Consider the following three-player team production problem. Simultaneously and independently, each player chooses between exerting effort (E) or not exerting effort (N). Exerting effort imposes a cost of 2 on the player who exerts effort.
If two or more of the players exert effort, each player receives a benefit of 4 regardless of whether she herself exerted effort. Otherwise, each player receives zero benefit. The payoff to each player is her realized benefit less the cost of her effort (if she exerted effort).
For instance, if player 1 selects N and players 2 and 3 both select E, then the payoff vector is (4, 2, 2). If player 1 selects E and players 2 and 3 both select N, then the payoff vector is (-2, 0, 0).
(a) Is there a pure-strategy equilibrium in which all three players exert effort?
Explain why or why not. (b) Find a symmetric mixed-strategy Nash equilibrium of this game. Let p denote the probability that an individual player selects N.