### ECO290E: Game Theory

### Review of Lecture 9 ^{}

Committing to some strategy by reducing the set of available actions may benefit.

A player has larger bargaining power if she

makes a take-it-or-leave-it offer in the final period

has a larger discount factor.

⇒_{ }These advantages shrink as T becomes larger.

If the stage game has a unique NE, then for any T, the finitely repeated game has a unique SPNE: the NE is played in every stage irrespective of the history of play.

### Repeated Games

### A repeated game is played over time, t = 1,2,…,T

*where T can be a finite number or can be infinity. *

### The same static game, called a “stage game,” is played

### in each period.

### The players observe the history of play, i.e., the

### sequence of action profiles from the first period

### through the previous period.

### SPNE in Repeated Games

### After all history of play, each player cannot

### become better off by unilaterally changing her

### strategy.

### This is equivalent to…

### After all history of play and for every player,

### immediate gains by a deviation must be smaller

### than future losses triggered by deviation.

### One-shot un-improvability.

### Finitely Repeated Bertrand Games

### If the Bertrand games are played only finitely, i.e., ends

*in period T (<* ^{∞} *), then can collusion be sustained? *

No, Price war happens in every period!

### There is unique Nash equilibrium in the Bertrand

### Game, price = marginal cost.

### Choosing price = marginal cost in every period is a

### unique subgame perfect Nash equilibrium.

### Remarks

If the stage game has a unique NE, then for any T, the finitely repeated game has a unique SPNE: the NE of the stage game is played in every stage irrespective of the history of play.

If the stage game has multiple NE, then for any T, any

sequence of those equilibrium profiles can be supported
as the outcome of a SPNE. Moreover, non-NE strategy
*profiles might be sustained as a SPNE. *

### Impossibility of Collusion ^{}

### By backward induction, firms end up getting into price

### wars in every period.

In the last period (t = T), no firm has an incentive to collude since there is no future play. The only possible outcome is a price war irrespective of the past history of the play.

In the second to the last period (t = T-1), no firm has an

incentive to collude since the future play will be a price war no matter how each firm plays in period T-1.

This logic continues…, then

⇒_{ }

### Collusion can NEVER be sustained if the repetition is

### finite.

### Bertrand Puzzle: Review

Firms receive 0 profit under the (one-shot) Bertrand competition.

But the actual firms engaging a price competition, e.g., gas stations locating next to each other, seem to earn positive profits.

⇒_{ } How can/do firms achieve positive profits?

Product differentiation

Capacity constraints

Dynamic interaction (Collusion/Cartel)

Finitely many interactions => Price war

Infinitely repeated interactions => ???

### Infinitely Repeated Bertrand Games ^{}

The following “trigger” strategy achieves collusion if the
*discount factor is high enough, δ ≥ 1/2. *

Each firm charges a monopoly price until someone undercuts the price.

If such deviation happens, the firms start taking a stage game Nash equilibrium, i.e., set a price equal to the marginal cost c. (get into a price war)

**t ** **t+1 ** **t+2 ** _{… }

_{… }

**collusion ** _{π } _{π } _{π } _{… }

_{π }

_{π }

_{π }

_{… }

### How to Derive the Condition

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3 2

2

For calculation, you can use the following formula.

Formula: Arithmetic Series

### Repeated Prisoner’s Dilemma ^{}

If repeated only finitely many times, then

Players mutually defect for every period.

When can cooperation be sustained as an SPNE?

** Player 2 **
**Player 1 **

** Cooperation ** ** Defection **

**Cooperation ** ** 2 **
**2 **

** 3 **
**-1 **

**Defection ** ** -1 ** ** 0 **

### Repeated Prisoner’s Dilemma ^{}

Consider the following trigger strategy:

Play (C, C) at the beginning.

Continue to play (C, C) unless someone has ever deviated.

Play (D, D) forever after somebody has deviated.

To sustain cooperation, future loss triggered by defection
must outweighs its immediate gain.^{
}

### 3

### /

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### Folk Theorem (Nash Reversion) ^{}

When discount factor is large enough, any payoff

combinations which Pareto dominate the stage game Nash equilibrium can be achieved by some subgame perfect Nash equilibrium.

1._{ } Set a (dynamic) strategy profile yielding some target payoffs
(which Pareto dominate the payoff in NE).

2._{ } Each player follows this dynamic strategy as long as all other
players follow their target strategies.

3._{ } If someone deviates, then all players switch to play the Nash
equilibrium (trigger strategy).

### Folk Theorem: Illustration ^{}

## (2, 2) ^{
}

## (0, 0) ^{
}

## (-1, 3) ^{
}

## (3, -1) ^{
}

## Sustainable Payoffs ^{
}

### Long-term Relationship

### Firms need some devises to prevent them

### from deviation, i.e., cutting its own price.

### Contracts (explicit cartels): If deviation

### happens, a deviator must be punished by a

### court or a third party.

⇒_{ }

### Illegal by antitrust law.

### Long-term relationship (implicit cartels): Firms

### collude until someone deviates. After deviation,

### firms engage in a price war.

### Remarks

### Long-term relationship has an advantage over

### contracts when

### Deviation is difficult to be detected by a court.

### The definition of “cooperation” is vague.

### Verifying a cheat to the court is costly .

### There is NO court, e.g., medieval history, developing

### countries, global warming.

⇒_{ }

### The best way to study the interaction

### between immediate gains and long-term

### incentives is to examine a repeated game .

### Finite Repetition with Multiple NE ^{}

Suppose the following game will be played TWICE.

Find all pure strategy Nash equilibria.

Can (A, X) be sustained in the first period?

**1 ╲ 2
** **X
** **Y
** **Z
**

### A ^{
} 7, 7 ^{
} 4, 8 ^{
} 1, 9 ^{
}

### B ^{
} 8, 4 ^{
} 5, 5 ^{
} 0, 0 ^{
}

### There Are 3 Nash Equilibria ^{}

Symmetric eqm. (B, Y); Asymmetric ones (A, Z), (C, X)

In the second period, one of these equilibria must be played.

Which equilibrium is played can be conditional on the past play.

**1 ╲ 2
** **X
** **Y
** **Z
**

### A ^{
} 7, 7 ^{
} 4, 8 ^{
} 1, 9 ^{
}

### B ^{
} 8, 4 ^{
} 5, 5 ^{
} 0, 0 ^{
}

### C ^{
} 9, 1 ^{
} 0, 0 ^{
} 0, 0 ^{
}

### Using Different NE as a Punishment ^{}

In the first period, players play (A, X).

In the second period,

If P1 alone deviates, play (A, Z).

If P2 alone deviates, play (C, X).

Otherwise, play (B, Y).

Check if it constitutes an SPNE.

If a player follows the above strategy: 7 + 5 = 12

If she deviates (and play best reply): 9 + 1 = 10

### Further Exercises ^{}

Examine a repeated prisoner’s dilemma whose stage game payoffs are not symmetric between players, and derive the condition to sustain cooperation.

Suppose that the chicken game is played finitely many times. Then, can (Chicken, Chicken) be sustained?

Consider whether a payoff combination that does not Pareto dominate the Nash equilibrium payoff can be sustained or not.

If you know the answer, check the (general) folk theorem.