{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Lecture16-new

Lecture16-new - 6.254 Game Theory with Engineering...

Info icon This preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon
6.254 : Game Theory with Engineering Applications Lecture 16: Repeated Games – II Asu Ozdaglar MIT April 13, 2010 1
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Game Theory: Lecture 16 Introduction Outline Repeated Games – perfect monitoring Folk Theorems Repeated Games – imperfect monitoring Price-trigger strategies Reference: Fudenberg and Tirole, Section 5.1 and 5.5. 2
Image of page 2
Game Theory: Lecture 16 Repeated Games Repeated Games By repeated games , we refer to a situation in which the same stage game (strategic form game) is played at each date for some duration of T periods. More formally, imagine that I players playing a strategic form game G = hI , ( A i ) i ∈I , ( g i ) i ∈I i for T periods (infinitely-repeated game if T = ). At each period, the outcomes of all past periods are observed by all players perfect monitoring The notation a = { a t } t = 0 denotes the (infinite) sequence of action profiles. The payoff to player i for the entire repeated game is then u i ( a ) = ( 1 - δ ) t = 0 δ t g i ( a t i , a t - i ) where δ [ 0, 1 ) . We have seen that grim trigger strategies can sustain “cooperation” in infinitely repeated games. This motivated the question what payoffs are achievable in equilibrium when players are sufficiently patient (i.e., when δ 1). 3
Image of page 3

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Game Theory: Lecture 16 Folk Theorems Folk Theorems for Infinitely Repeated Games We started last time talking about folk theorems which study equilibrium payoffs that can be obtained in infinitely repeated games. Recall that we use v i to denote the minmax payoff of player i , i.e., v i = min α - i max α i g i ( α i , α - i ) . The strategy profile m i - i denotes the minmax strategy of opponents of player i and m i i denotes the best response of player i to m i - i , i.e., g i ( m i i , m i - i ) = v i . The set V * R I denotes the set of feasible and strictly individually rational payoffs. We have seen that equilibrium payoffs of repeated games are bounded from below by minmax payoffs. 4
Image of page 4
Game Theory: Lecture 16 Folk Theorems Folk Theorems Our first theorem is the Nash folk theorem which shows that any feasible and strictly individually rational payoff vector can be achieved if the player is sufficiently patient. Theorem (Nash Folk Theorem) If ( v 1 , . . . , v I ) is feasible and strictly individually rational, then there exists some δ < 1 such that for all δ > δ , there is a Nash equilibrium of G ( δ ) with payoffs ( v 1 , · · · , v I ) . The Nash folk theorem states that essentially any payoff can be obtained as a Nash Equilibrium when players are patient enough. However, the corresponding strategies involve this non-forgiving punishments, which may be very costly for the punisher to carry out (i.e., they represent non-credible threats).
Image of page 5

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 6
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern