Infinitely repeated games

1
Infinitely repeated games

• Game Theory, Fall 2006/07
• Steffen Hoernig, FEUNL

2
Why infinitely repeated games

• Players often do not foresee the end of their
  relationship
• Discount factor can be interpreted as probability
  of continuing
• In this interpretation every play of the game is
  finite, but length is unknown!
• Players actually play as if there was no end
• Here discount factor has usual interpretation

3
Strategies and Information structures

• Open-loop only repetitions of stage game NE
• Closed-loop any function of history!
• Can obtain very complicated strategies
• In practice only simple ones are used
• Repetitions of stage game NE are still SPE

4
Trigger or Grim Strategy

• A famous and simple strategy
• Start by cooperating and continue to cooperate
  if there has been no deviation
• After a deviation punish forever
• Called Trigger strategy because deviation
  triggers punishment
• Called Grim strategy because punishment lasts
  forever
• There will be many other SPE, but this strategy
  is really simple!

5
Infinitely repeated prisonners dilemma

• Trigger strategy
• Play n if noboday has played c before
• Play c otherwise
• Check
• deviation from cooperation
• deviation from punishment
• do both assuming that players follow strategies
  in the future (one-deviation principle)

6
IRPD (2)

• Result Trigger strategy for both players is SPE
  if ? is high enough
• gt Players fear punishment because they are
  patient!
• Any other SPE?
• Try tit-for-tat start with n, and then copy
  the other players previous action
• check deviation from copying
• if other player has played n
• if other player has played c
• gt is not SPE!

7
The Folk Theorem

• Result on the sustainability of cooperation in
  infinitely repeated games
• Many different versions by many people, and known
  before formal proof appeared
• Three concepts first
• Feasible vector of payoffs
• Reservation utility or minmax value (know!)
• Individually rational vector of payoffs

8
Concepts, and the theorem

• Feasible vector of payoffs
• Some convex combination of stage game payoffs
• Individually rational vector of payoffs
• Every payoff is above reservation utility
• Folk theorem (roughly)
• Every feasible and individually rational payoff
  vector can be supported in an SPE if ? is close
  enough to 1.

9
Intuition

• Can define all kinds of individually rational
  behavior cycles, which produce convex
  combinations of payoffs
• These can be supported by punishment
• Good lots of possibilities for cooperation!
• Bad Infinitely many SPE!
• Even worse bootstrap nature of SPE

10
How to choose between SPE?

• Focal points
• symmetric game consider symmetric SPE
• consider Pareto frontier
• Repeated PD
• symmetric Pareto optimum (3,3)
• supportable by trigger strategy!

11
Formal Folk Theorems

• Nash Threat Folk Theorem
• Friedman (1971)
• SPE based on NE as threat
• Minmax Folk Theorem
• NE based on minmax payoff as threat
• Perfect Folk Theorem
• Fudenberg und Maskin (1986)
• SPE based on minmax payoff as threat

12
Readings and Exercises

• Readings Dutta, ch. 17, 18
• Exercises Dutta, 15.13-15.19, assuming that the
  action space is 1,,6