In game theory, a Perfect Bayesian Equilibrium (PBE) is a solution with Bayesian probability to a turn-based game with incomplete information.
More specifically, it is an equilibrium concept that uses Bayesian updating to describe player behavior in dynamic games with incomplete information.
Perfect Bayesian equilibria are used to solve the outcome of games where players take turns but are unsure of the "type" of their opponent, which occurs when players don't know their opponent's preference between individual moves.
Equilibrium 1 exists, a pooling equilibrium in which both types of sender choose the same action: The sender prefers the payoff of 0 from not giving to the payoff of -1 from sending and not being accepted.
Thus, Give has zero probability in equilibrium and Bayes's Rule does not restrict the belief Prob(Friend|Give) at all.
from accepting, so the requirement that the receiver's strategy maximize his expected payoff given his beliefs necessitates that Prob(Friend|Give)
, a second pooling equilibrium exists as well as Equilibrium 1, based on different beliefs: The sender prefers the payoff of 1 from giving to the payoff of 0 from not giving, expecting that his gift will be accepted.
In equilibrium, Bayes's Rule requires the receiver to have the belief Prob(Friend|Give) = p, since both types take that action and it is uninformative about the sender's type in this equilibrium.
This shows how pessimistic beliefs can result in an equilibrium bad for both players, one that is not Pareto efficient.
These beliefs seem unrealistic, though, and game theorists are often willing to reject some perfect Bayesian equilibria as implausible.
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 their payoff (since there is no gift anyway).
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.
To summarize: in this variant of the gift game, there are two SPEs: either the sender always gives and the receiver always accepts, or the sender always does not give and the receiver always rejects.
There is a recent application of this concept in Poker, by Loriente and Diez (2023).
The following game[4]: section 6.2 is a simple representation of the free-rider problem.
It is only known that each cost is drawn independently at random from some probability distribution.
The only connection between the games is that, by playing in the first day, the players may reveal some information about their costs, and this information might affect the play in the second day.
, we work backwards and analyze the players' actions in day 2.
Their actions depend on the history (= the two actions in day 1), and there are three options: It is possible to calculate the expected payoff of the "threshold player" (a player with cost exactly
In an open-outcry English auction, the bidders can raise the current price in small steps (e.g. in $1 each time).
However, often there is jump bidding - some bidders raise the current price much more than the minimal increment.