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Game Theory

Game Theory. Topic 7 Information. “A little knowledge is a dangerous thing. So is a lot.”. - Albert Einstein. Strategic Use of Information. Incentive Schemes

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Game Theory

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  1. Game Theory Topic 7Information “A little knowledge is a dangerous thing. So is a lot.” - Albert Einstein

  2. Strategic Use of Information • 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.

  3. Moral Hazard & Roulette $50

  4. 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

  5. Compensation Schemes • Benchmarks: • Fixed Payment • Observable Effort • Result-contingent bonus scheme

  6. Incentive Scheme 1: Fixed Payment • A fixed payment must be high enough to get the firm to accept the project • No amount of fixed payment can change the firm’s behavior once it accepts the project

  7. Incentive Scheme 1: Fixed Payment • For any fixed payment, effort will be low: • Payment - $100,000 > Payment - $150,000 • Optimal Payment • Lowest possible • Firm requires at least $100,000 • Payment = $100,000 • Expected Profit • Value of project — payment = (.6) $600K — $100K = $360 - $100 = $260K

  8. Incentive Scheme 2: Observable Effort • If we can observe effort, contracts are simple: • Work as hard as we tell you to, or you are fired • Only question: How hard do we want employees to work? • Remember, salary must be commensurate with level of effort, or no one will take the job

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

  10. Problems • Fixed payment scheme offers no incentives for high effort • High effort is more profitable • Worst case scenario: $260K • Effort-based scheme cannot be implemented • Cannot monitor firm effort • Best case scenario: $330K • Question: how close can we get to best case scenario if effort is unobservable?

  11. Incentive Scheme 3: Fixed Payment 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 probabilisticallyto effort • Imperfect but positive information

  12. Observable Outcome • Incentive Compatibility (high > low) • Putting in high effort must be better than putting in low effort • Participation Constraint (high > none) • Putting in high effort must be better than not taking the contract

  13. Incentive Compatibility • Compensation Package (f, b) • f: fixed base payment • b: bonus if the project succeeds

  14. Incentive Compatibility • If Firm puts in high effort • 80% chance of bonus, cost of $150K • Profit: f + (0.8)b – 150K • If Firm puts in low effort • 60% chance of bonus, cost of $100K • Profit: f + (0.6)b – 100K • If Firm does not take contract • Profit: 0

  15. Incentive Compatibility • (high > low) • Firm will put in high effort if f + (0.8)b - 150,000 ≥ f + (0.6)b - 100,000 • (0.2)b ≥ 50,000 marginal benefit of effort > marginal cost • b ≥ $250,000

  16. Incentive Compatibility • Firm will put in high effort if b ≥ $250,000 • To maximize profit, set b as low as possible b = $250,000 • Next, solve participation constraint

  17. Participation • (high > none) • The total compensation should be good enough for the firm to accept the job. • Want firm to prefer high effort to none. • The firm will accept the job if: f + (0.8)b -150,000 ≥ 0

  18. Participation • Firm will accept contract if expected pay is greater than cost f + (0.8)b ≥ 150,000 • Solution • Substitute minimum bonus: f + (0.8)250,000 ≥ $150,000 f + $200,000 ≥ $150,000 f ≥ – $50,000

  19. Negative Fixed Payment? • Certainly not for normal employees • Ante in gambling • Law firms / partnerships • Work bonds / construction • Startup funds • Interpretation: • Capital the firm must put up for the project • Fine the firm must pay if the project fails. • Risk premium

  20. Negative Fixed Payment? • Fixed payment not always negative, but: • Enough outcome-contingent incentive (bonus) to provide incentive to work hard. • Enough certain base wage (salary) to provide incentive to work at all. • Implicitly charging a “risk premium” to party with greatest control.

  21. Benchmarks • Fixed payment scheme: • Worst case scenario: $260K • Effort-based scheme: • Best case scenario: $330K • Fixed payment and bonus: Exp. Profit = (.8)600,000 – (.8)b – f = (.8)600,000 – (.8)250,000 + 50,000 = $330,000 • Same as with observable effort!!!

  22. Moral Hazard 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.

  23. Moral Hazard CAVEAT In the presence of uncertainty: Assign the risk to the better informed party. Efficiency and greater profits result. BUT If done imprecisely, may be better not to bother.

  24. Negative Fixed Payment? • Bonus depends on • difference between low and high effort costs • Fixed payment depends on • absolute magnitude of costs • Example: Firm has cost of effort • cost of routine effort = $250,000 (+$150K) • cost of high effort = $300,000 (+$150K) • BONUS UNCHANGED (b = $250,000) • FIXED INCREASES (f = $100,000)

  25. Summary • Can perfectly compensate for information asymmetry • Let employee take the risk • Use two part contracts (salary and bonus) • Often, this is unreasonable • Employees unwilling to assume risks • Contracts must be perfectly balanced • May be better to settle for low effort

  26. Risk Aversion Risk Risk Risk Seeking Neutral Averse Lottery Corporations (small stakes) one-time deals Multiple Insurance Gambles (big stakes)

  27. Biased coin flip: 52%-48% Would you bet $1000 on it? Chance of Loss 48%

  28. Biased coin flip: 52%-48% Would you bet $100 on it 10 times? Chance of Loss 33%

  29. Biased coin flip: 52%-48% Would you bet $1 on it 1000 times? Chance of Loss 10%

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