Get documents about "later Recall
Document Sample
Part A-III
(continued)
Microeconomic Theory
Review (continued)
09/22/09 Intro_E 1
Economics One-on-One
The limitations of the Market Model
Game Theory
A model of Bargaining
09/22/09 Intro_E 2
To begin with
Remember trading is ‘good’
Voluntary exchange of anything between any
two agents improves the wellbeing of both
agents (people, firms – parties)
Voluntary exchange generally represents a
Pareto Improvement
Generally? Why generally? Remember the concept of market failures:
-Monopoly (monopsony) -Externalities
-Public goods -Severe informational asymmetries
09/22/09 Intro_E 3
Market failures aside,
One primary goal of any economic system is to
structure itself so as to facilitate voluntary
exchange among the economic agents.
By ‘facilitate’ we mean
Make it ‘easy and cheap’
- minimize transaction costs
More on this later
09/22/09 Intro_E 4
Recall, a market is any coming together of buyers and sellers
Markets help facilitate voluntary exchange (trading) and trading
generally increases wellbeing – they lower the cost of trading
In first and second year you learn about the ‘market model’.
This is one particular model of how trading occurs.
The ‘market model’ describes the process of trading
- coordinating the utility and profit maximizing behaviour of
many independent of individual agents
09/22/09 Intro_E 5
The ‘market model’ is intended to explain how
trading takes place when there are many
relatively small individual agents coming together
to trade in a relatively large and very impersonal
environment.
- a farmer selling wheat
- you buying apples
No individual buyer or seller can affect the outcome
– each agent acts independently.
Welfare improving trades occur at a price set by
‘the market’
09/22/09 Intro_E 6
The market model
Market for Apples
Price of apples
Supply of apples
(all potential apple producers)
Pe
Demand for apples
(all potential apple consumers)
Qe
Quantity of apples
09/22/09 Intro_E 7
Limitations of the market model
Many buyers and/or sellers
All buyers are buying and all sellers are
selling the same (or similar) item
Perfect (very good) information
Price is known
Price is ‘given’ (accepted) by buyers and
sellers (or at least one of the buying side
or selling side)
09/22/09 Intro_E 8
But what if ?
There is only one buyer and one seller (or just a few decisions
makers)
The item being sold is more or less unique
There is no established market price set prior to trading
Can voluntary exchange (trading) still take place? - YES
Will it be welfare improving? – GENERALLY
What can the market model say about the process of
trading? – NOT MUCH
What can economic theory say about the process of
trading and the outcome? - LOTS
09/22/09 Intro_E 9
Economics One-on-One
- few decision makers (agents)
- the optimal decision of each agent depends on the
decision of the other agents - interdependence
In such a situation, each agent must formulate a
‘strategy’ which accounts for the possible decisions of
the other agents.
This is a ‘game’ - poker, football, bridge, dating,
politics, war, whatever.
09/22/09 Intro_E 10
Courts and lawyers (The Law) frequently deal with
situations in which there are a few decision
makers making decisions that are
interdependent.
Game theory is a formal method of analysis which
allows us to understand (and predict) the
behaviour of rational economic agents when
there are a few agents making interdependent
decisions.
09/22/09 Intro_E 11
Basic Game Theory
The Basics of a game:
- the players (economic agents)
- the strategies of each player
- the payoffs to each player under each strategy
- determining each player’s optimal strategy and
the game outcome
09/22/09 Intro_E 12
Example: Mary and John each want to
sell something
Situation: - two potential sellers of the same good
(home owners, car owners, merchants
contemplating a sale, etc.)
- if Mary offers her good for sale and John does
not, then Mary will earn $1.
- if Mary offers her good for sale and John offers
his good for sale, Mary will suffer a $2 loss.
09/22/09 Intro_E 13
- if John offers his good for sale and Mary does
not, then John will earn $1.
- if John offers his good for sale and Mary
offers her good for sale, John will suffer a $2
loss.
- if either seller does not sell, then that seller
will break even.
09/22/09 Intro_E 14
In the above ‘Game’
Players: Mary and John
Strategies: Sell or don’t sell
Payoffs to each player under each strategy:
(listed in the last two slides)
The decisions of Mary and John are clearly interdependent.
Each must consider what the other agent is going to do
before they make their own decision
Can we predict the outcome? What will Mary
and John do?
09/22/09 Intro_E 15
Let’s consider what we know in the form of a ‘payoff matrix’:
the value to each player for each of the possible outcomes
PAYOFF MATRIX
Mary’s decision
SELL DON’T SELL
John’s decision
SELL (-$2, -$2) ( $1, $0)
DON’T SELL ( $0, $1) ( $0, $0)
(first entry in the round brackets represents John's payoff
second entry represents Mary's payoff)
09/22/09 Intro_E 16
What will each player do? Consider Mary’s options
and the resulting outcomes – the ‘extensive form’
or a ‘decision tree’
Mary’s payoff
If John sells
and Mary sells -$2
and Mary does not sell $0
If John does not sell
and Mary sells $1
and Mary does not sell $0
09/22/09 Intro_E 17
Assume that Mary and John do not cooperate –
neither will tell the other agent what they intend to
do
Mary has no information on what John will do so
she must assume that there is a 50/50 chance of
him selling or not selling. In this case:
the expected value to Mary if she sells is
.5(-$2) + .5($1) = -$0.50
the expected value to Mary if she does not sell is
.5($0) + .5($0) = $0
09/22/09 Intro_E 18
Prediction: Mary will not sell
What about John?
How does he view the situation?
In this particular non-cooperative game with no
information – exactly the same way
09/22/09 Intro_E 19
John’s payoff
If Mary sells
and John sells -$2
and John does not sell $0
If Mary does not sell
and John sells $1
and John does not sell $0
09/22/09 Intro_E 20
If John has no information on what Mary will do so
he also must assume that there is a 50/50
chance of her selling or not selling. In this case:
the expected value to John if he sells is
.5(-$2) + .5($1) = -$0.50
the expected value to John if he does not sell is
.5($0) + .5($0) = $0
09/22/09 Intro_E 21
Prediction: Both Mary and John will decide
not to sell
Is this a good outcome?
It is the best that Mary and John can do in the
situation: no information/no co-operation
We would say that each player has a dominant
strategy
‘ the optimal move for a player to make
irrespective of the other player’s decision’
09/22/09 Intro_E 22
In a sense, this is a pretty amazing result.
Neither agent knows anything about the other
agent’s expected payoffs (the nature of their
utility or profit function) nor the other agent’s
likely strategy. Each agent only knows their own
utility or profit function.
Yet in this particular non-cooperative game with
no information we can predict how each agent
will behave and the game outcome
09/22/09 Intro_E 23
Dominant strategies and equilibrium
Since each player has a dominant strategy we can predict
what each player will do (in this game – it is ‘not sell’)
This leads to a prediction that the equilibrium for this game
is such that neither Mary not John will offer their item for
sale
In this particular case the equilibrium is a Nash Equilibrium
‘no individual player can do any better by changing
their decision so long as the other player does not
change their decision’
For a non-repeated game this is it?
09/22/09 Intro_E 24
But is there a better outcome for Mary and
John and for society?
Is the solution in the above game a Pareto
Optimum?
Can Society as a whole do better?
The game which we just described is a non-co-
operative game with no information
But what if Mary and John decide to meet and co-
operate?
09/22/09 Intro_E 25
With no co-operation Mary and John ended up at the ($0, $0)
payoff
PAYOFF MATRIX
Mary’s decision
SELL DON’T SELL
John’s decision
SELL (-$2, -$2) ( $1, $0)
DON’T SELL ( $0, $1) ( $0, $0)
Are there better outcomes for John and Mary (and
society)
09/22/09 Intro_E 26
What if John and Mary decide to co-
operate (jointly maximize profits)
Mary and John meet and decide that one of them
will sell and one will not sell. Together they will
make $1 in profit and then they will split the profit
(say $0.50 each).
PAYOFF MATRIX
Mary’s decision
SELL DON’T SELL
John’s decision
SELL (-$2, -$2) ( $1, $0)
DON’T SELL ( $0, $1) ( $0, $0)
09/22/09 Intro_E 27
Solution to the game if there is co-
operation
Mary and John are each better off and society is
better off – the co-operative outcome is a Pareto
Improvement (at least one person is made better
off and no one is made worse off). Actually three
people can be made better off (Mary, John and
the buyer).
Trading is good and in this case co-operation
among the players leads to trading which
otherwise would not take place.
09/22/09 Intro_E 28
Variations on the ‘Mary and John want
to sell something game’
Go back to no co-operation state but what if one
player has information on the other player’s likely
strategy?
How could this happen?
- reputation (watched this player play before)
- espionage (industrial spying)
- John (mistakenly) ‘sent’ a signal
09/22/09 Intro_E 29
Suppose Mary has reason to believe that there is a
75% chance that John will not sell
Then Mary believes that the probability of John
selling or not selling is 25/75. In this case:
the expected value to Mary if she sells is
.25(-$2) + .75($1) = $0.25
the expected value to Mary if she does not sell is
.25($0) + .75($0) = $0
09/22/09 Intro_E 30
What about John? If John has not acquired any new
information then his best belief on what Mary will
do remains that there is a 50/50 chance of her
selling or not selling. In this case nothing has
changed for John:
the expected value to John if he sells is
.5(-$2) + .5($1) = -$0.50
the expected value to John if he does not sell is
.5($0) + .5($0) = $0
09/22/09 Intro_E 31
Prediction: Mary will sell and earn $1 profit,
John will not sell
Mary’s dominant strategy has changed to sell while
John’s remains don’t sell. So the model predicts
that Mary will sell and John will not sell.
PAYOFF MATRIX
Mary’s decision
SELL DON’T SELL
John’s decision
SELL (-$2, -$2) ( $1, $0)
DON’T SELL ( $0, $1) ( $0, $0)
09/22/09 Intro_E 32
Is this still a Nash Equilibrium? Yes. Neither John
nor Mary can improve their situation if the other
player’s decision remains unchanged.
Is it a Pareto Improvement? Yes. Trading takes
place: poor John/fortunate Mary but economics
does not care who wins, as long as no one loses.
Two people have been made better off (Mary and
the buyer) John is no worse off.
09/22/09 Intro_E 33
Variations on the ‘Mary and John want
to sell something game’
What if the players value the outcomes differently?
How could this happen?
- Different taste (selling is more important to one than the
other)
- Different profit functions (one agent stands to gain more
if they sell)
- Different risk functions (one agent can deal with a loss if
it occurs better than the other agent)
09/22/09 Intro_E 34
Suppose that John stands to gain $3 if he sells and
Mary only $1. Then the payoff matrix is the
following:
PAYOFF MATRIX
Mary’s decision
SELL DON’T
SELL
John’s decision
SELL (-$2, -$2) ( $3, $0)
DON’T SELL ( $0, $1) ( $0, $0)
09/22/09 Intro_E 35
John’s payoff
If Mary sells
and John sells -$2
and John does not sell $0
If Mary does not sell
and John sells $3
and John does not sell $0
09/22/09 Intro_E 36
If John has no information on what Mary will do so
he also must assume that there is a 50/50
chance of her selling or not selling. In this case:
the expected value to John if he sells is
.5(-$2) + .5($3) = $0.50
the expected value to John if he does not sell is
.5($0) + .5($0) = $0
09/22/09 Intro_E 37
John will sell but Mary will not sell. Again a Pareto
Improvement outcome but John and the buyer
are the ‘winners’
PAYOFF MATRIX
Mary’s decision
SELL DON’T
SELL
John’s decision
SELL (-$2, -$2) ( $3, $0)
DON’T SELL ( $0, $1) ( $0, $0)
09/22/09 Intro_E 38
Another Game: The Prisoner’s Dilemma
(why they always get confessions on NYPD
BLUE and Law & Order - or how the Boulder
Police Department might have screwed-up.)
Two individuals (Jack and Jill) are
suspected of involvement in a crime
09/22/09 Intro_E 39
Situation: - two parents (Jack and Jill) are suspected
of involvement in a crime, say child murder
- Police only have circumstantial evidence and this
it not enough to get a conviction.
- if one (or both) of the suspects confess, then the
prosecutor will get a conviction (possibly two
convictions) The police/prosecutors need a
confession.
- no matter what, the child was harmed and the
parents can be convicted of child neglect (a lesser
crime
09/22/09 Intro_E 40
Here’s what the police do: First, each suspect is
taken to a different room – no co-operation
Jack is told
- if you confess and implicate your partner and
your partner does not confess, then
- she will be charged with murder (25 years in
prison)
- you will be charged with a lesser crime (no
time in prison).
- if you confess and implicate your partner and
your partner also confesses and implicates
you, then
- you will both be charged with manslaughter
and each of you will spend 10 years in prison.
09/22/09 Intro_E 41
- If neither of you parents confess, the prosecutor
threatens to harass them publicly and ultimately
charge them with child neglect for which they will
spend up to 1 year in jail.
Jill is told exactly the same thing
09/22/09 Intro_E 42
In the above ‘Game’
Players: Jack and Jill
Strategies: Confess or don’t confess
Payoffs’ to each player under each strategy:
(listed in the last two slides)
The decisions of Jack and Jill are clearly interdependent. Each
must consider what the other agent is going to do before
they make their own decision
What will Jack and Jill do? What are their options?
09/22/09 Intro_E 43
What does the payoff matrix look like ?
PAYOFF MATRIX
Jill’s decision
Confess Don’t confess
Jack’s decision
Confess (10 yrs, 10 yrs) (0 yrs, 25 yrs)
Don’t confess (25 yrs, 0 yrs) (1 yr, 1 yr)
(first entry in the round brackets represents Jack's payoff
second entry represents Jill's payoff)
09/22/09 Intro_E 44
What will each player do? Consider Jill’s options
and the resulting outcomes – the ‘extensive form’
or a ‘decision tree’
Jill’s payoff
If Jack confesses
and Jill confesses 10 yrs
and Jill does not confess 25 yrs
If Jack does not confess
and Jill confesses 0 yrs
and Jill does not confess 1 yr
09/22/09 Intro_E 45
Jack and Jill can not communicate so they cannot
cooperate – neither can tell the other person what
they intend to do
If Jack confesses then Jill’s best response is to
confess
If Jack does not confess then Jill’s best option is to
confess
The game has been carefully constructed by the
police/prosecutor so that a rational individual will
choose to confess.
In this game we need not consider the probabilities.
09/22/09 Intro_E 46
Prediction: Jill will confess and since Jack
faces the exact same options his dominant
strategy is also to confess
They each end up spending 10 years in jail
Jill’s decision
Confess Don’t confess
Jack’s decision
Confess (10 yrs, 10 yrs) (0 yrs, 25 yrs)
Don’t confess (25 yrs, 0 yrs) (1 yr, 1 yr)
09/22/09 Intro_E 47
What if Jack and Jill are allowed to co-operate,
then what are they likely to do?
They are likely to agree that neither will confess
and each will spend 1 year in prison.
Does guilt or innocence matter in determining the
outcome? Yes and no. It must be the case that
each feels that the threat of conviction on a
lesser offence is real.
Loyalty and the general nature of the relationship
between the two suspects gets mixed in.
09/22/09 Intro_E 48
Recall the Ramsey Case in Colorado
– Jon Benet Ramsey
what if Mr. Ramsey committed the murder, Mrs.
Ramsey knows he did since she arrived at the
scene just after the crime was committed?
Her natural tendency would be not to confess, since
she did not commit the crime.
However, what if Mr. Ramsey is a real creep and
she believes that he might confess, implicating
her and getting himself off free, while she spends
25 years in prison. Maybe she should confess, 10
years in prison is better than 25 years in prison.
09/22/09 Intro_E 49
What if Mr. and Mrs. Ramsey are both guilty but
they care deeply for each other. Then neither
would confess and both would spend 1 year in
prison. But this is really just co-operation
between the suspects, whether or not they are
interviewed separately. The police always try to
create mistrust among the suspects.
In the good old days the Mafia got cooperation
without any direct communication. If you
squealed you died. The Mafia can be thought of a
way of ensuring cooperative outcomes among a
group of criminals.
09/22/09 Intro_E 50
The `divide and conquer’ technique is likely to be more
successful the more distrust there is among the suspects.
What if both are completely innocent and neither confesses.
They will both spend some time in jail, 1 year. This always
has to be the case. The prosecutor must set up the options
so that there is some positive reward for confessing
(alternatively stated, some punishment for not confessing).
Do innocent people confess to crimes they know they did not
commit?
Could you be put in a position in which your best option was
to admit to having committed a crime which you did not
commit?
09/22/09 Intro_E 51
What went wrong (or right!) in Boulder?
The Ramsey’s lawyers refused to allow the Ramseys be
interviewed separately.
Is this a reflection on their guilt or innocence?
No, we saw that once you become entangled in the Prisoners’
Dilemma, you will be punished to some extent whether or
not you are guilty.
That is one reason why prosecutors ‘leak’ embarrassing
information about suspects who refuse to play the game.
They are being punished for not playing (the situation
becomes a multi-level game - a bit more complicated).
09/22/09 Intro_E 52
Change the setting.
Two young Irishmen arrested in a pub in London England after
a bomb goes off just down the street.
Four teenagers, each with outstanding warrants for parole
violations, picked-up near the scene of a recent robbery.
Two Palestinians in Jerusalem, two Middle Easterners
anywhere in the world.
Would a law that states that a suspect cannot be convicted
solely on the testimony of an accomplice make sense?
Would a law that allows for a joint defense of jointly accused
defendants make sense?
09/22/09 Intro_E 53
