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Ia.Players: Firms 1 and 2Strategy Spaces: Quantity Spaces ( S 1 = S 2 = [0 ; 1 ))Payo Function for player i : [1 (q i + q j )]q i

The reaction (best response) function for each rm is q i = 12qj2 and the

unique Cournot-Nash equilibrium is q 1 = q 2 = 13 :The prots for each rm arei = 13 :

13 =

19 :

b. Trigger strategies: in even periods (starting at t = 0), rm 1 producesthe monopoly quantity and rm 2 produces nothing; and in odd periods rm 2produces the monopoly quantity and rm 1 produces nothing. If any rm hasdeviated, then both rms play the unique Nash equilibrium of the stage gameforever.

Let us analyze possible protable deviations from the trigger strategies - foreach possible history of the game:

- if a deviation has occurred in the past, no rm will have an incentiveto deviate because they are playing the Nash equilibrium of the stage game(regardless of the discount factor).

- if no deviation has occurred in the past and this is an even period, thestrategy species that rm 2 will produce 0 and rm 1 should produce themonopoly quantity q M = 12 (the one that solves Max q(1 q ):q ) that yields aprot of 12 :

12 =

14 : Firm 1 will have no incentive to deviate because q M =

12 is

rm 1s best response to the choice of 0 by rm 2 (q 1 = 1202 ):However, rm

2 may want to deviate in odd periods where q 1 =12 and rm 2s best responsewoud be q 2 = 12

1=22 =

14 : A deviation for rm 2 would result in a prot (this

period) of 14 :14 =

116 and both would revert to the Cournot-Nash equilibrium in

subsequent periods. A deviation will not occur i 0 + 14 + 20 + 3 14 + :::

116 +

19 +

2 19 +

3 19 + ::: ,

14

11 2

116 +

19

11 :

- if no deviation has occurred in the past and this is an odd period, thestrategy species that rm 1 will produce 0 and rm 2 should produce themonopoly quantity q M = 12 that yields a prot of

12 :

12 =

14 : Firm 2 will have no

incentive to deviate because q M = 12 is rm 2s best response to the choice of 0by rm 1 (q 2 = 12

02 ): However, rm 1 may want to deviate in odd periods where

q 2 = 12 and rm 1s best response woud be q 1 =12

1=22 =

14 : A deviation for rm

1 would result in a prot (this period) of 14 :

14 =

116 and both would revert to theCournot-Nash equilibrium in subsequent periods. A deviation will not occur i

0 + 14 + 20 + 3 14 + :::

116 +

19 +

2 19 +

3 19 + ::: ,

14

11 2

116 +

19

11 :

The values of the discount factor for which these trigger strategies still forma subgame-perfect Nash equilibrium are those for which 14

11 2

116 +

19

11 .

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1-

[1-p]

[p]

2

ND

D1, b

c, 0

1, 0

d, 1

0, 0

d, b

B

B

OJ

type H [ 1-q]

N2

type DND

D

ND

D

ND

D0, 1

c, 0

[q]

OJ

c. Denitions of NE and SPNE.

IIa.Let the payos for player 2 be 1 if Duel the H type, b if Duel the D type and

0 if No Duel, where 1 > b > 0:Let the payos for player 1 type H be 1 for No Duel and Orange Juice, 0

for Duel and Beer, c for No Duel and Beer; and d for Duel and Orange Juice,where 1> c> 0 and 1> d> 0.

Let the payos for player 1 type D be 0 for No Duel and Orange Juice, 1for Duel and Beer, c for No Duel and Beer; and d for Duel and Orange Juice,where 1> c> 0 and 1> d> 0.

The representation in extensive form is above.

b. Perfect Bayesian Equilibrium (signaling game).

c. Separating equilibria:- If 1 plays (H chooses OJ , D chooses B ), 2s beliefs are q = 0 , p = 1 . 2s

best response is ( D; D ) and no type will have an incentive to deviate.- If 1 plays (H chooses B , D chooses OJ ), 2s beliefs are q = 1 , p = 0 . 2s

best response is ( D; D ) but both types would have an incentive to deviate (typeH would get d > 0 and type D would get 1 > d ): There is no separating PBEwhere H chooses B and D chooses OJ:

The unique separating PBE is [(OJ if H; B if D ); (D; D ); p = 1 ; q = 0] :

IIIa. The eort levels are observable.In order to achieve eH ; the patient solves:Maxw H 34 E +

14 F wH

s.t. w1=2H 34 0

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and sets wH = 916 :In order to achieve eL ; the patient solves:Maxw L F wLs.t. w1=2L 0 0and sets wL = 0 :

b. The patient would choose eH because 34 E +14 F

916 F 0 , E F +

34

c. In order to achieve eH ; the patient will set the base wage wF in state Fto the lowest possible value (0) and wage wE will be such that the doctor willprefer to exert high eort i.e. such that 34 w

1=2E +

14 0

1=2 34 0

1=2 0 , wE 1:The patient will then set wF = 0 and wE = 1 :

d. In order to achieve low eort, it would be enough to set the wages equal

to 0 - and the patient would get F: With high eort, the patient would get34 (E wE ) +

14 (F wF ) =

34 E +

14 F

34 :The patient will choose high eort i

34 E +

14 F

34 F , E F + 1 :

(We only know that E > F + 34 but E F + 1 may or may not hold.)

e. It is now possible for the patient to prefer low eort (if E < F + 1) :Moreover, even if the patient prefers high eort, the expected payment to theagent is now 34 instead of

916 :

f. If the patient were risk-averse, in the symmetric information case, thewages for each eort level would be the same but the choice of the eort levelwould be dierent. Ceteris paribus, high eort would become less appealingbecause its outcome involves risk.

IVa. The auctioneers problem is:MaxP ( ) ;P ( ) ;M ( ) ;M ( ) 34 M ( ) +

14 M ( )

s.t.IC P ( ) M ( ) P ( ) M ( )IC P ( ) M ( ) P ( ) M ( )IR P ( ) M ( ) 0IR P ( ) M ( ) 0

b.1. IC + IR ! IR (we can ignore IR )2. IR binding (otherwise increase M ( ) and M ( ) by " , all constraints still

satised and revenue increases)3. IC binding (otherwise increase M ( ) by " , all constraints still satised

and revenue increases)4. IC not binding (IC and IR met with equality)

Therefore, P ( ) = M ( ) and P ( ) M ( ) = P ( ) M ( )

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i.e. M ( ) = P ( ) and M ( ) = P ( ) ( )P ( )

The principals problem is Max P ( ) ;P ( ) 34 P ( ) + 14 [ P ( ) ( )P ( )] =14 P ( ) + ( 4 )P ( )

Since P ( ) = 14 P ( ; ) +34 P ( ; ) and P ( ) =

34 P ( ; ) +

14 P ( ; ), at

the optimum the auctioneer will set P ( ; ) = 12 (the highest possible value),P ( ; ) = 1 (and P ( ; ) = 1 P ( ; )) and P ( ; ) = 0 (because 4 < 0 byassumption).

c. The auctioneer will never sell to a type - and this could be replicatedby introducing a minimum bid above in a second-price sealed bid auction.

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