Economics & Evolution

1. Economics & Evolution Number 3

2. The replicator dynamics (in general)

3. The Replicator Dynamics in the Generalized Rock Scissors, Paper A = ? ? ?

4. The Replicator Dynamics in the Generalized Rock Scissors, Paper A =

5. The Replicator Dynamics in the Generalized Rock Scissors, Paper A =

6. S3 S2 S1 The Replicator Dynamics in the Generalized Rock Scissors, Paper A = Intersection of a hyperbola with the triangle

7. S3 S2 S1 The Replicator Dynamics in the Generalized Rock Scissors, Paper A = Does not converge to equlibrium ?

8. S3 S2 S1 The Replicator Dynamics in the Generalized Rock Scissors, Paper A = Moves away from equlibrium

9. S3 S2 S1 The Replicator Dynamics in the Generalized Rock Scissors, Paper A = Moves towards equlibrium

10. Let ξ(t, x0) denote the replicator dynamics of a given game, beginning at x0. Replicator Dynamics and Strict Dominance (here it is used that x0is in the interior)

11. Replicator Dynamics and Weak Dominance

12. Replicator Dynamics and Weak Dominance e1vanishes !!!

13. Replicator Dynamics and Weak Dominance x1/x2 increases as long as x3 > 0.

14. S2 S3 S1 Replicator Dynamics and Weak Dominance x1/x2 increases as long as x3 > 0. x1/x2 = 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 x0 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 S3 S2 S1 Example: A stable point need not be Lyapunov stable. On the edges: There are therefore close trajectories:

22. Stability Concepts A population plays the strategyp, 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 (s,s)is not an ESS,tcan invade and does better !! t is like s against s, but earns more against itself. (t,t)is an ESS,tis the unique best response to itself. (t,t)is a strict Nash Equilibrium

28. ESS does not always exist Rock, Scissors, Paper The only equilibrium is α =(⅓, ⅓, ⅓) But αcan 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 of player 1’s payoffs in a symmetric game GM. Obtain a matrix N by adding to each column of M a constant. (Nij=Mij+cj) Show that the two games: GM ,GN have the same eqilibria, same ESS, and the same Replicator Dynamics

30. ESS of 2x2 games Given a symmetric game, c1 = -a21 c2 = -a12

31. ESS of 2x2 games P.D If: b1 > 0, b2 < 0 The first strategy is the unique equilibrium of this game, and it is a strict one. Hence it is the unique ESS.

32. ESS of 2x2 games Coordination If: b1 > 0, b2 > 0 Both pure equilibria are strict. Hence they are ESS. The mixed strategy equilibrium: is not an ESS.

33. ESS of 2x2 games Chicken If: b1 < 0, b2 < 0 The only symmetric equilibrium is the mixed one. 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 2x2 games Chicken If: b1 < 0, b2 < 0 (-)

35. ESS of 2x2 games Chicken If: b1 < 0, b2 < 0

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 againsttwo mutants !!! Chicken

41. Q.E.D.

42. Q.E.D.

43. Q.E.D.