Osborne, Nash equilibria and the correctness of beliefs 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?
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.



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.