A Game

L

R

T M

B

Lecture 7

Rationalizability

14.123 Microeconomic Theory III Muha met Yildiz

Assume

L

R

T M

B

Player 1 is rational Player 2 is rational

Player 2 is rational and

Knows that Player 1 is rational

Player 1 is rational,

knows that 2 is rational knows that 2 knows that

1 is rational

Assume

P1 is rational

2

1

L

m

R

P2 is rational and

knows that P1 is rational

T

P1 is rational and

knows all these

M

B

Rationalizability

Elim ina t e a l l the s t r i c t ly dom inated s t rateg i es.

Yes

Any dom i nated strategy In the new gam e ?

No

The play is rationalizable, provided that …

Rationalizable strategies

Formally,

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

 N = set of players

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

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

 Belief = a probability distribution  -i on S -i

 Mi xed strategy = a probability distribution  i on S i

 Notation: u i ( s i ,  -i ), u i (  i , s -i ), etc.

 s i is a best response to  -i  u i ( s i ,  -i ) ≥ u i ( s i ’,  -i )  s i ’.

 B i (  -i ) = se t of best respon se s to  -i

 σ i strictly d o minate s s i  u i (  i , s -i ) > u i ( s i , s -i ) for all s -i .

 s i is strictl y domi n ated  so m e σ i stri ctly dominate s s i

Rationality & Dominance

Theorem: s i * is never a best reply to a belief  -i  s i * is

strictly dominated.

Proof:









(=>) Assume s i *  B i (  -i ).

  s i , u i ( s i *,  -i ) ≥ u i ( s i ,  -i )

  i , u i ( s i *,  -i ) ≥ u i (  i ,  -i )

 No  i strictly do minates s i *.

Separating - Hyperplan e T h eorem: F o r a n y convex, n on-emp t y and disjoint C a nd D with C closed,  r :  x  cl ( D )  y  C , r  x ≥ r  y.

(<=) Assume s i * is not strictly domin ated. Define

C = { u i (  i ,.)|  i is a mixed strategy of i }

C and D are disjoint, con v ex and non -e mpty with C closed. By SHT,   -i :  i , u i ( s i *,  -i ) ≥ u i (  i ,  -i )

D = { x | x k > u i ( s i *, s k )  k }.

-i





Iterated strict dominance & Rationalizability

 S 0 = S

 S i = B i (  ( S -i ))

m

m- 1

 where  ( S -i m- 1 ) = beli e fs with support on S -i m- 1

 Previous Theorem:

S i = S i \ { s i |  i : u i (  i , s -i ) > u i ( s i , s -i )  s -i  S -i }

m

m

m- 1

 ( Corre lated ) Rationalizable strategies:



S  S



i



k  0

k i

Foundations of rationalizability

 If the game and rationality are common knowledge, then each player plays a rationalizabl e strategy.

 Each rationalizable strategy profile is the outcome of a situation in which the game and rationality are common knowledge.

 In any “adaptive” learning model the ratio of players who play a non-rationalizable strategy goes to zer o as the system evol ves.

Rationalizability in Cou r not Duopoly

Simultaneously, each firm i  {1,2} produc es q i units at marginal cost c ,

and sells it at price P = max{0,1- q 1 - q 2 }.

q 2

1-c

1  c

2

q 1

1  c

2

1-c

Rationalizability in Cou r not duopoly

 If i knows that q j  q, then q i  (1-c-q)/2.

 If i knows that q j  q, then q i  (1-c-q)/2.

 We know t hat q j  q 0 = 0.

 Then, q i  q 1 = (1-c-q 0 )/2 = (1-c)/2 for each i;

 Then, q i  q 2 = (1-c-q 1 )/2 = (1-c)(1-1/2)/2 for each i;

 …

 Then, q n  q i  q n+1 or q n+1  q i  q n wher e

q n+1 = (1-c-q n )/2 = (1-c)(1-1/2+1/4-…+(-1/2) n )/2.

 As n  , q n  (1-c)/3.