Market | What Next? 1
Uncertainty
All the models considered so far have one thing in common,There is no uncertainty.
This is a very restrictive assumption,Often in economic situations there is less than perfect information,Both
production and consumption often involve unknown variables that afiect the proflts and utility of the agents.
When an agent makes a decision (about consumption for example) the utility they will receive may be uncertain.
How can decision making under uncertainty be modelled?
Random chance of this sort in economics is referred to as nature taking an action.
Nature decides which state of the world occurs,For example,a consumer is considering whether or not to purchase
a car,Unfortunately cars can break down,If the consumer purchases the car,nature \decides" whether the car
breaks down or not,These are the two possible states of the world.
How can the consumer compare the utility from purchasing the car with the utility from keeping their money?
Market | What Next? 2
Lotteries
Economists model this sort of decision problem as a choice between lotteries.
One lottery involves buying the car (which costs 500) and then facing the uncertainty of a possible break down
(which occurs 20% of the time),The other lottery involves keeping the money for sure.
...............
...............
...............
...............
...............
..............
...............
...............
...............
...............
...............
............
..................................................................................uc
1000
0
un 500
0.2
0.8
1
Buying the car Keeping the money
Suppose a car is worth 1500 to the agent,and the cost of repairing after a break down is 1000.
How can a consumer compare these two lotteries c and n? Consumers have preferences over lotteries,Either n,c
or c,n,Representing this preference relation as a utility function,Either un? uc or uc? un.
Recall consumer preferences over bundles,An analogous argument for the case of preferences over lotteries can be
made,Preferences that satisfy certain consistency requirements can be represented by an expected utility function.
Market | What Next? 3
Expected Utility
The consistency requirements include re exivity,completeness and transitivity as well as two others.
A lottery x is a set of possible outcomes fx1;,,,; xng and a set of probabilities for each outcome fp1;,,,; png.
The representation theorem states that a preference relation over lotteries,”,that satisfles these conditions can be
represented by a von-Neumann Morgernstern expected utility function:
x,y ()
nX
i=1
piu(xi)?
mX
i=1
qiu(yi)
The vNM expected utility for lottery x is U(x)= Pni=1 piu(xi) where u(xi) is the regular utility from outcome xi.
In the earlier example,suppose the utility from each outcome is simply the money payofi,The probability of each
outcome is known,If the consumers preferences can be represented by such a vNM expected utility function then:
U(c)=(0:8￡1000)+(0:2￡0) and U(n)=(1￡500) so U(c) > U(n) thus c,n
In this example the consumer prefers the lottery c and will buy the car.
Market | What Next? 4
Risk Aversion
The consumer has taken the action involving uncertainty,Do they then like risk? Not necessarily.
A common assumption is that of risk aversion,A consumer is risk averse when their utility function is concave.
Suppose there is just one good | money m,The consumer gets regular utility u(m) from money m.
......
.
......
.
......
.
......
.
......
.
......
.
......
.
......
.
......
.
......
.......,.
.......,......,......,......,......,......,......,......,......,......,......
......
.
......
.
......
.
......
.
......
.
......
........,......,......,......,......,......
.......,......,......,......,......,......
......
......
......
.....
......
......
......
......
......
.....
......
.....
......
......
......
.....
......
......
......
......
......
......
...
......................................................................................................................................................................................................
............
............
............
............
............
............
............
............
............
............
............
........
.......
.......
.....
....
......
......
......
.......
.......
...........
............
.............
...............
.................
..................
.......................
.............................
...................
0
u
m
u(m)
1 5 9
u(1)
u(5)
u(9)
1
2u(1)+
1
2u(9)
In the above diagram the consumer prefers 5 for sure than 9 with probability half and 1 with probability half,The
expected value of the latter lottery is 5,But u(5)? 12u(1)+ 12u(9),This is because u is concave | risk aversion.
In the earlier example the consumer had u(m)= m,This is a case of risk neutrality.
Market | What Next? 5
Asymmetric Information
What happens when individuals have difierent information? This is called a situation of asymmetric information.
The less informed individual is called the principal (P),the more informed is called the agent (A).
Economists think about asymmetric information problems using \principal-agent" models.
For example,a second-hand car salesman knows the quality of a particular used car,but the buyer does not,The
buyer is the principal and the salesman is the agent,Asymmetric information is an important type of market failure.
Typically the agent takes a hidden action or has hidden information,The principal tries to learn about the action
or information whilst the agent tries to conceal or reveal the action or information.
How does a market fail in the presence of asymmetric information?
Market | What Next? 6
Adverse Selection
Nature chooses how good a particular car is,It can be bad or good with equal probability,The principal (buyer)
ofiers a price,The agent (seller) can either accept or reject this price.
.............
.............
.............
.............
.............
...........
.............
.............
.............
.............
.............
..........
....................................................................,.............
.............
.............
.............
.............
...........
.............
.............
.............
.............
.............
..........
......
......
.....................
.....
......
.......
..................
......
......
......
.....................
.....
......
.......
..................
......
......
......
.....................
.....
......
.......
.................
......
.
.....
......
.
.....
.....
......
.
.....
N P A
Good
Bad
Contract
Accept
Reject
A good car is worth 1000,A bad car is worth 200,The principal might think it equally likely for a car to be good
or bad,Therefore they might ofier a price 600,The agent,however,knows the quality of the car.
If the car is good,they will not sell it for 600,(It is worth 1000),If it is bad they will sell it,as it is worth 200.
Only bad cars will be sold for a price below 1000,The principal knows this and ofiers a price 200.
There is no equilibrium in which good cars are sold,Half the market collapses | a serious failure.
Market | What Next? 7
Moral Hazard
Consider a farmer trying to hire a worker,The farmer (principal) ofiers a contract which the worker (agent) can
accept or reject,If the worker joins the farm they can be lazy or work hard.
The weather can be good or bad,If it is good,crop yields will be higher than if it is bad,The farmer observes the
yield but not the efiort of the worker,They do not know whether the good (or bad) yield is due to efiort or weather.
....................................................................,.............
.............
.............
.............
.............
...........
.............
.............
.............
.............
.............
..........
....................................................................,.............
.............
.............
.............
.............
...........
.............
.............
.............
.............
.............
............
.......
...........................
.....
......
.........................
...,..
......
...........................
....
......
..........................
...
......
.......
..................
......
......
.......
..................
......
......
......
.....................
.....
......
.......
.................
......
.
......
......
.
......
.
......
.
......
.
.....P A
1
A2 N
Good
Bad
Efiort
Contract
Accept
Reject?
This is an example of moral hazard,Once given the job,the worker might have an incentive to do nothing and
\blame it on the weather",The principal needs to give the worker the right incentives to work hard.
Notice the previous example was one of hidden information,Here,it is the action that is hidden from the principal.
Again,there is market failure in the sense that a less than e–cient amount of efiort is provided.
Market | What Next? 8
Contracts and Incentives
What are the solutions to these types of market failure? In the case of moral hazard,the principal needs to draw up
an efiective contract to induce the worker to put in high efiort levels.
The principal faces two constraints,(i) a participation constraint (the agent must accept the contract) and (ii) an
incentive compatibility constraint (the agent must choose efiort to maximise their payofi).
In the case of adverse selection one possible solution is a signal | a warranty in the car example.
.............
.............
.............
.............
.............
...........
.............
.............
.............
.............
.............
..........
....................................................................,...................................................................,.............
.............
.............
.............
.............
...........
.............
.............
.............
.............
.............
..........
......
......
.....................
.....
......
.......
..................
......
......
......
.....................
.....
......
.......
..................
......
......
......
.....................
.....
......
.......
..................
......
......
......
.....................
.....
......
.......
.................
......
.
.....
......
.
.....
.....
......
.
.....
N A1 A2P
Good
Bad
Signal Contract
Accept
Reject
Of course,the agent needs to convince the principal that the signal is reliable,The salesman can’t simply claim the
car is good,In fact the warranty needs to be more costly to obtain for the bad car than it is for the good car.
Signals can rescue markets from the serious collapses due to adverse selection | good cars can be sold!
Market | What Next? 9
Signalling,Pooling and Separating
The following example illustrates some of the di–culties with signalling.
An individual (the agent) is either clever (C) or stupid (S),They know this,but an employer (the principal) does
not,They can either take a degree (D) or not (O),The employer observes whether they take a degree.
Nature decides whether the individual is clever,50% of the time they are clever.
......
......
......
......
....
......
......
......
......
....
....................................................................
........................................................................................................................................
......................................................................................
..................
.......
..................
..................
......
..................
..................
.......
..................
..................
.....
..................
..................
.......
..................
..................
.....
..................
..................
.......
..................
..................
.....
......
......
.....................
.....
......
.......
..................
......
......
......
...................
......
......
......
...................
......
......
......
.....................
.....
......
.......
..................
......
......
.......
..................
......
......
.......
..................
......
......
.......
..................
......
......
.......
..................
......,...........
...................
......
......
.......
................
......
.
......
.......
................
......
.....
......
.......................
....
......
......
.
......
.
......
.
......
.
......
.
......
......
......
.
......
.
......
.
......
.
......
.
......
N
AC
AS
PD
PD
PO
PO
Hire
Hire
Hire
Hire
Not hire
Not hire
Not hire
Not hire?
Degree
Degree
No degree
No degree
0.5
0.5
Notice that the payofis are omitted,the decision of the employer is to hire or not,The dotted lines represent an
information set | the principal does not know which node they are at.
There may be a separating equilibrium or a pooling equilibrium.
Market | What Next? 10
Auctions
An important (and successful) application of game theory is to auctions.
Auctions are mechanisms for selling goods,they difier from the market mechanism presented in earlier lectures.
There are many difierent types of auction available for modelling,Here are some:
1,English,(Or Japanese),ascending bids,last bidder left wins the object at the price they bid.
2,Dutch,Descending price,winner is the flrst to make a bid (stop the auction),They pay that price.
3,First price sealed bid,Write bids down,put in sealed envelope,Highest bidder wins and pays that price.
4,Second price sealed bid,Same procedure,Highest bidder wins but pays price of second highest bidder.
An auction can be used to sell one or many goods,The flrst and last above have the same strategic structure,The
second and third have the same strategic structure,Why? All four auctions raise the same expected revenue.
What is the optimal auction to sell a particular good? Optimal in the sense of e–ciency | the highest valuation
agent wins the object,and optimal in the sense of raising the highest revenue.
Market | What Next? 11
Dynamic Games and Equilibrium
Dynamic games can be represented in normal form,Consider the entry game of lecture 10.
..................
..................
..................
..................
..................
..................
..................
.................
..................
..................
..................
..................
..................
..................
..................
...............
..................
..................
..................
..................
..................
..................
..................
.................
..................
..................
..................
..................
..................
..................
..................
................?
E
M
Fight
Don’t Fight
Enter
Don’t Enter (0;5)
(1;2)
(?1;0)
F DF
E 0
1
2
1
DE 5
0
5
0
There are two pure Nash equilibria | fDE; Fg and fE; DFg,In fact,there are inflnitely many mixed equilibria
also | where the entrant plays DE and the monopolist plays F with at least probability half.
Market | What Next? 12
Credibility
However,working from the back of the game in extensive form,the entrant will always choose to enter given they
know that if they do,the monopolist will not flght,The threat to flght is not credible.
An equilibrium is subgame perfect if it is a Nash equilibrium of each subgame and of the whole game.
A subgame is simply a game starting at any particular node,Subgame perfection rules out incredible threats.
The only subgame perfect equilibrium in this game is fE; DFg,This is a reflnement of Nash equilibrium.
Can the monopolist ever deter entry? If the game is repeated many (many) times then,yes.
The folk theorems prove that in an inflnitely repeated game almost anything is subgame perfect,For example,
flghting in the above game,collusion in oligopoly and even cooperation in the repeated prisoners’ dilemma.
Uncertainty
All the models considered so far have one thing in common,There is no uncertainty.
This is a very restrictive assumption,Often in economic situations there is less than perfect information,Both
production and consumption often involve unknown variables that afiect the proflts and utility of the agents.
When an agent makes a decision (about consumption for example) the utility they will receive may be uncertain.
How can decision making under uncertainty be modelled?
Random chance of this sort in economics is referred to as nature taking an action.
Nature decides which state of the world occurs,For example,a consumer is considering whether or not to purchase
a car,Unfortunately cars can break down,If the consumer purchases the car,nature \decides" whether the car
breaks down or not,These are the two possible states of the world.
How can the consumer compare the utility from purchasing the car with the utility from keeping their money?
Market | What Next? 2
Lotteries
Economists model this sort of decision problem as a choice between lotteries.
One lottery involves buying the car (which costs 500) and then facing the uncertainty of a possible break down
(which occurs 20% of the time),The other lottery involves keeping the money for sure.
...............
...............
...............
...............
...............
..............
...............
...............
...............
...............
...............
............
..................................................................................uc
1000
0
un 500
0.2
0.8
1
Buying the car Keeping the money
Suppose a car is worth 1500 to the agent,and the cost of repairing after a break down is 1000.
How can a consumer compare these two lotteries c and n? Consumers have preferences over lotteries,Either n,c
or c,n,Representing this preference relation as a utility function,Either un? uc or uc? un.
Recall consumer preferences over bundles,An analogous argument for the case of preferences over lotteries can be
made,Preferences that satisfy certain consistency requirements can be represented by an expected utility function.
Market | What Next? 3
Expected Utility
The consistency requirements include re exivity,completeness and transitivity as well as two others.
A lottery x is a set of possible outcomes fx1;,,,; xng and a set of probabilities for each outcome fp1;,,,; png.
The representation theorem states that a preference relation over lotteries,”,that satisfles these conditions can be
represented by a von-Neumann Morgernstern expected utility function:
x,y ()
nX
i=1
piu(xi)?
mX
i=1
qiu(yi)
The vNM expected utility for lottery x is U(x)= Pni=1 piu(xi) where u(xi) is the regular utility from outcome xi.
In the earlier example,suppose the utility from each outcome is simply the money payofi,The probability of each
outcome is known,If the consumers preferences can be represented by such a vNM expected utility function then:
U(c)=(0:8￡1000)+(0:2￡0) and U(n)=(1￡500) so U(c) > U(n) thus c,n
In this example the consumer prefers the lottery c and will buy the car.
Market | What Next? 4
Risk Aversion
The consumer has taken the action involving uncertainty,Do they then like risk? Not necessarily.
A common assumption is that of risk aversion,A consumer is risk averse when their utility function is concave.
Suppose there is just one good | money m,The consumer gets regular utility u(m) from money m.
......
.
......
.
......
.
......
.
......
.
......
.
......
.
......
.
......
.
......
.......,.
.......,......,......,......,......,......,......,......,......,......,......
......
.
......
.
......
.
......
.
......
.
......
........,......,......,......,......,......
.......,......,......,......,......,......
......
......
......
.....
......
......
......
......
......
.....
......
.....
......
......
......
.....
......
......
......
......
......
......
...
......................................................................................................................................................................................................
............
............
............
............
............
............
............
............
............
............
............
........
.......
.......
.....
....
......
......
......
.......
.......
...........
............
.............
...............
.................
..................
.......................
.............................
...................
0
u
m
u(m)
1 5 9
u(1)
u(5)
u(9)
1
2u(1)+
1
2u(9)
In the above diagram the consumer prefers 5 for sure than 9 with probability half and 1 with probability half,The
expected value of the latter lottery is 5,But u(5)? 12u(1)+ 12u(9),This is because u is concave | risk aversion.
In the earlier example the consumer had u(m)= m,This is a case of risk neutrality.
Market | What Next? 5
Asymmetric Information
What happens when individuals have difierent information? This is called a situation of asymmetric information.
The less informed individual is called the principal (P),the more informed is called the agent (A).
Economists think about asymmetric information problems using \principal-agent" models.
For example,a second-hand car salesman knows the quality of a particular used car,but the buyer does not,The
buyer is the principal and the salesman is the agent,Asymmetric information is an important type of market failure.
Typically the agent takes a hidden action or has hidden information,The principal tries to learn about the action
or information whilst the agent tries to conceal or reveal the action or information.
How does a market fail in the presence of asymmetric information?
Market | What Next? 6
Adverse Selection
Nature chooses how good a particular car is,It can be bad or good with equal probability,The principal (buyer)
ofiers a price,The agent (seller) can either accept or reject this price.
.............
.............
.............
.............
.............
...........
.............
.............
.............
.............
.............
..........
....................................................................,.............
.............
.............
.............
.............
...........
.............
.............
.............
.............
.............
..........
......
......
.....................
.....
......
.......
..................
......
......
......
.....................
.....
......
.......
..................
......
......
......
.....................
.....
......
.......
.................
......
.
.....
......
.
.....
.....
......
.
.....
N P A
Good
Bad
Contract
Accept
Reject
A good car is worth 1000,A bad car is worth 200,The principal might think it equally likely for a car to be good
or bad,Therefore they might ofier a price 600,The agent,however,knows the quality of the car.
If the car is good,they will not sell it for 600,(It is worth 1000),If it is bad they will sell it,as it is worth 200.
Only bad cars will be sold for a price below 1000,The principal knows this and ofiers a price 200.
There is no equilibrium in which good cars are sold,Half the market collapses | a serious failure.
Market | What Next? 7
Moral Hazard
Consider a farmer trying to hire a worker,The farmer (principal) ofiers a contract which the worker (agent) can
accept or reject,If the worker joins the farm they can be lazy or work hard.
The weather can be good or bad,If it is good,crop yields will be higher than if it is bad,The farmer observes the
yield but not the efiort of the worker,They do not know whether the good (or bad) yield is due to efiort or weather.
....................................................................,.............
.............
.............
.............
.............
...........
.............
.............
.............
.............
.............
..........
....................................................................,.............
.............
.............
.............
.............
...........
.............
.............
.............
.............
.............
............
.......
...........................
.....
......
.........................
...,..
......
...........................
....
......
..........................
...
......
.......
..................
......
......
.......
..................
......
......
......
.....................
.....
......
.......
.................
......
.
......
......
.
......
.
......
.
......
.
.....P A
1
A2 N
Good
Bad
Efiort
Contract
Accept
Reject?
This is an example of moral hazard,Once given the job,the worker might have an incentive to do nothing and
\blame it on the weather",The principal needs to give the worker the right incentives to work hard.
Notice the previous example was one of hidden information,Here,it is the action that is hidden from the principal.
Again,there is market failure in the sense that a less than e–cient amount of efiort is provided.
Market | What Next? 8
Contracts and Incentives
What are the solutions to these types of market failure? In the case of moral hazard,the principal needs to draw up
an efiective contract to induce the worker to put in high efiort levels.
The principal faces two constraints,(i) a participation constraint (the agent must accept the contract) and (ii) an
incentive compatibility constraint (the agent must choose efiort to maximise their payofi).
In the case of adverse selection one possible solution is a signal | a warranty in the car example.
.............
.............
.............
.............
.............
...........
.............
.............
.............
.............
.............
..........
....................................................................,...................................................................,.............
.............
.............
.............
.............
...........
.............
.............
.............
.............
.............
..........
......
......
.....................
.....
......
.......
..................
......
......
......
.....................
.....
......
.......
..................
......
......
......
.....................
.....
......
.......
..................
......
......
......
.....................
.....
......
.......
.................
......
.
.....
......
.
.....
.....
......
.
.....
N A1 A2P
Good
Bad
Signal Contract
Accept
Reject
Of course,the agent needs to convince the principal that the signal is reliable,The salesman can’t simply claim the
car is good,In fact the warranty needs to be more costly to obtain for the bad car than it is for the good car.
Signals can rescue markets from the serious collapses due to adverse selection | good cars can be sold!
Market | What Next? 9
Signalling,Pooling and Separating
The following example illustrates some of the di–culties with signalling.
An individual (the agent) is either clever (C) or stupid (S),They know this,but an employer (the principal) does
not,They can either take a degree (D) or not (O),The employer observes whether they take a degree.
Nature decides whether the individual is clever,50% of the time they are clever.
......
......
......
......
....
......
......
......
......
....
....................................................................
........................................................................................................................................
......................................................................................
..................
.......
..................
..................
......
..................
..................
.......
..................
..................
.....
..................
..................
.......
..................
..................
.....
..................
..................
.......
..................
..................
.....
......
......
.....................
.....
......
.......
..................
......
......
......
...................
......
......
......
...................
......
......
......
.....................
.....
......
.......
..................
......
......
.......
..................
......
......
.......
..................
......
......
.......
..................
......
......
.......
..................
......,...........
...................
......
......
.......
................
......
.
......
.......
................
......
.....
......
.......................
....
......
......
.
......
.
......
.
......
.
......
.
......
......
......
.
......
.
......
.
......
.
......
.
......
N
AC
AS
PD
PD
PO
PO
Hire
Hire
Hire
Hire
Not hire
Not hire
Not hire
Not hire?
Degree
Degree
No degree
No degree
0.5
0.5
Notice that the payofis are omitted,the decision of the employer is to hire or not,The dotted lines represent an
information set | the principal does not know which node they are at.
There may be a separating equilibrium or a pooling equilibrium.
Market | What Next? 10
Auctions
An important (and successful) application of game theory is to auctions.
Auctions are mechanisms for selling goods,they difier from the market mechanism presented in earlier lectures.
There are many difierent types of auction available for modelling,Here are some:
1,English,(Or Japanese),ascending bids,last bidder left wins the object at the price they bid.
2,Dutch,Descending price,winner is the flrst to make a bid (stop the auction),They pay that price.
3,First price sealed bid,Write bids down,put in sealed envelope,Highest bidder wins and pays that price.
4,Second price sealed bid,Same procedure,Highest bidder wins but pays price of second highest bidder.
An auction can be used to sell one or many goods,The flrst and last above have the same strategic structure,The
second and third have the same strategic structure,Why? All four auctions raise the same expected revenue.
What is the optimal auction to sell a particular good? Optimal in the sense of e–ciency | the highest valuation
agent wins the object,and optimal in the sense of raising the highest revenue.
Market | What Next? 11
Dynamic Games and Equilibrium
Dynamic games can be represented in normal form,Consider the entry game of lecture 10.
..................
..................
..................
..................
..................
..................
..................
.................
..................
..................
..................
..................
..................
..................
..................
...............
..................
..................
..................
..................
..................
..................
..................
.................
..................
..................
..................
..................
..................
..................
..................
................?
E
M
Fight
Don’t Fight
Enter
Don’t Enter (0;5)
(1;2)
(?1;0)
F DF
E 0
1
2
1
DE 5
0
5
0
There are two pure Nash equilibria | fDE; Fg and fE; DFg,In fact,there are inflnitely many mixed equilibria
also | where the entrant plays DE and the monopolist plays F with at least probability half.
Market | What Next? 12
Credibility
However,working from the back of the game in extensive form,the entrant will always choose to enter given they
know that if they do,the monopolist will not flght,The threat to flght is not credible.
An equilibrium is subgame perfect if it is a Nash equilibrium of each subgame and of the whole game.
A subgame is simply a game starting at any particular node,Subgame perfection rules out incredible threats.
The only subgame perfect equilibrium in this game is fE; DFg,This is a reflnement of Nash equilibrium.
Can the monopolist ever deter entry? If the game is repeated many (many) times then,yes.
The folk theorems prove that in an inflnitely repeated game almost anything is subgame perfect,For example,
flghting in the above game,collusion in oligopoly and even cooperation in the repeated prisoners’ dilemma.
