2. Game Theory The Problem of Strategic Behavior
What I do depends on what he does and …
Vice versa

3. What Von Neumann Was Trying to Do
A general solution to strategic behavior
how each player should play
And will play, being rational
And assuming the other players are
A solution that would cover
Economics
Politics
foreign policy
poker, …
But what he actually did was

4. Two Player Fixed Sum Game
Fixed Sum: What helps me hurts you
Strategy: A full description of what I will do in any situation
Including “flip a coin, if heads do A, if tails do B”
Consider ”scissors paper stone” where being predictable loses
Solution concept: A pair of strategies
such that each is best against the other
Does not include the benefit of stealing candy from babies
Von Neumann demonstrated how to find the solution for any such game
Provided, of course, that you have unlimited computing power to do it with

5. Scissors Paper Stone The solution:
Roll a die out of sight of your opponent
1-2 scissors, 3-4 paper, 5-6 stone
Whatever your strategy, I win 1/3rd, lose 1/3rd, tie 1/3rd
Average payout zero
If you follow the same strategy, whatever I do gets the same average payout
So a Von Neumann solution
And it does not matter if you know my strategy
As long as you can’t see the die
And similarly if I know yours
Which is true in general of a VN solution

6. Many Player Not Fixed Sum
VN Solution concept: A set of outcomes (who gets what)
Such that any outcome not in the set is dominated by one in the set
Where one outcome is dominated by another if
The people who prefer it (get more in it)
Are sufficient, working together, to get it
There may be many different solutions
Each containing many outcomes
So a “solution” in a very weak sense

7. Three Player Majority Vote: Allocating $1
Solution: (.5,.5,0), (0,.5,.5) (.5,0,.5)
Consider any other allocation of the dollar
There is always one of these that two people prefer
So every other allocation is dominated by one of these
Solution: (.1, x, .9-x) for all values of 0>x>.9
Also a solution, but one that includes
An infinite number of allocations
Try to find an allocation that isn’t dominated by one member of either the first or the second set of allocations

8. Bilateral Monopoly Selling an apple Putting a child to bed
A Doomsday Machine

9. The Human Doomsday Machine
Defer to me or I beat you up
A fight hurts both of us, but …
You don’t want to be hurt, so I don’t have to beat you up
Hawk/Dove game
Equilibrium number of hawks
Why crimes of passion can be deterred

10. Economics of Vice and Virtue
Economics of vice: The bully strategy
Economics of virtue
Why are there people who won’t steal
Even if they are sure nobody is looking?
What if your utility function was written on your forehead?
The cost to me of hiring someone who will steal from me
Is greater than the benefit to him of stealing from me
So I will pay the honest man more than enough more so that honesty pays
Your utility function is written on your forehead
With a fuzzy pencil
So honesty pays
Unless you are a very talented con man

11. Implication of the economics
The bully strategy only works for involuntary interactions
If you announce at the employment interview that you beat people up if they don’t do what you want
You don’t get the job
The virtue strategy only works for voluntary interactions
So a society where more interaction is voluntary will have less vice and more virtue.
Nicer people

12. Prisoner’s Dilemma Mike Confess Say Nothing Joe 2 years, 2 years
3 months, 5 years
5 years, 3 months
6 months, 6 months
Dominance as a solution concept
if choice A is better for me than choice B
Whatever the other player does
Then A dominates B
“Confess” dominates “Say Nothing” for both criminals
So they get 2 years each instead of 6 months each

13. Repeated Prisoner’s Dilemma
With a fixed number of plays
Both Criminals know they will betray on the last round
So …
A good reason not to get into that game

14. Consider Plea Bargaining as a Real Prisoner’s Dilemma
If most defendants cop a plea
The D.A. can spend lots of resources on those who don’t
Which raises the risk to those who don’t
Which makes those who do willing to accept worse terms
They might all be better off going to trial
Since the D.A. doesn’t have the resources to convict them all

15. Some More Approaches to Solving Games
Nash Equilibrium
Each player is acting optimally, given what every other player is doing
This assumes we can freeze the other players while I adjust my strategy
But what does “freeze” mean?
Consider Oligopoly: A few firms in competition
When I increase my output
The other firms cannot keep both price and quantity fixed
Because a larger quantity (due to my increase) implies a lower price
If we assume they hold quantity fixed we get one equilibrium
If we assume they hold price fixed, a different equilibrium
So the outcome depends on how we define a strategy

16. Subgame Perfect Equilibrium
A sequential game
First I make a choice then you make a choice then
Could be more than two players
And each sequence of choices ends with a payoff for each player
Go to the last choice, assume the chooser takes the branch that gives him the higher payoff. Cross off the other branch. Now the game is one step shorter
Repeat as you go back up the diagram
But …

17. The Tantrum Game

18. Commitment Strategies Matter
And kids have them too

19. Implications We have solved nothing harder than scissors paper stone
But learned some useful ways of thinking about strategic behavior
Commitment strategies matter
Bilateral monopoly with a wide range may be expensive
Crimes of passion are to some degree deterable
Plea bargaining may make average sentences higher

20. The Value of Life We measure value to you by our choices
You would not sell your heart for a million dollars
Not even a billion
Does that mean that the value of life is infinite?
If so, or not, what are the implications?
For building bridges and highways
Paying for medical procedures
Awarding damages for tortious loss of life
And living

21. Infinite Relative to What?
Relative to something of value, no
Or we would all have much safer lives than we do
Also more boring
Relative to dollars to a dead man, yes
The issue is not multiplying by infinity
But dividing by zero
1,000,000/0 = ∞

22. How Might We Measure the Value of Life?
Measured by revealed preference
The choices we make
Income premium for a risk of profession
Price people will pay for additional safety
What if the best surgeon costs more than the good surgeon?

23. How Might We Use the Information?
Highway design
Death panels–What Medicare will pay for
FDA drug approval? Tradeoffs: More life, less money, risks
Tort damages for tort loss of life
How we now do it
Based on the value of your life to everyone but you
How we should do it—standard Pigouvian argument
Hedonic damages

24. Tort Damages for Loss of Life: The Problem
A damage payment plays two roles
The disincentive to the injurer
Compensation to the victim
There is no reason to expect the optimal sum to be the same for both
In the case of loss of life, “compensation” is life insurance
The fact that I don’t want any (no wife or children, or I am already wealthy)
Does not mean that my life is of no value to me
The solution
Base damages on the value of my life to me and
Let me sell life insurance on myself
I can then spend the money in the future where I am not killed
So the value is measured in dollars when I am alive, and finite

25. A Market in Inchoate Torts
What I am selling is only life insurance if I am tortuously killed
So it is really my tort claim for a tort that has not yet happened
There are two reasons why this can’t be done at present
Damages awarded are not based on the value of my life to me
And tort claims cannot be sold in advance
Or sold at all, save in a few limited cases
Contingency fee for a lawyer
Selling the claim to the defendant—an out of court settlement

26. Other Advantages to Making Tort Claims Marketable
Protecting the poor—who cannot afford a lawyer
That was the system in saga period Iceland
Where all prosecution and enforcement were private
Kill someone, his relatives sue you
Our legal system is only a thousand years behind theirs
Finding the right lawyer
The poor plaintiff might find a firm to take the case on a contingency basis
But how does he know which firm will do the best job?
The one that offers the highest price for his claim
More advantages to be discussed later

27. Damages for Injury Losing your vision lowers both
Your utility and
Your ability to turn dollars into utiles
Fully compensating you ex post
May be impossible
And transfers from high MUI to low MUI
The blind billionaire problem
But we could fully compensate ex ante
Just as with loss of life
By letting you sell as much of the damages as you want in advance
Keep the rest as insurance

28. Damages for Injury in Jewish and Islamic Law
How much is your right arm worth?
What we need is a market
And they had one
How much would you be worth if sold as a slave with both arms?
How much less with only one arm?
The difference is the damage for the loss to you of the arm
Maimonides included a term for pain and suffering
If condemned to have your hand cut off
How much would you pay to have it done under drugs (opium?)
That’s how much more the person who cut your hand owes you

29. Thinking About the Unthinkable
People don’t like the idea of setting a value on life
A car company that could make the car safer and doesn’t
Because it costs $100,000,000 per life saved
And it doesn’t think the customers value their lives that high
Would be wise not to leave evidence that it made that decision
Lots of government decisions have an implicit value of life
When is it worth banking a curve on a highway, for example
What medical care will Medicaid fund
If you calculate the implicit value, it varies enormously from one decision to another
Possibly because politicians and bureaucrats also find it prudent not to be too explicit that that is what they are doing
Why “Death Panels” was effective rhetoric

30. To Think About Why is Marriage Different?
Why are patent law and copyright law so different?
Why are some torts strict liability, some negligence?
How, in principle, do you set the punishment for a crime?
Why do we have both criminal law and tort law?