1. Game Theory II – Repeated Games
Managerial Economics
Kyle Anderson

2. Repeated Games
Does the outcome of a game change if the game is played multiple times?
Repeated games:
Multiple players
Simultaneous moves
Perfect information
Game to be repeated:
Finite number of times
Infinite number of times
Kyle J. Anderson

3. Repeated Pricing Game 0, 0 40, -30 -30, 40 10, 10 Perry low price
Perry high price
Anderson low price
0, 0
40, -30
Anderson high price
-30, 40
10, 10
What if this game were to be played 5 times?

4. Repeated Prisoners’ Dilemma
Perry low price
Perry high price
Anderson low price
0, 0
40, -30
Anderson high price
-30, 40
10, 10
The NE in a one-time game is low-low (Prisoners’ Dilemma).
Repeat game five times.
Backward induction: what will happen in the fifth game?
Work backwards
In any finitely repeated game, the NE of each playing is the same as the NE of the one time game.

5. Infinitely Repeated Prisoners’ Dilemma
If a Prisoners’ Dilemma game is repeated an infinite number of times, is there another Nash Equilibrium?
No, the Nash Eq. of the one time game is just repeated forever.
Yes, firms should always cooperate.
Depends on the Federal Reserve.

6. Infinitely Repeated Prisoners’ Dilemma
Perry low price
Perry high price
Anderson low price
0, 0
40, -30
Anderson high price
-30, 40
10, 10
Trigger Strategy:
Anderson will cooperate (high) in the first period.
In the second period, cooperate if Perry cooperated previously.
If Perry plays low, then play low forever after.
Should Perry play the same strategy?

7. Infinitely Repeated Prisoners’ Dilemma
Perry low price
Perry high price
A low price
0, 0
40, -30
A high price
-30, 40
10, 10
Trigger Strategy:
Anderson will cooperate (high) in the first period.
In the second period, cooperate if Perry cooperated previously.
If Perry plays low, then play low forever after.
Perry’s Payouts:
Cheat (low) first period: …
Trigger Strategy: …
Is … > 40

8. Infinitely Repeated Prisoners’ Dilemma
Perry low price
Perry high price
A low price
0, 0
40, -30
A high price
-30, 40
10, 10
Trigger Strategy:
Perry’s Payouts:
(40-10)/(10-0)<1/i
3 < 1/i
If the interest rate is less than 33.3%, the Trigger Strategy is a Nash Equilibrium.

9. Franchising: an application
Business Traveler trying to find a hotel:
Hotel – decent quality
Hotel – low quality
Stay
5, 5
-10, 10
Don’t stay
0, -5
0, 0
Cheat: 10, Coop: 5, NE: 0
If the Interest rate is less than 100% then it is better to provide decent quality.
Kyle J. Anderson

10. Can Cooperative Pricing Occur:
Standardized pricing
High market concentration
Reaction speed and detection lags
Information about transactions
Number of buyers
Volatility of demand
High firm symmetries
Price sensitivity of buyers
Kyle J. Anderson

11. Big Finish Finitely repeated games
Infinitely repeated games have other NE if the interest rate is sufficiently low.
In practice, cooperation is harder due to a number of reasons.
Kyle J. Anderson