1 Economics & Evolution Number 3

2 The replicator dynamics (in general)

3 The Replicator Dynamics in the Generalized Rock Scissors, Paper 12+a A = ? ? ?

4 The Replicator Dynamics in the Generalized Rock Scissors, Paper 12+a A =

5 The Replicator Dynamics in the Generalized Rock Scissors, Paper 12+a A =

6 The Replicator Dynamics in the Generalized Rock Scissors, Paper 12+a A = S2S2 S1S1 S3S3 Intersection of a hyperbola with the triangle

7 The Replicator Dynamics in the Generalized Rock Scissors, Paper 12+a A = S2S2 S1S1 S3S3 Does not converge to equlibrium ?

8 The Replicator Dynamics in the Generalized Rock Scissors, Paper 12+a A = S2S2 S1S1 S3S3 Moves away from equlibrium

9 The Replicator Dynamics in the Generalized Rock Scissors, Paper 12+a A = S2S2 S1S1 S3S3 Moves towards equlibrium

10 Let ξ(t, x 0 ) denote the replicator dynamics of a given game, beginning at x 0. (here it is used that x 0 is in the interior) Replicator Dynamics and Strict Dominance

11 Replicator Dynamics and Weak Dominance

12 Replicator Dynamics and Weak Dominance e 1 vanishes !!!

13 Replicator Dynamics and Weak Dominance x 1 /x 2 increases as long as x 3 > 0.

14 Replicator Dynamics and Weak Dominance x 1 /x 2 increases as long as x 3 > 0. S3S3 S1S1 S2S2 x 1 /x 2 = Constant

15 Replicator Dynamics and Nash Equilibria

16 Replicator Dynamics and Nash Equilibria

17 Replicator Dynamics and Nash Equilibria Q.E.D

18 Replicator Dynamics and Stability Lyapunov: If the process starts close, it remains close.

19 Replicator Dynamics and Stability Lemma: If x 0 is Lyapunov stable then it is a N.E.

20 Replicator Dynamics and Stability Example: A stable point need not be Lyapunov stable.

21 Replicator Dynamics and Stability Example: Example: A stable point need not be Lyapunov stable. S2S2 S1S1 S3S3 On the edges: There are therefore close trajectories:

22 Stability Concepts A population plays the strategy p, A small group of mutant enters, playing the strategy q The population is now (1-ε)p+ εq The fitness of p is: The fitness of q is:

23 Definition:

24 Lemma: ESS  Nash Equilibrium ( If p is an ESS then p is the best response to p ) ( If p is a strict equilibrium then it is an ESS.)

25 Proof: Q.E.D.

26 Proof: Q.E.D.

27 ESS is Nash Equilibrium, But not all Nash Equilibria are ESS st s 0,01, 0 t 0, 12, 2 (s,s) is not an ESS, t can invade and does better !! t is like s against s, but earns more against itself. (t,t) is an ESS, t is the unique best response to itself. (t,t) is a strict Nash Equilibrium

28 ESS does not always exist RSP R 0, 01, -1-1, 1 S 0, 01, -1 P -1, 10, 0 Rock, Scissors, Paper The only equilibrium is α = (⅓, ⅓, ⅓) But α can be invaded by R There is no distinction between α, R There is no ESS (the only candidate is not an ESS)

29 Exercise: Given a matrix M MM M of player 1’s payoffs in a symmetric game G GG GM. Obtain a matrix N NN N by adding to each column of M MM M a constant. (N ij =M ij +c j ) Show that the two games: G GG GM,GN have the same eqilibria, same ESS, and the same Replicator Dynamics

30 ESS of 2 x 2 games a 11, a 11 a 12, a 21 a 21, a 12 a 22, a 22 a 11 a 12 a 21 a 22 a 11 + c 1 a 12 + c 2 a 21 + c 1 a 22 + c 2 b1b1 0 0b2b2 Given a symmetric game, c 1 = -a 21 c 2 = -a 12

31 ESS of 2 x 2 games b1b1 0 0b2b2 If: b 1 > 0, b 2 < 0 The first strategy is the unique equilibrium of this game, and it is a strict one. Hence it is the unique ESS. P.D

32 ESS of 2 x 2 games b1b1 0 0b2b2 If: b 1 > 0, b 2 > 0 Both pure equilibria are strict. Hence they are ESS. Coordination The mixed strategy equilibrium: is not an ESS.

33 ESS of 2 x 2 games b1b1 0 0b2b2 If: b 1 < 0, b 2 < 0 The only symmetric equilibrium is the mixed one. Chicken is ESS. All strategies get the same payoff against x To show that x is an ESS we should show that for all strategies y :

34 ESS of 2 x 2 games b1b1 0 0b2b2 If: b 1 < 0, b 2 < 0 Chicken (-)

35 ESS of 2 x 2 games b1b1 0 0b2b2 If: b 1 < 0, b 2 < 0 Chicken

36 How many ESS can there be? Q.E.D.

37 It can be shown that there is a uniform invasion barrier.

38 ESS is not stable against two mutants !!! Chicken

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