Lecture 8

Correlated & Sequential Equ ilibri a

14.123 Microeconomic Theory III

Muha met Yildiz

Correlated Equilibrium

Correlated Equilibrium

 '  I i (  )

 Correlated Equilibrium w.r.t. (  , I 1 ,…, I n , p ) is an adapted strategy profile ( s 1 ,… , s n ) s.t.

 u i  i (  ), s  i (  ' )  p (  ' | I i (  ))   u i  s i , s  i (  ' )  p (  ' | I i (  ))

 '  I i (  )

for all  , i,s i .

 Equivalently, for all i and adapted s i ’,

  

 u i  s (  )  p (  )   u i  s i ' (  ), s  i (  )  p (  )

  

Definitions

 Game G = ( N , S 1 ,…, S n ; u 1 ,…, u n ), where

 N = set of player s

 S i = set of all strategies of player i ,

 u i : S 1 ×…× S n → R is i's v NM utility function.

 Informa tion Structure (  , I 1 ,…, I n , p ) wh ere

 (  , p ) is a f inite probability space

 I i is an information partition of 

 Adapted strategy profile (wrt (  , I 1 ,…, I n , p )) ( s 1 ,…, s n ) s.t.

 s i : Ω→ S i

 s i ( ω ) = s i ( ω′ ) whenev er I i ( ω ) = I i ( ω′ ).

Correlated Equilibrium Distribution

 Correlated Equilibrium (distribution) is a probability distribution p on S such that

 u i  s i , s  i  p ( s  i | s i )   u i  s i ' , s  i  p ( s  i | s i )

s  i  S  i s  i  S  i

for all i,s i , s i ’.

 Equivalently, for all i and d i : S i → S i

  

 u i  s  p (  )   u i  d i  s i  , s  i  p (  )

  

 The two definitions are equivalent!

Relation to Other Solution Concepts

 If (  1 ,… ,  n ) is a Nash Equilibrium then  1  …

  n is a Correlated Equilibrium distribution

 If ( s 1 ,… , s n ) is a correlated equilibrium w.r.t. (  , I 1 ,… , I n , p ), s i  is rationalizable for i .

 Correlated Equilibrium =

Common Knowledge of Rationality + Common Prior Assumption

Sequential Equilibrium

What is wrong with this SPE?

X

1

(2, 6 )

T

B

2

L

(0, 1 )

R

(3,2)

L

(-1,3)

R

( 1,5)

Sequential Rationality

 A player is sequentially rational (at a history) if he plays a best reply to a belief conditional on being at that history.

Beer – Q uiche

0

1

1

1

beer

2

0

quiche

{.1}

t w

3

0

1

0

t s

0

0

beer

{.9}

quiche

3

1

2

1

Sequential Equilibrium

 An assessment : (  ) w here  is a strategy profile and  is a b e lief sy stem ,  ( h )  (h) f or ea ch h .

 An assessment (  ) i s sequentia lly rational if at e a ch h i ,  i is a best reply to  -i gi ve n  ( h ).

 (  ) i s con s istent if ther e is a seque nce (  m ,  m ) → (  ) wher e

 m is “completely mixed” and  m is co mputed from  m by Bayes rule:

 An assessment (  ) i s a sequential equilibrium if it i s sequ enti a lly rational an d consiste nt.

Centipede game with irrationality

1

2

1

(1,-5)

.9

.1

(4,4)

(3,3)

1

(5,2)

2

1

(0,-5)

(-1,4)

(0,2)

(-1,3)