Three game theorists work on a team project. Assume they work independently, and it is not possible for them to monitor each other. No one likes working, and the cost of efforts is measured in dollars. Each player can work up to 5 hours, and the cost per working hour is $6. The quality and hence the profit of the project is determined by the joint efforts (total number of working hours) of the three players. For each additional hour that the team invests in the project, up to 10 hours, the total profit will increase by $15. After 10 hours, efforts will not increase profits. Assume players divide the profit equally among them. Hence, each player's payoff is his share of the profit minus his cost. There is common knowledge of rationality.
(a) What is the strategy space (the set of strategies) for each player?
(b) What is the payoff for each player as a function of the strategy profiles?
(c) Compute the set of rationalizable effort levels for each player.
(d) Suppose one player is kicked out of the team, and the remaining two players still share the profit equally. Compute the set of rationalizable effort levels.