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# Game Theory: Review - PowerPoint PPT Presentation

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

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

• 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

• 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.

• 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

• 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.

• 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)

• 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

• 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.

• 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

• 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?

• 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

• 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

• 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

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

• 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.”

• 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.

• 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

• 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

• 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!

• 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

• 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

• 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?

• 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?

• 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?