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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.

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in period T (<∞), then can collusion be sustained?

  No, Price war happens in every period!

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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.

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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

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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 => ???

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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)

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2 2

(

3 2

2

  For calculation, you can use the following formula.

Formula: Arithmetic Series

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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

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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.

2 2

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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).

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Sustainable Payoffs

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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?

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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.

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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

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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.

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