EC319 Economic Theory and Its Applications, Part II: …econ.lse.ac.uk/staff/lfelli/teach/EC319...

EC319 Economic Theory and Its Applications,Part II: Lecture 1

Leonardo Felli

NAB.2.14

16 January 2014

Course Outline

Games of Incomplete Information

I Lecture 1: Static Bayesian Games: Examples and Definition.

I Lecture 2: Bayesian Nash Equilibrium, Auctions andPurification of Mixed Strategy.

I Lecture 3: Revelation Principle.

I Lecture 4: Adverse Section and Price Discrimination.

I Lecture 5: Dynamic Bayesian Games: Definition and PBE.

Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 16 January 2014 2 / 28

Course Outline (cont’d)

I Lecture 6: Dynamic Bayesian Games: Strong PBE.

I Lecture 7: Signaling Games.

I Lecture 8: Optimal Auctions.

I Lecture 9: Double Auctions and Efficient Trade.

I Lecture 10: Principal Agent: Moral Hazard.


Admin

I My coordinates: 32L.4.02, x7525, [email protected]

I PA: Katharine Buckle, 32L.1.03, [email protected].

I Office Hours:I Monday 13:00-14:00, Thursday 11:00-12:30I or by appointment (e-mail [email protected]).

I Course Material: available at:http://econ.lse.ac.uk/staff/lfelli/teaching


http://econ.lse.ac.uk/staff/lfelli/teaching

References:

I Robert Gibbons, A Primer in Game Theory, London:Harvester-Wheatsheaf, 1992.

I Martin Osborne, An Introduction to Game Theory, Oxford:Oxford University Press, 2009.

I Steve Tadelis, Game Theory: An Introduction, Princeton:Princeton University Press, 2013.

I Bernard Salanie, The Economics of Contracts: A Primer,Cambridge: The MIT Press, 2nd Edition, 2005.


Static Bayesian Games: Example 1.

Consider the following simple normal form game:

1\2 L R

U 1, 1 0, 0

D 0, 0 6, 6

This game has two pure strategy Nash equilibria (U, L) and (D,R)with payoffs respectively (1, 1) and (6, 6).

There also exists one mixed strategy Nash equilibrium (6/7, 6/7)with payoffs (6/7, 6/7).


Static Bayesian Games: Example 1 (cont’d)

Assume now that player 2 has some ‘small’ uncertainty whencasting her choice of strategy in the game.

With probability9

10she thinks she is playing Game A:

1\2 L R

U1 1, 1 0, 0

D1 0, 0 6, 6

while with probability1

10she thinks she is playing Game B:

1\2 L R

U2 1, 0 0, 10

D2 0, 0 6, 6



We assume that:

I Player 2 does not discover which game she is playing untilafter she has decided her strategy;

I Player 1 has perfect information and knows exactly whichgame he is playing.

This is a game of incomplete information.

The purpose of our analysis is to obtain predictions on what will bethe outcome of this game.



This is a simultaneous move game hence Nash equilibrium shouldbe the basic tool to get a prediction.

In particular we can determine the strategy choice of player 1 as afunction of what he expects player 2 to do as best reply:

I If the game played is A then player 1’s best reply if he expectsplayer 2 to play L is U1, while his best reply if he expectsplayer 2 to play R is D1.

I If the game played is B then player 1’s best reply if he expectsplayer 2 to play L is U2, while his best reply if he expectsplayer 2 to play R is D2.

This is not a surprise since the payoffs to player 1 are the same inboth games.



To be able to determine player 2 best reply however strategies andpayoffs are not enough.

We need to take into account player 2’s beliefs about which gameis played.

As mentioned in the description of the game player 2 believes that:

I with probability 9/10 the game played is A, and

I with probability 1/10 the game played is B.

We need to compute player 2’s expected payoffs in the gamegiven her beliefs on which game is played.



If player 2 expects player 1 to play U1 or U2 her expected payoffsare:

Π2(U1,U2; L) = 1

(9

10

)+ 0

(1

10

)=

9

10

Π2(U1,U2;R) = 0

(9

10

)+ 10

(1

10

)= 1

The best reply is therefore R.

If player 2 expects player 1 to play D1 or D2 her expected payoffsare:

Π2(D1,D2, L) = 0

(9

10

)+ 0

(1

10

)= 0

Π2(D1,D2,R) = 6

(9

10

)+ 6

(1

10

)= 6

The best reply is therefore R.



If player 2 expects player 1 to play U1 or D2 her expected payoffsare:

Π2(U1,D2; L) = 1

(9

10

)+ 0

(1

10

)=

9

10

Π2(U1,D2,R) = 0

(9

10

)+ 6

(1

10

)=

6

10

The best reply is therefore L.

Finally if player 2 expects player 1 to play D1 or U2 her expectedpayoffs are:

Π2(D1,U2; L) = 0

(9

10

)+ 0

(1

10

)= 0

Π2(D1,U2,R) = 6

(9

10

)+ 10

(1

10

)=

64

10

The best reply is therefore R.Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 16 January 2014 12 / 28


Recall that player 1’s best reply is:

I if player 2 chooses L to play U1 and U2,

I if player 2 chooses R to play D1 and D2.

The unique Bayesian Nash equilibrium of the game is therefore:

I player 2 chooses R and

I player 1 chooses D1 if the game is A and D2 if the game is B.

All other strategies yield a profitable deviation by at least one ofthe players given player 2’s beliefs on which game is played.


Static Bayesian Games: Cournot Duopoly

Consider now the following Cournot Duopoly game with imperfectinformation.

Firm 1’s cost function is:

c(q1) = c q1.

Firm 2 can be a high cost firm:

cH(q2) = cH q2 where cH > c ;

or it can be a low cost firm:

cL(q2) = cL q2 where cL < c ;


Static Bayesian Games: Cournot Duopoly (cont’d)

We assume that:

Firm 2 knows whether its variable cost is high cH or low cL,

Firm 1 does not know the variable cost of firm 2 but believes that:

I with probability1

2the cost is high cH ,

I with probability1

2the cost is low cL.



The inverse demand function faced by both firms is:

P(q1 + q2) = a− (q1 + q2)

where cH < a.

Notice that firm 1 will choose a unique quantity.

Notice that firm 2 will choose a different quantity depending onwhether its cost is high or low.

We label these quantities: qH2 or qL2 .



Depending on the value of the cost firm 2’s profit functions are:

ΠH2 (q1, q

H2 ) = qH2

[a− (q1 + qH2 )− cH

]or

ΠL2(q1, q

L2 ) = qL2

[a− (q1 + qL2 )− cL

]

while firm 1’s expected profit function is:

Π1(q1, qH2 , q

L2 ) =

1

2q1[a− (q1 + qH2 )− c

]+

+1

2q1[a− (q1 + qL2 )− c

]



For any given quantity chosen by firm 1, q1, firm 2’s best reply is,depending on the value of the cost:

maxqH2 ∈R+

qH2

[a− (q1 + qH2 )− cH

]or

qH2 =1

2(a− q1 − cH) ,

andmaxqL2∈R+

qL2

[a− (q1 + qL2 )− cL

]or

qL2 =1

2(a− q1 − cL) .



For any given quantity chosen by the two types of firm 2 qH2 andqL2 , firm 1’s best reply is:

maxq1∈R+

1

2q1[a− (q1 + qH2 )− c

]+

1

2q1[a− (q1 + qL2 )− c

]

or

q1 =1

2

[1

2

(a− qH2 − c

)+

1

2

(a− qL2 − c

)].



In other words the best reply of the two types of firm 2 and of firm1 are:

qH2 =1

2(a− q1 − cH) ,

qL2 =1

2(a− q1 − cL) ,

q1 =1

2

[1

2

(a− qH2 − c

)+

1

2

(a− qL2 − c

)].



The solution to this system of equations is:

q1 =(a− 2c + 1

2 cH + 12 cL)

3

and

qH2 =(a− 2cH + c)

3+

1

12(cH − cL)

and

qL2 =(a− 2cL + c)

3− 1

12(cH − cL)

where qL2 > qH2 .



This is the unique Bayesian Nash equilibrium of this Cournot gamewith asymmetric information.

Useful question is what is the difference with the Cournot modelwith perfect information.

Consider the same model in which everything is the same exceptthat firm 1 knows the costs of firm 2.

In this case firm 1 will choose two different quantities dependingon whether it is competing with the high cost firm 2, qH1 , or thelow cost firm 2, qL1 .



These quantities are the solution to the following two problems:

maxqH1 ∈R+

qH1

[a− (qH1 + qH2 )− c

]

with best reply:

qH1 =1

2

(a− qH2 − c

),

andmaxqL1∈R+

qL1

[a− (qL1 + qL2 )− cL

]or

qL1 =1

2

(a− qL2 − c

).



Firm 2’s best reply are instead:

qH2 =

(a− qH1 − cH

)3

, qL2 =

(a− qL1 − cL

)3

.

The Nash equilibria of this two Perfect Information Cournot gamesare then:

qH1 =(a− 2c + cH)

3qH2 =

(a− 2cH + c)

3

and

qL1 =(a− 2c + cL)

3qL2 =

(a− 2cL + c)

3



Compare now the quantities chosen by firm 2 in the presence ofcomplete vs. asymmetric information:

qH2 =(a− 2cH + c)

3+

1

12(cH − cL) > qH2

and

qL2 =(a− 2cL + c)

3− 1

12(cH − cL) < qL2

while for firm 1:

q1 =1

2qH1 +

1

2qL1

orqH1 > q1 > qL1


Static Bayesian Game: Definition

A game of incomplete information is defined by:

I The set of players N, in the Cournot model

N = 1, 2.

I The set of states Ω, in the Cournot model

Ω = cH , cL

I The action space for every player Ai , in the Cournot model:

Ai = R+.

I The set of types for every player Ti , in the Cournot model:

T1 = c, T2 = cH , cL.


Static Bayesian Game: Definition (cont’d)

I The believes of every player on the types of the opponent: µia conditional probability distribution on T−i , in the Cournotmodel:

µ1(cH) =1

2, µ1(cL) =

1

2, µ2(c) = 1.

I The payoff function for every player contingent on the player’stype ui , in the Cournot model:

u2(q1, qH2 | cH) = qH2

[a− (q1 + qH2 )− cH

]u2(q1, q

L2 | cL) = qL2

[a− (q1 + qL2 )− cL

]u1(q1, q

H2 , q

L2 ) =

1

2q1[a− (q1 + qH2 )− c

]+

+1

2q1[a− (q1 + qL2 )− c

]Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 16 January 2014 27 / 28

Static Bayesian Game: Definition (cont’d)

A strategy for each player is then an action choice for any type ofthe player:

si (ti ) ∈ Ai .

In the Cournot model:

I player 1’s strategy: q1:

I player 2’s strategy: q2(cH) = qH2 , q2(cL) = qL2 .

Therefore the following is a Game of Incomplete Information:

Γ = N,Ω,Ai ,Ti , µi , ui


EC319 Economic Theory and Its Applications, Part II: …econ.lse.ac.uk/staff/lfelli/teach/EC319...

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