Osborne, Nash equilibria and the correctness of beliefs

Question

In Osborne's An Introduction to Game Theory Nash equilibrium is described as follows (p. 21–22):

First, each player chooses her action according to the model of rational choice, given her beliefs about the other players' actions. Second, every player's belief about the other players' actions is correct.

It seems to me that this definition is not completely equivalent to the usual definition of the Nash equilibrium as a strategy profile where each player's strategy is a best response to the strategies of the others.

The usual definition says nothing about beliefs and therefore allows for the possibility that beliefs might be incorrect.

To take a trivial possibility, consider the Prisoner's Dilemma. Suppose each player believes that the other player will not confess. Since confessing is a dominant strategy each player would still confess. So the actions constitute a Nash equilibrium even though the players' beliefs are completely the opposite of the actual equilibrium actions.

Am I right in this understanding that Osborne's definition characterizes something other than Nash's equilibrium?

Answer 1

Introducing the language of beliefs here is slightly strange, given that beliefs do have a very specific meaning in other parts of game theory.

Indeed, Osborne's description is reminiscent of a Bayes Nash Equilibrium. We could introduce the notion of beliefs into the normal form of a complete information game as follows: suppose that with probability $a_i$ each player, $i$, is a "strategic" type who will play according to (Nash) equilibrium, and with probability $1-a_i$ he will select some strategy uniformly at random (because, say, he is indifferent across all actions). We thus have a Bayesian game where thinking about beliefs is more natural.

The Bayes Nash solution concept then says that $i$'s strategy must be optimal given the expected play induced by the other players' strategies and the beliefs over their types implied by $\{a_j\}_{j\neq i}$. If we look at the limit as $a_i\rightarrow 1$ for all $i$ then the Bayes Nash equilibrium of this game will coincide with the solution concept described by Osborne.

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I guess the reason Osborne wrote it like this is a pedagogical one, given that this is an introductory text. When we introduce students to static games, we tell them that player $i$ best responds to the actions of the other players. Students naturally want to know "how can they respond to a strategy chosen simultaneously without knowing what that strategy will be?" This is, in many senses, a philosophical question. Common answers are

• If the game is one that is played often then (putting aside issues of other outcomes that can be sustained in repeated games) we can think of Nash as being an equilibrium in the sense that if we converge there we can develop a norm whereby people continue to play that equilibrium indefinitely (and expect others to do the same).
• If the game is really one-shot then we usually invoke the idea that players are going to try to predict what others will do—and our equilibrium notion embeds the idea that these predictions must be correct.

It seems that the predictions in the second point correspond to the "beliefs" invoked by Osborne. However, it is important to stress that these predictions/"beliefs", are merely an informal/intuitive tool for helping us to conceptualise what is going on in an equilibrium and are not part of the definition of such an equilibrium. The concept of Nash equilibrium itself is completely agnostic on the notion of beliefs (as you note in a comment, it is defined only over actions), which is why, when Osborne goes on to formally define Nash equilibrium, he does so without invoking the idea of beliefs at all.

Answer 2

Introducing belief makes the concept of NE comparable to other refinement concepts such as PBE and sequential equilibrium, but the meaning of NE is not changed.

The graduate micro textbook by Mas-Colell, Whinston, and Green (MWG) has a result for this

Proposition 9.C.1. A strategy profile $\sigma$ is a Nash equilibrium of an extensive form game $\Gamma_E$ if and only if there exists a system of beliefs $\mu$ such that

1. The strategy profile $\sigma$ is sequentially rational given belief system $\mu$ at all information sets $H$ such that $\Pr(H|\sigma)>0$.
2. The system of beliefs $\mu$ is derived from strategy profile $\sigma$ through Bayes' rule whenever possible.

Thus, the Prisoner's Dilemma example you give where players have beliefs opposite to what the opponent's actual strategy fails the second condition, which requires beliefs to be derived from Bayes' rule whenever possible. In fact, this is the mathematical equivalent of the second requirement of Osborne's definition: that a player's belief about the other players' actions is correct.

Answer 3

I might be repeating things which have been said before, but here is my take on this.

I think we face a usual problem when comparing two different models. What an "equivalence" means is not completely obvious because the two definitions lie in different worlds, or different models. However, if "equivalence" is properly define, I think one can make sense of Osborne definition and show that it is indeed "equivalent" to a NE.

The solution concept underlying the quoted section would be something like the following :

Belief equilibrium (BE) : > A strategy profile $s^*$ and a belief profile $b^*$ in which for all player $i$

$$ u_i ( s^*_i ~|~ s_{-i} = b^*_{i}) \geq u_i ( s' ~|~ s_{-i} = b^*_{i}) \text{ for all } s' \in S_i$$ and $$b^*_i = s^*_{-i}$$

Now the problem if we are to get to any "equivalence" statement is that on the one hand, we have the BE which "lives" in a world with... beliefs, and on the other the NE notion which lives in a world... exempt of beliefs. So what would an equivalence statement like "NE $\Leftrightarrow$ BE" possibly mean?

1) BE $\Rightarrow$ NE

This direction of the implication is probably uncontroversial, because we go from a more complex to a more simple model. "Every BE is a NE" should mean that if we look at the equilibrium strategy profile of a BE alone (that is without its supporting belief profile $p$), it should be a NE. One can check that this is the case.

2) NE $\Rightarrow$ BE

This is the tricky part. What does it mean that "Every NE is a BE"? Certainly not that "a NE plus any belief profile is a BE", as the OP showed with his counter-example. Yet, it is the case that "any NE can be made a BE for some belief profile". I think it is in this sense that one should understand Osborne's "equivalence" claim

Notice that we also have the following more "equivalence-like" statement : "An outcome of the game is a NE outcome if and only if it is a BE outcome".

Answer 4

Your prisoner's dilemma example only works because that is a game with dominant strategies. Osborne is correct.

To be best responding to another player's strategy, as in the definition you give, I must know their strategy. In other words, I must have beliefs about what they are doing, and those beliefs must be correct. This is a strengthening of the concept of rationalizability.

You make an interesting point about how you can get strange "equilibria" in games with dominant strategies. It amounts to outcome equivalent $(\sigma,\mu_1)$ and $(\sigma,\mu_2)$ where $\mu_2$ might be wrong and placing positive weight on nonrationalizable strategies. But, I have never seen a Nash equilibrium that included beliefs. The definitions I recall go, "a strategy profile $\sigma\in \Sigma$ is a Nash equilibrium if $\sigma_i\in B_i(\sigma_{-i})$..." I believe this to mean that defining the beliefs is unnecessary, because the beliefs are exactly a correct assessment of the strategy profile. Referencing, one of my books, it gives the usual definition with a Nash (1950) citation, and then goes on to discuss two underlying assumptions. One is correct beliefs and the other is rational play given those correct beliefs.

Answer 5

Page 3 of this paper explains it well: https://www.jstor.org/stable/pdf/2171725.pdf

Epistemic Conditions for Nash Equilibrium Robert Aumann and Adam Brandenburger