beyond nash: raising the flag of rebellion yisrael aumann university of haifa, 11 kislev 5771...

Beyond Nash:Raising the Flag of Rebellion

Yisrael Aumann

University of Haifa, 11 Kislev 5771 (18.11.10)

Based on:

Rational Expectations in Gamesby

Robert Aumann and Jacques Dreze

American Economic Review, March 08 http://www.ma.huji.ac.il/raumann/pdf/86.pdf

• The usual justification for Nash equilibrium is that if game theory is to recommend strategies to the players in a game, then the resulting strategy profile must be known, so each strategy must be a best reply to the others, so the strategies must be in equilibrium.

• What’s wrong with that?• Let’s first backtrack and ask three questions: 1. Why should decision making in games be different

from ordinary (one-person) decision making? Why not just maximize, given our belief about what the others do?

2. Isn’t something vital missing in the description of a game – namely, it’s context?

(Examples: – Coalition government formation – Driving on a one-lane road) 3. What about multiple equilibria? (Is Harsanyi-Selten equilibrium selection the answer?)

• Re 1: Suggested by Kadane & Larkey (Man. Sci., 1982).

K&L ignored the interactive nature of games,

but they didn’t have to.

We’ll show how to incorporate it.• Re 2: This suggests looking at game situations –

games with a context – rather than just games.• Re 3: The answer might be Question 2: Different

equilibria are associated with different contexts.

Formal Definition:

Game Situation :=

Game with belief hierarchies

Assumptions:

1. Common Knowledge of Rationality (CKR)

2. Common Priors (CP)

So now let’s return to our discussion. We said that • The usual justification for Nash equilibrium is that if game

theory is to recommend strategies to the players in a game, then the resulting strategy profile must be known, so each strategy must be a best reply to the others, so the strategies must be in equilibrium.

and we asked• What’s wrong with that?

The answer is that • Game Theory need not recommend any particular

strategy. It can—indeed should—recommend to each player simply to maximize given his private information.

Note that• Nash equilibrium results only in the special case when

the private information is commonly known—in particular, when each one knows what all believe.

Formal Definition:Game Situation :=

Game with belief hierarchies

Assumptions:

1. Common Knowledge of Rationality (CKR)

2. Common Priors (CP)

Definition:

A rational expectation of a player in a game G is

her expectation in some game situation based on G,

with CKR and CP.

Theorem A:

Every rational expectation in a two-person

zero-sum game is that game’s value.

Theorem B:

The rational expectations of a player in a game are

precisely her conditional payoffs (expected payoffs

to her individual pure strategies) when a correlated

equilibrium is played in the “doubled” game: that in

which each of her pure strategies is written twice.

Belief Hierarchies and Belief Systems

Definition: A belief system for a game consists of a set of

types for each player, where a type of player determines

i. his strategy, and

ii. his beliefs: probabilities on the other players’ types.

CKR obtains if all types of all players maximize given their

beliefs. CP obtains if the beliefs have a common prior.

Thm (Harsanyi, 1967). Every belief hierarchy is derived

from some belief system.

Example 1:

Conditional payoff to T = 4

Conditional payoff to B = 7

0,07,2

2,76,6

0⅓

⅓⅓

01

½½

L

L L

R

R R

T

TT

B

B

B

6,62,7

7,20,0

½½

⅞⅛

½⅞

½⅛

T

B

L R

T

B

7/227/22

7/221/22

T

B

L R

Note: Rational Expectations of different players may be mutually inconsistent.

Here the expectations for (B, R) are (6⅛, 6⅛), which is infeasible.

Example 2:

Original Game:

Doubled Game:

0,05,44,5

4,5

5,4

0,05,4

4,50,0

L M R

T

C

B

4,5

5,44,5

4,5

5,4

0,0

0,0

4,50,0

0,05,44,5

0,0

5,4

0,05,4

5,4

4,5

T1

T2

C1

C2

B1

B2

Note 1: The conditional payoffs change when the

game is doubled; there are then more such

payoffs. Thus in Example 2, in the original game

5 is not a conditional payoff, whereas in the

doubled game, it is.

4,5

5,44,5

4,5

5,4

0,0

0,0

4,50,0

0,05,44,5

0,0

5,4

0,05,4

5,4

000

0

1/12

1/6 1/6

1/61/6

1/12

0

00

4,5 1/6

Indeed, consider this correlated equilibrium of thedoubled game:

Here, 5 is the conditional payoff to T1.

0

0

0 0

T2

T1

B1

C1

C2

B2

L M R

Proof Outline for Theorem B:

Suppose there are just two players. A belief

hierarchy of a player can be represented by a type

of that player, a la Harsanyi; each type of each

player is characterized by a pure strategy of that

player, and probabilities for the other player’s

types. Having a CP (common prior) means that

these probabilities are conditionals that derive from

a single distribution on pairs of types.

In Example 2, the situation might look like this:

The rows and columns are types; the entries in the matrix are probabilities that add to 1 overall—the CP. Requiring CKR means that it is optimal for each type to play the pure strategy that that type specifies.

L1L2M1M2M3M4M5R1

T1

T2

T3

C1

C2

C3

C4

B1

B2

Hence, this is a correlated equilibrium of the game

;

the rows and columns are now pure strategies, whose conditional payoffs are the expectations of the corresponding types. “Amalgamating” the copies of each column (adding the corresponding probabilities)

L1L2M1M2M3M4M5R1

T10,00,04,54,54,54,54,5 5,4

T20,00,04,54,54,54,54,55,4

T30,00,04,54,54,54,54,55,4

C15,45,40,00,00,00,00,04,5

C25,45,40,00,00,00,00,04,5

C35,45,40,00,00,00,00,04,5

C45,45,40,00,00,00,00,04,5

B14,54,55,45,45,45,45,40,0

B24,54,55,45,45,45,45,40,0

yields a correlated equilibrium of the game

; note that the conditional payoffs to the row player remain unchanged. Amalgamating

LMR

T10,04,5 5,4

T20,04,55,4

T30,04,55,4

C15,40,04,5

C25,40,04,5

C35,40,04,5

C45,40,04,5

B14,55,40,0

B24,55,40,0

LMR

T10,04,55,4

T2 ,T30,04,55,4

C5,40,04,5

B4,55,40,0

rows as indicated yields a correlated equilibrium of

. The conditional payoff to strategy T1 is the same as the expectation of type T1 in the original type space.

By doubling C and B, and assigning 0 probabilities

to the new rows, we conclude that the expectation

of type T1 is a conditional payoff to a correlated

equilibrium in the doubled game. Similarly for all

types. But the expectations of the types are

precisely all the rational expectations in the given

game. QED

☺

In Economics, “a rational expectation is one that is the sameas the prediction of the relevant economic theory” (Muth, 1961).

Slightly rephrased: the players know the relevant theory (andof course, that it applies to the situation at hand).

In games, the relevant theory takes all players to be rational.

So all players know that all are rational.

So all know that

So all know that…So, CKR.______________________________________________

Next, the “relevant” theory may be thoughtof as yielding a probability distribution p on profiles of beliefs of the players.

But each player knows her own beliefs.

So her beliefs are the conditional of p givenher knowledge.

That is CP.

Discussion of Theorem A

Traditional arguments for the minmax value v of a 2-person 0-sum game:

Guaranteed Value: In expectation, the row playercan guarantee at least v, and the column playercan guarantee paying at most v. “So” --- rationalplayers must end up expecting precisely v.

Equilibrium

Rational Expectations as Benchmarks

!תודה