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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. Strategies for the less informed Incentive Schemes Screening Strategies for the more informed Signaling Signal Jamming.

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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 • Strategies for the less informed • Incentive Schemes • Screening • Strategies for the more informed • Signaling • Signal Jamming 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 unobservable traits. Game Theory - Mike Shor

  4. Less-informed Players Screen Incentive Game Theory - Mike Shor

  5. $50 Moral Hazard & Roulette Game Theory - Mike Shor

  6. Risk Aversion Risk Risk Risk Seeking Neutral Averse Lottery Corporations (small stakes) one-time deals Multiple Insurance Gambles (big stakes) Game Theory - Mike Shor

  7. Multiple Gambles(biased) coin flip 52%-48% Chance of Loss 48% Game Theory - Mike Shor

  8. 10 tosses Chance of Loss 32% Game Theory - Mike Shor

  9. 1000 tosses Chance of Loss 9% Game Theory - Mike Shor

  10. 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

  11. Compensation Schemes I. Fixed Payment Scheme II. Observable Effort III. Bonus Scheme IV. Franchise Scheme Game Theory - Mike Shor

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

  13. 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

  14. Problems • Fixed payment scheme offers no incentives for high effort • High effort is more profitable • Effort-based scheme cannot be implemented • Cannot monitor firm effort Game Theory - Mike Shor

  15. Incentive Scheme 3 Wage and Bonus • Suppose effort can not be observed • Incentive-Compatible compensation • Compensation contract must rely on something that can be directly observed and verified. • Project’s success or failure • Related probabilistically to effort • Imperfect but positive information Game Theory - Mike Shor

  16. 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

  17. Incentive Compatibility • Compensation Package (s, b) • s: base salary • b: bonus if the project succeeds • Firm’s Expected Earnings • routine effort: s + (0.6)b • high-quality effort: s + (0.8)b Game Theory - Mike Shor

  18. Incentive Compatibility • Firm will put in high effort if 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

  19. Participation • The total compensation should be good enough for the firm to accept the job. • If incentive compatibility condition is met, the firm prefers the high effort level. • The firm will accept the job if: s + (0.8)b ≥ 150,000 Game Theory - Mike Shor

  20. Enticing High Effort • Incentive Compatibility: • Firm will put in high effort if b ≥ $250,000 • Participation: • Firm will accept contract if s + (0.8)b ≥ 150,000 • Solution • Minimum bonus: b = $250,000 • Minimum base salary: s = 150,000 – (0.8)250,000 = -$50,000 Game Theory - Mike Shor

  21. Negative Salaries ? • Ante in gambling • Law firms / partnerships • Work bonds / construction • Startup funds • Risk Premium !!! Game Theory - Mike Shor

  22. 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

  23. Negative Salaries? • Not always a negative salary, but: • Enough outcome-contingent incentive (bonus) to provide incentive to work hard. • Enough certain base wage (salary) to provide incentive to work at all. • Optimally, charge implicit “risk premium” to party with greatest control. Game Theory - Mike Shor

  24. Incentive Scheme 4 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

  25. Summary of Incentive Schemes • Observable Effort • Expected Profit: 330K • Expected Salary: 150K • Salary and Bonus • Expected Profit: 330K • Expected Salary: 150K • Franchising • Expected Profit: 330K • Expected Salary: 150K Game Theory - Mike Shor

  26. Uncertainty and Risk COMMANDMENT In the presence of uncertainty: Assign the risk to the better informed party. Efficiency and greater profits result. The more risks are transferred to the well-informed party, the more profit is earned. Game Theory - Mike Shor

  27. 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 Game Theory - Mike Shor

  28. 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 Game Theory - Mike Shor

  29. 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

  30. 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

  31. 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

  32. 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

  33. 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

  34. 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

  35. 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

  36. 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

  37. 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

  38. 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

  39. How to Screen • Want to know an unobservable trait • Identify an action that is more costly for “bad” types than “good” types • Ask the person (are you “good”?) • But… attach a cost to the answer • Cost • high enough so “bad” types don’t lie • Low enough so “good” types don’t lie Game Theory - Mike Shor

  40. Screening • Education as a signaling and screening device • Is there value to education? • Good types: less hardship cost Game Theory - Mike Shor

  41. 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

  42. “High” Quality Workers If I get an MBA (signal “good” type): 1,600,000 – 5,000N If I don’t get an MBA ( “bad” type): 1,000,000 1,600,000 – 5,000N > 1,000,000 600,000 > 5,000N 120 > N Game Theory - Mike Shor

  43. “Low” Quality Workers If I get an MBA (signal “good” type): 1,600,000 – 20,000N If I don’t get an MBA ( “bad” type): 1,000,000 1,600,000 – 20,000N < 1,000,000 600,000 < 20,000N 30 < N Game Theory - Mike Shor

  44. Signaling • The opportunity to signal may prevent some types from hiding their characteristics • Pass / Fail grades (C is passing) Game Theory - Mike Shor

  45. 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 and C’s. • This leaves B’s and C’s taking the course pass/fail. Now, B students have incentive to take the course for a letter grade to separate from C’s. • Ultimately, only C students take the course pass/fail. • If employers are rational – will know how to read pass/fail grades. C students cannot hide! Game Theory - Mike Shor

  46. 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

  47. 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

  48. 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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