Bayesian Games - Computer Science · 2008-10-01 · Perfect Bayesian equilibrium is a similar...

$: Bayesian Games - Computer Science · 2008-10-01 · Perfect Bayesian equilibrium is a similar concept to sequential equilibrium, both trying to achieve some sort of \subgame perfection".$
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 relevantinformation about each other. Such games are known as games with

complete information.


Games with Incomplete Information

Bayesian Games = Games with Incomplete Information

Incomplete Information: Players have private information aboutsomething relevant to his decision making.

I Incomplete information introduces uncertainty about the gamebeing played.

Imperfect Information: Players do not perfectly observe theactions of other players or forget their own actions.

We will see that Bayesian games can be represented as extensive-formgames with imperfect information.


Example 1: A Modified Prisoner’s Dilemma GameWith probability λ, player 2 has the normal preferences as before(type I), while with probability (1− λ), player 2 hates to rat on hisaccomplice and pays a psychic penalty equal to 6 years in prison forconfessing (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


Simultaneous-Move Bayesian Games

A simultaneous-move Bayesian game is (N ,A,Θ,F , u)I N = {1, ..., n} is the set of playersI A = {A1,A2, ...,An} is the set of actions

Ai = {Cooperation, Defection}.I Θ = {Θ1,Θ2, ...,Θn} is the set of types. θi ∈ Θi is a realization of

types for player i .Θ2 = {I, II}.

I F : Θ→ [0, 1] is a joint probability distribution, according to whichtypes of players are drawn

p(θ2 = type I) = λI u = {u1, u2, ..., un} where ui : A×Θ→ R is the utility function of

player i

Two assumptionsI All possible games have the same number of agents and the same

action spaces for each agentI Agents have common prior. The different beliefs of agents are

posteriors.


Imperfect-Information Extensive-FormRepresentation of Bayesian Games

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

Nature

λ 1− λ

C D DC

C D C D DCDC

1

2 2

(5, 5) (0, 8) (8,0) (1, 1) (5, 5) (0, 2) (8, 0) (1, -5)


Strategies in Bayesian Games

A pure strategy si : Θi → Ai of player i is a mapping from everytype player i could have to the action he would play if he hadthat 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 iftype 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 overhis pure strategies.


Best Response and Bayesian Nash EquilibriumWe use pure strategies to illustrate the concepts. But they hold thesame 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 maxsi (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 ∀isi(θi) ∈ BRi .


Bayesian Nash Equilibrium: Example 1

Playing D is a dominant strategy for type I player 2; playing C is adominant 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 isλ× 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


Example 2: An Exchange Game

Each of two players receives a ticket t on which there is anumber in [0,1].

The number on a player’s ticket is the size of a prize that he mayreceive.

The two prizes are identically and independently distributedaccording to a uniform distribution.

Each player is asked independently and simultaneously whetherhe wants to exchange his price for the other player’s prize.

If both players agree than the prizes are exchanges; otherwiseeach player receives his own prize.


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 ofplayer 2?

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

Since players are symmetric, player 1’s best response is of thesame form.

Hence, at a Bayesian Nash equilibrium, both players are willing toexchange only when ti = 0.


Signaling (Sender-Receiver Games)

There are two types of works, bright and dull.

Before entering the job market a worker can choose to get aneducation (i.e. go to college), or enjoy life (i.e. go to beach).

The employer can observe the educational level of the worker butnot his type.

The employer can hire or reject the worker.


Example 3: Signaling

Nature

λ 1− λBright Dull

C B BC

H R H R RHRH

Worker Worker

EmployerEmployer

(2, 2) (-1, 0) (4,-1) (1, 0) (2, 1) (-1, 0) (4, -2) (1, 0)


Bayesian Extensive Games with Observable Actions

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

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

I Θ = {Θ1,Θ2, ...,Θn} is the set of types. θi ∈ Θi is a realization oftypes for player i .

Θ1 = {Bright, Dull}.I F : Θ→ [0, 1] is a joint probability distribution, according to which

types of players are drawnp(θ1 = Bright) = λ

I u = {u1, u2, ..., un} where ui : Z ×Θ→ R is the utility function ofplayer i . Z ∈ H is the set of terminal histories.


Best Responses for Example 3E.g. If the employer always plays H, then the best response forthe worker is B.But how to define best responses for the employer?

I Beliefs on information setsI Beliefs derived from strategies

Nature


C B BC

H R H R RHRH

Worker Worker

EmployerEmployer

(2, 2) (-1, 0) (4,-1) (1, 0) (2, 1) (-1, 0) (4, -2) (1, 0)


A Bayesian Nash Equilibrium of Example 3

Nature


C B BC

H R H R RHRH

Worker Worker

EmployerEmployer

(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


“Subgame Perfection”

The previous Bayesian Nash Equilibrium is not “subgameperfect”. When the information set College is reached, theemployer should choose to hire no matter what belief he has.

We need to require sequential rationality even foroff-equilibrium-path information sets.

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


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 hasbeliefs about the node that he is located given that theinformation set is reached.

(Sequential Rationality) At any information set of player i , therestriction of (σ, µ) to the continuation game must be a BayesianNash equilibrium.

(On-the-path beliefs) The beliefs for any on-the-equilibrium-pathinformation set must be derived from the strategy profile usingBayes’ Rule.

(Off-the-path beliefs) The beliefs at any off-the-equilibrium-pathinformation set must be determined from the strategy profileaccording to Bayes Rule whenever possible.


Perfect Bayesian Equilibrium

Perfect Bayesian equilibrium is a similar concept to sequentialequilibrium, both trying to achieve some sort of “subgameperfection”.

Perfect Bayesian equilibrium is defined for all extensive-formgames with imperfect information, not just for Bayesian extensivegames with observable actions.

Thm: For Bayesian extensive games with observable actions,every sequential equilibrium is a Perfect Bayesian equilibrium.


A Perfect Bayesian Equilibrium of Example 3

Nature


C B BC

H R H R RHRH

Worker Worker

EmployerEmployer

(2, 2) (-1, 0) (4,-1) (1, 0) (2, 1) (-1, 0) (4, -2) (1, 0)

β 1− βλ 1− λ

β ∈ [0, 1]


Summary of Non-Cooperative Game Theory

Normal-Form GamesI Nash Equilibrium (pure strategy and mixed strategy)

Extensive-Form Game with Perfect InformationI Subgame Perfect Nash Equilibrium

Extensive-Form Game with Imperfect InformationI Sequential Equilibrium

Bayesian gamesI Bayesian Nash EquilibriumI Perfect Bayesian Equilibrium


What’s Next

Wednesday: A lecture on background knowledge for predictionmarkets

Monday: Start reading research papers and student presentation

I Sign up for paper presentations ASAP and no later thanWednesday!

I Everyone is required to submit their comments and questions onpapers to [email protected] by midnight before the class,with the title of the paper as the subject line.

I Comments will be posted on the course website before class.


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, ordepends on the specific topic.)

Presentation: a short summary + a critique

You are asked to come to talk with the teaching staff before yourpresentation.

After the presentation, we will break into discussions under theassumption that everyone has read the paper.


Reading PapersWhen you read papers and write your comments, please think aboutthe 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, oroverly 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 thepaper?


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