Two individuals (i {1, 2}) work independently on a joint project. They each independently decide how much effort ei they put. Effort choice has to be any real number between 0 and 1 (ei [0, 1] not just 0 or 1). The cost of putting an amount of effort ei is n e2 i /2, where n is a parameter strictly greater than 2. If individual i puts effort ei , then he succeeds with probability ei and fails with probability 1 ei . The probability of success of the two agents are independent; this means that both succeed with probability e1 e2, 1 succeeds and 2 fails with probability e1 (1 e2), 1 fails and 2 succeeds with probability (1 e1) e2, and both fail with probability (1 e1) (1 e2). If at least one of the individuals succeeds then, independently of who did succeed, both individuals get a payoff of 1. If none of them succeeds, both individuals get 0. Therefore, each individual is affected by the action of the other. However, individuals choose the level of effort that maximizes their own expected utility (benefit minus cost of effort).

(a) Write down the expected utility of individuals 1 and 2 (note that the utility of 1 depends on the efforts of 1 and 2 and the utility of 2 depends on the efforts of 1 and 2). [Hint. The expected benefit of 1 is the probability that 1 and/or 2 succeed times the payoff if 1 and/or 2 succeed plus the probability that both 1 and 2 fail times the payoff if both 1 and 2 fail.]

(b) Find the Nash equilibrium of this game, that is, the optimal level of effort. Find the expected utility of each individual in equilibrium (use the first-order condition and make sure that the second-order condition is satisfied).