SLIDE 1
Road Map
- 1. Folk Theorem
- 2. Applications (Problems)
Lecture 14 Infinitely Repeated Games II 14.12 Game Theory Muhamet - - PDF document
Lecture 14 Infinitely Repeated Games II 14.12 Game Theory Muhamet - - PDF document
Lecture 14 Infinitely Repeated Games II 14.12 Game Theory Muhamet Yildiz 1 Road Map 1. Folk Theorem 2. Applications (Problems) 2 ~. Folk Theorem Definition: v = (v 1 ,v , . .. ,vn) is feasible iff v is a convex 2 combination of pure-strategy
SLIDE 2
SLIDE 3
Folk Theorem
Definition: v = (v1,v , . ..
2
,vn) is feasible iff v is a convex
combination of pure-strategy payoff-vectors: 1 2 m v = p1 u(a ) + p2u(a ) + ... + Pnu(a ), where P1 + P2 + ... + Pm = 1, and u(ai) is the payoff vector at strategy profile ai of the stage game. Theorem: Let x = (X ,X , ...
) 1 2
,xn be s feasible payoff vector, and e = (e ,e , . . .
1 2
,en) be a payoff vector at some equilibrium of the stage game such that Xi >
ei for each i. Then, there exist ~ < 1 and a strategy
profile s such that s yields x as the expected average-payoff vector and is a SPE whenever 8 >
~.
3
SLIDE 4
Folk Theorem in PO
C D
- A SPE with PV
C 5,5 0,6
(1.1,1.1)?
D 6,0 1,1
- With PV (1 .1
,5)?
- With PV (6,0)?
- With PV (5.9,0.1)?
4
SLIDE 5
Proof for a special case
- Assume x = u(a*) = (u1(a*), ...
, un(a*» for some a*.
- s*: Every player i plays ai* until somebody
deviates and plays ei thereafter.
- Average value of
i from s* is Xi = ul
a
*).
- s* is a SPE ¢:> {) > 8 where
5
SLIDE 6
Applications/Problems
6
SLIDE 7
2010 Midterm 2, P2
2.
- 1
- l
- t.
2
- 1
3
- ;L
:;
- i
D
7
SLIDE 8
Range of 8 for SPE
- Alice Hires and Bob and Colin both Work
until any of the workers Shirk; Alice Hires and Bob and Colin both Shirk thereafter.
- Alice Always Hires. Both workers Work at
t = 0. At any t > 0, each worker Works if the previous play is (Hire, Work, Work) or (Hire, Shirk, Shirk); each worker Shirks
- therwise.
8
SLIDE 9
2007 Midterm 2, P3
- Stage Game: Linear Bertrand Duopoly (c=O; Q=I-p)
- s*: They both charge 112 until somebody deviates;
they both charge 0 thereafter.
- s**: n + I modes: Collusion, WI, W2, ... , Wn. Game
starts at Collusion. Both charge 112 in the Collusion mode and p*<112 in WI,
... , Wn. Without deviation,
Collusion leads to Collusion, WI leads to W2, ... , Wn-I leads to Wn, and Wn leads to Collusion. Any deviation leads to WI.
9
SLIDE 10
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.