CHAPTER 28 – PERFECT BAYESIAN EQUILIBRIUM

• This chapter examines a class of games, where players have private information,
and they move sequentially.
• To study games with sequential moves and incomplete information, the
appropriate notion of equilibrium goes beyond subgame perfection to allow
sequential rationality to be applied to all information sets.
• Consider the version of the game pictured in the figure below which is a variant of
the gift game of chapter 24.
• In this variant player 2 enjoys opening gifts from both types of player 1.

• Because the gift game has sequential decisions, it seems appropriate to look for
subgame perfect equilibria. But the game has no proper subgames, so every Nash
equilibrium is subgame perfect.
• In particular, (NFNE, R) is a subgame perfect equilibrium of the game.
• In this equilibrium, both types of player 1 choose not to give a gift and player 2
plans to refuse gifts.
• One problem with the profile (NFNE, R) is that it prescribes behavior for player 2
that is clearly irrational conditional on the game reaching his information set.
• Regardless of player 1’s type, player 2 prefers to accept any gift offered. This
preference is not incorporated into the subgame perfect equilibrium because (i)
player 2’s information set is not reached on the path induced by (NFNE, R) and (ii)
player 2’s information set does not represent the start of a subgame.
• As this example shows, not all information sets are necessarily evaluated in a
subgame perfect equilibrium.
• In other words, the concept of subgame perfection does not sufficiently capture
sequential rationality (that players maximize their payoffs from each of their
information sets).
• To address sequential rationality better, we must employ an equilibrium concept
that isolates every information set for examination – Perfect Bayesian equilibrium
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• The key to this equilibrium concept is that it combines a strategy profile with a
description of beliefs that the players have at each of their information sets.
• The beliefs represent the players’ assessments about each other’s types,
conditional on reaching different points in the game.

CONDITIONAL BELIEFS ABOUT TYPES

• The gift game in the figure above illustrates the idea of a conditional belief.
• Recall that in this game (in chapter 24), player 2 does not observe nature’s
decision. Therefore, at the beginning of the game, player 2 knows only that player
1 is the friend type with probability p and the enemy type with probability 1 − p.
• This belief p is called player 2’s initial belief about player 1’s type.
• This is a belief about player 1’s type, not a belief about player 1’s strategy.
• Although player 2 does not observe nature’s decision, player 2 does observe
whether player 1 decided to give a gift. Player 2 might learn something about
player 1’s type by observing player 1’s action.
• As a result, player 2 will have an updated belief about player 1’s type.
• For example, suppose that you are player 2 in the gift game and suppose that
player 1 behaves according to strategy NFGE; thus, you expect only to receive a gift
from the enemy type. Hence, given player 1’s strategy, you should conclude that
player 1 is an enemy.
• In reference to the figure above, when your information set is reached, you
believe that you are playing at the lower of the two nodes in the information set.
• Player 2 has an updated belief about player 1’s type, conditional on arriving at
player 2’s information set (that is, conditional on receiving a gift).
• Player 2’s updated belief about player l’s type can be put in terms of a probability
distribution over the nodes in player 2’s information set.
• In the figure, this probability distribution is described by the numbers q and 1 − q
that appear beside the nodes. Literally, q is the probability that player 2 believes
he is at the top node when his information set is reached. Thus, q is the probability
that player 2 believes player 1 is the friend type, conditional on receiving a gift.

SEQUENTIAL RATIONALITY

• Taking account of conditional beliefs allows us to evaluate rational behavior at all

information sets, even those that may not be reached in equilibrium play.
• Consider, again, the variant of the gift game. Regardless of player l’s strategy,
player 2 will have some updated belief q at his information set.
• This number has meaning even if player 2 believes that player 1 adopts the
strategy NFNE (where neither type gives a gift).
• In this case, q represents player 2’s belief about the type of player 1 when the
“surprise” of a gift occurs. Given the belief q, we can determine player 2’s optimal
action at his information set.
2.
• You can readily confirm that action A is best for player 2, whatever is q. Thus,
sequential rationality requires that player 2 select A.
• For another example, consider the gift game pictured in figure below.

• Note that regardless of the probability q, player 2 receives a payoff of 0 if he

selects R at his information set.
• In contrast, if player 2 chooses A, then he gets a payoff of 1 with probability q (the
probability that his decision is taken from the top node in his information set), and
he gets a payoff of −1 with probability 1 − q.
• Player 2’s expected payoff of selecting A is therefore q + ( −1) (1 − q) = 2q − 1.
• Player 2 will select A if q > 1/2, he will select R if q < 1/2, and he will be indifferent
between A and R if q = 1/2.

CONSISTENCY OF BELIEFS

• In an equilibrium, player 2’s updated belief should be consistent with nature’s

probability distribution and player l’s strategy.
• For example, as noted earlier, if player 2 knows that player 1 adopts strategy NFGE,
then player 2’s updated belief should specify q = 0; that is, conditional on receiving
a gift, player 2 believes that player 1 is the enemy type.
• In general, consistency between nature’s probability distribution, player 1’s
strategy, and player 2’s updated belief can be evaluated by using Bayes’ rule.
• At player 2’s information set, his updated belief gives the relative likelihood that
player 2 thinks his top and bottom nodes have been reached.
• Let rF and rE be the probabilities of arriving at player 2’s top and bottom nodes,
respectively. That is, rF is the probability that nature selects F and then player 1
selects GF.
• Likewise, rE is the probability that nature selects E and then player 1 chooses GE.
• As an example, suppose that rF = 1/8 and rE = 1/16.
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• In this case, player 2’s information set is reached with probability 1/8 + 1/16 =
3/16, which is not a very likely event. But note that the top node is twice as likely
as is the bottom node.
• Thus, conditional on player 2’s information set actually being reached, player 2
ought to believe that it is twice as likely that he is at the top node than at the
bottom node.
• Because the probabilities must sum to 1, this updated belief is represented by a
probability of 2/3 on the top node and 1/3 on the bottom node.
• In general, the relation between rF, rE, and q is given by q = rF/(rF + rE).
• Or q is the probability of reaching the top node divided by the total probability of
reaching the top and the bottom nodes (the latter of which is the probability of
reaching the information set).
• Numbers rF and rE can be calculated from nature’s probability distribution and
player l’s strategy.
• Specifically, let αF and αE denote the probabilities that the friend and enemy types
of player 1 choose to give a gift. Then player 2’s top node is reached with
probability rF = pαF whereas player 2’s bottom node is reached with probability rE =
(1 – p)αE.
• Therefore, q = pαF/[ pαF+ (1 − p)αE]
• This fraction can be represented in a more intuitive way. Let Prob[G] denote the
overall probability that player 1 gives a gift, which is the denominator of the
fraction.
• The numerator is the probability that nature selects the friend type, Prob[F], times
the probability that the friend gives a gift, Prob[G | F]. The number q is the
probability that player 1 is a friend conditional on player 1 giving a gift.
• Substituting for the terms in the preceding fraction, we have
q = Prob[F | G] = Prob[G | F] Prob[F] / Prob[G]
which is the familiar Bayes’ rule expression.
• Note that Bayes’ rule cannot be applied if player 2’s information set is reached
with 0 probability, which is the case when player 1 employs strategy NFNE. In this
situation, q is still meaningful—it is the belief of player 2 when he is surprised to
learn that player 1 has given a gift—but q is not restricted to be any particular
number.
• In other words, any updated belief is feasible after a surprise event.

EQUILIBRIUM DEFINITION

• Perfect Bayesian equilibrium is a concept that incorporates sequential rationality

and consistency of beliefs.
• The beliefs must be consistent with the players’ strategy profile, and the strategy
profile must specify rational behavior at all information sets, given the players’
beliefs.
• More formally:
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 Consider a strategy profile for the players, as well as beliefs over the nodes
at all information sets. These are called a perfect Bayesian equilibrium
(PBE) if: (1) each player’s strategy specifies optimal actions, given his beliefs
and the strategies of the other players, and (2) the beliefs are consistent
with Bayes’ rule wherever possible.
• Two additional terms are useful in categorizing the classes of potential equilibria.
Specifically, we call an equilibrium separating if the types of a player behave
differently. In contrast, in a pooling equilibrium, the types behave the same.

• Steps for calculating perfect Bayesian equilibria:

1. Start with a strategy for player 1 (pooling or separating).
2. If possible, calculate updated beliefs (q in the example) by using Bayes’ rule. In
the event that Bayes’ rule cannot be used, you must arbitrarily select an
updated belief; here you will generally have to check different potential values
for the updated belief with the next steps of the procedure.
3. Given the updated beliefs, calculate player 2’s optimal action.
4. Check whether player 1’s strategy is a best response to player 2’s strategy. If so,
you have found a PBE.

• To solidify your understanding of the PBE concept, follow along with the
computation of equilibria in the gift game of the second figure above.
• Let’s find a perfect Bayesian equilibria for the above game following the steps
mentioned above. (Refer to the table at the end)
Remember q = Prob[F | G] = Prob[G | F] Prob[F] / Prob[G]

• The example shows that because the types of player 1 have the same preferences
over outcomes, there is no separating equilibrium. There is always a pooling
equilibrium in which no gift is given.
• In this equilibrium, player 2 believes that a gift signals the presence of the enemy.
• Finally, if there is a great enough chance of encountering a friend (so that p ≥ 1/2),
then there is a pooling equilibrium in which gifts are given by both types. In this
equilibrium, player 2 gladly accepts gifts.

5.
Step 1: Choose a GFGE GFNE NFGE NFNE
strategy for player
1
Type of Pooling (same actions for Separating (different Separating (different Pooling (same actions for both types – can’t
equilbrium both types – can’t action for different type action for different type – distinguish type based on action.)
distinguish type based on – action informs of the action informs of the
action.) type.) type.)
Step 2: Update For this strategy of For this strategy of For this strategy of player For this strategy of player 1, the node where
player 2’s beliefs player 1, if player 2 player 1, if player 2 1, if player 2 receives a player 2 moves is not achieved and Prob[G] =
(q) using Bayes’ receives a gift, he is still receives a gift, he will gift, he will update his 0. Thus, we cannot use the Bayes’ rule here.
rule. not sure of the type of update his belief of belief of player 1’s type However, we can still update player 2’s beliefs.
player 1. He updates is player 1’s type and and deduce he must be Both types of players prefer gifting to not
belief as per Bayes’ rule. deduce he must be the the Enemy type. Thus, gifting when the gift is accepted but prefer
Here, Prob[G | F] = 1, Friend type. Thus, updates his belief and not gifting in the likely possibility of rejection.
given the specified updates his belief and assigns probability 1 to R is optimal when player 2 believes that player
strategy of player 1. assigns probability 1 to enemy type. Thus, q = 0. 1 is more likely to be an enemy. That is q ≤ ½.
Prob[F] = p and Prob[G] = friend type. Thus, q = 1.
1. Thus, q = p.
Step 3: Find player Player 2 optimally selects Player 2’s optimal Player 2’s best response is Strategy R is optimal as long as q ≤ 1/2.
2’s optimal A iff p ≥ ½ where he strategy is A. R.
actions. believes player 1 is more
likely a friend. If p < 1/2,
player 2 selects R.
Step 4: Check if If p < 1/2, player 2 selects The enemy type of No, player 1 would strictly As long is R is optimal, both types of player 1
player 1’s strategy R, in which case neither player 1 would strictly prefer not to play GE prefer not to gift.
is a based type of player 1 wishes to prefer to play GE rather when of the enemy type.
response to player play G in the first place. If than NE
2’s actions. p ≥ ½, player 1 plays G
Step 5: PBE or Thus, there is no PBE of There is no PBE in There is no PBE in which Thus, for every q ≤ ½, there is a PBE in which
Not? If the answer this type when p < 1/2. which GFNE is played. NFGE is played. player 2’s belief is q and the strategy profile
to step 4 is yes, When p ≥ 1/2, there is a (NFNE, R) is played.
you have a PBE. PBE in which q = p and
(GFGE, A) is played.