Rationalizability 14.12 Game Theory Muhamet Yildiz 1 - ~ ~ - - - PDF document

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Lecture 5 Rationalizability 14.12 Game Theory Muhamet Yildiz 1 - ~ ~ - ~ ~- o v M = 0 V 2p-(I-p) = 3p-1 = T Recall: A Game VB = -p+2 (I-p) = 2-3p V L R 2 (2 , 0) (-1,1) T (0,10) (0,0) M (-1,-6) (2,0) B -I L- ______________

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Lecture 5 Rationalizability

14.12 Game Theory Muhamet Yildiz

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v

M = 0

V

3p-1

Recall: A

2p-(I-p) =

Game

VB = -p+2 (I-p) = 2-3p

V

L R

2 T

(2,0)

(-1,1)

M

(0,10) (0,0)

~

(-1,-6) (2,0)

I L-______________

p p I-p 2

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Recap: Rationality & Dominance

Belief: A probability distribution p_; on others' strategies;
Mixed Strategy: A probability distribution (Ji on own strategies;
Playing s;* is rational ~

s;* is a best response to a belief p_;: VSi

Ls_

i Ui(S;*,Lap-i(La ;:::: Ls_ i Ui(S;,Lap-i(Li)

(Ji dominates s;'* ~

VS_i

Lsi Ui (S;,La(Ji (Si) > Ui(S/*,Li)

Theorem: Playing s;* is rational ~

s;* is not dominated.

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Assume

Player I is rational

L R

Player 2 is rational Player 2 is rational and

T

(2,0)

(-1,1)

Knows that Player I is rational Player I is rational,

M

(0,10) (0,0)

knows that 2 is rational knows that 2 knows that

B (-1,-6) (2,0)

I is rational 4

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Assume

P I is rational 2 P2 is rational and

L m R

I knows that PI is rational

T (3,0) (1,1) (0,3)

P I is rational and knows all these

M (1,0) (0,10) (1,0) B (0,3) (1,1) (3,0)

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Rationalizability

Eliminate all the strictly dominated strategies. Yes y dominated strate In the new game? No Rationalizable strategies

The play is rationalizable, provided that ...

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Important

Eliminate only the strictly dominated

strategies

Ignore weak dominance
Make sure to eliminate the strategies

dominated by mixed strategies as well as pure

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Beauty Contest

There are n students.
Simultaneously, each student submits a

number Xi between 0 and 100.

The payoff of

student i is 100 - (Xi - 2i/3)2 where

n

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Rationalizability in Beauty Contest

If

Xi = Expected value of

sum of

Xj withJ:;ti, best strategy is

(2/3)X) (n-2/3) After Round 1:

0, 2 n -1 100]

[ 3 n-2/3

After Round 2:

0,(2 n-1 J2100] [

3 n-2/3

After Round k:

0,(2 n-1 Jk100] [

3 n -2/3

Rationalizability = {O}. 9

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with m mischievous students

Payoff for mischievous: (Xi - 2x/3)2 Round 1: only 0 and 100 survive for-mischievous; same as before for normal Rounds 2 to k(m,n )-1: no elimination for mischievous; same as before for normal Round k(m,n): eliminate 0 for mischievous; same as before for normal Round k> k(m,n):

Strategies for normal after round k = [Lk,Hkl

Lk = 2100m+(n-m-1)Lk_

H _ 2100m+(n-m-1)Hk_ l

3 n - 2/3 k - 3

n-2/3

Ratinalizability = mischievous 100, norma1200m/(n+2m)

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Matching pennies with perfect information

2 1

HH HT TH TT

Head (-1,1) -1 ,1) (1,

1) (1,-1)

Tail

(1,-1) -1,1) (1,-1) (-1,1)

Head Tai 2 2 head head tail , (-1 ,1) (1,-1 ) (1,-1) (-1,1) 11

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A summary

players are rational and cautious, they play the dominant-strategy equilibrium whenever it exists

But, typically, it does not exist
If

rationality is common knowledge, a rationalizable strategy is played

Typically, there are too many rationalizable

strategies

Nash Equilibrium: the players correctly guess

the other players' strategies (or conjectures).

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MIT OpenCourseWare http://ocw.mit.edu

14.12 Economic Applications of Game Theory

Fall 2012 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.