Consider the following n-player game. Simultaneously and independently, the players each select either X, Y, or Z. The payoffs are defined as follows. Each player who selects X obtains a payoff equal to g, where g is the number of players who select Z.
Each player who selects Y obtains a payoff of 2a, where a is the number of players who select X. Each player who selects Z obtains a payoff of 3b, where b is the number of players who select Y. Note that a + b + g = n.
(a) Suppose n = 2. Represent this game in the normal form by drawing the appropriate matrix.
(b) In the case of n = 2, does this game have a Nash equilibrium? If so, describe it.
(c) Suppose n = 11. Does this game have a Nash equilibrium? If so, describe an equilibrium and explain how many Nash equilibria there are.