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Intermediate Microeconomics

Lecture 13

Game Theory (3)

Static Bayesian Games

1

Outline

? Introduction to static games of incomplete information ? Normal-form (or strategic-form) representation of static Bayesian games ? Bayesian Nash equilibrium ? Auction

2

Static (or simultaneous-move) games of complete information

? A set of players (at least two players) ? For each player, a set of strategies/actions ? Payoffs received by each player for the combinations of the strategies, or for each player, preferences over the combinations of the strategies ? All these are common knowledge among all the players.

3

Static (or simultaneous-move) games of INCOMPLETE information

? Payoffs are no longer common knowledge

? Incomplete information means that ?At least one player is uncertain about some other player’s payoff function.

? Static games of incomplete information are also called static Bayesian games

4

What is Incomplete Information?

? When some players don’t know others’ payoffs

– This is often true in the real world, of course.

? The notion of ―type‖ is very helpful in this setting

– A player’s ―type‖ , typically private information, is typically associated with his cost. – More generally, the ―type‖ of a player embodies any private information relevant to the player’s decision making

? By ―private information‖: info that is not common knowledge ? In addition to the player’s payoff function this may include his beliefs about other players’ payoff functions, his beliefs about what other players believe his beliefs are, and so on.

5

How do we handle games of incomplete information?

? Harsanyi: introduce a prior move by nature that determines the player(s)’s type(s).

? In the transformed game, player 2’s incomplete information about player 1’s cost becomes imperfect information about nature’s moves.

? This transformed game can be analyzed with standard techniques.

6

What is a Bayesian Nash Equilibrium?

? The Nash Equilibrium of the imperfectinformation game

– A Bayesian Equilibrium is a set of strategies such that each player is playing a best response, given a particular set of beliefs about the move by nature. – All players have the same prior beliefs about the probability distribution on nature’s moves.

– So for example, all players think the odds of player 1 being of a particular type is p, and the probability of her being the other type is 1-p

7

Prisoners’ dilemma of complete information

? Two suspects held in separate cells are charged with a major crime. However, there is not enough evidence. ? Both suspects are told the following policy: ? If neither confesses then both will be convicted of a minor offense and sentenced to one month in jail. ? If both confess then both will be sentenced to jail for six months. ? If one confesses but the other does not, then the confessor will be released but the other will be sentenced to jail for nine months. Prisoner 2 Mum Prison er 1 Mum Confess

-1 ,

-1

-9 ,

0

Confess

0 ,

-9

-6 ,

-6

8

Prisoners’ dilemma of incomplete information

? Prisoner 1 is always rational (selfish). ? Prisoner 2 can be rational (selfish) or altruistic, depending on whether he is happy or not. ? If he is altruistic then he prefers to mum and he thinks that ―confess‖ is equivalent to additional ―four months in jail‖. ? Prisoner 1 can not know exactly whether prisoner 2 is rational or altruistic, but he believes that prisoner 2 is rational with probability 0.8, and altruistic with probability 0.2. Payoffs if prisoner 2 is altruistic Prisoner 1 Mum Confess Prisoner 2 Mum -1 , -1 0 , -9 Confess -9 , -4 -6 , -10

9

Prisoners’ dilemma of incomplete information cont’d

? Given prisoner 1’s belief on prisoner 2, what strategy should prison 1 choose? ? What strategy should prisoner 2 choose if he is rational or altruistic?

Payoffs if prisoner 2 is rational Prisoner 1 Mum Prisoner 2 Mum -1 , -1 Confess -9 , 0

Confess

0 ,

-9

-6 ,

-6

Payoffs if prisoner 2 is altruistic Prisoner 1 Mum

Prisoner 2 Mum -1 , -1 Confess -9 , -4

Confess

0 ,

-9

-6 , -10

10

Prisoners’ dilemma of incomplete information cont’d

? Solution: ? Prisoner 1 chooses to confess, given his belief on prisoner 2 ? Prisoner 2 chooses to confess if he is rational, and mum if he is altruistic ? This can be written as (Confess, (Confess if rational, Mum if altruistic)) ? Confess is prisoner 1’s best response to prisoner 2’s choice (Confess if rational, Mum if altruistic). ? (Confess if rational, Mum if altruistic) is prisoner 2’s best response to prisoner 1’s Confess ? A Nash equilibrium called Bayesian Nash equilibrium

11

Example: Incomplete Information

PLAYER 2

P L A Y E R 1

PLAYER 2

Enter Build 0, -1

Don’t Enter 2, 0

Don’t Build

2, 1

3, 0

P L A Y E R 1

Enter Build Don’t Build 3, -1 2, 1

Don’t Enter 5, 0 3, 0

Payoffs if 1’s building costs are high

State of the World SH

Payoffs if 1’s building costs are low State of the World SL

When does player 2 choose to enter?

In SH, Pl 1 doesn’t build (dominant strat) and player 2 enters: payoffs are (2, 1) In SL, Pl 1 builds (dominant strat) and player 2 doesn’t enter: payoffs are (5, 0) Let p1 denote prior probability player 2 assigns to p(SH); implies p(SL) = 1-p1 Decision is based on p1 *1 + (1- p1 )*(-1) > p1 *0 + (1- p1 )*(0)

12

Example, Modified in SL

PLAYER 2

P L A Y E R 1

PLAYER 2

Enter Build Don’t Build 0, -1 2, 1

Don’t Enter 2, 0 3, 0

P L A Y E R 1

Enter

Don’t Enter

Build Don’t Build

1.5, -1 2, 1

3.5, 0 3, 0

Payoffs if 1’s building costs are high State of the World SH

Payoffs if 1’s building costs are low State of the World SL

What is Player 1’s optimal strategy?

In SH, Pl 1 doesn’t build (dominant strat), player 2 enters: payoffs are (2, 1) In SL, Pl 1 does not build if Player 2 is likely to enter; let y be p(player 2 enters) Player 1 must have prior probability about pl 2’s behavior to choose own action. Let y = player 2’s probability of entry Strategy: SH don’t build; If SL, build if 1.5*y +3.5*(1-y) > 2y+3*(1-y) 13

Let’s Transform the Game

PLAYER 2

P L A Y E R 1

PLAYER 2

Enter

Build Don’t Build 0, -1 2, 1

Don’t Enter

2, 0 3, 0

P L A Y E R 1

Enter

Build Don’t Build 1.5, -1 2, 1

Don’t Enter

3.5, 0 3, 0

Payoffs if 1’s building costs are high State of the World SH High Build Enter Don’t Don’t Build Enter Don’t

Payoffs if 1’s building costs are low

N

State of the World SL

Low Build Enter Don’t Don’t Build Enter Don’t

Transformed into a Game of Imperfect Information

14

The Modified Example

PLAYER 2

P L A Y E R 1

PLAYER 2

Enter Build Don’t Build 0, -1 2, 1

Don’t Enter 2, 0 3, 0

P L A Y E R 1

Enter Build Don’t Build 1.5, -1 2, 1

Don’t Enter 3.5, 0 3, 0

Payoffs if 1’s building costs are high State of the World SH

Payoffs if 1’s building costs are low State of the World SL

What is the Bayesian Nash Equilibrium?

Let p1 denote prior probability player 2 assigns to p(SH); implies p(SL) = 1-p1 Let x = player 1’s probability of building when her cost is low; Let y = player 2’s probability of entry What is Player 2’s optimal strategy? Enter (y =1) if p1(1) +(1-p1)[-x+(1-x)] > 0 This is equivalent to enter if x < 1/(2(1-p1)

15

Modifying the Example

PLAYER 2

P L A Y E R 1

PLAYER 2

Enter

Build 0, -1

Don’t Enter

2, 0

P L A Y E R 1

Enter Build Don’t Build 1.5, -1 2, 1

Don’t Enter 3.5, 0 3, 0

Don’t Build

2, 1

3, 0

Payoffs if 1’s building costs are high State of the World SH

Payoffs if 1’s building costs are low

State of the World SL

What is the Bayesian Nash Equilibrium?

Let p1 denote prior probability player 2 assigns to p(SH); implies p(SL) = 1-p1 Let x = player 1’s probability of building when her cost is low; Let y = player 2’s probability of entry Already shown that the best response for Low-Cost Player 1 is identified by: Build (x = 1) if 1.5*y +3.5*(1-y) > 2y+3*(1-y), so when y < ?

16

Modifying the Example

P L A Y E R 1

PLAYER 2

PLAYER 2

Enter

Don’t Enter

Build Don’t Build

0, -1 2, 1

2, 0 3, 0

P L A Y E R 1

Enter Build Don’t Build 1.5, -1 2, 1

Don’t Enter 3.5, 0 3, 0

Payoffs if 1’s building costs are high State of the World SH

Payoffs if 1’s building costs are low

State of the World SL

3 Bayesian Nash Equilibrium

Recall Player 2 indifferent when x = 1/(2(1-p1), Player 1 indifferent when SL, y = ? Mixed-strategy Bayesian Eqm: x and y as above, for all p1 in (0, ?), pl 1 in SL never builds Pure strategy Bayesian Nash Equilibria: (x = 0, y = 1 for any p1), (x = 1, y = 0 for for all p1 in (0, ?)

17

Cournot duopoly model of complete information

? The normal-form representation:

? Set of players: ? Sets of strategies: { Firm 1, Firm 2} S1=[0, +∞), S2=[0, +∞)

? Payoff functions: u1(q1, q2)=q1(a-(q1+q2)-c), u2(q1, q2)=q2(a-(q1+q2)-c)

? All these information is common knowledge

18

Cournot duopoly model of incomplete information (version I)

? A homogeneous product is produced by only two firms: firm 1 and firm 2. The quantities are denoted by q1 and q2, respectively. ? They choose their quantities simultaneously. ? The market price: P(Q)=a-Q, where a is a constant number and Q=q1+q2. ? Firm 1’s cost function: C1(q1)=cq1. ? All the above are common knowledge

19

Cournot duopoly model of incomplete information (version I) cont’d

? Firm 2’s marginal cost depends on some factor (e.g. technology) that only firm 2 knows. Its marginal cost can be ? HIGH: cost function: C2(q2)=cHq2. ? LOW: cost function: C2(q2)=cLq2. ? Before production, firm 2 can observe the factor and know exactly which level of marginal cost is in. ? However, firm 1 cannot know exactly firm 2’s cost. Equivalently, it is uncertain about firm 2’s payoff. ? Firm 1 believes that firm 2’s cost function is ? C2(q2)=cHq2 with probability ?, and ? C2(q2)=cLq2 with probability 1–?. ? All the above are common knowledge

20

Cournot duopoly model of incomplete information (version I) cont’d

A solution for the Cournot duopoly model of incomplete information Firm 2 knows exactly its marginal cost is high or low. ? If its marginal cost is high, i.e. C2 (q2 ) ? cH q2 , then, for any given q1 , it will solve

Max s.t.

q2 [a ? (q1 ? q2 ) ? cH ] q2 ? 0

1 ? FOC: a ? q1 ? 2q2 ? c H ? 0 ? q2 (c H ) ? ( a ? q1 ? c H ) 2 ? q2 (cH ) is firm 2's best response to q1 , if its marginal cost is

high.

21

Cournot duopoly model of incomplete information (version I) cont’d

Firm 2 knows exactly its marginal cost is high or low. ? If its marginal cost is low, i.e. C2 (q2 ) ? cL q2 , then, for any given q1 , it will solve

Max s.t. q2 [a ? (q1 ? q2 ) ? cL ] q2 ? 0

1 q2 (cL ) ? ( a ? q1 ? cL ) 2

? FOC: a ? q1 ? 2q2 ? c L ? 0 ?

? q2 (cL ) is firm 2's best response to q1 , if its marginal cost is low.

22

Cournot duopoly model of incomplete information (version I) cont’d

? Firm 1 knows exactly its cost function C1 (q1 ) ? cq1. ? Firm 1 does not know exactly firm 2's marginal cost is high or low. ? But it believes that firm 2's cost function is C2 (q2 ) ? cH q2 with probability ? , and C2 (q2 ) ? cL q2 with probability 1 ? ? ? Equivalently, it knows that the probability that firm 2's quantity is q2 (cH ) is ? , and the probability that firm 2's quantity is q2 (cL ) is 1 ? ? . So it solves

Max ? ? q1[a ? (q1 ? q2 (cH )) ? c] ? (1 ? ? ) ? q1[a ? (q1 ? q2 (cL )) ? c] s.t. q1 ? 0

23

Cournot duopoly model of incomplete information (version I) cont’d

? Firm 1's problem:

Max ? ? q1[ a ? ( q1 ? q2 (cH )) ? c ] ? (1 ? ? ) ? q1[ a ? ( q1 ? q2 (cL )) ? c ] s.t.

? FOC:

q1 ? 0

? [a ? 2q1 ? q2 (cH ) ? c] ? (1 ? ? )[a ? 2q1 ? q2 (cL ) ? c] ? 0

Hence, q1 ?

? [a ? q2 (cH ) ? c] ? (1 ? ? )[ a ? q2 (cL ) ? c]

2

? q1 is firm 1's best response to the belief that firm 2 chooses q2 (cH ) with probability ? , and q2 (cL ) with probability 1 ? ?

24

Cournot duopoly model of incomplete information (version I) cont’d

? Now we have

1 q2 (cH ) ? ( a ? q1 ? cH ) 2 1 q2 (c L ) ? ( a ? q1 ? c L ) 2 ? [a ? q2 (cH ) ? c] ? (1 ? ? )[ a ? q2 (cL ) ? c] q1 ? 2

? We have three equations and three unknowns. Solving these gives us

25

Cournot duopoly model of incomplete information (version I) cont’d

1 1?? * q 2 ( c H ) ? ( a ? 2c H ? c ) ? (c H ? c L ) 3 6 1 ? * q 2 ( c L ) ? ( a ? 2c L ? c ) ? ( c H ? c L ) 3 6 a ? 2 c ? ?cH ? (1 ? ? ) cL * q1 ? 3 * ? Firm 1 chooses q1

* * ? Firm 2 chooses q2 (cH ) if its marginal cost is high, or q2 (cL ) if its marginal cost is low. * * * ? This can be written as ( q1 , ( q2 (cH ) , q2 (cL ) ))

? One is the best response to the other ? A Nash equilibrium solution called Bayesian Nash equilibrium.

26

Cournot duopoly model of incomplete information (version II)

? A homogeneous product is produced by only two firms: firm 1 and firm 2. The quantities are denoted by q1 and q2, respectively. ? They choose their quantities simultaneously. ? The market price: P(Q)=a-Q, where a is a constant number and Q=q1+q2.

? All the above are common knowledge

27

Cournot duopoly model of incomplete information (version II) cont’d

? Firm 2’s marginal cost depends on some factor (e.g. technology) that only firm 2 knows. Its marginal cost can be ? HIGH: cost function: C2(q2)=cHq2. ? LOW: cost function: C2(q2)=cLq2. ? Before production, firm 2 can observe the factor and know exactly its marginal cost is high or low. ? However, firm 1 cannot know exactly firm 2’s cost. Equivalently, it is uncertain about firm 2’s payoff. ? Firm 1 believes that firm 2’s cost function is ? C2(q2)=cHq2 with probability ?, and ? C2(q2)=cLq2 with probability 1–?.

28

Cournot duopoly model of incomplete information (version II) cont’d

? Firm 1’s marginal cost also depends on some other independent factor that only firm 1 knows. Its marginal cost can be ? HIGH: cost function: C1(q1)=cHq1. ? LOW: cost function: C1(q1)=cLq1. ? Before production, firm 1 can observe the factor and know exactly its marginal cost is high or low.

? However, firm 2 cannot know exactly firm 1’s cost. Equivalently, it is uncertain about firm 1’s payoff.

? Firm 2 believes that firm 1’s cost function is ? C1(q1)=cHq1 with probability ?, and ? C1(q1)=cLq1 with probability 1–?.

29

Cournot duopoly model of incomplete information (version II) cont’d

? Firm 1 knows exactly its marginal cost is high or low before production. ? Firm 1 does not know exactly firm 2's marginal cost is high or low. ? But it believes that firm 2's cost function is ? C2 (q2 ) ? cH q2 (in which it chooses q2 (cH ) ) with probability ? . ? C2 (q2 ) ? cL q2 (in which it chooses q2 (cL ) ) with probability 1 ? ? . ? Now we solve firm 1's problem, given its belief on firm 2.

30

Cournot duopoly model of incomplete information (version II) cont’d

Firm 1 knows exactly its marginal cost is high or low. ? If its marginal cost is HIGH, i.e. C1 (q1 ) ? cH q1, then, given its belief on firm 2, it will solve Max ? ? q1[a ? (q1 ? q2 (cH )) ? cH ] ? (1 ? ? ) ? q1[a ? (q1 ? q2 (cL )) ? cH ]

s.t.

q1 ? 0

? FOC:

Hence, q1(c H ) ?

? [a ? 2q1 ? q2 (cH ) ? cH ] ? (1 ? ? )[a ? 2q1 ? q2 (cL ) ? cH ] ? 0 ? [a ? q2 (c H ) ? cH ] ? (1 ? ? )[a ? q2 (cL ) ? cH ]

2 ? q1 (cH ) is firm 1's best response to the belief that firm 2 chooses q2 (cH ) with probability ? , and q2 (cL ) with probability 1 ? ? , if firm 1's marginal cost is HIGH.

31

Cournot duopoly model of incomplete information (version II) cont’d

Firm 1 knows exactly its marginal cost is high or low. ? If its marginal cost is LOW, i.e. C1 (q1 ) ? cL q1, then, given its belief on firm 2, it will solve Max ? ? q1[a ? (q1 ? q2 (cH )) ? cL ] ? (1 ? ? ) ? q1[a ? (q1 ? q2 (cL )) ? cL ]

s.t.

? FOC:

q1 ? 0

Hence, q1(c L ) ?

? [a ? 2q1 ? q2 (cH ) ? cL ] ? (1 ? ? )[a ? 2q1 ? q2 (cL ) ? cL ] ? 0 ? [a ? q2 (c H ) ? cL ] ? (1 ? ? )[a ? q2 (cL ) ? cL ]

2 ? q1 (cL ) is firm 1's best response to the belief that firm 2 chooses q2 (cH ) with probability ? , and q2 (cL ) with probability 1 ? ? , if firm 1's marginal cost is LOW.

32

Cournot duopoly model of incomplete information (version II) cont’d

? Firm 2 knows exactly its marginal cost is high or low before production. ? Firm 2 does not know exactly firm 1's marginal cost is high or low. ? But it believes that firm 1's cost function is ? C1 (q1 ) ? cH q1 (in which it chooses q1 (cH ) ) with probability ? . ? C1 (q1 ) ? cL q1 (in which it chooses q1 (cL ) ) with probability 1?? . ? Now we solve firm 2's problem, given its belief on firm 1.

33

Cournot duopoly model of incomplete information (version II) cont’d

Firm 2 knows exactly its marginal cost is high or low.

? If its marginal cost is HIGH, i.e. C2 (q2 ) ? cH q2 , then, given its belief on firm 1, it will solve Max ? ? q2 [a ? (q1 (cH ) ? q2 ) ? cH ] ? (1 ? ? ) ? q2 [a ? (q1 (cL ) ? q2 ) ? cH ]

s.t.

q2 ? 0

? FOC:

Hence, q2 (c H ) ?

? [a ? q1 (cH ) ? 2q2 ? cH ] ? (1 ? ? )[a ? q1 (cL ) ? 2q2 ? cH ] ? 0 ? [a ? q1 (c H ) ? cH ] ? (1 ? ? )[a ? q1 (cL ) ? cH ]

2 ? q2 (cH ) is firm 2's best response to the belief that firm 1 chooses q1 (cH ) with probability ? , and q1 (cL ) with probability 1 ? ? , if firm 2's marginal cost is HIGH.

34

Cournot duopoly model of incomplete information (version II) cont’d

Firm 2 knows exactly its marginal cost is high or low.

? If its marginal cost is LOW, i.e. C2 (q2 ) ? cL q2 , then, given its belief on firm 1, it will solve Max ? ? q2 [a ? (q1 (cH ) ? q2 ) ? cL ] ? (1 ? ? ) ? q2 [a ? (q1 (cL ) ? q2 ) ? cL ]

s.t.

? FOC:

q2 ? 0

Hence, q2 (c L ) ?

? [a ? q1 (cH ) ? 2q2 ? cL ] ? (1 ? ? )[a ? q1 (cL ) ? 2q2 ? cL ] ? 0 ? [a ? q1 (c H ) ? cL ] ? (1 ? ? )[a ? q1 (cL ) ? cL ]

2 ? q2 (cL ) is firm 2's best response to the belief that firm 1 chooses q1 (cH ) with probability ? , and q1 (cL ) with probability 1 ? ? , if firm 2's marginal cost is LOW.

35

Cournot duopoly model of incomplete information (version II) cont’d

? Now we have

q1(c H ) ?

? [a ? q2 (c H ) ? cH ] ? (1 ? ? )[a ? q2 (cL ) ? cH ]

2 ? [a ? q2 (c H ) ? cL ] ? (1 ? ? )[a ? q2 (cL ) ? cL ] q1(c L ) ? 2 ? [a ? q1 (c H ) ? cH ] ? (1 ? ? )[a ? q1 (cL ) ? cH ] q2 (c H ) ? 2 ? [a ? q1 (c H ) ? cL ] ? (1 ? ? )[a ? q1 (cL ) ? cL ] q2 (c L ) ? 2

? This is a symmetric model. So q1(cH ) ? q2 (cH ) and q1(cL ) ? q2 (cL ) . Solving

these four equations with four unknowns gives us. 1 1?? * * q1 (c H ) ? q2 (c H ) ? (a ? c H ) ? (c H ? c L ) 3 6 1 ? * * q1 (c L ) ? q2 (c L ) ? (a ? c L ) ? (c H ? c L ) 3 6

36

Cournot duopoly model of incomplete information (version II) cont’d

* * * * ? This can be written as (( q1 (cH ) , q1 (cL ) ), ( q2 (cH ) , q2 (cL ) ))

* ? If firm 1's marginal cost is HIGH then it chooses q1 (cH ) which is its * * best response to firm 2's ( q2 (cH ) , q2 (cL ) ) (and the probability).

* ? If firm 1's marginal cost is LOW then it chooses q1 (cL ) which is its * * best response to firm 2's ( q2 (cH ) , q2 (cL ) ) (and the probability).

* ? If firm 2's marginal cost is HIGH then it chooses q2 (cH ) which is its * * best response to firm 1's ( q1 (cH ) , q1 (cL ) ) (and the probability). * ? If firm 2's marginal cost is LOW then it chooses q2 (cL ) which is its * * best response to firm 1's ( q1 (cH ) , q1 (cL ) ) (and the probability).

? A Nash equilibrium solution called Bayesian Nash equilibrium.

37

Battle of the sexes (version I)

? At the separate workplaces, Chris and Pat must choose to attend either an opera or a prize fight in the evening. ? Both Chris and Pat know the following:

? Both would like to spend the evening together. ? But Chris prefers the opera. ? Pat prefers the prize fight.

Pat Opera Prize Fight Opera Prize Fight

Chris

2 , 0 ,

1 0

0 , 1 ,

0 2

38

Battle of the sexes with incomplete information (version I)

? Now Pat’s preference depends on whether he is happy. ? If he is happy then his preference is the same. ? If he is unhappy then he prefers to spend the evening by himself and his preference is shown in the following table. ? Chris cannot figure out whether Pat is happy or not. But Chris believes that Pat is happy with probability 0.5 and unhappy with probability 0.5

Payoffs if Pat is unhappy

Chris Opera

Pat

Opera Prize Fight

2, 0

0, 2

Prize Fight

0, 1

1, 0

39

Battle of the sexes with incomplete information (version I) cont’d

? How to find a solution ?

Payoffs if Pat is happy with probability 0.5 Chris Pat Opera Prize Fight

Opera

Prize Fight

2, 1

0, 0

0, 0

1, 2 Pat Opera 2, 0 0, 1 Prize Fight 0, 2 1, 0

40

Payoffs if Pat is unhappy with probability 0.5 Chris Opera Prize Fight

Battle of the sexes with incomplete information (version I) cont’d

? Best response

? If Chris chooses opera then Pat’s best response: opera if he is happy, and prize fight if he is unhappy ? Suppose that Pat chooses opera if he is happy, and prize fight if he is unhappy. What is Chris’ best response?

?If Chris chooses opera then she get a payoff 2 if Pat is happy, or 0 if Pat is unhappy. Her expected payoff is 2?0.5+ 0?0.5=1 ?If Chris chooses prize fight then she get a payoff 0 if Pat is happy, or 1 if Pat is unhappy. Her expected payoff is 0?0.5+ 1?0.5=0.5 ?Since 1>0.5, Chris’ best response is opera

? A Bayesian Nash equilibrium: (opera, (opera if happy and prize fight if unhappy))

41

Battle of the sexes with incomplete information (version I) cont’d

? Best response

? If Chris chooses prize fight then Pat’s best response: prize fight if he is happy, and opera if he is unhappy ? Suppose that Pat chooses prize fight if he is happy, and opera if he is unhappy. What is Chris’ best response?

?If Chris chooses opera then she get a payoff 0 if Pat is happy, or 2 if Pat is unhappy. Her expected payoff is 0?0.5+ 2?0.5=1 ?If Chris chooses prize fight then she get a payoff 1 if Pat is happy, or 0 if Pat is unhappy. Her expected payoff is 1?0.5+ 0?0.5=0.5 ?Since 1>0.5, Chris’ best response is opera

? (prize fight, (prize fight if happy and opera if unhappy)) is not a Bayesian Nash equilibrium.

42

Cournot duopoly model of incomplete information (version III)

? A homogeneous product is produced by only two firms: firm 1 and firm 2. The quantities are denoted by q1 and q2, respectively. ? They choose their quantities simultaneously. ? The market price: P(Q)=a-Q, where a is a constant number and Q=q1+q2. ? All the above are common knowledge

43

Cournot duopoly model of incomplete information (version III) cont’d

? Firm 2’s cost depends on some factor (e.g. technology) that only firm 2 knows. Its cost can be ? HIGH: cost function: C2(q2)=cHq2. ? LOW: cost function: C2(q2)=cLq2. ? Firm 1’s cost also depends on some other (independent or dependent) factor that only firm 1 knows. Its cost can be ? HIGH: cost function: C1(q1)=cHq1. ? LOW: cost function: C1(q1)=cLq1.

44

Cournot duopoly model of incomplete information (version III) cont’d

? Firm 1's quantity depends on its cost. It chooses ? q1 (cH ) if its cost is HIGH ? q1 (cL ) if its cost is LOW ? Firm 2's quantity also depends on its cost. It chooses ? q2 (cH ) if its cost is HIGH ? q2 (cL ) if its cost is LOW

45

Cournot duopoly model of incomplete information (version III) cont’d

? Before production, firm 1 knows exactly its cost is HIGH or LOW. ? However, firm 1 cannot know exactly firm 2’s cost. Equivalently, it is uncertain about firm 2’s payoff. ? Firm 1 believes that if its cost is HIGH then firm 2’s cost function is ? C2 (q2 ) ? cH q2 with probability p1 (c2 ? cH | c1 ? cH ) , and ? C2 (q2 ) ? cL q2 with probability p1 (c2 ? cL | c1 ? cH ) . ? Firm 1 believes that if its cost is LOW then firm 2’s cost function is ? C2 (q2 ) ? cH q2 with probability p1 (c2 ? cH | c1 ? cL ) , and ? C2 (q2 ) ? cL q2 with probability p1 (c2 ? cL | c1 ? cL ) . ? Example: p1 (c2 ? cH | c1 ? cH ) ? p1 (c2 ? cH | c1 ? cL ) ? ? p1 (c2 ? cL | c1 ? cH ) ? p1 (c2 ? cL | c1 ? cL ) ? 1 ? ? as in version two.

46

Cournot duopoly model of incomplete information (version III) cont’d

? Before production, firm 2 knows exactly its cost is HIGH or LOW. ? However, firm 2 cannot know exactly firm 1’s cost. Equivalently, it is uncertain about firm 1’s payoff. ? Firm 2 believes that if its cost is HIGH then firm 1’s cost function is ? C1 (q1 ) ? cH q1 with probability p2 (c1 ? cH | c2 ? cH ) , and ? C1 (q1 ) ? cL q1 with probability p2 (c1 ? cL | c2 ? cH ) . ? Firm 1 believes that if its cost is LOW then firm 2’s cost function is ? C1 (q1 ) ? cH q1 with probability p2 (c1 ? cH | c2 ? cL ) , and ? C1 (q1 ) ? cL q1 with probability p2 (c1 ? cL | c2 ? cL ) . ? Example: p2 (c1 ? cH | c2 ? cH ) ? p2 (c1 ? cH | c2 ? cL ) ? ? p2 (c1 ? cL | c2 ? cH ) ? p2 (c1 ? cL | c2 ? cL ) ? 1 ? ? as in version two.

47

Cournot duopoly model of incomplete information (version III) cont’d

Firm 1 knows exactly its cost is high or low. ? If its cost is HIGH, i.e. C1 (q1 ) ? cH q1, then, given its belief on firm 2, it will solve u1(q1, q2(cH); cH)

Max s.t. p1 (c2 ? c H | c1 ? c H ) ? q1[a ? (q1 ? q2 (c H )) ? cH ] ? p1 (c2 ? c L | c1 ? c H ) ? q1[a ? (q1 ? q2 (cL )) ? cH ] q1 ? 0

? FOC:

u1(q1, q2(cL); cH)

p1 (c2 ? cH | c1 ? cH )[a ? 2 q1 ? q2 (cH ) ? cH ] ? p1 (c2 ? cL | c1 ? cH )[a ? 2 q1 ? q2 (cL ) ? cH ] ? 0

Hence,

q1(c H ) ? a ? cH ? p1 (c2 ? c H | c1 ? c H ) q2 (c H ) ? p1 (c2 ? c L | c1 ? c H ) q2 (cL ) 2

? q1 (cH ) is firm 1's best response to its belief (probability) on firm 2's ( q2 (cH ) , q2 (cL ) ) if firm 1's cost is HIGH.

48

Cournot duopoly model of incomplete information (version III) cont’d

Firm 1 knows exactly its cost is high or low. ? If its cost is LOW, i.e. C1 (q1 ) ? cL q1, then, given its belief on firm 2, it will solve u1(q1, q2(cH); cL)

Max s.t. p1 (c2 ? c H | c1 ? c L ) ? q1[a ? (q1 ? q2 (c H )) ? cL ] ? p1 (c2 ? c L | c1 ? c L ) ? q1[a ? (q1 ? q2 (cL )) ? cL ] q1 ? 0

? FOC:

u1(q1, q2(cL); cL)

p1 (c2 ? cH | c1 ? cL )[a ? 2 q1 ? q2 (cH ) ? cL ] ? p1 (c2 ? cL | c1 ? cL )[a ? 2 q1 ? q2 (cL ) ? cL ] ? 0

Hence,

q1(c L ) ? a ? cL ? p1 (c2 ? c H | c1 ? c L ) q2 (c H ) ? p1 (c2 ? c L | c1 ? c L ) q2 (cL ) 2

? q1 (cL ) is firm 1's best response to its belief (probability) on firm 2's ( q2 (cH ) , q2 (cL ) ) if firm 1's cost is LOW.

49

Cournot duopoly model of incomplete information (version III) cont’d

Firm 2 knows exactly its cost is high or low. ? If its cost is HIGH, i.e. C2 (q2 ) ? cH q2 , then, given its belief on firm 1, it will solve u2(q1(cH), q2; cH)

Max p2 (c1 ? c H | c2 ? c H ) ? q2 [a ? (q1 (c H ) ? q2 ) ? cH ] s.t. q2 ? 0 ? p2 (c1 ? c L | c2 ? c H ) ? q2 [a ? (q1 (cL ) ? q2 ) ? cH ]

? FOC:

u2(q1(cL), q2; cH)

p2 (c1 ? cH | c2 ? cH )[a ? q1 (cH ) ? 2 q2 ? cH ] ? p2 (c1 ? cL | c2 ? cH )[a ? q1 (cL ) ? 2 q2 ? cH ] ? 0

Hence,

q2 (c H ) ? a ? cH ? p2 (c1 ? c H | c2 ? c H ) q1 (c H ) ? p2 (c1 ? c L | c2 ? c H ) q1 (cL ) 2

? q2 (cH ) is firm 2's best response to its belief (probability) on firm 1's ( q1 (cH ) , q1 (cL ) ) if firm 2's cost is HIGH.

50

Cournot duopoly model of incomplete information (version III) cont’d

Firm 2 knows exactly its cost is high or low. ? If its cost is LOW, i.e. C2 (q2 ) ? cL q2 , then, given its belief on firm 1, it will solve u2(q1(cH), q2; cL)

Max p2 (c1 ? cH | c2 ? cL ) ? q2 [a ? (q1 (cH ) ? q2 ) ? cL ] s.t. q2 ? 0 ? p2 (c1 ? cL | c2 ? cL ) ? q2 [a ? (q1 (cL ) ? q2 ) ? cL ]

u2(q1(cL), q2; cL)

? FOC:

p2 (c1 ? cH | c2 ? cL )[a ? q1 (cH ) ? 2 q2 ? cL ] ? p2 (c1 ? cL | c2 ? cL )[a ? q1 (cL ) ? 2 q2 ? cL ] ? 0

Hence,

q2 (c L ) ? a ? cL ? p2 (c1 ? c H | c2 ? c L ) q1 (c H ) ? p2 (c1 ? c L | c2 ? c L ) q1 (cL ) 2

? q2 (cL ) is firm 2's best response to its belief (probability) on firm 1's ( q1 (cH ) , q1 (cL ) ) if firm 2's cost is LOW.

51

Cournot duopoly model of incomplete information (version III) cont’d

? Now we have four equations with four unknowns.

a ? cH ? p1 (c2 ? c H | c1 ? cH ) q2 (c H ) ? p1 (c2 ? c L | c1 ? c H ) q2 (cL ) 2 a ? cL ? p1 (c2 ? cH | c1 ? cL ) q2 (c H ) ? p1 (c2 ? cL | c1 ? cL ) q2 (cL ) q1 (cL ) ? 2 a ? cH ? p2 (c1 ? c H | c2 ? cH ) q1 (c H ) ? p2 (c1 ? c L | c2 ? c H ) q1 (cL ) q2 (c H ) ? 2 a ? cL ? p2 (c1 ? c H | c2 ? c L ) q1 (c H ) ? p2 (c1 ? c L | c2 ? c L ) q1 (cL ) q2 (c L ) ? 2 q1 (c H ) ?

? Solving these gives us the following Bayesian Nash equilibrium.

?q1* (cH ), q1* (cL )? * * ?q2 (cH ), q2 (cL )?

52

Cournot duopoly model of incomplete information (version III) cont’d

* * * * ? The Bayesian Nash equilibrium: (( q1 (cH ) , q1 (cL ) ), ( q2 (cH ) , q2 (cL ) ))

* ? If firm 1's marginal cost is HIGH then it chooses q1 (cH ) which is its best * * response to firm 2's ( q2 (cH ) , q2 (cL ) ) (and the probability).

* ? If firm 1's marginal cost is LOW then it chooses q1 (cL ) which is its best * * response to firm 2's ( q2 (cH ) , q2 (cL ) ) (and the probability).

* ? If firm 2's marginal cost is HIGH then it chooses q2 (cH ) which is its best * * response to firm 1's ( q1 (cH ) , q1 (cL ) ) (and the probability). * ? If firm 2's marginal cost is LOW then it chooses q2 (cL ) which is its best * * response to firm 1's ( q1 (cH ) , q1 (cL ) ) (and the probability).

53

Normal-form representation of static Bayesian games

? The normal-form representation of an n-player static game G of incomplete information specifies: ? A finite set of players {1, 2, ..., n}, ? players’ action sets A1, A2 , A3 , ..., An and ? their payoff functions ? more ? Remark: a player's payoff function depends on not only the n players' actions but also her TYPE. ? Ti is player i 's type set. ? Example: T1 ? {cH , cL }, T2 ? {cH , cL }

54

Normal-form representation of static Bayesian games : payoffs

? Player i's payoff function is represented as: ui (a1 , a2 , ..., an ; ti ) for a1 ? A1 , a2 ? A2 , ..., an ? An , ti ? Ti . ? Example: u1 (q1, q2 ; cH ) ? q1[a ? (q1 ? q2 ) ? cH ]

u1 (q1, q2 ; cL ) ? q1[a ? (q1 ? q2 ) ? cL ]

? Each player knows her own type. Equivalently, she knows her own payoff function. ? Each player may be uncertain about other players' types. Equivalently, she is uncertain about other players' payoff functions.

55

Normal-form representation of static Bayesian games : beliefs (probabilities)

? Player i has beliefs on other players' types, denoted by

pi (t1 , t2 , ..., ti ?1 , ti ?1 , ..., tn | ti ) for t1 ? T1 , t2 ? T2 , ..., tn ? Tn . or pi (t?i | ti ) wheret?i ? (t1 , t2 , ..., ti ?1 , ti ?1 , ..., tn ), t1 ? T1 , t2 ? T2 , ..., tn ? Tn .

? Player i's beliefs are conditional probabilities ? Example:

p1 (c2 ? cL | c1 ? cH )

p1 (c2 ? cH | c1 ? cL )

p1 (c2 ? cH | c1 ? cH )

p1 (c2 ? cL | c1 ? cL )

56

Strategy

? In a static Bayesian game, a strategy for player i is a function si ( ti ) for each ti ? Ti . ? si ( ti ) specifies what player i does for her each type ti ? Ti ? Example: ( q1 (cH ) , q1 (cL ) ) is a strategy for firm 1 in the Cournot model of incomplete information (version three).

57

Bayesian Nash equilibrium: 2-player

? In a static Bayesian 2-player game { A1, A2 ; T1, T2 ; p1, p2 ; u1, u2 }, * * the strategies s1 (?), s2 (?) are pure strategy Bayesian Nash equilibrium if ?

* for each of player 1's types t1 ? T1, s1 (t1 ) solves

Max

a1?A1

t2?T2

? u1 (a1 , s2 (t 2 ); t1 ) p1 (t 2 | t1 )

*

?

* and for each of player 2's types t2 ? T2 , s2 (t 2 ) solves

a2?A2

Max

t1?T1

? u2 ( s1 (t1 ), a2 ; t2 ) p2 (t1 | t 2 )

*

58

Bayesian Nash equilibrium: 2-player

? In a static Bayesian 2-player game { A1, A2 ; T1, T2 ; p1, p2 ; u1, u2 }, the * * strategies s1 (?), s2 (?) are pure strategy Bayesian Nash equilibrium if for each i and j, (assume T1 ? {t11, t12 , ....}, T2 ? {t21, t22 , ....})

* s1 (t11 )

* s1 (t12 )

In the sense of expectation based on her belief

* s2 (t21 )

? ?

* s1 (t1i )

player 2’s best response if her type is t2j

* s2 (t22 )

? * s2 (t2 j ) ?

* s2 (t2 n )

?

* s1 (t1n )

player 1’s best response if her type is t1i

In the sense of expectation based on her belief

?

?

59

What is Bayes’ Rule?

? A mathematical rule of logic explaining how you should change your beliefs in light of new information (updating) ? Bayes’ rule is essential knowledge for all rigorous scientific methodologies ? Bayes’ Rule: P(A|B) = P(B|A)*P(A)/P(B)

– Typically, P(A|B) is your scientific inquiry, with A as your quantity of interest—maybe A is your regression coefficient, and you want to know if it equals zero.* – To use Bayes’ Rule, you need to know a few things: – You need to know P(B|A) – You also need to know the probabilities of A and B

60

A Bayes’ Rule Exercise

? Situation: There are 2 States of the World ? We have ―priors‖ on the causal mechanisms ? Priors about the true data-generating process ? State 1: Unstable political order: occurs with probability r ? State 2: Stable political order: occurs with probability 1 - r ? Many citizens get independent signals of which state it is ? If State 1 ? Probability(stable) = v, probability(unstable) = (1-v) ? If State 2 ? Probability(stable) = q, probability(unstable) = (1-q)

61

A Bayes’ Rule Exercise

? Many citizens get independent signals of which state it is ? If State 1 ? Probability(stable) = v, probability(unstable) = (1-v) ? If State 2 ? Probability(stable) = q, probability(unstable) = (1-q) ? If a “stable” signal is received, what is the probability that it is State 1? ? P(State1|stable) = P(stable|State1)*P(State1)/P(stable) = v*r/P(stable) = v*r/[P(stable & State1)+ P(stable & State2)] = v*r/[rv+ (1-r)q]

62

A Bayes’ Rule Exercise

? Many citizens get independent signals of which state it is ? If State 1 ? Probability(stable) = v, probability(unstable) = (1-v) ? If State 2 ? Probability(stable) = q, probability(unstable) = (1-q) ? If 3 stable signals and 2 unstable signals are received, what is P(State1)? Notation: Call 3 stable signals and 2 unstable signals <3,2> ? P(State1|<3,2>) = P(<3,2>|State1) *P(State1)/P<3,2> = [v3*(1-v)2 ] *(r)/ P<3,2> = [v3*(1-v)2 ] *(r)/[(P<3,2>& State1) +(P<3,2>& State2)] = [v3*(1-v)2 ] *(r)/{r*[v3(1-v)2] +(1-r)*[q3(1-q)2]}

63

Battle of the sexes (version II)

? At the separate workplaces, Chris and Pat must choose to attend either an opera or a prize fight in the evening. ? Both Chris and Pat know the following: ? Both would like to spend the evening together. ? But Chris prefers the opera. ? Pat prefers the prize fight.

Pat Opera Chris Opera Prize Fight Prize Fight

2 , 0 ,

1 0

0 , 1 ,

0 2

64

Battle of the sexes with incomplete information (version II)

? Pat’s preference depends on whether he is happy. If he is happy then his preference is the same. ? If he is unhappy then he prefers to spend the evening by himself. ? Chris cannot figure out whether Pat is happy or not. But Chris believes that Pat is happy with probability 0.5 and unhappy with probability 0.5 ? Chris’ preference also depends on whether she is happy. If she is happy then her preference is the same. ? If she is unhappy then she prefers to spend the evening by herself. ? Pat cannot figure out whether Chris is happy or not. But Pat believes that Chris is happy with probability 2/3 and unhappy with probability 1/3.

65

Battle of the sexes with incomplete information (version II) cont’d

Chris is happy Pat is happy Chris Pat Opera Fight Chris is happy Pat is unhappy Chris Opera Fight Pat Opera 2, 0 0, 1 Fight 0, 2 1, 0 Pat Oper a 0, 0 1, 1

Opera

Fight

2, 1

0, 0

0, 0

1, 2 Pat

Chris is unhappy Pat is happy

Chris Opera Fight

Opera 0, 1 1, 0

Fight 2, 0 0, 2

Chris is unhappy Pat is unhappy Chris Opera Fight

Fight

2, 2 0, 0

? Check whether ((Opera if happy, Opera if unhappy), (Opera if happy, Fight is unhappy)) is a Bayesian NE

66

Battle of the sexes with incomplete information (version II) cont’d

? Check whether ((Opera if happy, Opera if unhappy), (Opera if happy, Fight is unhappy)) is a Bayesian Nash equilibrium. ? Chris' best response to Pat's (Opera if happy, Fight is unhappy) if Chris is HAPPY ? If Chris chooses Opera then she gets a payoff 2 if Pat is happy (probability 0.5), or a payoff 0 if Pat is unhappy (probability 0.5). Her expected payoff=2?0.5+0?0.5=1 ? If Chris chooses Fight then she gets a payoff 0 if Pat is happy (probability 0.5), or a payoff 1 if Pat is unhappy (probability 0.5). Her expected payoff=0?0.5+1?0.5=0.5 ? Hence, Chris' best response is Opera if she is HAPPY.

67

Battle of the sexes with incomplete information (version II) cont’d

? Check whether ((Opera if happy, Opera if unhappy), (Opera if happy, Fight is unhappy)) is a Bayesian Nash equilibrium. ? Chris' best response to Pat's (Opera if happy, Fight is unhappy) if Chris is UNHAPPY ? If Chris chooses Opera then she gets a payoff 0 if Pat is happy (probability 0.5), or a payoff 2 if Pat is unhappy (probability 0.5). Her expected payoff=0?0.5+2?0.5=1 ? If Chris chooses Fight then she gets a payoff 1 if Pat is happy (probability 0.5), or a payoff 0 if Pat is unhappy (probability 0.5). Her expected payoff=1?0.5+0?0.5=0.5 ? Hence, Chris' best response is Opera if she is UNHAPPY.

68

Battle of the sexes with incomplete information (version II) cont’d

? Check whether ((Opera if happy, Opera if unhappy), (Opera if happy, Fight is unhappy)) is a Bayesian Nash equilibrium. ? Pat's best response to Chris' (Opera if happy, Opera if unhappy) if Pat is HAPPY ? If Pat chooses Opera then he gets a payoff 1 if Chris is happy (probability 2/3), or a payoff 1 if Chris is unhappy (probability 1/3). His expected payoff=1?(2/3)+1?(1/3)=1 ? If Pat chooses Fight then he gets a payoff 0 if Chris is happy (probability 2/3), or a payoff 0 if Chris is unhappy (probability 1/3). His expected payoff=0?(2/3)+0?(1/3)=0 ? Hence, Pat's best response is Opera if he is HAPPY.

69

Battle of the sexes with incomplete information (version II) cont’d

? Check whether ((Opera if happy, Opera if unhappy), (Opera if happy, Fight is unhappy)) is a Bayesian Nash equilibrium. ? Pat's best response to Chris' (Opera if happy, Opera if unhappy) if Pat is UNHAPPY ? If Pat chooses Opera then he gets a payoff 0 if Chris is happy (probability 2/3), or a payoff 0 if Chris is unhappy (probability 1/3). His expected payoff=0?(2/3)+1?(1/3)=0 ? If Pat chooses Fight then he gets a payoff 2 if Chris is happy (probability 2/3), or a payoff 2 if Chris is unhappy (probability 1/3). His expected payoff=2?(2/3)+2?(1/3)=2 ? Hence, Pat's best response is Fight if he is UNHAPPY. ?Hence, ((Opera if happy, Opera if unhappy), (Opera if happy, Fight is unhappy)) is a Bayesian Nash equilibrium.

70

Battle of the sexes with incomplete information (version II) cont’d

Chris believes that Pat is happy with probability 0.5, unhappy 0.5

Pat (0.5, 0.5) Chris is happy (O,O) (O,F) (F,O) (F,F)

Chri s O 2 0 1 1/2 1 1/2 0 1

Chris is Pat (0.5, 0.5) unhappy (O,O) (O,F) (F,O) (F,F) Chris O 0 1 1 2

F

F

1

1/2

1/2

0

Chris’ expected payoff of playing Fight if Chris is happy and Pat plays (Opera if happy, Fight if unhappy)

71

Battle of the sexes with incomplete information (version II) cont’d

Pat believes that Chris is happy with probability 2/3, unhappy 1/3 Pat is happy O (O,O) 1 Pat F 0 Pat is unhappy Pat O F

(O,O)

(O,F) Chris (2/3, 1/3) (F,O) (F,F)

0

1/3 2/3 1

2

4/3 2/3 0

(O,F) Chris (2/3, 1/3) (F,O)

(F,F)

2/3

1/3 0

2/3

4/3 2

Pat’s expected payoff of playing Opera if Pat is unhappy and Chris plays (Fight if happy, Fight if unhappy)

72

Battle of the sexes with incomplete information (version II) cont’d

? Check whether ((Fight if happy, Opera if unhappy), (Fight if happy, Fight is unhappy)) is a Bayesian Nash equilibrium.

? Answer: Yes

73

First-price sealed-bid auction

? A single good is for sale. ? Two bidders, 1 and 2, simultaneously submit their bids. ? Let b1 denote bidder 1's bid and b2 denote bidder 2's bid ? ? ? ? The higher bidder wins the good and pays the price she bids The other bidder gets and pays nothing In case of a tie, the winner is determined by a flip of a coin Bidder i has a valuation vi ?[0, 1] for the good. v1 and v2 are independent. ? Bidder 1 and 2's payoff functions: ?v1 ? b1 if b1 ? b2 ?v2 ? b2 if b2 ? b1 ?v ? b ?v ? b u1 (b1 , b2 ; v1 ) ? ? 1 1 if b1 ? b2 u2 (b1 , b2 ; v2 ) ? ? 2 2 if b2 ? b1 ? 2 ? 2 if b1 ? b2 if b2 ? b1 ?0 ?0

74

First-price sealed-bid auction cont’d

? Normal form representation: ? Two bidders, 1 and 2 ? Action sets (bid sets): A1 ?[0, ?) , A2 ?[0, ?) ? ? Type sets (valuations sets): T1 ?[0, 1] , T2 ?[0, 1] Beliefs: Bidder 1 believes that v2 is uniformly distributed on [0, 1]. Bidder 2 believes that v1 is uniformly distributed on [0, 1]. v1 and v2 are independent.

if b2 ? b1 if b2 ? b1 if b2 ? b1

75

Bidder 1 and 2's payoff functions: ?v1 ? b1 if b1 ? b2 ?v2 ? b2 ? v1 ? b1 ? v2 ? b2 u1 (b1 , b2 ; v1 ) ? ? if b1 ? b2 u2 (b1 , b2 ; v2 ) ? ? ? 2 ? 2 if b1 ? b2 ?0 ?0

?

First-price sealed-bid auction cont’d

? A strategy for bidder 1 is a function b1 (v1 ) , for all v1 ? [0, 1]. ? A strategy for bidder 2 is a function b2 (v2 ) , for all v2 ?[0, 1]. ? Given bidder 1's belief on bidder 2, for each v1 ? [0, 1], bidder 1 solves 1 Max (v1 ? b1 )Prob{b1 ? b2 (v2 )} ? (v1 ? b1 )Prob{b1 ? b2 (v2 )} b1?0 2 ? Given bidder 2's belief on bidder 1, for each v2 ?[0, 1], bidder 2 solves 1 Max (v2 ? b2 )Prob{b2 ? b1 (v1 )} ? (v2 ? b2 )Prob{b2 ? b1 (v1 )} b2 ?0 2

76

First-price sealed-bid auction cont’d

v v * * ? Check whether ? b1 (v1 ) ? 1 , b2 (v2 ) ? 2 ? is Bayesian Nash equilibrium. ? ? 2 2? ? ? Given bidder 1's belief on bidder 2, for each v1 ? [0, 1], bidder 1's best * response to b2 (v2 ) solves 1 * * Max (v1 ? b1 )Prob{b1 ? b2 (v2 )} ? (v1 ? b1 )Prob{b1 ? b2 (v2 )} b1?0 2 v 1 v Max (v1 ? b1 )Prob{b1 ? 2 } ? (v1 ? b1 )Prob{b1 ? 2 } b1?0 2 2 2 1 Max (v1 ? b1 )Prob{v2 ? 2b1} ? (v1 ? b1 )Prob{v2 ? 2b1} b1?0 2 Max (v1 ? b1 )2b1

b1?0

FOC:

2v1 ? 4b1 ? 0 ?

b1 (v1 ) ?

v1 2

77

First-price sealed-bid auction cont’d

* ? Hence, for each v1 ? [0, 1], b1 (v1 ) ? * 2's b2 (v2 ) ?

v1 is bidder 1's best response to bidder 2

v2 . 2 v2 is bidder 2's best response 2

* ? By symmetry, for each v2 ?[0, 1], b2 (v2 ) ? * to bidder 1's b1 (v1 ) ?

v1 . 2

v v ? ? * * ? Therefore, ? b1 (v1 ) ? 1 , b2 (v2 ) ? 2 ? is Bayesian Nash equilibrium. 2 2? ?

78

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