Game Theory: Review

1
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

2
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

3
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 Yand 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 Yand if you know
  I am doing Y, you don't do X. Unstable.
  Scissors/paper/stone.

4
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

5
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.

6
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)

7
Schelling aka Focal Point

• Two people have to coordinate without
  communicating
• Either cant communicate (students to meet)
• Or cant 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.

8
Conclusion

• Game theory is helpful as a way of thinking
• Since I know his final choice will be, I should
  
• Commitment strategies
• Prisoners dilemmas and how to avoid them
• Mixed strategies where you dont want the
  opponent to know what you will do
• Nash equilibrium
• Schelling points
• But it doesnt 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

9
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?
• Doesnt 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 its a lousy deal for good risks
• So good risks dont get insured
• even if they would be willing to pay a price
• At which it is worth insuring them

10
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 didnt burn

11
Risk Aversion a Misleading Term

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

12
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(.01x10,000,000100,000gt10,000)
• 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 hereprecautions worth taking
  that are not getting taken, because I pay for
  them but the gain goes mostly to the insurance
  company.

13
Solutions?

• Require precautions (signs in car repair shopsno
  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
  dont
• 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

14
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

15
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!

16
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

17
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 dont want to risk paying even 10 of a
  million dollar procedure
• But maybe its intended
• To transfer to the insurance company
• The incentive to find me a good doctor
• Negotiate a good price
• Robin Hansons 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

18
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 problemdue to
  asymmetric information
• Make the information symmetricinspect the car.
  Or
• Transfer the risk to the party with the
  informationseller insures the car
• What problems does the latter solution raise?

19
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?

20
Why insurance matters?

• Most of you wont 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?