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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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