Perfect Bayesian equilibrium

Perfect Bayesian Equilibrium
A solution concept in game theory
Relationship
Subset of	Subgame perfect equilibrium, Bayesian Nash equilibrium
Significance
Proposed by	Cho and Kreps
Used for	Dynamic Bayesian games
Example	signaling game

In game theory, a Perfect Bayesian Equilibrium (PBE) is an equilibrium concept relevant for dynamic games with incomplete information (sequential Bayesian games). A PBE is a refinement of both Bayesian Nash equilibrium (BNE) and subgame perfect equilibrium (SPE). A PBE has two components - strategies and beliefs:

The strategies and beliefs should satisfy the following conditions:

Every PBE is both a SPE and a BNE, but the opposite is not necessarily true.

A signaling game is the simplest kind of a dynamic Bayesian game. There are two players, one of them (the "responder") has only one possible type, and the other (the "sender") has several possible types. The sender plays first, then the receiver.

To calculate a PBE in a signaling game, we consider two kinds of equilibria: a separating equilibrium and a pooling equilibrium. In a separating equilibrium each sender-type plays a different action, so the sender's action gives information to the receiver; in a pooling equilibrium, all sender-types play the same action, so the sender's action gives no information to the receiver.

Consider the following gift game:

To analyze PBE in this game, let's look first at the following potential separating equilibria:

We conclude that in this game, there is no separating equilibrium.

Now, let's look at the following potential pooling equilibria:

To summarize:

In the following example, the set of PBEs is strictly smaller than the set of SPEs and BNEs. It is a variant of the above gift-game, with the following change to the receiver's utility:

Note that in this variant, accepting is a dominant strategy for the receiver.

Similarly to example 1, there is no separating equilibrium. Let's look at the following potential pooling equilibria:

Note that option 3 is a Nash equilibrium! If we ignore beliefs, then rejecting can be considered a best-response for the receiver, since it does not affect her payoff (since there is no gift anyway). Moreover, option 3 is even a SPE, since the only subgame here is the entire game! Such implausible equilibria might arise also in games with complete information, but they may be eliminated by applying subgame perfect Nash equilibrium. However, Bayesian games often contain non-singleton information sets and since subgames must contain complete information sets, sometimes there is only one subgame—the entire game—and so every Nash equilibrium is trivially subgame perfect. Even if a game does have more than one subgame, the inability of subgame perfection to cut through information sets can result in implausible equilibria not being eliminated.

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