1

2. Static (simultaneous) games of complete information. Players decisions are simultaneous (or without information on the others’ players

moves).

Players know the opponents’ payoffs

2

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

ui is the VNM utility function of player i

Assumption: G is common knowledge.

Definition: A player i is rational iff he tries to maximize the expected value of ui given

his beliefs.

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remarks:

1. Player 1 knows the strategy set and the payoff functions of the other players; he is

aware that the other players also know his set of strategies and payoff functions, etc.

- games of complete information and,

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

2. We should distinguish outcomes and payoffs:

- if ( )ni ssu ,...,1 is money, then we usually are assuming that players are risk-neutral

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

functions

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An example: Matching pennies (Mas-Colell, p. 220)

There are two players, player 1 and player 2

Each player simultaneously puts a penny down, either heads up or tails up (two

strategies, heads up or tails up)

Outcomes: If the two pennies match, player 1 pays 1 euro to player 2; otherwise,

player 2 pays 1 euro to player 1.

If players’ payoffs are in money (after all, the amounts are small, hence the risk is

small), then we have the following game in normal form:

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1. Matching pennies

Player 2 Head Tails Player 1 Head -1, 1 1, -1

Tails 1, -1 -1, 1

In the first cell (the pennies match), -1 is the payoff for player 1, 1 the payoff for player

2.

(hence we consider expected profits for evaluating utilities and payoffs).

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2. The rational pigs (McMillan, page 13)

Two pigs, one dominant and the other subordinate, are put in a box. There is a lever at

one end of the box which, when pressed, dispenses food at the other end. Thus the pig

that presses the lever must run to the other end; by the time it gets there, the other pig

has eaten most but not all, of the food. The dominant pig is able to prevent the

subordinate pig from getting any of the food when both are at the food. Assuming the

pigs can reason like game theorists, which pig will press the lever? Numbers:

6 units of grain are delivered whenever the lever is pushed. If the subordinate pig presses the lever, the

dominant pig eats all 6 units; but if the dominant pig pushes the lever, the subordinate pig eats 5 of the 6

units before the dominant pig pushes it away. If the two pigs press simultaneously, the subordinate pig is

faster getting to the other end and eats 2 units. Finally, suppose pressing the lever and running to the other

requires some effort, equivalent to one-half unit of food.

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Dominant pig Press Don’t press

Press 1.5, 3.5 -0.5, 6 Subordinate pigDon’t press 5, 0.5 0, 0

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How to solve a simultaneous game of complete information?

1. Dominant strategy

2. Iterated elimination of strictly dominated strategies

3. Nash-equilibrium

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1. Dominant strategy

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

strategy 'is if

( ) ( )' "1 1 1 1 1 1,... , , ,..., ,... , , ,..., , i i i i n i i i i n iu s s s s s u s s s s s S− + − + −< ∀

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

rationality; see Brandenburger)

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3. The prisoners’ dilemma

Player 2

confess don't confessPlayer 1 confess 2, 2 10, 0

don't confess 0, 10 6, 6 Expected outcome?

Efficiency?

This is a clear example of conflict.

What information is needed by the players?

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Definition: A strategy profile ( )1

,..., ,...,i

d d d dNs s s s= is a dominant strategy equilibrium, if and only if sdi is a

dominant strategy for each player i.

( ) ( )'1 1 1 1,..., , ,..., ,..., , ,..., , .

i i

di i N i i N iu s s s s u s s s s S i− − −> ∀ ∀

If a game has a dominant strategy equilibrium, then this prediction is quite strong, however…

Player 2 Left Middle Right Player 1 Up 1, 0 1, 2 0, 1

Down 0, 3 0, 1 2, 0

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2. Iterated elimination of strictly dominated strategies

If we ask for a little more, common knowledge of rationality, we can narrow the prediction in games that were no solvable using dominant strategy equilibrium.

4. 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

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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

14

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

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5. Example in Gibbons (figure 1.1.4)

L C R

T 0, 4 4, 0 5, 3

M 4, 0 0, 4 5, 3

B 3, 5 3, 5 6, 6

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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'

+−∈

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Best responses in game 3?

( ) ( ) topmiddleRleftR == 11

( ) bottomrightR =1

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

( ) ....2 =R

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3. 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 +−∈

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i.e., no player has any incentive to deviate if he knows what the others play.

If players’ rationality and their conjectures about what the others play are mutually

known, then their conjectures must form a Nash equilibrium.

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6. 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?

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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 ( )1,...,i i iKp pσ = , where 10 ≤≤ ikp and

11

=∑=

K

kikp .

A pure strategy is, for instance, the “mixed” strategy ( )1,0,...,0iσ = .

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

From Mas-Colell et al. (pages 248-250):

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)

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7. 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.

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8. 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: - - Is the Nash-equilibrium (top, left) self-enforcing? Aumann, Farrell discuss it.

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9. 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= ?

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Nash equilibrium when we consider mixed strategies: ⎟⎠⎞

⎜⎝⎛=

31,

32*

1p , ⎟⎠⎞

⎜⎝⎛=

43,

41*

2p

10. Computation of equilibria with mixed strategies

Player 2 Left Right Player 1 Top 4, 7 10, 6

Bottom 12, 8 5, 10

{ } { }1 2, , ,S T B S L R= =

If these payoffs are utilities, we assume that take the expected utility form.

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Player 1

T if :

( ) ( )e x p e c t e d u t i l i t y e x p e c t e d u t i l i t yf r o m T f r o m B

54 1 0 1 1 2 5 11 3

q q q q q+ − ≥ + − ⇒ ≤

( )

( ) [ ]

( )

*

*

*

5 1 chooses T 1135therefore, if 1 is indifferent between T and B 0,1

135 1 chooses B 0

13

q p q

q p q

q p q

⎧ ≤ ⇒ ⇒ =⎪⎪⎪ = ⇒ ⇒ ∈⎨⎪⎪ > ⇒ ⇒ =⎪⎩

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Best reply correpondance from player 1:

p

1

2/3

q 0 5/13 1

29

Player 2:

Chooses L if: ( ) ( )

321106187 ≥⇒−+≥−+ ppppp

p

1

2/3

q 0 5/13 1

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Equilibria? Points where the best-reply correspondences from players 1 and 2 intersect.

In our case, there is one equilibrium: ( ) ( )1 252, ,3 13σ σ =

p

1

2/3

q 0 5/13 1

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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 =−−

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Mixed strategies and iterated elimination of strictly dominated strategies

A strategy ( )i iSσ ∈ ∆ is stricltly dominated for player I if there exists another strategy

( ) ( )such that ,i i i j i jS Sσ σ− ≠′∈∆ ∀ ∈∏ ∆

( ) ( ), , .i i i i i iu uσ σ σ σ− −′ >

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11. Example (Mas Colell, 241)

Player 2 L R

U 10, 1 0, 4 M 4, 2 4, 3 Player 1 D 0, 5 10, 2

Strategy M is not strictly dominated.

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10 uR

uL

10

5

54

4

11 1,0,2 2

σ ⎛ ⎞= ⎜ ⎟⎝ ⎠

M

U

D

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

Player 3 chooses boxes.

Box 1

Player 2 Left Middle Player 1 Up 1, 0, 1 1, 2, 1 Down 0, 3, 2 0, 1, 1

Box 2

Player 2 Left Middle Player 1 Up 2, 0, 2 3, 2, 2 Down 1, 3, 1 0, 1, 2

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

2 examples:

13. 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?

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14. 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

( ) 1 2 1 21 2

1 2

00 0

A q q whenever q qp q q

whenever q q− − + >⎧

+ =⎨ + <⎩

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

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(Previously, let’s look at the behavior of a monopolist and a cartel (decision problem):

( ) ( ) cQQQAQ Q

−−=πmax

First order condition (FOC): Find Q that solves ( ) 0)()(' =−+=∂

∂ cQPQQP QQπ

( )c-AQmon21

=

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If firms are able to collude (there is not strategic behavior!), they solve:

( ) ( ) ( ) ( ) 22211121212211 ,, cqq qqAcqq qqAqqqq max −−−+−−−=+ππ

FOC: ( ) { } 0)(')()(' 221211211

21 =++−+++=∂+∂ qqqPcqqPqqqP qππ

Solution: monQqq =+ 21

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Now, multidecision problem (strategic behavior): Firm i solves:

( ) ( ) iijijiiq

cqq qqAqq i

−−−=,maxπ subject to 0≥iq .

FOC: Find qi that solves ( )

0)()(',

21121 =−+++=∂

∂cqqPqqqP

qqq

i

jiπ

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

02 ,2

2<−=

∂

∂

i

ji

q

qqπ: the profit

function is a concave function on qi).

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From ( )

0 ,

=∂

∂

i

jiq

qqπ we obtain the best response function

( ) ( )⎪⎩

⎪⎨⎧ −<−−=

otherwise

cAqwhenevercqAqR jjji0

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

( ) ( )( )1221 , qRqR :

( ) ( )( ) ( )*2

*1

*12

*21 ,, qqqRqR = = ( ) ( )1 1,

3 3A c A c⎛ ⎞− −⎜ ⎟

⎝ ⎠

42

Iterated elimination in the Cournot model (Fudenberg and Tirole, 47)

Now, observe that under certain conditions, we can reach the same result by iterated

elimination of strictly dominated strategies.

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A-c

( ) ( )2 22 0mA c q q− = =

q1 1mq

Strictly dominated at stage 1

Strictly dominated at stage 1

22q

21q

( )2 1q q

( )1 2q q

Strictly dominated at stage 2

Strictly dominated at stage 2

q2

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Rationalizable strategies

We argued that a rational player should never play a dominated strategy. Further, we

could say that rational players with common knowledge of each others’ rationality and

the game’s structure, allows us to go from iterated elimination of dominated strategies

to rationalizable strategies.

Definition: In a game ( ){ } ( ){ }, ,i iI S u⎡ ⎤∆ ⋅⎣ ⎦ , the strategies in ( )iS∆ that survive the iterated

removal of strategies that are never a best response are known as player i’s

rationalizable strategies.

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15. Example (Mas Colell et al. pag. 244)

Player 2 B1 B2 B3 B4

A1 0, 7 2, 5 7, 0 0, 1 A2 5, 2 3, 3 5, 2 0, 1 A3 7, 0 2, 5 0, 7 0, 1 Player 1

A4 0, 0 0, -2 0, 0 10, -1 Player 1: If B1, then A3. If B2, then A2. If B3 then A1. If B4, then A4.

Player 2: If A1, B1, if A2B2, if A3B3, if A4 B1 or B3.

B4 is never a best response. It is strictly dominated by 0.5B1+0.5B3.

Once we rule out B4, A4 is strictly dominated by A2.

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In two player games, the set that survive iterative elimination of strictly dominated

strategies and the set of rationalizable strategies coincide.

In two player games a mixed strategy is a best response whenever it is not strictly

dominated.