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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 more informed

- Strategies for the less informed
- Incentive Schemes
- Screening

- Signaling
- Signal Jamming

Game Theory - Mike Shor

Less-informed Players Screening

- Incentive Schemes
- Creating situations in which observable outcomes reveal the unobservable actions of the opponents.

- Creating situations in which the better-informed opponents’ observable actions reveal their unobservable traits.

Game Theory - Mike Shor

Risk Aversion

Risk Risk Risk

Seeking Neutral Averse

Lottery Corporations

(small stakes) one-time deals

Multiple Insurance

Gambles (big stakes)

Game Theory - Mike Shor

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

- Probability of success depends on firm’s effort

Game Theory - Mike Shor

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 If firm puts in high effort: Firm puts in low effort! Optimal Payment: lowest possible. Expected Profit

- If firm puts in routine effort:
- Utility = Payment - $100,000

- Utility = Payment - $150,000

- Value of project = (.6)600,000 = $360,000

- Payment = $100,000

= $360 - $100 = $260K

Game Theory - Mike Shor

Incentive Scheme 2 Observable Effort Pay additional $50K for high effort: If effort is observable, pay for high effort Expected Profit = $330K

- Firm puts in the effort level promised, given its pay
- Pay for routine effort:
- E[Profit] = (.6)600,000 – 100,000
= $260,000

- E[Profit] = (.6)600,000 – 100,000

- E[Profit] = (.8)600,000 – 150,000
= $330,000

Game Theory - Mike Shor

Problems Effort-based scheme cannot be implemented

- Fixed payment scheme offers no incentives for high effort
- High effort is more profitable

- Cannot monitor firm effort

Game Theory - Mike Shor

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

- Compensation contract must rely on something that can be directly observed and verified.

Game Theory - Mike Shor

Observable Outcome Participation Constraint

- Incentive Compatibility
- It must be in the employee’s best interest to put in high effort

- Expected salary must be high enough for employee to accept the position

Game Theory - Mike Shor

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 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

- (0.2)b ≥ 50,000

Game Theory - Mike Shor

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

- Incentive Compatibility:
- Firm will put in high effort if
b ≥ $250,000

- Firm will put in high effort if
- Participation:
- Firm will accept contract if
s + (0.8)b ≥ 150,000

- Firm will accept contract if
- Solution
- Minimum bonus: b = $250,000
- Minimum base salary:
s = 150,000 – (0.8)250,000 = -$50,000

Game Theory - Mike Shor

Negative Salaries ?

- Ante in gambling
- Law firms / partnerships
- Work bonds / construction
- Startup funds
- Risk Premium !!!

Game Theory - Mike Shor

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 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 Charge franchise fee equal to highest expected profit Expected Profit: $330K

- Charge the firm f regardless of profits
- Contractee takes all the risks and becomes the “residual owner” or franchisee

- Routine effort: .6(600K)-100K = 260K
- High effort: .8(600K)-150K = 330K

Game Theory - Mike Shor

Summary of Incentive Schemes Salary and Bonus Franchising

- Observable Effort
- Expected Profit: 330K
- Expected Salary: 150K

- Expected Profit: 330K
- Expected Salary: 150K

- Expected Profit: 330K
- Expected Salary: 150K

Game Theory - Mike Shor

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 Franchising

- Employees (unlike firms) are rarely willing to bare high risks
- Salary and Bonus
- 0.8 chance: 200K
- 0.2 chance –50K

- 0.8 chance: 270K
- 0.2 chance –330K

Game Theory - Mike Shor

Summary Usually, this is unreasonable

- Can perfectly compensate for information asymmetry
- Let employee take the risk (franchising)
- Make employee pay you a “moral hazard premium” (salary and bonus)

- 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

- 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 The test is accurate You test positive! Are you worried?

- A new test has been invented for a horrible, painful, terminal disease
- The disease is rare
- One in a million people are infected

- 95% correct

Game Theory - Mike Shor

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

- 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 Signal-jamming

- Signaling
- Using actions that other players would interpret in a way that would favor you in the game play

- 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

- 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 Offer $5,000 premium contract The company is trying to “pool” high and low risk drivers Will it succeed?

- An insurance company can offer a single insurance contract
- Expected cost of accidents:
- (½ .9 + ½ .1 )10,000 = $5,000

Game Theory - Mike Shor

Self-Selection Low-risk drivers: Only high risk drivers “self-select” into the contract to buy insurance

- High risk drivers:
- Don’t buy insurance: (.9)(-10,000) = -9K
- Buy insurance: = -5K
- High risk drivers buy insurance

- Don’t buy insurance: (.1)(-10,000) = -1K
- Buy insurance: = -5K
- Low risk drivers do not buy insurance

Game Theory - Mike Shor

Adverse Selection Expected cost of accidents among insured Cannot ignore this “adverse selection” If only going to have high risk drivers, might as well charge more ($9,000)

- Expected cost of accidents in population
- (½ .9 + ½ .1 )10,000 = $5,000

- .9 (10,000) = $9,000
- Insurance company loss: $4,000

Game Theory - Mike Shor

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

- 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

Screening

- Education as a signaling and screening device
- Is there value to education?
- Good types: less hardship cost

Game Theory - Mike Shor

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

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

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

- The opportunity to signal may prevent some types from hiding their characteristics
- Pass / Fail grades (C is passing)

Game Theory - Mike Shor

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 I make a take-it-or-leave it offer

- 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

- ½ of cars are “mint”

- 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 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

Summary If more informed If market suffers from lack of info

- 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

- Invest in signals to demonstrate traits

- Create mechanisms for information revelation

Game Theory - Mike Shor

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