1

Syllabus

1. Preliminaries. Role of game theory in economics. Normal and extensive form of a game. Game-tree. Information partition. Perfect recall. Perfect and imperfect information. Strategy. 2. Static games of complete information. Dominant strategies. Nash equilibrium. 3. Games with continuous-space strategies 4. Dynamic games of complete information. Backward induction. Subgame perfect equilibrium. Forward induction. 5. Repeated games. 6. Static and dynamic games of incomplete information. 7. Behavioral game theory 8. Cooperative games.

2

references Robert Gibbons, A primer in game theory Chapters on game theory in Mas-Colell, Whinston and Green Other references –Game theory classes by Yildiz at Sloan BS (MIT), by Brandenburger at Stern BS (NYU) –Other books on game theory: Myerson, Gardner, Fudengerg&Tirole –Applications to business strategy: Branderburger and Nalebuff, Coopetition

3

preliminaries

–Game theory is about decision with interaction: more than one agent decides. Each agent’s payoff depends on the behavior of other agents –Compare: a small consumer and a monopoly (decision theory) with a duopoly (interaction) –Interaction leads to some difficulties for agents: 1. Even if they agree on cooperation, coordination issues 2. Conflict

4

Gibbons

–Model building 1: an informal description is translated into a formal problem

–Model building 2: instead of adding all the relevant issues of a problem, the formal

representation emphasizes a few aspects of the problem to analyze them

–Then evidence (through empirical investigation) must be found about its relevance

–Similar issues arise in different areas of economics, and the same game-theoretic

tools can be applied in different settings

5

Myerson, Journal of economic literature, 1999 –Economic theory was transformed by Nash’s ideas –What is economics?

1. Before Nash, a specialized social science concerned with the production and

allocation of material goods.

2. Now, economics is about the analysis of incentives in all social institutions.

(Obviously, Nash’s contributions is just one –very important- among others)

–To analyze any social interaction, we need:

1.A description of the kind of interaction we want to analyze

2.A prediction of individuals’ likely behavior

6

Normal-form game

with n players G = { S1,…, Sn , u1,…un }

Strategy sets S1,…, Sn (a concrete strategy of player i is denoted si).

Payoffs functions ( ) ( )nnn ssussu ,...,,...,,..., 111 :

ℜ→×× ni SSu ...: 1

7

remarks:

Players know the strategy set and the payoff functions of the other players; they are

aware the other players also know their own set of strategies and payoff functions, etc

- games of complete information and

- common knowledge, assumptions on rationality, etc. (see papers by Brandenburger)

We should distinguish outcomes and payoffs:

- if ( )ni ssu ,...,1 is an outcome (money), then we usually assume players are risk-

neutral

- if ( )ni ssu ,...,1 is a payoff (utility, player's preferences), then we use expected utility

functions

8

How to solve a game (in normal-form)?

1. Iterated elimination of strictly dominated strategies

2. Nash-equilibrium

9

1. Iterated elimination of strictly dominated strategies

Definition: In a normal-form game, for player i the strategy "is strictly dominates

strategy 'is if for each combination of strategies of other players

( ) ( )niiiiniiii sssssusssssu ,...,,,,...,...,,,,... 1"

111'

11 +−+− <

a "rational" player never plays dominated strategies (it could be a definition of

rationality; see Brandenburger)

10

An example of iterated elimination of strictly dominated strategies

(Gibbons, figure 1.1.1)

Player 2 Left Middle Right Player 1 Up 1.0 1.2 0.1

Down 0.3 0.1 2.0

11

Delete strategy right for player 2

Player 2 Left Middle

Player 1 Up 1.0 1.2 Down 0.3 0.1

Delete strategy down for player 1

Player 2 Left Middle

Player 1 Up 1.0 1.2

12

Delete strategy left for player 1

Player 1 Middle

Player 2 Up 1.2

Caveats about solving a game this way:

1. It's not enough for player 1, say, to be rational; player 2 must be rational and player 1 must expect player 2 to be rational and so on.

2. For most of the games we do not obtain a unique prediction, we at best discard

some outcomes (see for instance the game in figure 1.1.4 in Gibbons).

13

Best response

Definition: In a normal-form game, for player i the strategy 'is is a best response to

strategies ( )nii ssss ,...,,,... 111 +− by its rivals if

( ) ( )niiiiniiii sssssusssssu ,...,,,,...,...,,,,... 1111'

11 +−+− ≥

for any other feasible strategy ii Ss ∈ .

The following notation is used often: ( )niiii ssssRs ,...,,,... 111'

+−∈

14

Best responses in the previous game?

( ) ( ) topmiddleRleftR == 11

( ) bottomrightR =1

( ) ...3211 =×+×+× rightpmiddlepleftpR

( ) ....2 =R

15

The Prisoner's dilemma

Player 2 confess don't confessPlayer 1 confess 2.2 10.0

don't confess 0.10 6.6

Expected outcome?

This is a clear example of conflict.

16

2. Nash equilibrium

Definition: In a normal-form game, the strategies ( )**1 ,..., nss are a Nash equilibrium if

for each player i, the strategy *is is a best response to ( )****

1 ,...,,,..., nii ssss +− , that is, for

any si on Si,

( ) ( )*****1

****1 ,...,,,,...,,...,,,,..., niiiiniiii sssssusssssu +−+− ≤

In term of the best response correspondence, ( )****1

* ,...,,,..., niiii ssssRs +−∈

17

An example (from Gibbons)

Player 2 Left Middle Right Top 0.4 4.0 5.3 Player 1 Middle 4.0 0.4 5.3 Bottom 3.5 3.5 6.6

Which is the Nash-equilibrium in pure strategies?

18

The previous definition is not very accurate. The Nash equilibrium also considers

mixed strategies:

Mixed strategies (Gibbons 1.3.A)

Suppose the space of strategies of player i is { }iKii ssS ,...,1= . Then a mixed strategy for player i is a probability distribution ( )iKii ppp ,...,1= , where 10 ≤≤ ikp and

11

=∑=

K

kikp .

A pure strategy is, for instance, ( )0,...,0,1=ip .

19

How to justify the Nash-equilibrium (NE) concept as the solution of a game?

From Mas-Colell et al.:

1. Rational inference? 2. If there is only one solution of the game, it must be a NE. 3. Focal points 4. It is a self-enforcing agreement. Compelling argument when there is only one NE. With more than one NE, see below. 5. Stable social convention (convergence to NE, evolutionary economics, John Maynard Smith in biology)

20

Battle of sexes

Player 2 opera football Player 1 opera 3.5 0.0

football 0.0 5.3 2 Nash-equilibria in pure strategies (what about mixed strategies?) There is (maybe) a problem of coordination.

21

Stag-hunt game

Player 2 left right Player 1 top 9.9 0.8

bottom 8.0 7.7 2 Nash-equilibria in pure strategies There is a problem of coordination as before and moreover: - If the cells denote outcomes, not payoffs, (bottom, right) seems less risky. - Is the Nash-equilibrium (top, left) self-enforcing? Aumann, Farrell discuss it.

22

An example without Nash equilibrium in pure strategies (Gibbons, exercise 1.12)

Player 2 Left Right Player 1 Top 2.1 0.2

Bottom 1.2 3.0

Best responses of player 1 to left and to right?

Best responses of player 2 to top and to bottom?

Best responses of player 1 to ( )µµ −= 1,2

p or ( )rightleft ppp 22 ,2= ?

Nash equilibrium when we consider mixed strategies:

=

31,

32*

1p ,

=

43,

41*

2p

23

When players have finite sets of strategies, a Nash equilibrium always exists (it is not

necessarily unique, and maybe it is an equilibrium in mixed strategies).

The proof amounts to show that there is always a fixed point in the correspondence

( ) ( )( )nn sRsR −− ,...,11 :

( ) ( )**1

**11 ,...,)(),...,( nnn sssRsR =−−

24

An example of a game in normal form with 3 players.

Player 3 chooses boxes.

Box 1

Player 2 Left MiddlePlayer 1 Up 1.0.1 1.2.1 Down 0.3.2 0.1.1

Box 2

Player 2 Left MiddlePlayer 1 Up 2.0.2 3.2.2 Down 1.3.1 0.1.2

25

The space of strategies was finite until now. It could be infinite.

2 examples:

1. Second price auction.

2 players have valuations { }21,vv . Each player submits simultaneously a bid { }21,bb .

The player that submits the higher bid gets the object and pays the lower bid:

( )

<

=−

>−

=

21

2121

2121

211

02

,

bbif

bbifbvbbifbv

bbu

Dominant strategy for player 1?

26

2. A duopoly that competes in quantities: the Cournot game (Gibbons, application

1.2.A).

There is a linear demand, ( ) pApD −= ; firms have constant marginal costs of

production c, and choose q1, q2. Assume the market clears (demand = supply):

21 qqQ += , and as a consequence

( )

<+<+−−

=+0 00

21

212121 qqwhenever

qqwheneverqqAqqp

Strategy space Si of firm i: 0≥iq (I write qi instead of si; qi for "quantities").

27

Firm i solves: ( ) ( ) iijijiq

cqqqqAqqi

−−−= , maxπ subject to 0≥iq .

First order condition (FOC): Find qi that solves ( )

0 ,

=∂

∂

i

jiq

qqπ

(This is indeed a solution: Second order condition ( )

02 ,2

2<−=

∂

∂

i

ji

q

qqπ: the profit

function is a concave function on qi).

From ( )

0 ,

=∂

∂

i

jiq

qqπ we obtain the best response function

( ) ( )

−<−−=

otherwise

cAqwhenevercqAqR jjji0

21

28

The Nash equilibrium amounts to finding the fixed point in the correspondence

( ) ( )( )1221 , qRqR :

( ) ( )( ) ( )*2

*1

*12

*21 ,, qqqRqR =