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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

Game Theory

“A little knowledge is a dangerous thing.

So is a lot.”

- Albert Einstein

Mike Shor

Lectures 9&10


Strategic manipulation of information
Strategic Manipulation of Information

  • Strategies for the less informed

    • Incentive Schemes

    • Screening

  • Strategies for the more informed

    • Signaling

    • Signal Jamming

  • Game Theory - Mike Shor


    Less informed players
    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


    Less informed players1
    Less-informed Players

    Screen

    Incentive

    Game Theory - Mike Shor


    Moral hazard roulette

    $50

    Moral Hazard & Roulette

    Game Theory - Mike Shor


    Risk aversion
    Risk Aversion

    Risk Risk Risk

    Seeking Neutral Averse

    Lottery Corporations

    (small stakes) one-time deals

    Multiple Insurance

    Gambles (big stakes)

    Game Theory - Mike Shor


    Multiple gambles biased coin flip 52 48
    Multiple Gambles(biased) coin flip 52%-48%

    Chance of Loss

    48%

    Game Theory - Mike Shor


    10 tosses
    10 tosses

    Chance of Loss

    32%

    Game Theory - Mike Shor


    1000 tosses
    1000 tosses

    Chance of Loss

    9%

    Game Theory - Mike Shor


    Moral hazard
    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


    Compensation schemes
    Compensation Schemes

    I. Fixed Payment Scheme

    II. Observable Effort

    III. Bonus Scheme

    IV. Franchise Scheme

    Game Theory - Mike Shor


    Incentive scheme 1 fixed payment scheme
    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


    Incentive scheme 2 observable effort
    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


    Problems
    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


    Incentive scheme 3 wage and bonus
    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


    Observable outcome
    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


    Incentive compatibility
    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


    Incentive compatibility1
    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


    Participation
    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


    Enticing high effort
    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


    Negative salaries
    Negative Salaries ?

    • Ante in gambling

    • Law firms / partnerships

    • Work bonds / construction

    • Startup funds

    • Risk Premium !!!

    Game Theory - Mike Shor


    Interpretation
    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


    Negative salaries1
    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


    Incentive scheme 4 franchising
    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


    Summary of incentive schemes
    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


    Uncertainty and risk
    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


    Risks
    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


    Summary
    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


    Cost of information asymmetry
    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


    A horrible disease
    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


    Bayes rule
    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


    Example hmos
    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


    Better informed players
    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


    Screening signaling
    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


    Pooling
    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


    Self selection
    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


    Adverse selection
    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


    Screening
    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


    How to screen
    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


    Screening1
    Screening

    • Education as a signaling and screening device

    • Is there value to education?

    • Good types: less hardship cost

    Game Theory - Mike Shor


    Example mbas
    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


    High quality workers
    “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


    Low quality workers
    “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


    Signaling
    Signaling

    • The opportunity to signal may prevent some types from hiding their characteristics

    • Pass / Fail grades (C is passing)

    Game Theory - Mike Shor


    Screening2
    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


    Market for lemons
    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


    Market for lemons1
    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


    Summary1
    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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