Beyond Nash: Raising the Flag of Rebellion Yisrael Aumann University of Haifa, 11 Kislev 5771 ( )

Based on: Rational Expectations in Games by Robert Aumann and Jacques Dreze American Economic Review, March

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 0,00,07,27,2 2,72,76,66,6 0 ⅓ ⅓⅓ 01 ½½ L LL R RR T T T B B B

2,72,76,66,6 0,00,07,27,2 ½½ ⅛⅞ ⅞½ ⅛½ T B LR T B 7/22 1/227/22 T B LR 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: 0,00,05,45,44,54,5 4,54,5 5,45,4 0,00,05,45,4 4,54,50,00,0 LMR T C B 4,54,5 5,45,44,54,5 4,54,5 5,45,4 0,00,0 0,00,0 4,54,50,00,0 0,00,0 5,45,44,54,5 0,00,0 5,45,4 0,00,0 5,45,4 5,45,4 4,54,5 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.

4,54,5 5,45,44,54,5 4,54,5 5,45,4 0,00,0 0,00,0 4,54,50,00,0 0,00,0 5,45,44,54,5 0,00,0 5,45,4 0,00,0 5,45,4 5,45, /12 1/6 1/ ,54,51/6 Indeed, consider this correlated equilibrium of the doubled game: Here, 5 is the conditional payoff to T T2 T1 B1 C1 C2 B2 LMR

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. R1R1 M5M5 M4M4 M3M3 M2M2 M1M1 L2L2 L1L1 T1T1 T2T2 T3T3 C1C1 C2C2 C3C3 C4C4 B1B1 B2B2

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) R1R1 M5M5 M4M4 M3M3 M2M2 M1M1 L2L2 L1L1 5,44,54,54,54,54,54,54,54,54,54,50,0 T1T1 5,45,44,54,54,54,54,54,54,54,54,54,5 T2T2 5,45,44,54,54,54,54,54,54,54,54,54,5 T3T3 4,54,5 5,45,45,45,4 C1C1 4,54,5 5,45,45,45,4 C2C2 4,54,5 5,45,45,45,4 C3C3 4,54,5 5,45,45,45,4 C4C4 5,45,45,45,45,45,45,45,45,45,44,54,54,54,5 B1B1 5,45,45,45,45,45,45,45,45,45,44,54,54,54,5 B2B2

yields a correlated equilibrium of the game ; note that the conditional payoffs to the row player remain unchanged. Amalgamating RML 5,44,54,50,0 T1T1 5,45,44,54,5 T2T2 5,45,44,54,5 T3T3 4,54,5 5,45,4 C1C1 4,54,5 5,45,4 C2C2 4,54,5 5,45,4 C3C3 4,54,5 5,45,4 C4C4 5,45,44,54,5 B1B1 5,45,44,54,5 B2B2

RML 5,45,44,54,50,00,0T1T1 5,45,44,54,50,00,0T 2,T 3 4,54,50,00,05,45,4C 0,00,05,45,44,54,5B rows as indicated yields a correlated equilibrium of. The conditional payoff to strategy T 1 is the same as the expectation of type T 1 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 T 1 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 same as the prediction of the relevant economic theory” (Muth, 1961). Slightly rephrased: the players know the relevant theory (and of 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, CKR. ______________________________________________

Next, the “relevant” theory may be thought of 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 given her 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 player can guarantee at least v, and the column player can guarantee paying at most v. “So” --- rational players must end up expecting precisely v. Equilibrium

Rational Expectations as Benchmarks

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