Epistemic Conditions for Nash Equilibrium

Abstract

Game theoretic reasoning has been widely applied in economics in recent years. Undoubtedly, the most commonly used tool has been the strategic equilibrium of Nash (Ann Math 54:286–295, 1951), or one or another of its so-called “refinements.” Though much effort has gone into developing these refinements, relatively little attention has been paid to a more basic question: Why consider Nash equilibrium in the first place?

Appendix: Extensions and Converses

We start with some remarks on Theorem 41.3 and its proof. First, the conclusions of Theorem 41.3 continue to hold under the slightly weaker assumption that the common prior assigns positive probability to φ = φ being commonly known, and there is a state at which φ = φ is commonly known and g = g and the rationality of the players are mutually known.

Second, note that the rationality assumption is not used until the end of Theorem 41.3’s proof, after (41.3) is established. Thus if we assume only that there is a common prior that assigns positive probability to the conjectures φ ⁱ being commonly known, we may conclude that all players i have the same conjecture σ _j for other players j, and that each φ ⁱ is the product of the σ _j with j ≠ i; that is, the n − 1 conjectures of each player about the other players are independent.

Third, if in Theorem 41.3 we assume that the game being played is commonly (not just mutually) known, then we can conclude that also the rationality of the players is commonly known.³⁷ That is, we have

Proposition 41.A1

Suppose that at some state s, the game g and the conjectures φ ⁱ are commonly known and rationality is mutually known. Then at s, rationality is commonly known. (Note that common priors are not assumed here.)

Proof

Set G := [g], F := [φ], R _j := [j is rational], and R := [all players are rational] = R ₁∩⋯∩R _n . In these terms, the proposition says that CK(G ∩ F) ∩ K ¹ R ⊂ CKR. We assert that it is sufficient for this to prove

$$ {K}^{{}^2}\!\!\left(G\cap F\right)\cap {K}^1R\subset {K}^2R. $$

(41.A1)

Indeed, if we have (A1), an inductive argument using Lemma 41.5 and that E ⊂ E′ implies K ¹(E) ⊂ K ¹(E′) (which follows from Lemma 41.2) yields K ^m(G ∩ F) ∩ K ¹ R ⊂ K ^m R for any m; so taking intersections, CK (G ∩ F) ∩ K ¹ R ⊂ CKR follows.

Let j be a player, B _j the set of actions a _j of j to which the conjecture φ ⁱ of some other player i assigns positive probability. Let E _j := [a _j ∈ B _j ] (the event that the action chosen by j is in B _j ). Since the game g and the conjectures φ ⁱ are commonly known at s, they are a fortiori mutually known there; so by Lemma 41.6, each action in B _j maximizes g _j against $\varphi^{j}$. Hence E _j ∩ G ∩ F ⊂ R _j . At each state in F, each player other than j knows that j’s action is in B _j ; that is, F ⊂ ∩ _i≠j K _i E _j . So G ∩ F ⊂ (∩ _i≠j K _i E _j ) ∩ (G ∩ F). So Lemmas 41.2 and 41.5 yield

$$ \begin{array}{l}{K}^{{}^2}\left(G\cap F\right)\subset {K}^2\left({\cap}_{i\ne j}{K}_i{E}_j\right)\cap {K}^2\left(G\cap F\right)\subset {K}^1\left({\cap}_{i\ne j}{K}_i{E}_j\right)\cap {K}^2\left(G\cap F\right)\subset \\ {K}^1\left({\cap}_{i{\ne} j}{K}_i{E}_j\right)\cap {K}^1\left({\cap}_{i{\ne} j}{K}_i\left(G\cap F\right)\right){=}{K}^1\left({\cap}_{i{\ne} j}{K}_i\left({E}_j\cap G\cap F\right)\right)\subset\\ {} \times {K}^1 \left({\cap}_{i\ne j}{K}_i{R}_j\right).\end{array} $$

Hence using R ⊂ R _j and R _j  = K _j R _j (Corollary 41.2), we obtain

$$ \begin{array}{l}{K}^2\left(G\cap F\right)\cap {K}^{{}^1}R\subset {K}^1\left({\cap}_{i\ne j}{K}_i{R}_j\right)\cap {K}^1R\subset {K}^1\left({\cap}_{i\ne j}{K}_i{R}_j\right)\cap {K}^1{R}_j= \\ {K}^1\left({\cap}_{i\ne j}{K}_i{R}_j\right)\cap {K}^1{K}_j{R}_j={K}^2{R}_j.\end{array} $$

Since this holds for all j, Lemma 41.2 yields³⁸ (A.2). ■

Our fourth remark is that in both Theorems 41.2 and 41.3, mutual knowledge of rationality may be replaced by the assumption that each player knows the others to be rational; in fact, all players may themselves be irrational at the state in question. (Recall that “know” means “ascribe probability 1”; thus a player may be irrational even though another player knows that he is rational.)

We come next to the matter of converses to our theorems. We have already mentioned (at the end of section “Description of the results”) that the conditions are not necessary, in the sense that it is quite possible to have a Nash equilibrium even when they are not fulfilled. In Theorem 41.1, the action n-tuple a(s) at a state s may well be a Nash equilibrium even when a(s) is not mutually known, whether or not the players are rational. (But if the actions are mutually known at s and a(s) is a Nash equilibrium, then the players are rational at s; cf. Remark 41.2.) In Theorem 41.2, the conjectures at a state s in a two-person game may constitute a Nash equilibrium even when, at s, they are not mutually known and/or rationality is not mutually known. Similarly for Theorem 41.3.

Nevertheless, there is a sense in which the converses hold: Given a Nash equilibrium in a game g, one can construct a belief system in which the conditions are fulfilled. For Theorem 41.1, this is immediate: Choose a belief system where each player i has just one type, whose action is i’s component of the equilibrium and whose payoff function is g _i . For Theorems 41.2 and 41.3, we may suppose that as in the traditional interpretation of mixed strategies, each player chooses an action by an independent conscious randomization according to his component σ _i of the given equilibrium σ. The types of each player correspond to the different possible outcomes of the randomization; each type chooses a different action. All types of player i have the same theory, namely, the product of the mixed strategies of the other n − 1 players appearing in σ, and the same payoff function, namely g _i . It may then be verified that the conditions of Theorems 41.2 and 41.3 are met.

These “converses” show that the sufficient conditions for Nash equilibrium in our theorems are not too strong, in the sense that they do not imply more than Nash equilibrium; every Nash equilibrium is attainable with these conditions. Another sense in which they are not too strong—that the conditions cannot be dispensed with or even appreciably weakened—was discussed in sections “The main counterexamples” and “Additional counterexamples.”