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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

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

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

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.

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

- One could imagine a real world situation where

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.

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)

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.

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

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

- Someone comes running into your office

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?

- By pooling risks, we reduce 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

“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.”

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.

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

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

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!

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

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

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?

- Assumptions
- 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?

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?

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?

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