Is a Nash equilibrium anything more than what it is?

Question

(Sorry for the fuzzy title, could not think of something more informative. Feel free to suggest improvements)

This question is somewhat of a generalization of "Osborne, Nash equilibria and the correctness of beliefs". Consider the normal form game

$G = \langle P, S, U \rangle$ with

$P = \{1,\dots, m\}$ the set of players,

$S =\{S_1,\dots,S_m\}$ the $m$-tuple of pure strategies for players in $P$

$U = \{u_1,\dots,u_m\}$ the $m$-tuple of payoff functions for players in $P$

The structure $G$ is rich enough to define the notion of a Nash equilibrium (NE).

However, authors sometimes describe NE based on richer models. For instance, the description by Osborne discussed in "Osborne, Nash equilibria and the correctness of beliefs" hinges on a structure

$\hat{G} = \langle P, S, B, U \rangle$ with

$P = \{1,\dots, m\}$ the set of players,

$S =\{S_1,\dots,S_m\}$ the $m$-tuple of pure strategies for players in $P$

$B = \{B_1,\dots,B_m\}$ the $m$-tuple of possible beliefs about each other's actions for players in $P$

$U = \{u_1,\dots,u_m\}$ the $m$-tuple of payoff functions for players in $P$

In this setting a NE is equivalent to

A strategy profile $s^*$ and a belief profile $b^*$ in which for all $i\in P$

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

Based on this equivalence, Osborne describes a NE as a situation in which

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.

Now in other settings, I have seen justifications of NE based on other structures richer than $G$. For instance, one could have a game with incomplete information

$\tilde{G} = \langle P, \Theta, p , S, U \rangle$ with

$P = \{1,\dots, m\}$ the set of players,

$\Theta = \{\Theta_1,\dots,\Theta_m\}$ the $m$-tuple of possible types for players in $P$

$p$, a joint probability distribution over types

$S =\{S_1,\dots,S_m\}$ the $m$-tuple of pure strategies for players in $P$

$U = \{u_1,\dots,u_m\}$ the $m$-tuple of payoff functions for players in $P$

In $\tilde{G}$, a NE is equivalent to

A Bayesian-Nash equilibrium in which agent know each other types for sure, that is $p$ is degenerate.

In this case, one could therefore describe a NE as a situation in which

Players know each other's type for sure and play according to a Bayesian-Nash equilibrium

(See for instance "Featherstone, C., & Niederle, M. (2008). Ex ante efficiency in school choice mechanisms: an experimental investigation")

Now these two examples puzzle me. One seems to recommend to view NE as a situation in which beliefs about actions are correct, whereas the other suggests that NE be viewed as situations in which agent have perfect information about each other's preferences.

So my question is :

• Is any of these two argument better than the another in any meaningful way? Is any of the two generalization of $G$ (to $\tilde{G}$ or $\hat{G}$) any more helpful than the other in terms of understanding the necessary conditions for a NE to prevail?

My impression is that the concept of NE stands alone, independently of these generalizations and that there is no "correct" way of enriching the NE framework. As I understand it, "a NE is a situation in which agents are rational and beliefs are correct" is not more or less true than "a NE is a situation in which agents have perfect information on each other's payoffs". But as I see such somewhat conflicting claims showing up again and again, I am worried that I might be missing something.

Answer 1

The conventional definition of Bayesian Nash equilibrium (BNE) is as follows:

A pure strategy BNE is a profile of type-contingent strategies $$(s_i(\theta_i),s_{-i}(\theta_{-i}))=(s_1(\theta_1),\dots,s_{i-1}(\theta_{i-1}),s_i(\theta_i),s_{i+1}(\theta_{i+1}),\dots,s_m(\theta_m))$$ such that for each $i\in P$, $$ s_i(\theta_i)\in\arg\max_{s_i'\in S_i} \sum_{\theta_{-i}}p(\theta_{-i}\mid \theta_i) u_i(s_i'(\theta_i),s_{-i}(\theta_{-i}),(\theta_i,\theta_{-i})). $$

If we want to define BNE with the language of beliefs about the other player's actions, we may propose the following definition:

A pure strategy BNE is a profile of type contingent strategies $(s_i(\theta_i),s_{-i}(\theta_{-i}))$ and a profile of beliefs $b^*=\{b_i(\theta_{-i})\}_{i\in P}$, where $b_i(\cdot)$ is a distribution over $S_{-i}(\cdot)$ conditional on $\theta_{-i}$, such that

1. $s_i(\theta_i)$ maximizes $i$'s expected utility (expectation over $\theta_{-i}$) given $b_i(\cdot)$ and a prior on types; and
2. $b_i(\cdot)=s_{-i}(\cdot)$, i.e. beliefs in $b^*$ are correct.

Thus if the type distribution is degenerate, the definitions of BNE and NE coincide. Nevertheless, the correctness of beliefs (about the opponents' strategies) are present in both definitions. The difference is that in BNE, players' beliefs are over a set of functions (due to the fact that in Bayesian games, strategies are type-contingent).