{"id":1712,"date":"2023-07-07T18:38:13","date_gmt":"2023-07-07T09:38:13","guid":{"rendered":"https:\/\/saraheee.com\/?p=1712"},"modified":"2023-07-07T20:56:04","modified_gmt":"2023-07-07T11:56:04","slug":"game-theory-10-chap18-perfect-bayesian-equilibrium","status":"publish","type":"post","link":"https:\/\/saraheee.com\/ko\/2023\/07\/game-theory-10-chap18-perfect-bayesian-equilibrium\/","title":{"rendered":"Game Theory 10 \u2013 chap18. Perfect Bayesian Equilibrium"},"content":{"rendered":"

Part IV: Analysis of Dynamic Bayesian Games<\/h3>\n\n\n\n

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<\/p>\n\n\n\n

Perfect Bayesian Equilibrium<\/h3>\n\n\n\n

A pair \u27e8\\(\\sigma, \\mu\\)\u27e9 consisting of a strategy profile \\(\\sigma\\) and beliefs \\(\\mu\\) is called an assessment<\/em><\/p>\n\n\n\n

DEFINITION 5.1. Assessment \u27e8\\(\\sigma, \\mu\\)\u27e9 is a (weak) perfect Bayesian equilibrium (PBE) if<\/p>\n\n\n\n

(i) \\(\\mu\\) is Bayesian given \\(\\sigma\\);
(ii) \\(\\sigma\\) is sequentially rational given \\(\\mu\\).<\/p>\n\n\n\n

Existence and Structure of Equilibrium*<\/h3>\n\n\n\n

THEOREM 5.2 (KREPS AND WILSON (1982)). A finite extensive-form game has at least one perfect Bayesian equilibrium.<\/p>\n\n\n\n

Sequential Eqbm* \u2192 Subgame Perfect Eqbm \/ Perfect Bayesian Eqbm \u2192 Nash Eqbm<\/pre>\n\n\n\n
Figure: Relationship among equilibrium concepts<\/p>\n\n\n\n
For the games we will be studying, there is no difference between Sequential Eqbm and PBE (except for one).<\/mark><\/p>\n\n\n\n
Revisit Example 5.1<\/mark>:<\/p>\n\n\n\n
SPE is (T, L), (B, R)<\/mark><\/p>\n\n\n\n
\"\"<\/figure>\n\n\n\n
Given any beliefs \\(\\mu_2\\) strategy L is sequentially rational, and thus the second condition of PBE eliminates the Nash eqbm (B, R)<\/p>\n\n\n\n
The Bayesian belief under (T, L) is \\(\\mu_2\\) = (1, 0). Therefore, the unique PBE of this game is<\/p>\n\n\n\n
\u27e8(T, L), \\(\\mu_2(x)\\) = 1\u27e9
strategy, beliefs<\/p>\n\n\n\n
Revisit Example 5.2<\/mark>:<\/p>\n\n\n\n
\"\"<\/figure>\n\n\n\n
When player 1 chooses T, player 2’s Bayesian belief is \\(\\mu_2\\) = (1, 0) under which action L is sequenially rational \u2192
\u27e8(T, L), \\(\\mu_2(x)\\) = 1\u27e9 is a PBE<\/p>\n\n\n\n
When player 1 chooses B, player 2’s Bayesian belief is never reached so \\(\\mu_2\\) are unrestricted \u2192
\u27e8(B, R), \\(\\mu_2(x)\\) < 1\/2\u27e9 is another PBE<\/p>\n\n\n\n
EXAMPLE 5.5<\/mark>. Find NE, SPE and PBE of the next three-player game.<\/p>\n\n\n\n
The game can be represented by the two payoff matrices,<\/p>\n\n\n\n
\"\"<\/figure>\n\n\n\n
For Nash eqbm, we check if there is a profitable deviation in each strategy profile:<\/p>\n\n\n\n
(1) (D, T, L) \u2192 P3 deviates to R \u2192 Not a NE
(2) (D, T, R)<\/mark> \u2192 No one deviates \u2192 NE
(3) (D, B, L) \u2192 P1 deviates to A \u2192 Not a NE
(4) (D, B, R) \u2192 P1 deviates to A \u2192 Not a NE
(5) (A, T, L)<\/mark> \u2192 No one deviates \u2192 NE
(6) (A, T, R) \u2192 P1 deviates to D \u2192 Not a NE
(7) (A, B, L)<\/mark> \u2192 No one deviates \u2192 NE
(8) (A, B, R)<\/mark> \u2192 No one deviates \u2192 NE<\/p>\n\n\n\n
For subgame perfect eqbm, note that the game has one proper subgame:<\/p>\n\n\n\n
\"\"<\/figure>\n\n\n\n
This proper subgame has a unique Nash eqbm, (T, R). Anticipating their behaviors,
player 1 chooses D. Therefore, (D, T, R)<\/mark> is the unique SPE.<\/p>\n\n\n\n
Lastly, for perfect Bayesian eqbm, we first compute player 3’s BR under \\(\\mu_3\\):<\/p>\n\n\n\n
expected payoff from L given \\(\\mu_3\\) is 1\u2219\\(\\mu_3(x)\\) + 2\u2219(1-\\(\\mu_3(x)\\)) = 2-\\(\\mu_3(x)\\)<\/mark>
expected payoff from R given \\(\\mu_3\\) is 3\u2219\\(\\mu_3(x)\\) + 1\u2219(1-\\(\\mu_3(x)\\)) = 1+2\\(\\mu_3(x)\\)<\/mark><\/p>\n\n\n\n
2-\\(\\mu_3(x)\\) \u2265 1+2\\(\\mu_3(x)\\) \u21d2 3\\(\\mu_3(x)\\) \u2264 1<\/mark> \u21d2 \\(\\mu_3(x)\\) \u2264 1\/3<\/mark><\/p>\n\n\n\n
Hence action L is a BR if and only if \\(\\mu_3(x)\\) \u2264 1\/3. For player 2, strategy B is strictly dominated so it cannot be played in any PBE.<\/p>\n\n\n\n
We then examine the Nash eqba:<\/p>\n\n\n\n
(1) (D, T, R) \u2192 \\(I_3\\) is reached \u2192 \\(\\mu_3\\) = (1, 0) is Bayesian
\u2234 \u27e8(D, T, R), \\(\\mu_3(x)\\) = 1\u27e9<\/mark><\/p>\n\n\n\n
(2) (A, T, L) \u2192 \\(I_3\\) is not reached \u2192 No restrictions on \\(\\mu_3\\) \u2192 L is player 3’s BR under \\(\\mu_3(x)\\) \u2264 1\/3
\u2234 \u27e8(A, T, L), \\(\\mu_3(x)\\) \u2264 1\/3\u27e9<\/mark><\/p>\n\n\n\n
(3) (A, B, L) \u2192 eliminated by (ii) of PBE<\/mark>
(4) (A, B, R) \u2192 eliminated by (ii) of PBE<\/mark><\/p>\n\n\n\n
<\/p>\n\n\n\n