Part 1 Equilibrium Selection Consider the following version of the example in section 2 of Kandori et al (1993). “Learn-ing, mutation, and long run equilibria in games” with students meeting in pairs every period to work together with each using one of two computer systems, si or 82. There are 9 students in the group and the payoff matrix is
Si
S2
S1 S2 3,3 0,0 0,0 1,1
During each period every student meets every other student once and at the end of each period, students observe the average payoffs from having each computer system. Those who have the system doing worse on average change their systems for the next period. Also at the beginning of each period, before meetings begin, each member of the group exits (mutates) with a probability E and is replaced by a student who has computer system Si with probability m and s2 with probability (1 — m). The meetings then take place with no opportunity to change computer systems during the period. Let z represent the state of the system which is the number of the nine students with computer system si and rj(z) the average payoff of a student with computer system si when the state is z.
(a) What are the three Nash equilibria of this game?
(b) Work out the values of z for which the system moves to coordination on si and for which it move to coordination on s2 (check this with average payoff calculations). Illustrate the basins of attraction on a line and indicate how many mutations are required to escape from each equilibrium.
(c) Give the intuition for why coordination on si is more likely in the long run when the mutation rate is sufficiently small and how this result is independent of m so long as m > 0.
(d) Briefly discuss how the approach in this paper is an improvement on other ways of addressing the equilibrium selection problem and also comment on any limitations.
