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This lecture explores the establishment of reputations and the role of leaders in directing self-enforcing agreements. Topics covered include multiple pure strategy Nash equilibriums, the threat of bankruptcy in industries, quality control, and offering partial refunds to consumers.
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Lecture 5Reputation and Leadership 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). Then we investigate the role of leaders in directing organizations into self enforcing contracts and equilibriums that support their goals. A pure strategy Nash equilibrium can be interpreted as a self enforcing agreement. When there is more than one equilibrium, a natural question to ask is which equilibrium, if any, will be played. This lecture develops the notion that leaders facilitate coordination, by recognizing, and implementing self enforcing agreements.
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.
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.
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.
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.
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?
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.
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.
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
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: • 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.
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
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
Why is the pure strategyNash 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?
Offering a partial refund We now modify the game slightly. If the customer buys a defective product, she receives partial compensation.
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).
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Leadership described • This previous slides motivate our study of games where there are multiple equilibrium, and indicate where there might be a role for leadership. • A good leader is someone who persuades other people to follow his or her suggestions when they would have behaved differently otherwise, even the the advice giver has no power of coercion. A poor leader is ignored. • Managers are routinely called upon to be leaders, directing activity without having the power to enforce their suggestions.
Competition in the Performing Arts • Suppose there are two firms in the classical performing arts, ballet and symphony, competing for a limited demand pool, as is the case in Pittsburgh. • Each firm simultaneously advertises its upcoming performance. There are two types of advertising: • Generic advertising increases the sales and net revenue of both firms. • Differential advertising increases the sales and net revenue of the differential advertiser at the expense of the other firm. Furthermore the net gain to the former is less than the net loss to the latter.
Multiplicity and mixing There are two pure strategy Nash equilibrium, yielding (6,1) and (1,6) respectively. There is also a mixed strategy Nash equilibrium. In fact one can prove that if a game has more than one pure strategy Nash equilibrium, then it also has at least one mixed strategy Nash equilibrium. We shall call the mixed strategy Nash equilibrium an uncoordinated solution.
Solving the mixed strategy solution • Let p denote the probability that Symphony does “generic” rather than “differential” advertising. • Ballet is indifferent between the choices if: 4p + (1 – p) = 6p => p = 1/3 • Similarly let q denote the probability that Ballet does “generic” rather than “differential” advertising. • By symmetry, setting q = 1/3 makes Symphony indifferent between the two choices. • Therefore the mixed strategy equilibrium solution is to set p = q = 1/3 , yielding an expected payoff to both firms of: 4*1/9 + 1*2/9 + 6*2/9 +0*2/9 = 2
Achieving coordinationwith repetition • Now suppose there are N performances in the season. Some coordinated solution outcomes might be: (6,1), (6, 1) . . . for the N rounds. (1,6), (1,6) . . . for the N rounds. (6,1),(1,6) . . . for the N rounds. (4,4), . . .,(4,4),(1,6),(6,1) N - 2 rounds of (4,4) followed by (1,6),(6,1).
Feasible average payoffs This area shows what average payoffs in a finitely repeated game are feasible given the firms’ strategy spaces. (1,6) Ballet average payoffs (4,4) (6,1) (0, 0) Symphony average payoffs
Individual rationality Ballet average payoffs (1,6) The area, bounded below by the dotted lines, gives each player an average payoff of at least 1. It is guaranteed by individual rationality. Individual rationality coordinate pair (1,1) (6,1) (1,1) (0, 0) Symphony average payoffs
Average payoffs in equilibrium Ballet average payoffs The theorem in the next slide states that every pair in the enclosed area represents average payoffs obtained in a solution to the finitely repeated game. (1,6) (4,4) (6,1) (1,1) (0, 0) Symphony average payoffs
Folk theorem Let w1 be the worst payoff that Player 1 receives in a solution to the one period kernel game, let w2 be the worst payoff that Player 2 receives in a solution to the one period kernel game, and define w = (w1, w2) In our example w = (1,1) Folk theorem for two players: Any point in the feasible set that has payoffs of at least w can be attained as an average payoff to the solution of a repeated game with a finite number of rounds.
How can Symphony and Ballet both earn more than 7 in a 3 period game? The outcome {(4,4), (1,6), (6,1)} comes from playing: (generic1, differential1) (generic2, differential2) (differential3, generic3). Is this history the outcome of a solution strategy profile to the 3 period repeated game?
Strategy for Symphony Round 1: generic1 Round 2: (…, generic1) generic2 otherwise differential2 Round 3: (differential1, …) generic3 otherwise differential3 Symphony should play generic in the first round. If Ballet plays generic in the first round, Symphony should play generic in the second round too. If Ballet plays differential in the first round, Symphony should play differential in the second. Symphony should play differential in the final round, unless it played differential in the first round.
Strategy for Ballet Round 1: generic1 Round 2: (…, differential1) generic2 otherwise differential2 Round 3: (differential1, …) differential3 otherwise generic3 Ballet should play generic in the first round. In that case, it should play differential in the second. However if it played differential in the first round, then it should play generic in the second round. If Symphony played differential in the first round, Ballet should be differential in the final round, but generic otherwise.
Verifying this strategy profile is a solution Note that the last two periods of play, taken by themselves, are solutions to the kernel game, and therefore strategic form solutions for all sub-games starting in period 2. To check whether generic is a best response for Symphony given that Ballet chooses according to its prescribed strategy in the first period, we compare the value of deviating from it versus following it.
Checking for deviations by Symphony in the first round Since 11 > 8 Symphony does not profit from deviating in the first period. A similar result holds for Ballet. Therefore, by symmetry, the strategy profile is a coordinated solution. Compare (generic1, generic1) 4 (generic2, differential2) 1 (differential3, generic3) 6 --- 11 with (differential1, generic1) 6 (generic2, differential2) 1 (generic3, differential3) 1 --- 8
Cleanup crew To illustrate some other solutions, we now consider an alternative interpretation of the game. Suppose Tom and Jerry are cleaners hired to take away trash, vacuum the floors and clean bathrooms each night after work . The players choose between shirking on the job or working diligently.
Overtime possibilities If they both work diligently, cleaning is completed without any overtime. If one shirks and the other is diligent, then the diligent worker is paid an extra 6 hours of overtime, while the shirker is paid just 1 extra hour. If they both shirk, they each collect 4 hours of overtime pay.
Net benefits to the crew We assume that each player’s benefits are proportional to the amount of overtime received.
Will the crew coordinate? The goal of the manager is to minimize the cost of cleaning the workplace. If he can prevent coordinated shirking, then the mixed strategy equilibrium for the crew cost 4 units per period. But if the crew coordinates then they can achieve a summed payoff of 7 each period.
Cost minimizing contracts What is the lowest sum of payoffs in a 3 period repeated game that can be supported by a coordinated solution? Consider the outcome of receiving (0,0) in the first period (both diligent), followed by (1,6), (6,1), in the final 2 periods. (Tom shirks and then Jerry.) It is induced by playing (work, work) followed by (shirk, work) and (work,shirk). Can this outcome be supported by a coordinated solution?
Confirming the solution strategy • Strategy for Tom: If Jerry plays deviates from the profile prescribed in previous slide, work diligently for all the remaining rounds. Otherwise follow prescription. Similarly for Jerry. • Using the same methods as before one can show this is a solution strategy profile for the repeated game.
Should the manager intervene? • The overtime hours cost for cleaning is: (0 + 0) + (6 + 1) + (6 + 1) = 14 • The expected cost from leaving Tom and Jerry to play an uncoordinated game is: (2 + 2) + (2 + 2) + (2 + 2) = 12 • Therefore the manager should not intervene unless Tom and Jerry are coordinating. • This result shows that a decentralized contract, leaving the workers to play a period by period game with each other, may be more profitable to the employer direct intervention.
Longer term employment • Now suppose the contract lasts 13 rounds. • The expected cost of an uncoordinated piece rate contract is 52 overtime hours, computed from 2 (crew)*2 (overtime hours per round)*13 (rounds). • Noting the individual rationality constraint means each crew member will receive an average of at least 1 unit of overtime for all remaining rounds, consider the following long term contract costing 28 overtime hours: (0,0) for the first 9 rounds and then (1,6), (6,1), (1,6), (6,1) to conclude the final 4 • This is a coordinated solution, supported by a penalty system that awards 1 hour overtime for the remaining rounds to the first player that deviates from the contract.
