Bayes-Nash equilibrium and correctness of beliefs

Question

Define a Bayesian game as follows: $$G = \left\langle I, \left(A_i,T_i,(p_{t_i})_{t_i \in T_i}, u_i \right)_{i \in I} \right\rangle$$

• $I$ is the set of players
• $A_i$ is the action set for player $i$,
• $T_i$ is the set of possible types for player $i$,
• $p_{t_i} \in \Delta(T_{-i})$ is player $i$'s beliefs regarding the types of the other players. $(T_{-i}=\times_{j \ne i}T_j)$
• $u_i : A \times T \rightarrow \mathbb{R}$ is player $i$'s utility function

Then a Bayes-Nash equilibrium is defined as follows:

A (pure) Bayes-Nash equilibrium is a profile of choice functions (or strategies) $(\sigma_i:T_i \rightarrow A_i)_{i \in I}$ such that, $\forall i \in I, \forall t_i \in T_i, \forall a_i \in A_i$,

$$\sum_{t_-{i}}p_{t_i}(t_{-i}) \cdot u_i(\sigma_i(t_i), \sigma_{-i}(t_{-i}); t_i, t_{-i}) \geq \sum_{t_-{i}}p_{t_i}(t_{-i}) \cdot u_i(a_i, \sigma_{-i}(t_{-i}); t_i, t_{-i}) $$ where, for every $t_{-i}, \sigma_{-i}(t_{-i}) = (\sigma_j(t_j))_{j\ne i}$.

And now my question: Am I correct that this implies that, in a BNE, every player (or rather, every type of every player) best responds given their beliefs about the types of other players, but that there is nothing in a BNE that pins down the beliefs a player (or type of player) has on the types of other players? That is to say, in a BNE, a player (or type of player) could have a degenerate belief (putting full probability on some $t_j^*$ for player $j$) in an equilibrium in which $t_j \ne t_j^*$? Put more simply, can a type of player's beliefs about the type of another player be wrong in a BNE?

Answer 1

I think your definition is incorrect, or at least incomplete.** Usually, in a Bayesian game, there is assumed to be a prior distribution on $T$ (where $T = \times_i T_i$). This distribution is called the "common prior" and it is assumed to be common knowledge that types are drawn according to this distribution. In this case, each player $i$'s belief $p_i$ is given by Bayesian updating on $T_i$ and this prior; and in BNE player $i$ must be best-responding to this belief.

The assumption of a common prior and Bayesian update rule (which is what gives this solution concept its name, after all) mean that players cannot be wrong, merely underinformed. In other words, a player with posterior belief $p_i$ is correct about the distribution of types of other players conditioned on her own, even though she does not know which realizations they have.

** Edit. Osborne and Rubenstein's text mentions that it is possible to define a more general game in which each player has a different prior distribution (there is no common prior). So your definition does match their most general definition. I suppose that in such a case two players may hold incompatible views, hence you could say that someone must be incorrect. That all being said, the vast majority of Bayesian games assume a common prior.

Answer 2

Your definition is missing that in every Bayesian game each player must have a prior $p_i\in\Delta(\prod_{i\in I}T_i)$ this can be the same for all players (common prior assumption) or different.

Note that the existence of a prior already imposes restrictions on the types of beliefs a player can entertain (this are called consistent hierarchies of beliefs in the literature, basically this restricts players to think that someone is of type A, but behaves as a type B, or other inconsistencies that are a bit hard to put in English), in any case, players beliefs cannot be schizophrenic.

The Bayesian part of it is that their interim beliefs (after they learn their own type) must be Bayesian. For example, they cannot have a prior belief that types are chosen by nature independently and with equal probability, and after learning his type assign probability 1 to some other player being of some type. This would not be Bayesian. Since you defined beliefs at the interim stage, it is hard to see how beliefs are being disciplined.

As other people mentioned, when there is a common prior, you impose further constraints on beliefs being consistent across players, this is not needed for defining a Bayesian game. To that extent, a Bayesian equilibrium only restricts beliefs to not being schizophrenic or inconsistent with learning your type. If the common prior is assumed, it also restricts players to have consistent beliefs with each other. But other than that there is no restriction of what prior beliefs can be, so they can be "wrong". However, most economists think that beliefs are beliefs and saying that they are wrong is inappropriate.

Answer 3

Suppose we have a Bayesian game involving two players, 1 and 2, with asymmetric priors. This scenario prompts two questions:

1. What is the source of the asymmetry in the priors?
2. How do the players know each other's priors?

In a common priors setting, these questions are not relevant, as each player knows their own priors and, by extension, the other player's priors as well.

To explain the asymmetry in priors, we could consider a prior-setting game. However, if the asymmetry in priors stems from an asymmetry in the priors of the prior-setting game, we encounter the same problem. Thus, this game must have common priors to provide meaningful answers to the posed questions.

If the prior-setting game has common priors and an asymmetry in payoffs or actions leads to asymmetric priors, then both questions are addressed. In this situation, we can integrate the prior-setting game into the subsequent game, resulting in a common priors game.

As a result, a game must have common priors to effectively address these two questions.

But, do we need to answer these questions?

If not, then the Bayesian Nash restriction loses its impact because, if the prior is truly unrestricted, the posterior is also unrestricted. This is because one can always find a prior that yields a desired posterior almost everywhere (in the sense of the likelihood), as the posterior equals the prior multiplied by the likelihood. Therefore, the desired prior equals the posterior divided by the likelihood, except on a set with a likelihood of zero. For the Bayesian Nash restriction to be meaningful, the prior must be suitably restricted. The natural restrictions involve how the priors are formed if they are informed priors and how they are communicated if they are asymmetric priors. Uninformed common priors, by their nature, do not require further restrictions.