [Bx ] i = [Bx ] j for i , j such that x i 6= 0 and x _j 6= 0.

Then, for the two results above, ˙ x _i = 0, ∀i .

Stability of st.st. and Nash equilibria

Lyapunov stable steady states of RE are Nash Equilibria of the game

Asymptotically stable steady states of RE are Nash Equilibria of the game

Evolutionary stable strategies of the game are asymptotically

stable steady states of RE

Figura: Steady states of RE and Nash equilibria of the game

Figura: Stability of steady states of RE and Nash equilibria of the game