A Note on Repeated Games

1
A Note on Repeated Games Trigger Strategies

• Recall calculation of utility over time
  discounting
• i gets payoff of a gt 0 today, a tomorrow, a the
  day after, etc.
• T
• EUit S(t0) at?t at0?t at1?t
  at2?t atT?T
• a ?0 ?1 ?2 ?T
• a 1/(1- ?)
• a/(1- ?) where ? ? 0,1
• ? ? 0 impatience, myopia
• ? ? 1 patience, far-sightedness

2
Utility Streams over Time
3
Trigger Strategies

• We know that NE of a PD is Defect, Defect
• Players wish to obtain the Pareto superior
  outcome Co-op, Co-op
• Repeated play can sometimes help
• Grim Trigger Strategy i says to j, Ill
  cooperate as long as you do too, but if you
  defect, then Ill defect and never again
  cooperate.

4
Trigger Strategies

• So i is offering j a stream of 3s from now
  until forever
• UjC 3 ?0 ?1 ?2 ?T
• 3/1- ?

T ?
5
Trigger Strategies

• But if j defects at some point, then i ensures
  j gets a stream of 2s after j gets her one-time
  4
• UjC 4 2? 1 ?2 ?3 ?T-1
• 4 2? /(1- ?)

T ?
6
Trigger Strategies

• To see how j responds partition js utility
  stream into 4 parts
• A js cooperative payoff up to defection
• B js big defection bonus at t
• C js stream of punishment utility from t 1
  to T
• D js stream of forgone cooperative utility
  from t-1 to T

B
D
A
C
T ?
7
Trigger Strategies

• A has no impact on js response its common to
  both histories
• Comparison is B C vs. D
• If D (B C) gt 0 ? j cooperates

B
D
A
C
T ?
8
Trigger Strategies

• If D (B C) gt 0 ? j cooperates
• i.e., j cooperates iff 3/1-d gt 4 2d/(1-d)
• cooperate iff d gt .5

B
D
A
C
T ?
9
Trigger Strategies

• Punishing j forever seems like overkill
• Could i punish j for just N periods?
• if, e.g., dj s.t. 3 3d 3d2 ? 4 2d 2d2 ,
  then 2 period of punishment keep j cooperating

B
D
A
C
T ?
10
Finite Games Backward Induction

• What if the game were finite, but when it ended
  was uncertain?
• j weight future payoffs by p, probability game
  reaches t
• If ps independent equal Pr(tT) 1 p p2
  ... pT
• So every dt in js utility stream partnered
  with a pt
• EUj 1/(1-pd)

B
A
p
t T
11
Finite Games Backward Induction

• What if there were no game after B for certain?
• Then j does not sacrifice future utility at t
  T
• To make up for the suckers payoff of 0 at t
  T, i defects at T-1...
• Knowing end of game leads to defection via
  backward induction

B
A
t T