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 + 𝛿2 + ... = 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.