Part 2 Strategic Dynamics

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Part 2Strategic Dynamics
Lecture 5AReputation

• How are reputations established? In this lecture
  we explore three ways, through small changes in
  the payoffs (such as limited warranties),
  affecting the information set (such as
  monitoring), and through the mutual selection of
  the equilibrium strategy (when there is more than
  one).
• Read Chapter 14 of Strategic Play.

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What is a reputation?

• Small changes in the payoffs induced by product
  guarantees and quality verification can bring
  about large changes in solution outcomes that are
  associated with reputation.
• In this case not all the solutions to the overall
  game can be found by merely piecing together the
  solutions of the kernel games.
• Dynamic strategies that preserve long term
  incentives and cooperation with appropriate
  rewards and penalties are, under the right
  circumstances, more lucrative than the outcomes
  realized from players choosing say, dominant
  strategies each period.

3
Quality control

• Manufacturers do not consistently produce
  flawless products despite legions of consultants
  who have advised them against this policy.
• Retailers help guard against flawed products by
  returning some of the defective items sent, and
  lending their brand to the ones they retail.
• Consumers cannot judge product quality as well as
  retailers and producers, since each one
  experiences only a tiny fraction of the end
  product.
• What is an acceptable defect rate, how often
  should retailers return defective items, and what
  are the implications for consumer demand?

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Total quality management
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If the consumer always buys . .

• To solve this game we first remark that the
  consumer plays a mixed strategy in equilibrium.
  This claim can be established by contradicting
  the alternative hypothesis that she plays a pure
  strategy.
• Suppose she always buys the product.
• Then the retailer would never return a defective
  one, and the producer would specialize in
  producing defective products.
• Thus the consumer only purchases defective
  products, and this is not a best response to the
  strategies of the manufacturer and the retailer.

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If the consumer never buys . . .

• But if the customer never buys the product, the
  retailer would always return defective ones.
• In this case the manufacturer specializes in
  produced flawless products.
• It now follows that the strategy of not buying is
  not a best response
• Therefore the consumer follows a mixed strategy.

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Defining the probabilities in the TQM problem

• Let q denote the probability that the retailer
  offers a defective product item sale.
• Let r denote the probability the shopper buys the
  item.
• Let p be the probability a producer produces a
  flawless item.
• Both probabilities are strictly positive.
  receiving both kinds of products is strictly
  positive.
• If the shopper mixes between buying and not
  buying the product, then she must be indifferent
  between making either choice.

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A schematic
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Solving for r, the probability of buying

• If 0 lt q lt 1, then the retailer is indifferent
  between offering a defective product and
  returning it.
• In that case
• 3r - 2(1 - r) -1
• ? 3r 2 2r -1
• ? 5r 1
• ? r 0.2

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How to solve for p and q

• Once we substitute for r 0.2 in the shoppers
  decision, we are left with the diagram
• q is chosen so that the producer is indifferent
  between production methods
• p is chosen so that the shopper is indifferent
  between buying and not buying.

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Solving q,the probability of offering the product

• The producer will only mix between defective
  and flawless items if the benefit from both are
  equated
• 6r (1 - r)q - 3(1 - q) 3r (1- r)
• ? 2q 3 3q 1.4
• ? 5q 4.4
• ? q 0.88

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Solving for p, the probability of producing a
flawless product

• Investigating the cases above shows that in a
  mixed strategy equilibrium r 0.2 and q 0.88.
• Since the shopper is indifferent between buying
  the item versus leaving it on the shelf, there
  are no expected benefits of acquiring the item
• 9p - 10(1 - p)q 0 ? (9 10q)p 10q
• ? p 44/89

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Offering a partial refund
We now modify the game slightly. If the customer
buys a defective product, she receives partial
compensation.
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A different outcome

• In this case the manufacturer has a weakly
  dominant strategy of specializing in the
  production of flawless goods.
• Recognizing this, the shopper picks a pure
  strategy of buying.
• Realizing that the shopper will buy everything
  she is offered, the retailer never returns its
  merchandise to the manufacturer (and indeed there
  is never any reason too).

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Medical malpractice

• One problem health insurance providers face is
  fraudulent behavior by doctors who prescribe
  treatment for healthy clients.
• Consider the following extensive form game

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Strategic form of medical malpractice

• In the strategic form of the game, we see that
  ignore is a dominated strategy.
• Furthermore the best replies indicate that the
  game has a unique mixed strategy Nash equilibrium.

• Solving, the patient takes the treatment with
  probability 5/6, while the doctor prescribes
  treatment 1/4 of the time to healthy patients.
• Thus healthy patients receive the treatment with
  probability 5/24, about 20 of the time.

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Diagnostic test

• Is it profitable to administer a test that
  verifies whether someone is ill or not?

• Solving the perfect information game, the doctor
  will only prescribe treatment to sick patients,
  who will always take it.

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Gains from testing

• The solution to the perfect information game
  yields an expected benefit of 84 to the patient
  and 6 to the doctor.
• In the malpractice game, the expected benefit to
  the patient is
• (382 50 1584 544)/64 74
• while the expected benefit to the doctor is
• (32 - 38 156 514)/ 64 5.3
• Together the patient and doctor are willing to
  pay up to 10.6 to administer the diagnostic test.

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Light rail

• Alstom, a French company, and Bombardier, a
  Canadian company based in Quebec, are the worlds
  largest producers of light rail systems.
• They frequently compete against each other for
  contracts from local governments and airport
  authorities.
• This industry is characterized by flurries of
  contracts interspersed with relatively lean
  periods.
• For this reason we treat each flurry as a known
  number of rounds that occur independently of the
  last flurry.

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Bidding for light rail contracts

• The company charging the lowest price wins.
• If both companies tender the same price, they
  have the same probability of winning the
  contract.
• The payoff matrix illustrates such a
  configuration.

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The last round in a finite horizon game

• Consider the last round in a typical flurry.
• The dominant strategy for each producer is to cut
  is price.
• This is an example of the prisoners dilemma.

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The reduced subgame starting at second last
round

• Folding back, the strategic form of the reduced
  game starting at the penultimate round is
  depicted.
• It is obtained by adding (2,2), the solution
  payoffs for the final auction, to each cell.
• The dominant strategy of cutting price is not
  affected by this additive transformation.

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The reduced game at the beginning of the first
round

• Using an induction argument we can prove that in
  the first round, the expected revenue each firm
  will get from the remaining N 1 tenders is 2(N
  1).
• Again the dominance principle applies, and both
  firms cut price in their first tender.

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Solution

• The preceding discussion proves the unique
  solution is to always cut the price in this
  repeated game.
• The reason we obtain a tight characterization of
  the solution to the repeated game is that the
  solution to the kernel game is unique.
• Indeed if a game has a unique solution, then
  repeating the game a finite number of times will
  simply replicate the solution to the original
  kernel game.

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Repeated games

• Multiplicity is the existence of multiple
  solutions within a game (such as a signed
  contract that still leaves the bargaining parties
  discretion about its implementation)
• It sometimes arises when there are ongoing
  benefits from continuing a relationship and/or
  potential for repeated trade.
• If the solutions to all the kernels forming a
  finite stage game are unique, then the unique
  solution to the stage game is to play those
  kernel solutions.
• In these cases there is no scope for either
  leadership or reputation.

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An Infinite Horizon Extension

• But what if this game did not end at a fixed
  point in time?
• Consider the following implicit agreement
  between the two firms
• If neither of us cheat on each other from now on
  by cutting price, then we will continue to hold
  firm and collect (3,3) each period.
• If either of us ever cheat even once, then from
  then on we will always cut price.
• This is called a trigger strategy.

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When are trigger strategies self enforcing?

• The benefit from following this strategy is the
  discounted sum of receiving 3 per period.
• The discounted sum of breaking the agreement is
  receiving 4 in the first period and 2 from the
  next period onwards.
• The net benefit from breaking the agreement is
  therefore the gross of 1 received now, less the
  cost of 1 unit paid each period from next period
  onwards.
• If the interest rate is r, then the net benefit
  is
• 1 1/r .
• Unless the interest rate exceeds 100 percent the
  trigger strategy is self enforcing in this case.

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Lecture summary

• Small changes in the payoffs induced by product
  guarantees and quality verification can bring
  about large changes in solution outcomes that are
  associated with reputation.
• Repetition may also play a role. When there are
  multiple solutions to the kernel game stage, and
  in infinite horizon repeated games, not all the
  solutions to the overall game can be found by
  merely piecing together the solutions of the
  kernel games.
• Dynamic strategies that preserve long term
  incentives and cooperation with appropriate
  rewards and penalties are, under the right
  circumstances, more lucrative than the outcomes
  realized from players choosing say, dominant
  strategies each period.