Prisoners' Dilemma
Cooperate
Cooperate
1
2
Defect
Defect
2
2
3
0
0
3
1
1
There's only one Nash equilibrium of this game.
Does it feel right? Do you live in a world in which nobody ever cooperates?
pollev.com/chrismakler
Repeated Games
and Collusion
Christopher Makler
Stanford University Department of Economics
Econ 51: Lecture 14
Today's Agenda
Part 1: Discrete Strategies
Part 2: Continuous Strategies
Finitely-repeated games
Evaluating infinite payoffs
Infinitely-repeated games
Collusion in Cournot Duopoly
Big Ideas
The ability to do so relies on
the ability to credibly promise/threaten something in the future,
and a high enough value placed on future payoffs
In ongoing relationships,
you can achieve things you cannot achieve in one-shot games.
What are we modeling?
-
Players who have ongoing relationships:
- Delta Airlines and American Airlines
- Managers and employees
- Spouses, roommates, friends
- Members of political parties
- Decisions they make have an immediate impact but also an impact on the relationship.
- How much greater is GDP because of the level of social capital (trust and other investments in relationships) is higher? How much worse off are we if that trust is eroded?
Finitely Repeated Games
-
Suppose there is some stage game G, which is repeated for T periods, with the payoffs being the sum of the payoffs in each period.
-
Trivial result: it is always a SPNE if a NE is played in every period.
-
Nontrivial result: there may be SPNE in which non-NE strategy profiles are played in early periods.
-
Key insight: this requires multiple Nash equilibria, which gives the ability to credibly promise a reward in the last period
pollev.com/chrismakler
1
2
A
B
X
Y
Z
4
0
1
2
3
1
0
4
0
0
0
0
If the following game is just played once, what are the Nash Equilibria?
1
2
A
B
X
Y
Z
If the following game is just played once, what are the Nash Equilibria?
4
0
1
2
3
1
0
4
0
0
0
0
If the following game is just played once, what are the Nash Equilibria?
A
Z
B
Y
1
2
A
B
X
Y
Z
If the following game is just played once, what are the Nash Equilibria?
4
0
1
2
3
1
0
4
0
0
0
0
What are the strategy spaces of the two players if the game is played twice?
A
Z
B
Y
What to do in the first period
What to do after every possible outcome of the first period.
1
2
A
B
X
Y
Z
4
0
1
2
3
1
0
4
0
0
0
0
What to do in the first period
What to do after every possible outcome of the first period.
Player 1
Player 2
A
B
X
Y
Z
A
B
X
Y
Z
Suppose the game is played twice, and the payoffs are the sum of the payoffs in each period.
A
A
B
B
B
B
B
If (A,X) was played, play A.
If anything else was played, play B.
What is B's best response to this stragey for A?
If (A,X) was played, play Z.
If anything else was played, play Y.
Z
Y
Y
Y
Y
Y
Start with the second stage: best to play the best response
from the stage game to whatever A chooses
1
2
A
B
X
Y
Z
4
0
1
2
3
1
0
4
0
0
0
0
What to do in the first period
What to do after every possible outcome of the first period.
Player 1
Player 2
A
B
X
Y
Z
A
B
X
Y
Z
A
A
B
B
B
B
B
If (A,X) was played, play A.
If anything else was played, play B.
If (A,X) was played, play Z.
If anything else was played, play Y.
Z
Y
Y
Y
Y
Y
Given all of this, what should player 2 play in the first stage?
?
1
2
A
B
X
Y
Z
4
0
1
2
3
1
0
4
0
0
0
0
Player 1
Player 2
Play A in the first period.
If (A,X) was played, play A.
If anything else was played, play B.
If (A,X) was played, play Z.
If anything else was played, play Y.
?
Suppose player 2 plays Z.
In the first period, they get 4. Yay!
In the second period, since (A,X) was not played, player 1 plays B and player 2 plays Y,
and player 2 gets a payoff of 1.
Sum of payoffs = 4 + 1 = 5
Suppose player 2 plays X.
In the first period, they get 3. OK!
In the second period, since (A,X) was played, player 1 plays A and player 2 plays Z,
and player 2 gets a payoff of 4.
Sum of payoffs = 3 + 4 = 7
X
1
2
A
B
X
Y
Z
4
0
1
2
3
1
0
4
0
0
0
0
Player 1
Player 2
Play A in the first period.
If (A,X) was played, play A.
If anything else was played, play B.
If (A,X) was played, play Z.
If anything else was played, play Y.
X
What's going on?
There are two NE in this game; one is better for player 2 by 3 points.
Player 1 can't promise to play non-NE strategies in the second stage, but they can offer to "reward" player 2 by coordinating on the one that's better for them if player 2 plays X in the first period.
This doesn't hurt player 2 too much, so they go along with it!
Infinitely Repeated Games
Infinitely Repeated Games
- A stage game G is repeated an infinite number of times.
- Can't just sum up the payoffs (would be infinite!)
- Approach: use discounting, like we did in week 2
(rest of the slides on PowerPoint)