Lectures 16 Incomplete Information Static Case 14.12 Game Theory - - PDF document

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Lectures 16

Incomplete Information Static Case

14.12 Game Theory Muhamet Yildiz

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Road Map

• 1. Example
• 2. Bayesian Games
• 3. Bayesian Nash Equilibrium
• 4. More Examples
• 5. Bayes' Rule

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Incomplete information

• ne player knows something (relevant)

that some other player does not know.

Example

Work (2, 2) W Firm Hir ~ Highp (0, 1) Nature Do not hire (0, 0) War

(1 , 1)

W Hire ~ (-1 , 2) Low 1- Do not hire (0, 0)

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Bayesian Game (Normal Form)

A Bayesian game is a list G = {A 1,··· ,An; T1,···, Tn;P1,··· 'Pn;U1'··· 'Un} where

• A; is the action space of i (a; in A;)
• T; is the type space of i (t; in T;)
• p;(t;lt;) is fs belief about the other players
• u;(a1,

... ,an;t1, ... ,tn) is i's payoff.

Finn Highp Nattrre

Low

• An Example

Wor

(1, 2) W

~

Do no! (0,0) hire

Hire

Dono!

hire W Wor

~

(0, 0) (0, 1) (I, 1) (-1,2)

TFirm={tf};

Tw =

{High,Low} AFirm = {Hire, Don't}

Aw = {Work,Shirk}

PF(High) = P PF(Low) = 1-p

ffv{tr)? /

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Bayesian Nash equilibrium

A Bayesian Nash equilibrium is a Nash equilibrium of a Bayesian game (when each type has positive prob).

• Bayesian game

G = {A1,···,An;T1'···' Tn;P1, ... ,pn;u1, ... ,un}

• a strategy of i is any function s;: T; ---j- Ai;
• A strategy profile s* = (S1 ',

...

, S1 ') is a Bayesian

Nash equilibrium ~

s;' (t;) is a best response to s_;'

for each t; i.e.,

max .

~>J

S

: (tl

), ...

,S;·_I (tH

),Q;,S;

+ I

(t;

+ 1

), ... ,s:

(t,,};t )p;(t_

;

I tJ

OiEA; t_iE T_i

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An Example

Wor

(1, 2) W

TFirm={tf};

Finn

~

T w = {High, Low} AF

irm = {Hire, Don't} Highp (0, 1)

Aw = {Work, Shirk}

e

Do no! (0,0)

PF(High) = P >112

hire

PF(Low) = 1-p

(I, 1

) W Wor

Low

• SF*

Hire

= Hire

~

sw* (High) = Work

(-1,2)

Sw * (Low) = Shirk

Dono!

hire

Another

(0, 0)

equilibrium?

Nattrr

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Another example

• 8

, L

E {0 2}, known by Player 1

R

• Y

E {1,3}, known by Player 1

x 8,y

1,2

• All values are equally likely

Y

• 1,y
• T[

8,0

= {0,2}; T2 = {1,3}

• Bayesian Nash Equilibrium:
• s[(O)=
• s[(2)=
• S

2(1) =

• si3) =

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Bayes'Rule

Prob(A and B)

• Prob(AIB) =

Prob(B)

• Prob(AIB)Prob(B) = Prob(A and B) = Prob(BIA)Prob(A)

Prob(BIA)Prob(A)

• Prob(AIB)

Prob(B)

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Example

p

• Pr(WorkISuccess)

Work

=

Success

J.l

1-p 1-p

• Pr(WorkIFailure) =

Party Failure p

I- J.l

0.9

P(WIS)

0.8 0.7

u:- 0.6

5:

iL

• -' 0.5

P(WIF)

(fJ

~ iL 0.4

0.3 0.2

0.1

0.2

0.4

0.6 0.8

~

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14.12 Economic Applications of Game Theory

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