Game Theory 10 – chap18. Perfect Bayesian Equilibrium

Part IV: Analysis of Dynamic Bayesian Games

5. Dynamic Bayesian Games
5.1 Games of Imperfect Information and Sequential Rationality*
5.2 Perfect Bayesian Equilibrium
5.3 Dynamic Bayesian Games: Signaling
– Definition and Equilibrium (Chapter 28)
– Forward Induction and Cho-Kreps’ Intuitive Criterion*
– Application 1: Job-Market Signaling (Chapter 29, Spence (1979))
– Application 2: Pecking Order Theor* (Myers and Majluf (1984))
5.4 Screening Games: Adverse Selection*
– Second-Best Outcome: Optimal Nonlinear Pricing
– Tradeoff between Rent Extraction and Efficiency, and the Revelation Principle
5.5 Principal-Agent Problems: Moral Hazard
– Introduction: Incentives and Risk-Sharing
– The Binary Model
– First-Best Contract
– The Optimal Incentive Contract

Perfect Bayesian Equilibrium

A pair ⟨\(\sigma, \mu\)⟩ consisting of a strategy profile \(\sigma\) and beliefs \(\mu\) is called an assessment

DEFINITION 5.1. Assessment ⟨\(\sigma, \mu\)⟩ is a (weak) perfect Bayesian equilibrium (PBE) if

(i) \(\mu\) is Bayesian given \(\sigma\);
(ii) \(\sigma\) is sequentially rational given \(\mu\).

Existence and Structure of Equilibrium*

THEOREM 5.2 (KREPS AND WILSON (1982)). A finite extensive-form game has at least one perfect Bayesian equilibrium.

Sequential Eqbm* → Subgame Perfect Eqbm / Perfect Bayesian Eqbm → Nash Eqbm

Figure: Relationship among equilibrium concepts

For the games we will be studying, there is no difference between Sequential Eqbm and PBE (except for one).

Revisit Example 5.1:

SPE is (T, L), (B, R)

Given any beliefs \(\mu_2\) strategy L is sequentially rational, and thus the second condition of PBE eliminates the Nash eqbm (B, R)

The Bayesian belief under (T, L) is \(\mu_2\) = (1, 0). Therefore, the unique PBE of this game is

⟨(T, L), \(\mu_2(x)\) = 1⟩
strategy, beliefs

Revisit Example 5.2:

When player 1 chooses T, player 2’s Bayesian belief is \(\mu_2\) = (1, 0) under which action L is sequenially rational →
⟨(T, L), \(\mu_2(x)\) = 1⟩ is a PBE

When player 1 chooses B, player 2’s Bayesian belief is never reached so \(\mu_2\) are unrestricted →
⟨(B, R), \(\mu_2(x)\) < 1/2⟩ is another PBE

EXAMPLE 5.5. Find NE, SPE and PBE of the next three-player game.

The game can be represented by the two payoff matrices,

For Nash eqbm, we check if there is a profitable deviation in each strategy profile:

(1) (D, T, L) → P3 deviates to R → Not a NE
(2) (D, T, R) → No one deviates → NE
(3) (D, B, L) → P1 deviates to A → Not a NE
(4) (D, B, R) → P1 deviates to A → Not a NE
(5) (A, T, L) → No one deviates → NE
(6) (A, T, R) → P1 deviates to D → Not a NE
(7) (A, B, L) → No one deviates → NE
(8) (A, B, R) → No one deviates → NE

For subgame perfect eqbm, note that the game has one proper subgame:

This proper subgame has a unique Nash eqbm, (T, R). Anticipating their behaviors,
player 1 chooses D. Therefore, (D, T, R) is the unique SPE.

Lastly, for perfect Bayesian eqbm, we first compute player 3’s BR under \(\mu_3\):

expected payoff from L given \(\mu_3\) is 1∙\(\mu_3(x)\) + 2∙(1-\(\mu_3(x)\)) = 2-\(\mu_3(x)\)
expected payoff from R given \(\mu_3\) is 3∙\(\mu_3(x)\) + 1∙(1-\(\mu_3(x)\)) = 1+2\(\mu_3(x)\)

2-\(\mu_3(x)\) ≥ 1+2\(\mu_3(x)\) ⇒ 3\(\mu_3(x)\) ≤ 1 ⇒ \(\mu_3(x)\) ≤ 1/3

Hence action L is a BR if and only if \(\mu_3(x)\) ≤ 1/3. For player 2, strategy B is strictly dominated so it cannot be played in any PBE.

We then examine the Nash eqba:

(1) (D, T, R) → \(I_3\) is reached → \(\mu_3\) = (1, 0) is Bayesian
⟨(D, T, R), \(\mu_3(x)\) = 1⟩

(2) (A, T, L) → \(I_3\) is not reached → No restrictions on \(\mu_3\) → L is player 3’s BR under \(\mu_3(x)\) ≤ 1/3
⟨(A, T, L), \(\mu_3(x)\) ≤ 1/3⟩

(3) (A, B, L) → eliminated by (ii) of PBE
(4) (A, B, R) → eliminated by (ii) of PBE

  • Reference: Chang-Koo Chi, (37/50) Game Theory and Applications 10 – Perfect Bayesian Equilibrium, Jul 14, 2020,

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