Principles of Game Theory

1. Principles of Game Theory Lecture 16: Contract Design

2. Last time • Adverse Selection • Eg: insurance • Moral Hazard • Eg: contracting with unobservable effort

3. Contracting with Moral Hazard Example from last class: • A project with uncertain outcome • Probability of success depends on employee’s effort • P(success) = 0.6 if effort is routine • P(success) = 0.8 if effort is high • Employee has cost of effort • cost of routine effort = $100,000 • cost of high effort = $150,000 • Project outcome = $600,000 if successful

4. Compensation Schemes • Scheme 1: Fixed Salary: • If employee puts in routine effort: • Utility = Payment - $100,000 • If employee puts in high effort: • Utility = Payment - $150,000  Employee puts in low effort! • Optimal Payment: lowest possible. • Payment = $100,000 • Value of project = (.6)600,000 = $360,000 • Expected Profit = $360 - $100 = $260K

5. Compensation Schemes • If effort is easy to observe, contracts are simple: • Work as hard as we tell you or get fired. • Only question is how hard do we want employees to work? • Salary must be commensurate with level of effort, or no one will take the job

6. Compensation Schemes • Scheme 2: Observable effort • Employee puts in effort level promised, given it’s 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

7. Problems • Fixed payment scheme offers no incentives for high effort • Worst case scenario: $260K • Effort-based scheme cannot be implemented • “First-Best” case scenario: $330K • Question: how close can we get to best case scenario if effort is unobservable? • Incentive-Compatible compensation? • Compensation contract must rely on something that can be directly observed and verified. In this case, effort is not. • Project’s success or failure • Related probabilisticallyto effort • Imperfect but positive information

8. Contracting Constraints Two main types of constraints: • Incentive Compatibility (IC) • Putting in high effort must be better than putting in low effort • Participation (aka Individual Rationality – IR) • Putting in any effort must be better than not taking the position

9. Incentive Compatibility One such contract: • Compensation Package (f, b) • f: fixed base payment • b: bonus if the project succeeds • Firm’s Expected Earnings • routine effort: f + (0.6)b • high-quality effort: f + (0.8)b

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

11. Incentive Compatibility • Firm will put in high effort if b ≥ $250,000 • Our profit maximization means that we want to set b as low as possible • So the IC constraint will bind: b = $250,000 • Next, solve participation constraint

12. Participation • IR constraint: The total compensation should be good enough for the firm to accept the job. • We can’t force them to take the job • Need to pay them enough so that they choose to take the job • But everything that we give takes away from our profit. • IC constraint: If incentive compatibility condition is met, the firm prefers the high effort level. • The firm will accept the job if: f + (0.8)b ≥ 150,000

13. Participation • Firm will accept contract if expected pay is greater than cost f + (0.8)b ≥ 150,000 • Solution • Substitute minimum bonus founding using the IC constraint: f + (0.8)250,000 ≥ $150,000 f + $200,000 ≥ $150,000 f ≥ – $50,000 What the what?!

14. 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

15. 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.

16. 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!!!

17. Take away? In the presence of uncertainty: Assign the risk to the better informed party. Efficiency and greater profits result. • But be careful so that you don’t construct other “agency” problems.

18. Agency • Typically in relationships of asymmetric information – and especially in cases of contracting – we refer to one party as an agent and the other as the principle • Principle: less-informed player who needs the other player, the agent, to do something • Agent: more-informed player who will use the mechanism (contract, etc) given to him by the principle to maximize his own payoffs • Principle must anticipate the agent’s behavior and design an appropriate mechanism. • Mechanism Design: “how to deal with someone who knows more than you.” – The Economist

19. Price Discrimination • Price setting (in specific markets) can be thought of as a mechanism design problem in order to deal with Adverse Selection: • Principle: Firm owner • Agent: Consumer with different types of willingness to pay • If the firm could know your willingness to pay, what would they charge? • Once you start to look, you’ll see many screening devices set up to try to figure this out (eg: airline schedules)

20. Price DiscriminationAn example • Suppose an airline is selling tickets to two types of consumers: business and tourist • Doesn’t know which type it is (eg; want to travel on the same day, each is booking via a website, etc) • Each type of consumer has a different willingness to pay for tickets • 70% Tourist • 30% Business • Airline can offer two types of ticket: first class and economy.

21. Airline Price Discrimination • Assume 100 consumers (from the book: p546) • What should the airline charge for each ticket?

22. Airline Price Discrimination:Ideal solution • Ideal solution (baseline): if it could observe types • Sell only First Class tickets to Business and only Economy to Tourists • 150*30 +40*70 = $7300 • But first best isn’t feasible – Airline doesn’t observe type

23. Airline Price Discrimination: • What would happen if firm naively priced at the first best solution? • 140 for the Economy (for the Tourist) • 300 for the First Class (for the Business Traveler) • No Tourist would buy a First Class ticket (good) but neither would a Business traveler (bad). Why? • From the business consumer’s point of view, he’s better off buying an economy ticket: 175-140 > 300-300

24. Airline Price Discrimination • So how do we solve the Airline’s problem? • Set up the right IC and IR constraints and optimize.

25. Next time • More on Mechanism Design • Price Discrimination • More Contracting • Review Questions • must be sent to me by tomorrow at noon • “Optional” homework problems • Trust me when I tell you that the IC constraints hold • Chp 14: U2, U3, U9 (fyi: I hope to see you in my DADSS class next semester)