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Game Theory: Review. Problem: Strategic behavior What I should do depends on what you do And vice versa Abstract representations of games Decision tree for sequential games Strategy matrix for all games (2D for 2 player) Solution concepts Subgame perfect equilibrium Dominance

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Game theory review l.jpg

Game Theory: Review

Problem: Strategic behavior

What I should do depends on what you do

And vice versa

Abstract representations of games

Decision tree for sequential games

Strategy matrix for all games (2D for 2 player)

Solution concepts

Subgame perfect equilibrium

Dominance

Von Neumann solution to 2 player game

Nash equilibrium

Von Neumann solution to many player game


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Subgame perfect equilibrium

  • Treat the final choice (subgame) as a decision theory problem

    • The solution to which is obvious

    • So plug it in

    • Continue right to left on the decision tree

  • Assumes no way of committing and

  • No coalition formation

    • In the real world, A might pay B not to take what would otherwise be his ideal choice--

    • because that will change what C does in a way that benefits A.

    • One criminal bribing another to keep his mouth shut, for instance

  • But it does provide a simple way of extending the decision theory approach

    • To give an unambiguous answer

    • In at least some situations

    • Consider our basketball player problem


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Dominant Strategy

  • Each player has a best choice, whatever the other does

  • Might not exist in two senses

    • If I know you are doing X, I do Y—and if you know I am doing Y, you do X. Nash equilibrium. Driving on the right. The outcome may not be unique, but it is stable.

    • If I know you are doing X, I do Y—and if you know I am doing Y, you don't do X. Unstable. Scissors/paper/stone.


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Nash Equilibrium

  • By freezing all the other players while you decide, we reduce it to decision theory for each player--given what the rest are doing

  • We then look for a collection of choices that are consistent with each other

    • Meaning that each person is doing the best he can for himself

    • Given what everyone else is doing

  • This assumes away all coalitions

    • it doesn't allow for two ore more people simultaneously shifting their strategy in a way that benefits both

    • Like my two escaping prisoners

  • It ignores the problem of defining “freezing other players”

    • Their alternatives partly depend on what you are doing

    • So “freezing” really means “adjusting in a particular way”

    • For instance, freezing prize, varying quantity, or vice versa

  • It also ignores the problem of how to get to that solution

    • One could imagine a real world situation where

      • A adjusts to B and C

      • Which changes B's best strategy, so he adjusts

      • Which changes C and A's best strategies …

      • Forever …

    • A lot of economics is like this--find the equilibrium, ignore the dynamics that get you there


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Von Neumann Solution

  • aka minimax aka saddlepoint aka ….?

  • It tells each player how to figure out what to do

    • A value V

    • A strategy for one player that guarantees winning at least V

    • And for the other that guarantees losing at most V

  • Describes the outcome if each follows those instructions

  • But it applies only to two person fixed sum games.


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VN Solution to Many Player Game

  • Outcome--how much each player ends up with

  • Dominance: Outcome A dominates B if there is some group of players, all of whom do better under A (end up with more) and who, by working together, can get A for themselves

  • A solution is a set of outcomes none of which dominates another, such that every outcome not in the solution is dominated by one in the solution

  • Consider, to get some flavor of this, three player majority vote

  • A dollar is to be divided among Ann, Bill and Charles by majority vote.

    • Ann and Bill propose (.5,.5,0)--they split the dollar, leaving Charles with nothing

    • Charles proposes (.6,0,.4). Ann and Charles both prefer it, to it beats the first proposal, but …

    • Bill proposes (0, .5, .5), which beats that …

    • And so around we go.

    • One solution is the set: (.5,.5,0), (0, .5, .5), (.5,0,.5)


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Schelling aka Focal Point

  • Two people have to coordinate without communicating

    • Either can’t communicate (students to meet)

    • Or can’t believe what each says (bargaining)

  • They look for a unique outcome

    • Because the alternative is choosing among many outcomes

    • Which is worse than that

    • While the bank robbers are haggling the cops may show up

  • But “unique outcome” is a fact

    • Not about nature but

    • About how people think

  • Which implies that

    • You might improve the outcome by how you frame the decision

    • Coordination may break down on cultural boundaries

      • Because people frame decisions differently

      • Hence each thinks the other is unreasonably refusing the obvious compromise.


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Conclusion

  • Game theory is helpful as a way of thinking

    • “Since I know his final choice will be, I should …”

    • Commitment strategies

    • Prisoner’s dilemmas and how to avoid them

    • Mixed strategies where you don’t want the opponent to know what you will do

    • Nash equilibrium

    • Schelling points

  • But it doesn’t provide a rigorous answer

    • To either the general question of what you should do

    • In the context of strategic behavior

    • Or of what people will do


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Insurance

  • Risk Aversion

    • I prefer a certainty of paying $1000 to a .001 chance of $1,000,000

    • Why?

  • Moral Hazard

    • Since my factory has fire insurance for 100%

    • Why should I waste money on a sprinkler system?

  • Adverse selection

    • Someone comes running into your office

      • “I want a million dollars in life insurance, right now”

      • Do you sell it to him?

    • Doesn’t the same problem exist for everyone who wants to buy?

      • Buying signals that you think you are likely to collect

      • I.e. a bad risk

      • So price insurance assuming you are selling to bad risks

      • Which means it’s a lousy deal for good risks

    • So good risks don’t get insured

      • even if they would be willing to pay a price

      • At which it is worth insuring them


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Risk Aversion

  • The standard explanation for insurance

    • By pooling risks, we reduce risk

      • I have a .001 chance of my $1,000,000 house burning down

      • A million of us will have just about 1000 houses burn down each year

      • For an average cost of $1000/person/year

    • Why do I prefer to reduce the risk?

  • The more money I have, the less each additional dollar is worth to me

    • With $20,000/year, I buy very important things

    • With $200,000/year, the last dollar goes for something much less important

  • So I want to transfer money

    • To futures where I have little, because my house burned down

    • From futures where I have lots--house didn’t burn


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“Risk Aversion” a Misleading Term

  • Additional dollars are probably worth less to me the more I have

  • It doesn’t follow that (say) additional years of life are

    • Without the risky operation I live fifteen years

    • If it succeeds I live thirty, but …

    • Half the time the operation kills the patient

    • And I always wanted to have children

  • So really “risk aversion in money”

  • Aka “declining marginal utility of income.”


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Moral hazard

  • I have a ten million dollar factory and am worried about fire

    • If I can take ten thousand dollar precaution that reduces the risk by 1% this year, I will—(.01x$10,000,000=$100,000>$10,000)

    • But if the precaution costs a million, I won't.

  • insure my factory for $9,000,000

    • It is still worth taking a precaution that reduces the chance of fire by %1

    • But only if it costs less than …?

  • Of course, the price of the insurance will take account of the fact that I can be expected to take fewer precautions:

    • Before I was insured, the chance of the factory burning down was 5%

    • So insurance should have cost me about $450,000/year, but …

    • Insurance company knows that if insured I will be less careful

    • Raising the probability to (say) 10%, and the price to $900,000

  • There is a net loss here—precautions worth taking that are not getting taken, because I pay for them but the gain goes mostly to the insurance company.


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Solutions?

  • Require precautions (signs in car repair shops—no customers allowed in, mandated sprinkler systems)

    • The insurance company gives you a lower rate if you take the precautions

    • Only works for observable precautions

  • Make insurance only cover fires not due to your failure to take precautions (again, if observable)

  • Only insure for (say) 50% of value

    • There are still precautions you should take and don’t

    • But you take the ones that matter most

    • I.e. the ones where benefit is at least twice cost

    • So moral hazard remains, but is cost is reduced

    • Of course, you also now have more risk to bear


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A Puzzle

  • The value of a dollar changes a lot between $20,000/year and $200,000/year

  • But very little between $200,000 and $200,100

  • So why do people insure against small losses?

    • Service contract on a washing machine

    • Even a toaster!

    • Medical insurance that covers routine checkups

    • Filling cavities, and the like


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Is Moral Hazard a Bug or a Feature?

  • Big company, many factories, they insure

    • Why? They shouldn't be risk averse

    • Since they can spread the loss across their factories.

  • Consider the employee running one factory without insurance

    • He can spend nothing, have 3% chance of a fire

    • Or spend $100,000, have 1%--and make $100,000 less/year for the company

    • Which is it in his interest to do?

  • Insure the factory to transfer cost to insurance company

    • Which then insists on a sprinkler system

    • Makes other rules

    • Is more competent than the company to prevent the fire!


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Put Incentive Where It Does the Most Good

  • Insurance transfers loss from owner to the insurance company

  • Sometimes the owner is the one best able to prevent the loss

    • In which case moral hazard is a cost of insurance

    • To be weighed against risk spreading gain

  • Sometimes the insurance company is best able

    • In which case “moral hazard” is the objective

    • Sears knows more about getting their washing machines fixed than I do

    • So I buy a service contract to transfer the decision to them

  • Sometimes each party has precutions it is best at

    • So coinsurance--say 50%--gives each an incentive to take

    • Those precautions that have a high payoff


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Health Insurance

  • If intended as risk spreading

    • Should be a large deductible

    • So I pay for all minor things

      • Giving me an incentive to keep costs down

      • Since I am paying them

    • But cover virtually 100% of rare high ticket items

      • If my life is at stake, I want it

      • But I don’t want to risk paying even 10% of a million dollar procedure

  • But maybe it’s intended

    • To transfer to the insurance company

    • The incentive to find me a good doctor

    • Negotiate a good price

  • Robin Hanson’s version

    • I decide how much my life is worth

    • I buy that much life insurance, from a company that also

    • Makes and pays for my medical decisions

    • And now has the right incentive to keep me alive


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Adverse Selection

  • The problem: The market for lemons

    • Assumptions

      • Used car in good condition worth $10,000 to buyer, $8000 to seller

      • Lemon worth $5,000, $4,000

      • Half the cars are creampuffs, half lemons

    • First try:

      • Buyers figure average used car is worth $7,500 to them, $6,000 to seller, so offer something in between

      • What happens?

    • What is the final result?

  • How might you avoid this problem—due to asymmetric information

    • Make the information symmetric—inspect the car. Or …

    • Transfer the risk to the party with the information—seller insures the car

  • What problems does the latter solution raise?


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Plea Bargaining

  • A student raised the following question:

  • Suppose we include adverse selection in our analysis of plea bargaining

  • What does the D.A. signal by offering a deal?

  • What does the defendant signal by accepting?

  • Which subset of defendants end up going to trial?


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Why insurance matters?

  • Most of you won’t be working for insurance companies

  • Or even negotiating contracts with them

  • But the analysis of insurance will be important

    • Almost any contract is in part insurance

    • Do you pay salesmen by the month or by the sale?

    • Is your house built for a fixed price, or cost+?

    • Do you take the case for a fixed price, contingency fee, or hourly charge?