Perfect Bayesian Equilibrium
Economics 302 - Microeconomic Theory II: Strategic Behavior
Instructor: Songzi Du
(Chapters 28 and 29 in Watson (2013))
Simon Fraser University
March 29, 2016
ECON 302 (SFU)
Lecture 11
March 29, 2016
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Perfect Bayesian Equilibrium
Fix an extensive-form game.
Conditional belief on an information set: probabilities over nodes in
the information set.
ECON 302 (SFU)
Lecture 11
March 29, 2016
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Perfect Bayesian Equilibrium
Fix an extensive-form game.
Conditional belief on an information set: probabilities over nodes in
the information set.
A Perfect Bayesian Equilibrium is a strategy profile and a system of
conditional beliefs such that
1. (Sequential Rationality) for every player, at every information set,
the player’s strategy is a best response given the strategies of others
and his conditional belief at that information set.
2. (Consistency) the system of conditional beliefs is derived from the
strategy profile following Bayes’ Rule whenever possible.
Subgame perfect equilibrium + Bayes’ Rule
ECON 302 (SFU)
Lecture 11
March 29, 2016
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Beer-Quiche game
ECON 302 (SFU)
Lecture 11
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Used Car Game: seller proposes
ECON 302 (SFU)
Lecture 11
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Education as Signal
Players: high type student, low type student, and a firm.
The student, high or low type, chooses whether or not to go to
University. The firm observes this choice of education, and decides to
hire the student or not.
The firm cannot tell if the student is high or low type, believes that
the student is high type with probability 0.5.
ECON 302 (SFU)
Lecture 11
March 29, 2016
5 / 11
Education as Signal
Players: high type student, low type student, and a firm.
The student, high or low type, chooses whether or not to go to
University. The firm observes this choice of education, and decides to
hire the student or not.
The firm cannot tell if the student is high or low type, believes that
the student is high type with probability 0.5.
The firm gets 1 he hires the high type student, -1 if he hires the low
type student, 0 if not hiring.
ECON 302 (SFU)
Lecture 11
March 29, 2016
5 / 11
Education as Signal
Players: high type student, low type student, and a firm.
The student, high or low type, chooses whether or not to go to
University. The firm observes this choice of education, and decides to
hire the student or not.
The firm cannot tell if the student is high or low type, believes that
the student is high type with probability 0.5.
The firm gets 1 he hires the high type student, -1 if he hires the low
type student, 0 if not hiring.
The high type student prefers university; the low type student prefers
leisure (no university). The student, high or low type, gets 1 if hired,
0 otherwise; and an additional 10 if he follows his preferred
educational choice.
ECON 302 (SFU)
Lecture 11
March 29, 2016
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Education as Signal
ECON 302 (SFU)
Lecture 11
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Mixed-strategy PBE
ECON 302 (SFU)
Lecture 11
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Poker with Two Cards (Watson, 2013)
Two cards: Ace and King, randomly dealt to player 1 (with 50-50
probability). The card is player 1’s private information (type). Player
2 does not receive a card.
Player 1 chooses first to bid or fold; if he folds, player 1 loses his ante
of $1 to player 2.
If player 1 bids, then player 2 chooses to bid or fold; if he folds, player
2 loses his ante of $1 to player 1.
If both bid, the ante is raised to $b, player 1 wins $b if he has Ace
and loses $b if he has King. (b is an exogenous parameter.)
ECON 302 (SFU)
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Poker with Two Cards (b = 2)
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Poker with Three Cards (Watson, 2013)
Three cards: Ace, King and Queen, one card is randomly dealt to
player 1, and one card is randomly dealt to player 2. The card is a
player’s private information (type).
Player 1 chooses first to bid or fold; if he folds, player 1 loses his ante
of $1 to player 2.
If player 1 bids, then player 2 chooses to bid or fold; if he folds, player
2 loses his ante of $1 to player 1.
If both bid, the ante is raised to $b, the cards are revealed, and the
lower card player loses $b to the higher card player. (Ace > King >
Queen)
ECON 302 (SFU)
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Poker with Three Cards (b = 2)
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