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1. 1. Nash equilibrium
In this game, player 2 knows which game they are playing but player 1 does not. Thus, player 1
has two strategies available (T and B) regardless of which game she is playing and her decision will
be based on the expected payo¤s (Left with probability 1
2 and Right with probability 1
2 ). But, player
2 should choose one strategy each game (Left and Right). This game can be summarised in matrix as
below.
Player 2
A; C A; D B; C B; D
Player 1 T 2; (2; 2) 4; (2; 0) 1
2 ; (4; 2) 5
2 ; (4; 0)
B 1; (2; 0) 5
2 ; (2; 3) 1
2 ; (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 deviate
from 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 0
and 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 the
seller 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 accepting
the o¤er conveys bad news; speci…cally, it means the restaraunt is not as valuable as he might have
previously thought.
BAYESIAN NASH EQUILIBRIUM
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2. 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 strategy
of …rm 1 is qH
1 (when a = aH) and qL
1 (when a = aL). However, Firm 2 does not know the market
demand and wants to maximize its expected pro…t. Thus, the strategy of …rm 2 is q2. We also need
to consider that output should be nonnegative. That is, q 2 [0; 1):
Firm 1’s problem
Max
qH
1
(aH qH
1 q2)qH
1 cqH
1
@qH
1 : qH
1 =
aH c q2
2
(1)
Max
qL
1
(aL qL
1 q2)qL
1 cqL
1
@qL
1 : qL
1 =
aL c q2
2
(2)
Firm 2’s problem
Max
q2
[(aH qH
1 q2)q2 cq2] + (1 )[(aL qL
1 q2)q2 cq2]
@q2 : q2 =
(aH qH
1 ) + (1 )(aL qL
1 ) c
2
(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)
q2 =
aH + (1 )aL c
3
(6)
Finally, we will consider the nonnegativity condition. Because qL
1 < qH
1 and qL
1 < q2, it is enough
to 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
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3. 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; bLg
The beliefs for …rm i (or j) : pi(bHjbi = 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,
Max
pi(bH )
[a pi(bH) bHpj (bH)]pi(bH) + (1 )[a pi(bH) bHpj (bL)]pi(bH)
@pi(bH) : pi (bH) =
a bHpj (bH) (1 )bHpj (bL)
2
(7)
when bi = bL,
Max
pi(bL)
[a pi(bL) bLpj (bH)]pi(bL) + (1 )[a pi(bL) bLpj (bL)]pi(bL)
@pi(bL) : pi (bL) =
a bLpj (bH) (1 )bLpj (bL)
2
(8)
Firm j’s problem
when bj = bH,
Max
pj (bH )
[a pj(bH) bHpi (bH)]pj(bH) + (1 )[a pj(bH) bHpi (bL)]pj(bH)
@pj(bH) : pj (bH) =
a bHpi (bH) (1 )bHpi (bL)
2
(9)
when bj = bL,
Max
pj (bL)
[a pj(bL) bLpi (bH)]pj(bL) + (1 )[a pj(bL) bLpi (bL)]pj(bL)
@pj(bL) : pj (bL) =
a bLpi (bH) (1 )bLpi (bL)
2
(10)
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4. We need (11) and (12) conditions to de…ne a symmetric pure-strategy Bayesian NE.
p (bH) = pi (bH) = pj (bH) (11)
p (bL) = pi (bL) = pj (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)
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5. 5. Nash equilibrium (Bertrand)
Market demand Q = 100 P where P is the lowest price o¤ered by a …rm
Firm 1’s marginal cost 20
Firm 2’s marginal cost 40 with probability 1
5
70 with probability 4
5
y 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
Max
P1
(100 P1)P1 20(100 P1)
@P1 : Pm
1 = 60 (17)
We can get …rm 2’s monopoly price in the same way.
Pm
2(MC=40) = 70 (18)
Pm
2(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 > 60
P2 0:01 if 20:01 < P2 60
20:01 if P2 = 20:01
x (x 20) if P2 = 20
y (y P2 + 0:01) if P2 19:99
Firm 2 (with MC=40)
BR2(P1)=
8
>>>><
>>>>:
70 if P1 > 70
P1 0:01 if 40:01 < P1 70
40:01 if P1 = 40:01
x (x 40) if P1 = 40
y (y P1 + 0:01) if P1 39:99
Firm 2 (with MC=70)
BR2(P1)=
8
>>>><
>>>>:
85 if P1 > 85
P1 0:01 if 70:01 < P1 85
70:01 if P1 = 70:01
x (x 70) if P1 = 70
y (y P1 + 0:01) if P1 69:99
There is no undominated equilibrium even when prices are discrete. It cannot be an undominated
equilibrium 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) then
it 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 chooses
P2(MC=40) = P1 :01 20% of the time and P2(MC=70) 70 80% of the time, then …rm 1 does better
choosing 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.
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