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Game Theory

Game Theory. “ A little knowledge is a dangerous thing. So is a lot. ” - Albert Einstein Mike Shor Lectures 9&10. Strategic Manipulation of Information. One player may have some information that affects other players’ decisions. The better-informed player may want to:

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Game Theory

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  1. Game Theory “A little knowledge is a dangerous thing. So is a lot.” - Albert Einstein Mike Shor Lectures 9&10

  2. Strategic Manipulation of Information • One player may have some information that affects other players’ decisions. • The better-informed player may want to: • conceal information • reveal misleading information • reveal selected information truthfully • The less-informed player may want to: • elicit information or filter truth from lies • continue ignorance Game Theory - Mike Shor

  3. Less-informed Players • Incentive Schemes • Creating situations in which observable outcomes reveal the unobservable actions of the opponents. • Screening • Creating situations in which the better-informed opponents’ observable actions reveal their information Game Theory - Mike Shor

  4. Better-informed Players • Signaling • Using actions that other players would interpret in a way that would favor you in the game play • Signal-jamming • Using a mixed-strategy that interferes with the other players’ inference process • Other players would otherwise find out things about you that they could use to your detriment in the game. Game Theory - Mike Shor

  5. Moral Hazard • A project with uncertain outcome • Probability of success depends on firm’s effort • prob. of success = 0.6 if effort is routine • prob. of success = 0.8 if effort is high • Firm has cost of effort • cost of routine effort = $100,000 • cost of high effort = $150,000 • project outcome = $600,000 if successful Game Theory - Mike Shor

  6. Incentive Scheme 1: Pure Wage Scheme • If firm puts in routine effort: • Utility = Wage - $100,000 • If firm puts in high effort: • Utility = Wage - $150,000 • Firm puts in low effort! • Value of project = (.6)600,000 = $360,000 • Optimal salary: lowest possible. • Wage = $100,000 • Expected Profit = $260K Game Theory - Mike Shor

  7. Incentive Scheme 2 Observable Effort • Firm puts in the effort level promised, given its pay • Pay for routine effort: • E[Profit] = (.6)600,000 – 100,000 = $260,000 • Pay additional $50K for high effort: • E[Profit] = (.8)600,000 – 150,000 = $330,000 • If effort is observable, pay for high effort • Expected Profit = $330K Game Theory - Mike Shor

  8. Problems • Pure wage scheme offers no incentives for high effort • High effort is more profitable • Cannot monitor employee effort • Effort-based scheme cannot be implemented Game Theory - Mike Shor

  9. Incentive Scheme 3 Wage and Bonus • Suppose effort can not be observed • Incentive-Compatible compensation • Based upon observable outcome • Incentive Compatibility • It must be in the employee’s best interest to put in high effort • Participation Constraint • Expected salary must be high enough for employee to accept the position Game Theory - Mike Shor

  10. Observable Outcome • The salary contract must rely on something that can be directly observed and verified. • Project’s success or failure • project’s success related probabilistically to effort • provides information about effort (imperfect but positive) • Imperfect Information  Costs! Game Theory - Mike Shor

  11. Incentive Compatibility • Compensation Package (s, b) • s: base salary • b: bonus if the project succeeds • Expected Earnings • routine effort: s + (0.6)b • high-quality effort: s + (0.8)b • Incentive Compatibility Condition • s + (0.8)b - 150,000 ≥ s + (0.6)b - 100,000 • (0.2)b ≥ 50,000 (marginal benefit of effort > marginal cost) • b ≥ $250,000 Game Theory - Mike Shor

  12. Participation • The total compensation should be good enough for the manager to work at all. • Given that the incentive compatibility condition is met, so that the manager prefers the high effort level, the manager is willing to work if: • s + (0.8)b - 150,000 ≥ 0 Game Theory - Mike Shor

  13. Owner’s Optimization • Maximize Profits • (.8)600,000 – (.8)b – s • Given constraints • b ≥ $250,000 • s + (0.8)b - 150,000 ≥ 0 • Solution • Minimum total wage: s + (0.8)b = 150,000 • Minimum bonus: b = $250,000 • Minimum base salary: = 150,000 – (.8)250,000 = -50,000 Game Theory - Mike Shor

  14. Interpretation • -50,000 is the amount of capital the firm must put up for the project • -50,000 is the fine the firm must pay if the project fails. • Expected profit: • (.8)600,000 – (.8)b – s = (.8)600,000 – (.8)250,000 + 50,000 = $330,000 • Same as with observable effort!!! Game Theory - Mike Shor

  15. Incentive Scheme 4 Constrained s and b • Negative salaries might not be possible • If s < 0 is not possible, then set s = 0. • Minimum bonus, b = $250,000 • Total expected compensation: (0.8)250,000 = 200,000 > 150,000 over-fulfilling the participation condition • The extra $50,000 is the cost of information asymmetry borne by the less-informed party. • Expected Profit: $280K Game Theory - Mike Shor

  16. Incentive Scheme 5 Franchising • Charge the firm f regardless of profits • Contractee takes all the risks and becomes the “residual owner” or franchisee • Charge franchise fee equal to highest expected profit • Routine effort: .6(600K)-100K = 260K • High effort: .8(600K)-150K = 330K • Expected Profit: $330K Game Theory - Mike Shor

  17. Summary of Incentive Schemes • Salary and Bonus • Expected Profit: 330K • Expected Salary: 150K • Franchising • Expected Profit: 330K • Expected Salary: 150K • Constrained Salary and Bonus • Expected Profit: 280K • Expected Salary: 200K Game Theory - Mike Shor

  18. Uncertainty and Risk COMMANDMENT In the presence of uncertainty: Assign the risk to the better informed party. Efficiency and greater profits result. If tolerance to risk is low: minimize the cost of information asymmetry. Game Theory - Mike Shor

  19. Risks • Employees (unlike firms) are rarely willing to bare high risks • Salary and Bonus • 0.8 chance: 200K 0.2 chance –50K • Franchising • 0.8 chance: 270K 0.2 chance –330K • Constrained Salary and Bonus • 0.8 chance: 200K 0.2 chance 0K Game Theory - Mike Shor

  20. Summary • Can perfectly compensate for information asymmetry • Let employee take the risk (franchising) • Make employee pay you a “moral hazard premium” (salary and bonus) • Usually, this is unreasonable • To offer a reasonable incentive-compatible wage, must pay a cost of information asymmetry born by less-informed party • Is this worth it? • Sayeth the Economist: “It depends” Game Theory - Mike Shor

  21. Depends on What? • With perfect information: • Profit to firm: $330K • With a pure wage scheme: • Profit to firm: $260K • If firm can induce high effort: • Surplus: $70K • If the cost of information asymmetry is less than $70K, it is worth it Game Theory - Mike Shor

  22. Cost of Information Asymmetry • Want to reward high effort • Can only reward good results • How well are results tied to effort? • Bayes rule: • How well results are linked to effort depends on • Likelihood of success at different effort levels • Amount of each effort level in the population Game Theory - Mike Shor

  23. A Horrible Disease • A new test has been invented for a horrible, painful, terminal disease • The disease is rare • One in a million people are infected • The test is accurate • 95% correct • You test positive! Are you worried? Game Theory - Mike Shor

  24. Bayes Rule • What is the chance that you have the disease if you tested positive? Infected Not Infected 1 1,000,000 95% test pos. 5% test pos. ______________________________________________________________________________________  1 50,000 • Chance: 1 in 50,000 Game Theory - Mike Shor

  25. Example: HMOs • HMOs costs arise from treating members: routine visits and referrals • Moral hazard: “necessity of treatment” is unobservable • Incentive scheme: • Capitation: fixed monthly fee per patient • Withholds: charges to the doctor if referred patients spend more than in a “withhold fund,” bonuses if they spend less. Game Theory - Mike Shor

  26. Screening & Signaling • Auto Insurance • Half of the population are high risk drivers and half are low risk drivers • High risk drivers: • 90% chance of accident • Low risk drivers: • 10% chance of accident • Accidents cost $10,000 Game Theory - Mike Shor

  27. Pooling • An insurance company can offer a single insurance contract • Expected cost of accidents: • (½ .9 + ½ .1 )10,000 = $5,000 • Offer $5,000 premium contract • The company is trying to “pool” high and low risk drivers • Will it succeed? Game Theory - Mike Shor

  28. Self-Selection • High risk drivers: • Don’t buy insurance: (.9)(-10,000) = -9K • Buy insurance: = -5K • High risk drivers buy insurance • Low-risk drivers: • Don’t buy insurance: (.1)(-10,000) = -1K • Buy insurance: = -5K • Low risk drivers do not buy insurance • Only high risk drivers “self-select” into the contract to buy insurance Game Theory - Mike Shor

  29. Adverse Selection • Expected cost of accidents in population • (½ .9 + ½ .1 )10,000 = $5,000 • Expected cost of accidents among insured • .9 (10,000) = $9,000 • Insurance company loss: $4,000 • Cannot ignore this “adverse selection” • If only going to have high risk drivers, might as well charge more ($9,000) Game Theory - Mike Shor

  30. Screening • Offer two contracts, so that the customers self-select • One contract offers full insurance with a premium of $9,000 • Another contract offers a deductible, and a lower premium Game Theory - Mike Shor

  31. Screening • Education as a signaling and screening device • Is there value to education? Game Theory - Mike Shor

  32. Example: MBAs • How long should an MBA program be? • Two types of workers: • High and low quality • NPV of salary high quality worker: $1.6M low quality worker: $1.0M • Disutility per MBA class high quality worker: $5,000 low quality worker: $20,000 Game Theory - Mike Shor

  33. Screening MBAs • Want high quality workers to “signal” 1,600,000 – 5,000N > 1,000,000 5,000N < 600,000 N < 120 • Want low quality workers not to “signal” 1,600,000 – 20,000N < 1,000,000 20,000N > 600,000 N > 30 Game Theory - Mike Shor

  34. Signaling • The opportunity to signal may prevent some types from hiding their characteristics Game Theory - Mike Shor

  35. Screening • Suppose students can take a course pass/fail or for a letter grade. • An A student should signal her abilities by taking the course for a letter grade – separating herself from the population of B’s, C’s, and D’s. • This leaves B’s, C’s, D’s taking the course pass/fail. Now, B students have incentive to take the course for a letter grade to separate from C’s and D’s. • Ultimately, only D students take the course pass/fail. • If employers are rational – will know how to read pass/fail grades. D students cannot hide! Game Theory - Mike Shor

  36. Market for Lemons • Used car market • ½ of cars are “mint” worth $1000 to owner, $1500 to buyer • ½ of cars are “lemons” worth $100 to owner, $150 to buyer • I make a take-it-or-leave it offer • If I offer 100<p<1000, only “lemons” Might as well offer p=100 • If I offer p>1000 everyone accepts On average, car is worth: ½(1500) + ½(150) = 825 < 1000 Game Theory - Mike Shor

  37. Market for Lemons • Only outcome: offer $100 • Unraveling of markets • Inefficient resale price of slightly used cars • Need to create signals • Inspection by mechanics • Warranties • Dealer reputation Game Theory - Mike Shor

  38. Summary • If less informed • Provide contracts which encourage optimal behavior on the part of the well-informed • Create costly screens which allow the types to self-select different signals • If more informed • Invest in signals to demonstrate traits • If market suffers from lack of info • Create mechanisms for information revelation Game Theory - Mike Shor

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