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
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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
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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
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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
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It also ignores the problem of how to get to that solution
– One could imagine a real world situation where
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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
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
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“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?
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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
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
“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:
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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
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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
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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?
• 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?