Infinitely Repeated Games 14.12 Game Theory Muhamet Yildiz 1 3. - - PDF document

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Infinitely Repeated Games 14.12 Game Theory Muhamet Yildiz 1 3. - - PDF document

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Lecture 13 Infinitely Repeated Games 14.12 Game Theory Muhamet Yildiz 1 3. Examples dVDC Road Map 1. Definitions 2. Single-deviation principle 2 Infinitely repeated Games with observable actions e dVDC T= {O,1,2, .. . ,t, ... } G =

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1

Lecture 13 Infinitely Repeated Games

14.12 Game Theory Muhamet Yildiz

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Road Map

  • 1. Definitions
  • 2. Single-deviation principle
  • 3. Examples dVDC

2

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Infinitely repeated Games with

  • bservable actions
  • T= {O,1,2, .. .

,t, ... }

  • G = "stage game" = a finite game
  • At each t in T, G is played, and players

remember which actions taken before t;

  • Payoffs = Discounted sum of

payoffs in th stage game.

  • Call this game G(T).

e dVDC

3

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C=> 5+5dVd C=> 6+dVD C <=> d>1/5. 4

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Definitions

The Present Value of a given payoff stream n = ( no, n

1 , ••• ,nt,

... ) is PV(n;8) = no + t 8n l + ... + 8 nt +

...

The Average Value of a given payoff stream n is (1-8)PV(n;8) = (1-8)(no+ 8n l + ... + t 8 nt+···) The Present Value of a given payoff stream n at t is

PVl n;8) = nt + 8nt+

1 + ... + 8Snt+s +

.. .

A history is a sequence of past observed plays e.g. (C,D), (C,C), (D,D), (D,D) (C,C) dVDC

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Recall: Single-Deviation Principle

  • s = (S,h, ... ,sn) is a SPE
  • ~

it passes the following test

  • for each information set, where a player i moves,
  • fix the other players' strategies as in s,
  • fix the moves of

i at other information sets as in s;

  • then i cannot improve her conditional payoff at the

information set by deviating from Si at the information set only. dVDC

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Single-Deviation Principle: Reduced Game

  • s = (s

I

,S2'· .. ,S

n)' date t, and history h fixed

  • Reduced Game: For each terminal node a of

the stage game at t,

  • assume that s is played from t+

1 on given (h,a)

  • write PV(h,a,s,t+

I) for present value at t+ 1

  • Define utility of

each player i at the terminal node a as

ula)+ 8 PV(h,a,s,t+l)

  • Single-Deviation Principle: s is SPE ¢:> for every

hand t, s gives a SPE in the reduced game dVDC

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C=> 5+5dVd => 6+dVD <=> d>1/5. C C 8

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Is (Tit-for-tat,Tit-for-tat) a SPE?

  • Tit-for-Tat: Start with C; thereafter, play what the
  • ther player played in the previous round.
  • No!
  • Consider (C,C) at t-l and Player 1.
  • C => 5/(1-8)
  • D => 6 /(1-82)
  • No Deviation ~

8;::: 115.

  • Consider (C,D) at t- and Player 1.
  • C => 5/(1-8)
  • D => 6 /(1-82)
  • No Deviation ~

8 :::: 115.

  • Not SPE if

8 i- 115. dVDC

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Modified Tit-for-Tat

Start with C; if any player plays D when the previous play is (C,C), play D in the next period, then switch back to C. dVDC

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Infinite-period entry deterrence

Enter

2 A1x .

.------.----+

~

(1,1)

Strategy of Entrant:

x

Enter iff

Fight

Accomodated before. Strategy of Incumbent:

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

Accommodate iff accomodated before. dVDC

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Reduced Games

Accommodated before: 1 1

;--__

E_nt_e_r

_---;-2

1

__

A_cc_, __

.... 1 + 8/(1-8) 1 + 8/(1-8) X Fight

  • 1 + 8/(1-8)

0 + 8/(1-8)

  • 1 + 8/(1-8)

2 + 8/(1-8) Not Accommodated before: 1 Enter 2 Acc,

.-----------.-------+:: 1 + 8/(1-8)

1 + 8/( 1-8)

x

Fight

U'

~,

0 + 0

  • 1 + 0

2 + 28/(1-8)

  • 1 + 28/(1-8) dVDC

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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.