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Three game theorists work on a team project. Assume they work independently, andit is not possible for them to monitor each other. No one likes working, and the costof e§orts is measured in dollars. Each player can work up to 5 hours, and the costper working hour is $6: The quality and hence the proÖt of the project is determinedby the joint efforts (total number of working hours) of the three players. For eachadditional hour that the team invests in the project, up to 10 hours, the total proÖtwill increase by $15: After 10 hours, efforts will not increase profits. Assume playersdivide the profit equally among them. Hence, each playerís payoff is his share of theprofit 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 profles?
(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 stillshare the profit equally. Compute the set of rationalizable effort levels.

(a)每个参与者的策略空间为(0,5)
(b)设三个参与者的工作时间分别为x1,x2,x3，则参与人i的支付函数为分段函数：ui=5(x1+x2+x3)-6xi  (当x1+x2+x3<=10时）ui=50-6xi(当x1+x2+x3>=10时）
(c)当x1+x2+x3<=10时，任何一个参与者的支付函数关于自身策略（工作时间）的导数为-1，自身的支付随着工作时间的增加而减少，所以最佳工作时间为0；当x1+x2+x3>=10时，任何一个参与者的支付函数关于自身策略（工作时间）的导数为-6，自身的支付还是随着工作时间的增加而减少，所以最佳工作时间为0。总之，每个参与者的最佳工作时间都是0，没有人花时间工作。
(d)当只有两个参与者时，参与者i的支付函数变为ui=7.5(x1+x2)-6xi，此时，自身支付函数关于工作时间的导数为1.5，意味着，自身支付随着工作时间的增加而增加，所谓最佳策略就是选择工作时间为5小时。