14.15J | Spring 2018 | Undergraduate

# Networks

Lecture and Recitation Notes

## Recitation 10 Notes

### Topics

• Repeated games
• Infinitely repeated prisoner’s dilemma
• Finitely repeated prisoner’s dilemma

### Repeated Games (A Special Case of Dynamic Games)

In the real world, strategic interactions continue over a period of time. Dynamic games incorporate time structure into sets of strategies and sets of payoffs.

Recall the Prisoner’s Dilemma:

C  D
C (1, 1) (-1, 2)
D (2, -1) (0, 0)

Both cooperate (1,1): Cooperative outcome (against individual’s incentives?).
Both defect (0,0): Nash equilibrium outcome.

Question: When can “cooperative outcome” be sustained?
Example: Unilever and P&G price fixing in 2011 or a lysine cartel in the 1990’s.

Answer: When the game is played repeatedly, “cooperative outcomes” can be sustained as an equilibrium! This means no incentive to deviate!

### Infinitely Repeated Prisoner’s Dilemma

Folk Theorem: (C, C) → (C, C) → … can be a equilibrum outcome if players are patient enough.

 C  D  C D  C  C (1, 1) (-1, 2) → (1, 1) (-1, 2) →  … (2, -1) (0, 0) (2, -1) (0, 0)

Proof:

Consider the following strategy for player 1:

• t = 1: Play C.
• t ≥ 2: If player 2 has been playing C up to period t, play C; otherwise, play D.

Consider the same strategy for player 2.

Let’s check that this is an equilibrium:

1. At period t, suppose they have been playing (C, C).
By sticking to C, one obtains 1 + 𝛿 + 𝛿2 + … = 1/(1-𝛿 ).
By deviating to D, one obtains 2 + 0 + 0 + … = 2.
→ If 𝛿 ≥ ½, no incentive to deviate.
2. At period t, suppose either has deviated to D.
By sticking to D, one obtains 0 + 0 + … = 0.
By deviating to C, one obtains -1 + 0 + … = -1
→ No incentive to deviate.

Thus, this pair of strategies constitutes an equilibrium. Intuiton: Players can punish opponents’ deviation in future periods.

### Finitely Repeated Prisoner’s Dilemma

What if players can only play for a fixed T times? Cooperation is not sustainable.

Proof:

1. Consider similar strategies and suppose they have coopperated until T-1. At period T:
By sticking to C, one obtains 1.
By deviating to D, one obtains 2.
→ Incentive to deviate.
2. Consider similar stategies except for playing D at T. At T-1:
By playing C, one obtains 1 + 0 = 1.
By deviating to D, one obtains 2 + 0 = 2.
→ Incentive to deviate.

By induction, they have incentives to deviate unless they play (D, D) in all periods.

## Course Info

Spring 2018
##### Learning Resource Types
Problem Sets with Solutions
Exams with Solutions
Lecture Notes