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1. Nash equilibrium

In this game, player 2 knows which game they are playing but player 1 does not. Thus, player 1has two strategies available (T and B) regardless of which game she is playing and her decision willbe based on the expected payo¤s (Left with probability 1

2 and Right with probability12 ). But, player

2 should choose one strategy each game (Left and Right). This game can be summarised in matrix asbelow.

Player 2A;C A;D B;C B;D

Player 1 T 2; (2; 2) 4; (2; 0) 12 ; (4; 2)

52 ; (4; 0)

B 1; (2; 0) 52 ; (2; 3)

12 ; (1; 0) 2; (1; 3)

If player 1 chooses T , player 2 has no incentive to deviate from B to A and no incentive to deviatefrom C to D. And, if player 2 chooses fB;Cg, player 1 has no incentive to deviate from T to B.

) Pure strategy NE : (T; fB;Cg)

2. Restaurant

I own a restaurant and know the worth, but you know its value is evenly distributed between 0and 1. And, if the restaurant is worth X to me, then it is worth 1:5X to you.

De�ne price that you o¤er as p.

The person making the o¤er must calculate the expected value of the restaraunt conditional on theseller accepting. The seller only accepts a price of p if X � p. Therefore, E[Xjo¤er accepted] = p

2 .For any o¤er of p, either the o¤er is declined or the buyer makes an expected pro�t of 1:5E[Xjo¤er accepted]�p = 1:5p

2 � p < 0. Therefore, the buyer�s best o¤er is to o¤er p = 0, i.e. not to buy at all.

This is an illustration of the winner�s curse. The buyer must internalize that the seller acceptingthe o¤er conveys bad news; speci�cally, it means the restaraunt is not as valuable as he might havepreviously thought.

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3. Gibbons 3.2

Inverse demand P (Q) = a�Q where Q = q1 + q2(Uncertainty) aH : with probability �

aL : with probability 1� �(Asymmetricity) Firm 1 knows whether demand is high or not.

Firm 2 does not.Both �rms�total cost Ci(qi) = cqi

Firm 1 knows the market demand and wants to maximize its pro�t for each state. Thus, the strategyof �rm 1 is qH1 (when a = aH) and qL1 (when a = aL). However, Firm 2 does not know the marketdemand and wants to maximize its expected pro�t. Thus, the strategy of �rm 2 is q2. We also needto consider that output should be nonnegative. That is, q 2 [0;1):

Firm 1�s problem

MaxqH1

(aH � qH1 � q�2)qH1 � cqH1

@qH1 : qH�1 =aH � c� q�2

2(1)

MaxqL1

(aL � qL1 � q�2)qL1 � cqL1

@qL1 : qL�1 =

aL � c� q�22

(2)

Firm 2�s problem

Maxq2

�[(aH � qH�1 � q2)q2 � cq2] + (1� �)[(aL � qL�1 � q2)q2 � cq2]

@q2 : q�2 =

�(aH � qH�1 ) + (1� �)(aL � qL�1 )� c2

(3)

By using (1), (2) and (3), we can get the Bayesian Nash equilibrium.

qH�1 =(3� �)aH � (1� �)aL � 2c

6(4)

qL�1 =(2 + �)aL � �aH � 2c

6(5)

q�2 =�aH + (1� �)aL � c

3(6)

Finally, we will consider the nonnegativity condition. Because qL�1 < qH�1 and qL�1 < q�2 , it is enoughto assume that qL�1 � 0. Thus, our assumption is that �aH + 2c � (2 + �)aL.

) Bayesian NE : (4), (5) and (6) under �aH + 2c � (2 + �)aL


4. Gibbons 3.3

Demand for �rm i qi(pi; pj) = a� pi � bi � pj(Sensitivity) bH : with probability �

bL : with probability 1� �y Each �rm knows its own bi but not its competitor�s

Both �rms�cost Zero cost

The action spaces for �rm i (or j) : Ai = [0;1) = R+(* Price can be any nonnegative real number.)The type spaces for �rm i (or j) : Ti = fbH ; bLgThe beliefs for �rm i (or j) : pi(bH jbi = bH or bL) = �; pi(bLjbi = bH or bL) = 1� �The utility function for �rm i (or j) : Ui(pi; pj ; bi; bj) = pi(a� pi � bi � pj)The strategy spaces for �rm i (or j) : [0;1)� [0;1) = R2+(* Firm i�s strategy � (pi(bH); pi(bL)) 2 R2+)

Firm i�s problem

when bi = bH ,

Maxpi(bH)

�[a� pi(bH)� bHp�j (bH)]pi(bH) + (1� �)[a� pi(bH)� bHp�j (bL)]pi(bH)

@pi(bH) : p�i (bH) =

a� �bHp�j (bH)� (1� �)bHp�j (bL)2

(7)

when bi = bL,

Maxpi(bL)

�[a� pi(bL)� bLp�j (bH)]pi(bL) + (1� �)[a� pi(bL)� bLp�j (bL)]pi(bL)

@pi(bL) : p�i (bL) =

a� �bLp�j (bH)� (1� �)bLp�j (bL)2

(8)

Firm j�s problem

when bj = bH ,

Maxpj(bH)

�[a� pj(bH)� bHp�i (bH)]pj(bH) + (1� �)[a� pj(bH)� bHp�i (bL)]pj(bH)

@pj(bH) : p�j (bH) =

a� �bHp�i (bH)� (1� �)bHp�i (bL)2

(9)

when bj = bL,

Maxpj(bL)

�[a� pj(bL)� bLp�i (bH)]pj(bL) + (1� �)[a� pj(bL)� bLp�i (bL)]pj(bL)

@pj(bL) : p�j (bL) =

a� �bLp�i (bH)� (1� �)bLp�i (bL)2

(10)


We need (11) and (12) conditions to de�ne a symmetric pure-strategy Bayesian NE.

p�(bH) = p�i (bH) = p

�j (bH) (11)

p�(bL) = p�i (bL) = p

�j (bL) (12)

By using (7), (8), (9), (10), (11) and (12), we can get (13) and (14).

p�(bH) =a� �bHp�(bH)� (1� �)bHp�(bL)

2(13)

p�(bL) =a� �bLp�(bH)� (1� �)bLp�(bL)

2(14)

By using (13) and (14), we can get (15) and (16).

p�(bH) =a

2(1� bH

2 + �bH + (1� �)bL) (15)

p�(bL) =a

2(1� bL

2 + �bH + (1� �)bL) (16)

) Bayesian NE : (15) and (16)


5. Nash equilibrium (Bertrand)

Market demand Q = 100� P where P is the lowest price o¤ered by a �rmFirm 1�s marginal cost 20Firm 2�s marginal cost 40 with probability 1

570 with probability 4

5y Firm 2 knows its MC, but �rm 1 does not know �rm 2�s MC.

We will consider the discrete price case in this problem.

Firm 1�s monopoly price

MaxP1

(100� P1)P1 � 20(100� P1)

@P1 : Pm1 = 60 (17)

We can get �rm 2�s monopoly price in the same way.

Pm2(MC=40) = 70 (18)

Pm2(MC=70) = 85 (19)

Each �rm�s best response is as below (under no uncertainty).

Firm 1 (with MC=20)

BR1(P2)=

8>>>><>>>>:60 if P2 > 60P2 � 0:01 if 20:01 < P2 � 6020:01 if P2 = 20:01x (x � 20) if P2 = 20y (y � P2 + 0:01) if P2 � 19:99

Firm 2 (with MC=40)

BR2(P1)=


Firm 2 (with MC=70)

BR2(P1)=


There is no undominated equilibrium even when prices are discrete. It cannot be an undominatedequilibrium for �rm 1 to choose a price close to $40. At best it receives an expected pro�t of $1; 200.However, if it chooses $60 and �rm 2 plays an undominated strategy (P2 � 70 when MC = 70) thenit receives greater expected pro�ts (at least $1; 280 = $1; 600 � 4

5 ). If �rm 1 chooses any P1 > 40:01,then when �rm 2 has MC = 40, its best response is P2(MC=40) = P1 � :01. However, if �rm 2 choosesP2(MC=40) = P1 � :01 20% of the time and P2(MC=70) � 70 80% of the time, then �rm 1 does betterchoosing P1� :02. Therefore, there is not undominated equilibrium. However, the equilibria such that�rm 1 chooses any 20:01 � P1 � 40 and �rm 2 chooses P1+ :01 do work. Although it is an equilibrium,it is not one we like because �rm 2 playing P < MC is not reasonable.
