Bayesian Games Yiling Chen October 1, 2008 CS286r Fall08 Bayesian - - PowerPoint PPT Presentation

Bayesian Games

Yiling Chen October 1, 2008

CS286r Fall’08 Bayesian Games 1

So far

Up to this point, we have assumed that players know all relevant information about each other. Such games are known as games with complete information.

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Games with Incomplete Information

Bayesian Games = Games with Incomplete Information Incomplete Information: Players have private information about something relevant to his decision making.

◮ Incomplete information introduces uncertainty about the game

being played.

Imperfect Information: Players do not perfectly observe the actions of other players or forget their own actions. We will see that Bayesian games can be represented as extensive-form games with imperfect information.

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Example 1: A Modified Prisoner’s Dilemma Game

With probability λ, player 2 has the normal preferences as before (type I), while with probability (1 − λ), player 2 hates to rat on his accomplice and pays a psychic penalty equal to 6 years in prison for confessing (type II). λ 5, 5 0, 8 8, 0 1, 1 C D C D Type I 1 − λ 5, 5 0, 2 8, 0 1, -5 C D C D Type II

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Simultaneous-Move Bayesian Games

A simultaneous-move Bayesian game is (N, A, Θ, F, u)

◮ N = {1, ..., n} is the set of players ◮ A = {A1, A2, ..., An} is the set of actions

Ai = {Cooperation, Defection}.

◮ Θ = {Θ1, Θ2, ..., Θn} is the set of types. θi ∈ Θi is a realization of

types for player i. Θ2 = {I, II}.

◮ F : Θ → [0, 1] is a joint probability distribution, according to which

types of players are drawn p(θ2 = type I) = λ

◮ u = {u1, u2, ..., un} where ui : A × Θ → R is the utility function of

player i

Two assumptions

◮ All possible games have the same number of agents and the same

action spaces for each agent

◮ Agents have common prior. The different beliefs of agents are

posteriors.

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Imperfect-Information Extensive-Form Representation of Bayesian Games

Add a player Nature who has a unique strategy of randomizing in a commonly known way.

Nature λ 1 − λ C D D C C D C D D C D C 1 2 2 (5, 5) (0, 8) (8,0) (1, 1) (5, 5) (0, 2) (8, 0) (1, -5)

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Strategies in Bayesian Games

A pure strategy si: Θi → Ai of player i is a mapping from every type player i could have to the action he would play if he had that type. Denote the set of pure strategies of player i as Si. S1 = {{C}, {D}} S2 = {{C if type I, C if type II}, {C if type I, D if type II}, {D if type I, C if type II}, {D if type I, D if type II}} A mixed strategy σi : Si → [0, 1] of player i is a distribution over his pure strategies.

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Best Response and Bayesian Nash Equilibrium

We use pure strategies to illustrate the concepts. But they hold the same for mixed strategies. Player i’s ex ante expected utility is Eθ[ui(s(θ), θ)] =

• θi∈Θi

p(θi)Eθ−i[ui(s(θ), θ)|θi] Player i’s best responses to s−i(θ−i) is BRi = arg max

si(θi)∈Si Eθ[ui(si(θi), s−i(θ−i), θ)]

=

• θi∈Θi

p(θi)

• arg max

si(θi)∈Si Eθ−i[ui(si(θi), s−i(θ−i), θ)|θi]

• A strategy profile si(θi) is a Bayesian Nash Equilibrium iif ∀i

si(θi) ∈ BRi.

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Bayesian Nash Equilibrium: Example 1

Playing D is a dominant strategy for type I player 2; playing C is a dominant strategy for type II player 2. Player 1’s expected utility by playing C is λ×0+(1−λ)×5 = 5−5λ. Player 1’s expected utility by playing D is λ × 1 + (1 − λ) × 8 = 8 − 7λ > 5 − 5λ. (D, (D if type I, C if type II)) is a BNE of the game. λ 5, 5 0, 8 8, 0 1, 1 C D C D Type I 1 − λ 5, 5 0, 2 8, 0 1, -5 C D C D Type II

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Example 2: An Exchange Game

Each of two players receives a ticket t on which there is a number in [0,1]. The number on a player’s ticket is the size of a prize that he may receive. The two prizes are identically and independently distributed according to a uniform distribution. Each player is asked independently and simultaneously whether he wants to exchange his price for the other player’s prize. If both players agree than the prizes are exchanges; otherwise each player receives his own prize.

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A Bayesian Nash Equilibrium for Example 2

Strategies of player 1 can be describe as “Exchange if t1 ≤ k” Given player 1 plays such a strategy, what is the best response of player 2?

◮ If t2 ≥ k, no exchange ◮ If t2 < k, exchange when t2 ≤ k/2

Since players are symmetric, player 1’s best response is of the same form. Hence, at a Bayesian Nash equilibrium, both players are willing to exchange only when ti = 0.

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Signaling (Sender-Receiver Games)

There are two types of works, bright and dull. Before entering the job market a worker can choose to get an education (i.e. go to college), or enjoy life (i.e. go to beach). The employer can observe the educational level of the worker but not his type. The employer can hire or reject the worker.

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Example 3: Signaling

Nature λ 1 − λ Bright Dull C B B C H R H R R H R H Worker Worker Employer Employer (2, 2) (-1, 0) (4,-1) (1, 0) (2, 1) (-1, 0) (4, -2) (1, 0)

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Bayesian Extensive Games with Observable Actions

A Bayesian extensive game with observable actions is (N, H, P, Θ, p, u)

◮ (N, H, P) is the same as those in an extensive-form game with perfect

information

◮ Θ = {Θ1, Θ2, ..., Θn} is the set of types. θi ∈ Θi is a realization of

types for player i. Θ1 = {Bright, Dull}.

◮ F : Θ → [0, 1] is a joint probability distribution, according to which

types of players are drawn p(θ1 = Bright) = λ

◮ u = {u1, u2, ..., un} where ui : Z × Θ → R is the utility function of

player i. Z ∈ H is the set of terminal histories.

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Best Responses for Example 3

E.g. If the employer always plays H, then the best response for the worker is B. But how to define best responses for the employer?

◮ Beliefs on information sets ◮ Beliefs derived from strategies

Nature λ 1 − λ Bright Dull C B B C H R H R R H R H Worker Worker Employer Employer (2, 2) (-1, 0) (4,-1) (1, 0) (2, 1) (-1, 0) (4, -2) (1, 0)

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A Bayesian Nash Equilibrium of Example 3

Nature λ 1 − λ Bright Dull C B B C H R H R R H R H Worker Worker Employer Employer (2, 2) (-1, 0) (4,-1) (1, 0) (2, 1) (-1, 0) (4, -2) (1, 0) λ 1 − λ p(Bright|Beach) =

p(Bright)σ(Beach|Bright) p(Bright)σ(Beach|Bright)+p(Dull)σ(Beach|Dull) = λ·1 λ·1+(1−λ)·1 = λ

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“Subgame Perfection”

The previous Bayesian Nash Equilibrium is not “subgame perfect”. When the information set College is reached, the employer should choose to hire no matter what belief he has. We need to require sequential rationality even for

• ff-equilibrium-path information sets.

Then, beliefs on off-equilibrium-path information sets matter.

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Perfect Bayesian Equilibrium

A strategy-belief pair, (σ, µ) is a perfect Bayesian equilibrium if (Beliefs) At every information set of player i, the player has beliefs about the node that he is located given that the information set is reached. (Sequential Rationality) At any information set of player i, the restriction of (σ, µ) to the continuation game must be a Bayesian Nash equilibrium. (On-the-path beliefs) The beliefs for any on-the-equilibrium-path information set must be derived from the strategy profile using Bayes’ Rule. (Off-the-path beliefs) The beliefs at any off-the-equilibrium-path information set must be determined from the strategy profile according to Bayes Rule whenever possible.

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Perfect Bayesian Equilibrium

Perfect Bayesian equilibrium is a similar concept to sequential equilibrium, both trying to achieve some sort of “subgame perfection”. Perfect Bayesian equilibrium is defined for all extensive-form games with imperfect information, not just for Bayesian extensive games with observable actions. Thm: For Bayesian extensive games with observable actions, every sequential equilibrium is a Perfect Bayesian equilibrium.

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A Perfect Bayesian Equilibrium of Example 3

Nature λ 1 − λ Bright Dull C B B C H R H R R H R H Worker Worker Employer Employer (2, 2) (-1, 0) (4,-1) (1, 0) (2, 1) (-1, 0) (4, -2) (1, 0) β 1 − β λ 1 − λ

β ∈ [0, 1]

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Summary of Non-Cooperative Game Theory

Normal-Form Games

◮ Nash Equilibrium (pure strategy and mixed strategy)

Extensive-Form Game with Perfect Information

◮ Subgame Perfect Nash Equilibrium

Extensive-Form Game with Imperfect Information

◮ Sequential Equilibrium

Bayesian games

◮ Bayesian Nash Equilibrium ◮ Perfect Bayesian Equilibrium CS286r Fall’08 Bayesian Games 21

What’s Next

Wednesday: A lecture on background knowledge for prediction markets Monday: Start reading research papers and student presentation

◮ Sign up for paper presentations ASAP and no later than

Wednesday!

◮ Everyone is required to submit their comments and questions on

papers to cs286r@fas.harvard.edu by midnight before the class, with the title of the paper as the subject line.

◮ Comments will be posted on the course website before class. CS286r Fall’08 Bayesian Games 22

Paper Presentations

About 30 minutes per paper. (If we cover two papers in a class, the total presentation time may be about 45 to 50 minutes, or depends on the specific topic.) Presentation: a short summary + a critique You are asked to come to talk with the teaching staff before your presentation. After the presentation, we will break into discussions under the assumption that everyone has read the paper.

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Reading Papers

When you read papers and write your comments, please think about the following. (You don’t need to hit on all of them.)

What is the main contribution of the paper? Is it important? Why? What is the limitation of the paper? What was the main insight in getting the result? What assumptions were made? Are they reasonable, limiting, or

• verly simplified?

What applications might arise from the paper? How can the results be extended? What was unclear to you? How does the paper relate to other work that we have seen? Can you suggest a two-sentence project idea based on the ideas in the paper?

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