In
game theory, a Perfect Bayesian Equilibrium (PBE) is an
equilibrium concept
In game theory, a solution concept is a formal rule for predicting how a game will be played. These predictions are called "solutions", and describe which strategies will be adopted by players and, therefore, the result of the game. The most comm ...
relevant for
dynamic games with
incomplete information
In economics and game theory, complete information is an economic situation or game in which knowledge about other market participants or players is available to all participants. The utility functions (including risk aversion), payoffs, strategies ...
(sequential
Bayesian games). It is a refinement of
Bayesian Nash equilibrium (BNE). A perfect Bayesian equilibrium has two components -- ''strategies'' and ''beliefs'':
* The strategy of a player in given information set specifies his choice of action in that information set, which may depend on the history (on actions taken previously in the game). This is similar to a
sequential game
In game theory, a sequential game is a game where one player chooses their action before the others choose theirs. The other players must have information on the first player's choice so that the difference in time has no strategic effect. Sequ ...
.
* The belief of a player in a given information set determines what node in that information set he believes the game has reached. The belief may be a
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomeno ...
over the nodes in the information set, and is typically a probability distribution over the possible ''types'' of the other players. Formally, a belief system is an assignment of probabilities to every node in the game such that the sum of probabilities in any information set is 1.
The strategies and beliefs should satisfy the following conditions:
* Sequential rationality: each strategy should be optimal in expectation, given the beliefs.
* Consistency: each belief should be updated according to the equilibrium strategies, the observed actions, and
Bayes' rule
In probability theory and statistics, Bayes' theorem (alternatively Bayes' law or Bayes' rule), named after Thomas Bayes, describes the probability of an event, based on prior knowledge of conditions that might be related to the event. For exampl ...
on every path reached in equilibrium with positive probability. On paths of zero probability, known as ''off-equilibrium paths'', the beliefs must be specified but can be arbitrary.
A perfect Bayesian equilibrium is always a Nash equilibrium.
Examples of perfect Bayesian equilibria
Gift game 1
Consider the following game:
* The sender has two possible types: either a "friend" (with probability
) or an "enemy" (with probability
). Each type has two strategies: either give a gift, or not give.
* The receiver has only one type, and two strategies: either accept the gift, or reject it.
* The sender's utility is 1 if his gift is accepted, -1 if his gift is rejected, and 0 if he does not give any gift.
* The receiver's utility depends on who gives the gift:
** If the sender is a friend, then the receiver's utility is 1 (if he accepts) or 0 (if he rejects).
** If the sender is an enemy, then the receiver's utility is -1 (if he accepts) or 0 (if he rejects).
For any value of
Equilibrium 1 exists, a
pooling equilibrium in which both types of sender choose the same action:
:''Equilibrium 1.'' Sender: ''Not give'', whether of the friend type or the enemy type. Receiver: ''Do not accept'', with the beliefs that ''Prob(Friend, Not Give) = p'' and ''Prob(Friend, Give) = x,'' choosing a value
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. That belief must be pessimistic enough that the receiver prefers the payoff of 0 from rejecting a gift to the expected payoff of
from accepting, so the requirement that the receiver's strategy maximize his expected payoff given his beliefs necessitates that ''Prob(Friend, Give)''
On the other hand, ''Prob(Friend, Not give) = p'' is required by Bayes's Rule, since both types take that action and it is uninformative about the sender's type.
If
, a second pooling equilibrium exists as well as Equilibrium 1, based on different beliefs:
:''Equilibrium 2.'' Sender: ''Give'', whether of the friend type or the enemy type. Receiver: ''Accept,'' with the beliefs that ''Prob(Friend, Give) = p'' and ''Prob(Friend, Not give) = x'', choosing any value for
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. The out-of-equilibrium belief does not matter, since the sender would not want to deviate to ''Not give'' no matter what response the receiver would have.
Equilibrium 1 is perverse if
The game could have
so the sender is very likely a friend, but the receiver still would refuse any gift because he thinks enemies are much more likely than friends to give gifts. This shows how pessimistic beliefs can result in an equilibrium bad for both players, one that is not
Pareto efficient
Pareto efficiency or Pareto optimality is a situation where no action or allocation is available that makes one individual better off without making another worse off. The concept is named after Vilfredo Pareto (1848–1923), Italian civil engine ...
. These beliefs seem unrealistic, though, and game theorists are often willing to reject some perfect Bayesian equilibria as implausible.
Equilibria 1 and 2 are the only equilibria that might exist, but we can also check for the two potential
separating equilibria, in which the two types of sender choose different actions, and see why they do not exist as perfect Bayesian equilibria:
# Suppose the sender's strategy is: ''Give'' if a friend, ''Do not give'' if an enemy. The receiver's beliefs are updated accordingly: if he receives a gift, he believes the sender is a friend; otherwise, he believes the sender is an enemy. Thus, the receiver will respond with ''Accept''. If the receiver chooses ''Accept'', though, the enemy sender will deviate to ''Give'', to increase his payoff from 0 to 1, so this cannot be an equilibrium.
# Suppose the sender's strategy is: ''Do not give'' if a friend, ''Give'' if an enemy. The receiver's beliefs are updated accordingly: if he receives a gift, he believes the sender is an enemy; otherwise, he believes the sender is a friend. The receiver's best-response strategy is ''Reject.'' If the receiver chooses ''Reject'', though, the enemy sender will deviate to ''Do not give'', to increase his payoff from -1 to 0, so this cannot be an equilibrium.
We conclude that in this game, there is ''no'' separating equilibrium.
Gift game 2
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:
* If the sender is a friend, then the receiver's utility is 1 (if they accept) or 0 (if they reject).
* If the sender is an enemy, then the receiver's utility is 0 (if they accept) or -1 (if they reject).
Note that in this variant, accepting is a weakly
dominant strategy
In game theory, strategic dominance (commonly called simply dominance) occurs when one strategy is better than another strategy for one player, no matter how that player's opponents may play. Many simple games can be solved using dominance. The o ...
for the receiver.
Similarly to example 1, there is no separating equilibrium. Let's look at the following potential pooling equilibria:
# The sender's strategy is: always give. The receiver's beliefs are not updated: they still believe in the a-priori probability, that the sender is a friend with probability
and an enemy with probability
. Their payoff from accepting is always higher than from rejecting, so they accept (regardless of the value of
). This is a PBE - it is a best-response for both sender and receiver.
# The sender's strategy is: never give. Suppose the receiver's beliefs when receiving a gift is that the sender is a friend with probability
, where
is any number in