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

, 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

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
.