### 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 | (1, 1) | (-1, 2) | → | C | (1, 1) | (-1, 2) | → … | ||

D | (2, -1) | (0, 0) | D | (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:

- At period
*t*, suppose they have been playing (C, C).

By sticking to C, one obtains 1 + 𝛿 + 𝛿^{2}+ … = 1/(1-𝛿 ).

By deviating to D, one obtains 2 + 0 + 0 + … = 2.

→ If 𝛿 ≥ ½, no incentive to deviate. - 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:*

- 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. - 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.