TOC
1. Motivation
For some game, there is no sub-game (except the whole game itself) which means all NEs are SPNE, however, not all NEs of it satisfy the sequential rationality. Recall that one reason we introduce SPNE is to make sure equilibria satisfy sequential rationality. To handle this, we introduce the Perfect Bayesian Equilibrium (PBE).
A motivation example is shown as below
The normal-form representation of the game is
The pure-strategy NE are: and . Since there is no subgame (games beginning with nodes other than the root but do separate the information set), these two NEs are SPNEs. However, does not satisfy the sequentail rationality because it is based on a non-credible threat from player 2.
If player 1 believes player 2’s threat of playing , then player 1 should choose to end the game with payoff 1, which is larger than 0 by choosing or . If player 1 does not believe the threats and plays or , then when player 2 gets the move, he will indeed chosse , since strictly dominates for player 2. Thus, the threat of playing by player 2 is not credible.
We can strengthen the equilibrium concept the rule out some subgame-perfect NE like . A stronger equilibrium is Perfect Bayesian Equilibrium.
2. Perfect Bayesian Equilibrium
2.1 Definition
A Perfect Bayesian Equilibrium consists of strategies and beliefs satifying requirement 1 through 4.
Requirement 1
At each info set, the player with the move must have a belief about which node in the info set has been reached by the play of the game.
- For a nonsingleton information set, a belief is a probability distribution over the nodes in the information set;
- For a singleton information set, a belief puts probability one on the single decision node.
For example, the strategy profile and belief system of means player one is going to play and player 2 is going to play , and player 2 believe that player 1 will play . For another example, means player 1 is going to play or with equal probability while player 2 is going to play , and player 2 believes that player 1 will play .
Requirement 2
Given their beliefs, the players’ strategies must be sequentially rational. That is, at each information set, the action taken by the player with the move (and the player’s subsequent strategy) must be optimal, given the player’s belief at that information set and the other players’ subsequent strategies. (where a “subsequent strategy” is a complete plan of actions covering every contingency that might arise after the given information set has been reached.)
For example,
thus action of player 2 satisfies requirement 2.
For player 1:
which means, is not optimal given and
Therefore, the strategy profile satisfies the requirement 1 but does not statisfy the requirement 2.
For another example satisfies requirement 2.
Note that this requirement should be satisfied by all players, if player 2 fail, the strategy profile cannot be PBE no matter what player 1 plays.
Requirement 3
On the equilibrium path or not:
For a given equilibrium in a given extensive-form game, an information set
- is on the equilibrium path if it will be reached with positive probability if the game is played according to the equilibrium strategies
- if off the equilibrium path if it is definitely not to be reached if the game is played according to the equilibrium strategies
Here the equilibrium could refer to NE, SPNE, BNE or PBE.
Whether the information set is on the equilibrium path or not depends on the strategy profile. For example, if the strategy profile is , the information set is on the equilibrium set because the probability of be reached is positive (no matthe is). However, if the strategy profile is , the information set is not on the equilibrium path because the probability of it be reached is zero.
Here, we come with the Requirement 3:
At information sets on the equilirbrium path, beliefs are determined by Bayes’ rule (from the initial node) and the players’ equilibrium stategies. (belief should be consistent)
For example, cannot be PBE because it does not satisfy requirement 3 (and requirement 2) because according to the Bayes’ rule:
For another example, , has to be 1 to make this strategy profile satisfies requirement 3.
For another example, , the should be
For another example, , should be
Note that requirement 3 does not put any constraint on off-equilibrium-path strategy profiles, for example, for the strategy profile , can be anything.
Requirement 4
At information sets off the equilibrium path, beliefs are determined by Bayes’ rule and the players’ equilibrium strategies where possible.
For the strategy profile , can be anything.
Example 2
Another game example
is off the equilibrium path. However, once player 1 deivate from (while other players do not deivate), according to Bayes’ rule, should be . Therefore, to satisfy the requirement 4, should be 1.
The normal-form representation of the game
There are four pure-strategy NE: . The game has a unique subgame (beginning at player 2’s singleton information set), and the unique NE of this subgame is . Hence the unique SPNE of the game is . The other three NE are not subgame-perfect.
When =1, the stategy profile satisfies requirement 1~4 and thus constitute a PBE.
Example 3
There are three pure-strategy NE: .
For , to be a PBE, should be 1.
For , it satisfies re1 and re3, re4. But since player 3 will choose , player 2 should choose rather than . Thus it is not PBE. For , it’s not PBE because EU3 is larger when player 3 chooses (break the requirement 2).
2.2 Solve for PBE
There are two methods to find PBE:
- Write the normal-representation of a game, find the NE, and then verify whether each of them is PBE (according to Requirement 1~4)
- By using Requirement 1~4 to eliminate strategy profiles, and find PBE
Relationship between Different Equilibrium Concepts:
- PBE is a stronger equilibrium concept that refines different types of equilibria
- It refines BE in the same way as SPNE refines NE
- It strengthens SPNE by explicitly analyzing beliefs
- In addition, while a NE requires that no player chooses a strictly dominated strategy, a PBE requires no player’s strategy to be strictly dominated beginning at any information set
PBE corresponds to:
- NE (with appropriate beliefs) in static games of complete information
- BE in static games of incomplete information
- SPNE (with appropriate beliefs) in dynamic games of complete and perfect information (and also many dynamic games of complete but imperfect information)
Examples
3. Signaling games
3.1 Representation & Solution
Signaliing games are kind of dynamic game with incomplete information. The extensive form of it is shown as below (one example)
There are three types of Perfect Bayesian Equilibria of signaling games: separating equilibrium, pooling equilibrium and hybrid equilibrium. In this course we care about the separating equilibrium and pooling equilibrium.
For separating equilibrium, player 1’s strategy is different for different types of nature. For above example, the potential separating equilibria for player 1 is .
For pooling equilibrium, player 2’ strategy is the same for different types of nature. For above example, the potential pooling equilibria for player 1 is .
The method the find PBE is the same as before. For above example
A precise approach to solve signaling game is shown in the following video.
3.2 Cheap Talk Games
The timing of the simplest cheap talk game is identical to the timing of the simplest signaling game (only payoff functions differ):
- Nature draws a type for the Sender from a set of feasible types according to to a probability distribution , where for every and .
- The Sender observes and then chooses a message from a set of feasible messages
- The Receiver observes (but not ) and then chooses an action from a set of feasible actions
- Payoffs are given by and (independent of )
The key feature of the cheap talk game is that the message has no direct effect on the payoffs of the Sender and the Receiver. The message can only be informative by changing the Receiver’s belief about the Sender’s type. One key difference between these two games is that there always exist a pooling equilibrium in a cheap talk game.
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