Bonus Problem 3 (Optional, 20 marks)
We consider a N-person coordination games as follows: There are m (m ? 2) choices (say
C1, C2, ..., Cm) in the games and each player can choose one of these choices (so that his
strategic set is S? = {C1, C2, ..., Cm}. Every player will be awarded a positive payoff (denoted
by Vi(ck) > 0, i = 1,2,..., N and k = 1,2,..., m) if ALL players choose the same choice.
Otherwise, all players have zero payoffs.
We assume that the games is in dynamic setting which the players take turn choosing their
preference (say player 1 chooses first, player 2 chooses second and so on) and each player
knows the choice chosen by the players who move earlier.
(a) Show that all players will choose the same choice for any sequentially rational Nash
equilibrium.
(b) Show that the common choice $c^*$ chosen by the players must be the most preferable
choice for player 1. That is, $V_1(c^*) = \max_{c_i \in S_i} V_1(c_i)$.
