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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). A pure strategy Nash equilibrium can be interpreted as a self enforcing agreement. When there is more than one, a natural question to ask is which pure strategy equilibrium, if any, will be played. Week 5 Coordination and Reputation

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How many Nash equilibriums are there? A Nash equilibrium solution to a game can be found by writing down its strategic form. We have already noted that every game has at least one Nash equilibrium. However some games have more than one equilibrium.

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The threat of bankruptcy We consider an industry with weak board of directors, an organized workforce and an entrenched management. Workers and management simultaneously make demands on the firms resources. If the sum of their demands is less than or equal to the total resources of the firm, shareholders receive the residual. If the sum exceeds the firm’s total resources, then the firm is bankrupted by industrial action.

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Strategic form of bargaining game To achieve a bigger share of the gains from trade, both sides court disastrous consequences. This is sometimes called a game of chicken, or attrition.

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Extensive form of bargaining game

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Best responses illustrated

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Multiple pure strategy Nash equilibrium In this game, there are three pairs of mutual best responses. The parties coordinate on an allocation of the pie without excess demands. Shareholders get nothing. But any of the three allocations is an equilibrium. If labor and management do not coordinate on one of the equilibrium, the firm will bankrupt or shareholders will receive a dividend.

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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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TQM in strategic form There are two strategies for each player. Having derived the strategic form of the game, we can easily locate the pure strategy Nash equilibriums. There is a unique pure strategy Nash equilibrium, in which the producer only manufactures flawless products, the retailer only sells flawless products and the customer always buys the product.

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Why is the pure strategy Nash equilibrium unconvincing? But is this Nash equilibrium convincing? Sure you can’t eliminate any dominated strategies. If, however, the producer does manufacture a defective item, the retailer, but not the consumer will know, and makes more by offering the item for sale. Can the retailer convince consumers that they really will return defective products?

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Is there a mixed strategy equilibrium? Let q denote the probability that the retailer offers a defective product item sale. Let r denote the probability the customer buys the item. Let p be the probability of producing a flawless item.

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Solving for r, the probability of buying If 0 < q < 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 shopper’s decision, we are left with the diagram: 1.q is chosen so that the producer is indifferent between production methods; 2.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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Light rail Alstom, a French company, and Bombardier, a Canadian company based in Quebec, are the world’s 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 prisoner’s 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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Multiple equilibriums There is no role for coordination and leadership in situations where the solutions strategies for each player are uniquely defined. Thus opportunities for coordination and leadership arise when there are several solutions to a game, which we may describe as self enforcing contracts. In this case not all the solutions to the overall game can be found by merely piecing together the solutions of the kernel games.

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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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What is a 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.

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Summary There is no role for coordination in situations where the solutions strategies for each player are uniquely defined. However 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. Opportunities for coordination also arise when there are several solutions to a game, which we may describe as self enforcing contracts.

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EC941 - Game Theory Lecture 7 Prof. Francesco Squintani

EC941 - Game Theory Lecture 7 Prof. Francesco Squintani

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