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### Transcript of 14.126 Spring 2016 Bayesian Games Slides Lecture .Bayesian Games Mihai Manea MIT. Partly based on

• Bayesian Games

Mihai Manea

MIT

Partly based on lecture notes by Muhamet Yildiz.

• Bayesian Games

A Bayesian game is a list (N,A ,,T , u, p)I N: set of playersI A = (Ai)iN: set of action profilesI : set of payoff parametersI Ti : set of types for player i; T =

iN Ti

I ui : A R: payoff function of player iI pi(|ti) ( Ti): belief of type ti

Each player i knows his own type ti but does not necessarily know orother players types. . . belief pi(|ti).The game has a common prior if there exists ( T) such that

pi(|ti) = (|ti),ti Ti , i N.

Mihai Manea (MIT) Bayesian Games June 27, 2016 2 / 30

• Infinite Hierarchies of Beliefs

When a researcher models incomplete information, there is often noex-ante stage or explicit information structure in which players observesignals and make inferences. At the modeling stage, each player i has aninfinite hierarchy of beliefsI a first-order belief 1 ()i about payoffs (and other aspects of the

world)I a second-order belief 2i

1 N () \{i} about and other

players first-order beliefs ia 3

( )I third-order belief i about correlations in player is second-order

uncertainty 2i and other players second-order beliefs2i . . .

Mihai Manea (MIT) Bayesian Games June 27, 2016 3 / 30

• Formal DefinitionFor simplicity, consider two players.

Suppose that is a Polish (complete separable metric) space.

Player i has beliefs about , about others beliefs about ,. . .

X0 =

X1 = X0 (X0)...

Xn = Xn1 (Xn1)...

1 2i = ( , , . . .)i i n= (0 Xn): belief hierarchy of player i

Hi =

n= (0 Xn): set of is hierarchies of beliefs

Every Xn is Polish. Endow Xn with the weak topology.

Mihai Manea (MIT) Bayesian Games June 27, 2016 4 / 30

• Interpretation of Type Space

Harsanyis (1967) parsimonious formalization of incomplete informationthrough a type space (,T , p) naturally generates an infinite hierarchy ofbeliefs for each ti Ti , which is consistent by construction:

first-order belief: h1(i |ti) = margp(|ti) = p(, ti |ti)ti

second-order belief: h2(

h1

, |ti

)=

p(,i ti i |ti) . . .

t h1 t h1i | ( | i)=i i

A type tin

T in 1i a space (,T , p) models a belief hierarchy 2( , , . . .)i i ifh (i |t ni) = i for each n.

Mihai Manea (MIT) Bayesian Games June 27, 2016 5 / 30

• Coherency

How expressive is Harsanyis language?

Is there (T , p) s.t. {hi(|ti)|ti Ti} = Hi?Hierarchies should be coherent:

marg n n 1X = n .2 i i

Different levels of beliefs should not contradict one another.

H0i : set of is coherent hierarchies.

Proposition 1 (Brandenburger and Dekel 1993)

There exists a homeomorphism fi : H0i ( Hi) s.t.

marg nXn1 fi(|i) = ,i n 1.

Mihai Manea (MIT) Bayesian Games June 27, 2016 6 / 30

• Common Knowledge of CoherencyIs there (T , p) s.t. {hi(|ti)|ti Ti} = H0i ?We need to restrict attention to hierarchies of beliefs under which there iscommon knowledge of coherency:

H1 0=i {0I i Hi |fi(Hi |i) = 1}

I H2 =i {i H1i |f 1i(Hi |i) = 1}

I . . .I Hi

=

k0 Hki

Proposition 2 (Brandenburger and Dekel 1993)There exists a homeomorphism gi : Hi

( H ) s.t.i

marg nX ( ) = , .n1 gi | i i n 1

Hi: universal type space

Mihai Manea (MIT) Bayesian Games June 27, 2016 7 / 30

• The Interim Game

For any Ba( yesian game B = (N,A ,,T , u, p), define the interim gameIG(B) = N, S,U

),

N = iNTiSti = Ai

Uti (s) = Epi(|ti) [ui (, s)]

pi(, t i |ti)ui (, sti , st i ) ,t i N ,(,ti)

where s = (sti )tiN.

Assume finite T to avoid measurability issues.

Mihai Manea (MIT) Bayesian Games June 27, 2016 8 / 30

• Th Ex Ante Game

For a Bayesian game B = (N,A ,,T , u, ) with a common prior , theex-ante game G(B) = (N,S,U) is given by

Si = ATii 3 si : Ti Ai

Ui (s) = E[ui(, s(t))].

Mihai Manea (MIT) Bayesian Games June 27, 2016 9 / 30

• Bayesian Nash Equilibrium

Strategies of player i in B are mappings si : Ti Ai (measurable when Tiis uncountable).

Definition 1In a Bayesian game B = (N,A ,,T , u, p), a strategy profile s : T A isa Bayesian Nash equilibrium (BNE) if it corresponds to a Nash equilibriumof IG(B), i.e., for every i N, ti Ti

Epi( ti) [ui (, si (t s| i) , t E u a s t a Ai ( i))] pi(|ti) [ i (, i , i ( i))] , i i .

Interim rather than ex ante definition preferred since in models with acontinuum of types the ex ante game has many spurious equilibria thatdiffer on probability zero sets of types.

Mihai Manea (MIT) Bayesian Games June 27, 2016 10 / 30

• Connections to the Complete Information Games

When i plays a best-response type by type, he also optimizes ex-antepayoffs (for any probability distribution over Ti). Therefore, a BNE of B isalso a Nash equilibrium of the ex-ante game G (B).BNE(B): Bayesian Nash equilibria of B

Proposition 3For any Bayesian game B with a common prior ,

BNE (B) NE (G (B)) .

If (ti) > 0 for all ti Ti and i N, then

BNE (B) = NE (G (B)) .

Mihai Manea (MIT) Bayesian Games June 27, 2016 11 / 30

• Existence of Bayesian Nash Equilibrium

Theorem 1Let B = (N,A ,,T , u, p) be a Bayesian game in whichI Ai is a convex, compact subset of a Euclidean spaceI ui : A R is continuous and concave in aiI is a compact metric spaceI T is finite.

Then B has a BNE in pure strategies.

Concavity is used instead of quasi-concavity to ensure (quasi-)concavity ofpayoffs in the interim game. . . integrals (sums) of quasi-concave functionsare not always quasi-concave.

Upper-hemicontinuity of BNE with respect to parameters, including thebeliefs p, can be established similarly to the complete information case.

Mihai Manea (MIT) Bayesian Games June 27, 2016 12 / 30

• Ex-ante Rationalizability

In a Bayesian game B = (N,A ,,T , u, p), a strategy si : Ti Ai isex-ante rationalizable if si is rationalizable in the ex-ante game G(B).

Appropriate solution concept if the ex-ante stage is real. Imposesrestrictions on players interim beliefs: all player is types have the samebeliefs about other players actions.

Mihai Manea (MIT) Bayesian Games June 27, 2016 13 / 30

• An Example

={,

T1 = {t1, t1}},T2 = {t2}

p(, t1, t2) = p(, t1 , t2) = 1/2

L RU 1, 2, 0D 0, 0 0, 1

L RU 2, 1, 0D 0, 0 0, 1

G(B) :

L RUU 1/2, 1/2, 0UD 1/2, /2 1, 1/2DU 1, /2 1/2, 1/2DD 0, 0 0, 1

S(G(B)) = {(DU,R)}

Mihai Manea (MIT) Bayesian Games June 27, 2016 14 / 30

• Interim Independent Rationalizability

In a Bayesian game B = (N,A ,,T , u, p), an action ai is interimindependent rationalizable for type ti if ai is rationalizable in the interimgame IG(B).Implicit assumption on the interim game: common knowledge that thebelief of a player i about (, ti) is independent of his belief about otherplayers actions( . His) belief about (, ti , ai) is derived from pi(|ti)

T i ti for

some ti A i . We take expectations with respect to p i(|ti) indefining IG(B) before considering ti s beliefs about other players actions.ti believes that and aj are independent conditional on tj .

Mihai Manea (MIT) Bayesian Games June 27, 2016 15 / 30

• Example

={,

T1 = {t1, t1}}; T2 = {t2}

p(, t1, t2) = p(, t1 , t2) = 1/2

L RU 1, 2, 0D 0, 0 0, 1

L RU 2, 1, 0D 0, 0 0, 1

IG(B) : N = {t1, t1 , t2}; t1 chooses rows, t2 columns, and t1 matricesL R

U : U 1, ,2 2, 0, 1D 0, /2,2 0, 1/2, 1

L RD : U 1, /2, 0 2, 1/2, 0

D 0, 0, 0 0, 1, 0

St1 (IG(B)) = St 1

(IG(B)) = {U,D}; St2 (IG(B)) = {L ,R}.

Mihai Manea (MIT) Bayesian Games June 27, 2016 16 / 30

• Interim Correlated Rationalizability

Dekel, Fudenberg, and Morris (2007) allow ti to have correlated beliefsregarding and atj .

For each i N, ti Ti , set S0[ i] =i t Ai and define Sk [i ti] for k 1

ai Sk [i ti] ai arg max ui(, a ,i a i)d(, t i , a i)ai

for some ( Ti Ai) such that

marg p |t and(a Sk1 =T i) [i i( i t ] = 1.i i

The set of interim correlated rationalizable (ICR) actions

)for type ti is

S k[ ] = [ ].i ti

Si ti

k=0

Mihai Manea (MIT) Bayesian Games June 27, 2016 17 / 30

• ICR and Higher Order BeliefsProposition 4

Sk [ti] depends only on ti s k -order beliefs about i . ICR is invariant withrespect to belief hierarchies.

Sufficient to show that S1[ti] depends only on ti s marginal on i . . .If ai S1[i ti], there exists (Ti Ai) with marg T = p i i(|ti) s.t.

ai arg max

ui(, ai, a

ai

i)d(, ti , ai)

= arg maxai

ui(, ai

, a i)d(, a i)

= arg max

ui(, a, i())i d()

ai

where and are s marginals on A i and , resp. and i()represents theai arg maxa

marginal of on Ai conditional on . Conversely, ifui(, a, i())d()i and

= margpi(|ti). . .i Mihai Manea (MIT) Bayesian Games June 27, 2016 18 / 30

• BNE, IIR and Higher Order Beliefs = {, } ,N = {1, 2}. Actions and payoffs:

L RU 1, 0 0, 0D .6, 0 .6, 0

L RU 0, 0 1, 0D .6, 0 .6, 0

Type space T = {t1, t1} {t2, t2} with common prior

t2 t2

t1 1/6 1/12t 1 1/12 1/6

t2 t 2t1 1/12 1/6t 1 1/6 1/12

Every action can be played in a BNE, e.g.,

s (t1) =1 U, s1 t1 = D

s ( ) =2 t2 L , s2

( )(t2) = R .

Second BNE with flipped actions. Each action is played by each type in aBNE, so all actions are interim independent rationalizable.

Mihai Manea (MIT) Bayesian Games