Economics 100B Microeconomics

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Economics 100BMicroeconomics
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Announcements

• Solution to Problem Set 5 has been posted on the
  course website

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Course Outline
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Todays Plan (Chapter 15)

• Repeated Games
• Incomplete Information Games

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D. Repeated Games

• Repeated games expand the strategy sets available
  to the players
• Can reward and punish

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Ex Prisoners Dilemma

• One-time play
• the Nash equilibrium would be the expected outcome

Player B
Player A
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Ex Prisoners Dilemma

• Repeated play for T periods
• the only subgame perfect equilibrium is the Nash
  equilibrium outcome in every round

Player B
Player A
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Ex Prisoners Dilemma

• Infinite repetitions
• each player can announce a trigger strategy
• promise to play the cooperative strategy as long
  as the other player does, otherwise revert to the
  competitive strategy
• the twin trigger strategy is a subgame perfect
  equilibrium if the promise of cooperative play is
  credible

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Ex Prisoners Dilemma

• Infinite repetitions
• Assume A plays a trigger strategy
• The present value of continued cooperation is
• 2 ?2 ?22 22 ? /(1-?)
• The payoff from cheating is
• 3 ?1 ?21 3 ?/(1-?)
• Continued cooperation will be credible if
• 22 ? /(1-?) gt 3 ?/(1-?) ? gt ½

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Tacit Collusion

• Players in infinitely repeated games may be able
  to adopt subgame-perfect Nash equilibrium
  strategies that yield better outcomes than simply
  repeating a less favorable Nash equilibrium
  indefinitely

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Tacit Collusion

• With any finite number of replications, the
  Bertrand result will remain unchanged
• If the pricing game is repeated over infinitely
  many periods, twin trigger strategies become
  feasible
• each firm sets its price equal to the monopoly
  price (PM) providing the other firm did the same
  in the prior period

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Tacit Collusion

• Suppose that firm B is considering cheating
• by choosing PB lt PA PM it can obtain almost all
  of the single period monopoly profits (?M)
• by continuing to collude tacitly with A, it will
  earn its share of the profit stream
• (?M ???M ?2?M ?n?M )/2
• (?M /2)1/(1-??)

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Tacit Collusion

• Cheating will be unprofitable if
• ?M lt (?M /2)1/(1- ?)
• or if
• ? gt 1/2

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Ex Steel Producers

• Suppose only two firms produce steel bars for
  jailhouse windows
• constant AC and MC of 10
• the demand for bars is
• Q 5,000 - 100P

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Ex Steel Producers

• Bertrand competition
• each firm charges a price of 10 and 4,000 bars
  are sold
• Cartel
• the monopoly price is 30 quantity is 2,000
• Combined profits are 40,000 each period
• Any one firm gains from price cut if
• 40,000 gt 20,000 (1/1-?)

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Tacit Collusion

• More general models of tacit collusion allow for
• difficulty in monitoring other firms behavior
• other forms of punishment

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E. Incomplete Information Games

• Players do not know their opponents payoff
  functions with certainty
• equilibrium concepts involve straightforward
  generalizations of Nash and subgame-perfect
  notions

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Basics

• Each player may be one of a number of possible
  types (tA and tB) with differing payoff functions
• Each players conjectures about the opponents
  type are represented by belief functions fA(tB)

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Bayesian Equilibrium

• A strategy pair (a,b) will be a Bayesian-Nash
  equilibrium if a maximizes As expected utility
  when B plays b and vice versa

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Bayesian-Cournot Example

• Inverse demand is P 100 q1 q2
• MC1 10
• MC2 may be low MC24 or high MC216
• Suppose that firm 1 assigns equal probabilities
  0.5 to these two types for firm 2

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Bayesian-Cournot Example

• Firm 2 chooses q2 to maximize
• ?2 (P MC2)q2
• (100 MC2 q1 q2)q2
• The first-order condition for a maximum is
• q2 (100 MC2 q1)/2
• q2H (84 q1)/2
• q2L (96 q1)/2

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Bayesian-Cournot Example

• Firm 1s expected profit is
• ?1 ½(100 MC1 q1 q2H)(q1)
• ½ (100 MC1 q1 q2L)(q1)
• ?1 (90 q1 ½ q2H ½ q2L)(q1)
• The first-order condition for a maximum is
• q1 (90 ½ q2H ½ q2L)/2

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Bayesian-Cournot Example

• The Bayesian-Nash equilibrium is
• q1 30
• q2H 27
• q2L 33

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Assignment

• Finish reading Nicholson Chapter 15