Game Theory

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. Signaling • Definition • Using actions that other players would interpret in a way that would favor you in the game play • Requires • It is not in the best interest for people to signal falsely • Implies signaling must be costly!

4. Auto Insurance • A $1,000 deductible? • High risk drivers: • 30% chance of claim • Risk aversion: willing to pay $500 • Low risk drivers: • 10% chance of claim • Risk aversion: willing to pay $200

5. Pooling vs. Separating • A pooling equilibrium has all types taking the same action • Therefore, cannot distinguish types by the actions they take • A separating equilibrium has different types taking different actions • Therefore, can distinguish types by the actions they take

6. Cost of No Deductible • If the cost of avoiding a deductible is • Less than $200 • Both types buy • Pooling Equilibrium • Greater than $500 • Neither type buys • Pooling equilibrium • Between $200 and $500 • Only high risk drivers buy • Separating equilibrium

7. Pooling • Insurance company charges $200 • Both types buy • Expected profit for insurance company: • High risk drivers: $200 - (30%) $1,000 = $200 - $300 = -$100 • Low risk drivers: $200 - (10%) $1,000 = $200 - $100 = $100 • Profitable only if there are more low-risk drivers than high-risk drivers

8. Separating • Insurance company charges $500 • Only high-risk drivers buy • Expected profit for insurance company: • High risk drivers: $500 - (30%) $1,000 = $500 - $300 = $200 • Low risk drivers: $0 • Always profitable

9. Comparing Equilibria • Imagine that p proportion are high-risk • Insurance company charges $200 • Profit: $100 (1-p) - $100 p = $100-$200p • Insurance company charges $500 • Profit: $200p • Compare: • $200p > $100-$200p • p > ¼  better to separate than pool

10. Self-Selection • Only high risk drivers “self-select” into the contract to buy insurance • Screening sets up the proper incentives for individuals to self-select • Pooling has the danger of adverse selection

11. Adverse Selection • Imagine ½ of the population are high-risk drivers • Insurance company calculates expected cost of not having a deductible: • (1/2) (10%) $1000 + (1/2) (30%) $1000 = $200 • Add a 10% profit, charge $220 • Only high risk drivers sign up!

12. 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

13. Screening • Education as a signaling and screening device • Is there value to an economics degree? • Imagine not: • no effect on productivity, but is observed by employers • “Cost” of economics major varies

14. Example: Econ majors • How hard should an econ major be? • Two types of workers: • High and low quality • NPV of salary high quality worker: $900,000 low quality worker: $700,000 • Disutility per econ credit high quality worker: $4,000 low quality worker: $6,000

15. “High” Quality Workers • If I get an econ major: • Signal I am a high quality worker • Receive $900,000 - $4,000 N • If I don’t get an econ major • Signal I am a low quality worker • Receive $700,000 900,000 – 4,000 N > 700,000 200,000 > 4,000 N 50 credits > N

16. “Low” Quality Workers • If I get an econ major: • Signal I am a high quality worker • Receive $900,000 - $6,000 N • If I don’t get an econ major • Signal I am a low quality worker • Receive $700,000 900,000 – 6,000 N < 700,000 200,000 < 6,000 N 33 credits < N

17. Screening • To achieve a separating equilibrium: • Costly enough to deter low types • Not so costly as to deter high types High reward High reward – high-type cost – low-type cost > Low reward < Low reward

18. Screening • To achieve a separating equilibrium: • High types work for high reward • Low types accept low reward High reward High reward – Low reward – Low reward >high-type cost <low-type cost

19. Screening Solves Market Imperfections • Market for lemons (used cars) • Worth between $1000 and $3000 to buyers • Worth $200 less to sellers • Only seller knows true value • Buyer offers $2,000  Adverse selection • Only cars between $1,000 and $2,200 sold • Buyer offers $1,600  Adverse selection • Only cars between $1,000 and $1,800 sold • Market equilibrium price: $1,200 • Only worst 20% of cars are ever sold

20. Screening Solves Market Imperfections • Market for lemons • What about introducing a screen? • Extended warranty • Cheaper to provide for good cars than bad cars • Other examples • Coupons • Banks made of granite

21. Hiding from Signals • The opportunity to signal may prevent some types from hiding their characteristics • Examples: • Financial disclosures • GPA on résumé • Taking classes pass / fail

22. Hiding from Signals • 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!

23. Summary • Enticing high effort is hard work • Leakages • Global vs. individual incentives • Rewarding the right people • Screening • Identify unobservable cost differences • Exploit them (carefully)