This choice is binary and denoted by a with a 1 if the boss audits and a 0

This choice is binary and denoted by a with a 1 if

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, must choose whether to audit the effort exerted by the agent. This choice is binary and denoted by a , with a = 1 if the boss audits and a = 0 otherwise. Auditing is unpleasant (it costs c > 0, but allows the boss to recoup half of the amount of effort that the agent did not exert ( i.e. , it gains the boss 1 - e ). The agent incurs a fixed penalty of P > 0 if he is caught exerting less than full effort. The players’ payoffs are as follows: u agent ( e,a ) = 1 - e if a = 0 1 if a = 1& e = 1 1 - e - P if a = 1& e 1 u boss ( e,a ) = e + 1 - e 2 - c a (a) [5pts] What are the players’ strategy spaces? (b) [10pts] Are there any dominated strategies? (c) [10pts] Find all of the Nash equilibria of the game. 4
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7. Consider a simultaneous first price auction between two players. Each player i ’s type, t i , is privately observed by player i and is uniformly distributed between 0 and 1. The two players’ types are independently distributed. The players each simultaneously submit a bid, b i . Player i ’s payoff, for both i { 1 , 2 } and j 3 - i , is u i ( b i ,b j ) = t i - b i if b i > b j , t i - b i 2 if b i = b j , 0 if b j > b i . (a) [5pts] What are the dominated strategies in this game? (b) [10pts] Find a perfect Bayesian equilibrium of this game. 8. Consider a sequential first price auction between two players. Each player i ’s type, t i , is privately observed by player i and is uniformly distributed between 0 and 1. The two players’ types are independently distributed. Player 1 submits a bid, b 1 , which is observed by player 2, who then submits a bid, b 2 . Player i ’s payoff, for both i { 1 , 2 } and j 3 - i , is u i ( b i ,b j ) = t i - b i if b i > b j , t i - b i 2 if b i = b j , 0 if b j > b i .
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  • Spring '11
  • JohnPaddy
  • Game Theory, Nash equilibria, Nash

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