Mathematical Economics:
Some simple, useful examples
Herb Scarf
Whitney Humanities Center
February 25, 2009
I am a Mathematical Economist. What do I do for a living?
• I prove theorems about economic theory.
• What’s a theorem?
• What’s a proof?
• What’s economic theory?
Hard to say.
Some Examples
Will help.
I will try to select some examples that are important,
practically useful, and can be appreciated with simple
numerical illustrations.
But I would like to digress with some general themes in
economic theory.
One of the central themes of economic theory is that people
(countries?) exchange the goods and services that they happen
to own for goods and services that other economic agents own.
Exchange takes place because we like other peoples' goods
more than we like the goods we happen to possess.
Sometimes the goods that other agents own are the result of some
manufacturing process involving labor, raw materials or machinery
to produce.
For example, China exchanges clothing that it manufactures for oil
from Saudi Arabia.
The U.S. exchanges computer software and Caterpillar tractors for
grapes produced in Chile.
The U.S. exchanges Treasury Bills (claims on future consumption)
and wheat for automobiles produced in Japan.
Now let’s turn to the first elementary, useful, topic.
The Matching (Marriage) Problem
• We consider 3 men, Alan, Bob and Chuck
• and 3 women, Alice, Beth and Clair.
• The men have preferences for the women, and
• the women have preferences for the men.
• Who shall marry?
Preferences
Men 1 2 3 Suppose that the men and women
Alan: Alice Beth Clair are paired as follows:
Bob: Beth Alice Clair
Chuck Alice Beth Clair Alan Alice
Bob Beth
Women 1 2 3
Chuck Clair
Alice: Bob Alan Chuck
Beth: Bob Alan Chuck
Is this assignment stable?
Clair: Alan Bob Chuck
The assignment is unstable if a man and a woman prefer each other to their assigned
mates.
This assignment is unstable because Alan and Alice prefer each other to their
assigned partners.
A Stable Marriage
(with the same Preferences)
Men 1 2 3
Suppose that the men and women
Alan: Alice Beth Clair
are paired as follows:
Bob: Beth Alice Clair
Chuck Alice Beth Clair Alan Alice
Women 1 2 3 Bob Beth
Alice: Bob Alan Chuck Chuck Clair
Beth: Bob Alan Chuck
Clair: Alan Bob Chuck
This pairing is stable. Alan and Bob are assigned their first choice.
The only objection can be by Chuck and one of the women.
But neither Alice nor Beth prefer Chuck to her assigned partner.
Is there always a stable marriage? We’ll see.
The Roommate Problem
The Roommate Problem is the Marriage Is there a stable pairing?
Problem without Gender.
Suppose that there are 4 roommates who • If the assignments are
are to be paired to share two rooms. {Alan, Bob}, {Chuck, Dan}
The preferences of the roommates are as then Bob and Chuck will object.
follows:
• If the assignments are
{Alan, Chuck}, {Bob, Dan}
then Bob and Alan will object.
Alan: Bob Chuck Dan
• If the assignments are
Bob: Chuck Alan Dan {Alan, Dan}, {Bob, Chuck}
then Alan and Chuck will object.
Chuck: Alan Bob Dan • There is no stable assignment.
Dan: Alan Bob Chuck
Polygamous Matching
• An important generalization of the Matching Problem arises when
the two sets to be matched don't have the same size.
• For example, the two sets can consist of a large number of Residents
(R) who are to be assigned to a small number of Hospitals (H).
• Each Hospital has a limited number of places. For simplicity let us
assume that the number of places at the various Hospitals is equal to
the total number of Residents.
• We assume that each Resident ranks all of the Hospitals, and each
Hospital ranks all of the Residents.
• The problem is to assign each Resident to a Hospital in a stable
fashion.
Stability
• The assignment is stable if there is no Resident R and
Hospital H such that
– Resident R prefers Hospital H to the hospital to which she is
assigned, and
– Hospital H prefers Resident R to one of the residents who has been
assigned to H.
• Is there a stable assignment?
• Theorem: Yes! (Gale, Shapley, 1962)
• This includes the Marriage Problem as a special case.
Observations
The proof of the Theorem asserting that there always
is a stable matching is based on an elegant, but not
totally obvious, algorithm.
In general, there are many different matchings, some
of which favor the Residents and others favoring the
Hospitals.
There are many interesting variations:
A Resident may refuse to work at a particular Hospital. (An
easy modification.)
A pair of Residents (say a husband and wife) may insist on
being assigned to Hospitals in the same city. (There may
be no such matchings.)
Applications
• These techniques have been used for many years in the National
Residents Matching Program (MRMP).
• They have been used in the assignments of students to schools in
New York, Boston, Budapest and Singapore. And in college
admissions in Turkey.
• In the allocation of graduating students from Hebrew Union
College to congregations.
• The assignment of students to houses at MIT.
• Matching advertisers to advertising slots on Google.
• I know of applications to the assignment of packets on the
Internet.
• Now I would like to chat with you about a variation in the basic
model that has been used to discuss, among other things, kidney
transplants.
• The issue is to assign the kidneys offered by willing donors to
recipients in need of a functioning kidney.
• The problem is that the friendly donor’s kidney may be not be
suitable for the particular recipient and may have to be exchanged
for the kidney of another donor.
• “A Kidney Exchange Clearinghouse in New England,” Alvin E. Roth,
Tayfun Sönmez, M. Utku Ünver (2004)
• The model to be discussed is called the House Problem. It is very
simple, but it is a serious introduction to the economics of
exchange.
The House Problem
Preferences of Owners for Houses
O-1 H-1
O-2 H-2
O-1 O-2 O-3 O-4 O-5 O-3 H-3
O-4 H-4
H-1 5 5 4 2 2
O-5 H-5
H-2 2 4 2 1 4 1. Owner 1’s favorite house is H-3.
Let’s give it to him.
H-3 1 3 3 5 1
2. We have to compensate Owner
H-4 3 1 5 4 5 3. His favorite house is H-5.
Let’s give him H-5.
H-5 4 2 1 3 3
3. What shall we give Owner 5?
Let’s give him his favorite house
How shall we exchange these houses? H-3.
If Owners 3 and 5 exchange their houses, 4. We seem to have a cycle of
they will each receive their favorite house. size 2. (It could be size 3 or
They will certainly not object. more.)
How to Continue?
Preferences of Owners for Houses
Since Owners 3 and 5 exchange their
houses, let’s remove them.
O-1 O-2 O-3 O-4 O-5
H-1 5 5 4 2 2
H-2 2 4 2 1 4 O-1 O-2 O-4
H-1 5 5 2
H-3 1 3 3 5 1
H-2 2 4 1
H-4 3 1 5 4 5
H-4 3 1 4
H-5 4 2 1 3 3
How shall we distribute these 3 houses?
O-1 H-1
O-1 O-2 O-4
O-2 H-2
H-1 5 5 2
O-3 H-3
H-2 2 4 1
O-4 H-4
H-4 3 1 4
O-5 H-5
1. Owner 1’s favorite is House 2.
Give it to him. This sets up a second cycle
In which Owners 2 and 4
2. We have to compensate exchange their houses.
Owner 2. His favorite house is
H-4.
Give it to him. Owner 1 seems to be left with
3. But Owner 4’s favorite is H-2. his own House which he values
We give H-2 to Owner 4. less than any other House.
The Solution
Preferences of Owners for Houses
O-1 H-1
O-2 H-2
O-3 H-3
O-1 O-2 O-3 O-4 O-5 O-4 H-4
H-1 5 5 4 2 2 O-5 H-5
H-2 2 4 2 1 4
This assignment has the property
H-3 1 3 3 4 1
that no set of owners can make
H-4 3 1 5 4 5 themselves better off by exchanging
the houses they originally own in a
H-5 4 2 1 3 3 different way.
WHY?
We are given a set of Owners, each with a single House.
Each Owner has specific preferences for the Houses.
Is there always an assignment of the Houses to the Owners
that is stable in the sense that no collection of Owners can
make themselves better off by swapping their original Houses
in a different way?
Theorem: Yes. (Shapley, Scarf, 1974)
Proof: You’ve just seen it. (Gale)
Oddly enough, there is only one such assignment!
Even with 7000 Houses.
The particular assignment that we have reached is called the Core of the
House Market.
Gale’s algorithm constructs a Top Trading Cycle (of arbitrary size),
removing the Owners and Houses in this cycle,
solving the smaller House Market by finding a new Top Trading Cycle,
and repeating the procedure until all of the Houses have been assigned.
This is called a “mechanism”: We select an arbitrary set of Owners’
preferences for Houses and obtain a specific redistribution of the Houses.
Is it ever advantageous for a specific Owner to misrepresent his
preferences if this mechanism is used? Can he obtain a better outcome for
himself by lying?
Theorem: No. (Can you see the argument?)
We have the following very surprising result:
Theorem (JinPeng Ma, 1994): Gale’s procedure is the only mechanism
(subject to minor conditions) for which lying is disadvantageous!
Some Reflections
We have examined several elementary examples illustrating the exchange of
spouses, roommates and houses.
Example Existence of a Uniqueness of the
Solution Solution
Marriage Market Yes Typically Many
Solutions
Roommate Not Guaranteed
Problem
House Market Yes Unique
The existence of a solution and its uniqueness are basic questions to be
raised about any economic model.
These examples are very special. In a mature economy such as ours,
vast numbers of economic choices are being made at every moment
of time.
Economists see these choices as mediated by competitive prices which
simultaneously equate supply and demand in a great multiplicity
of markets.
(Our very special examples haven’t used prices at all, but they were
selected to be simple.)
Investment decisions and the levels of manufacturing outputs are
governed by considerations of profitability, using market prices.
Consumer choices depend on income and the prices of available
goods and services, today and in the future.
These decisions are linked together in an extraordinarily complex way.
This discussion can be captured by a formal mathematical model which
allows us to analyze the issues of existence of equilibrium prices,
uniqueness of market clearing prices, and the stability of the market
system after a disruption in its functioning.
There are algorithms for calculating market clearing prices, if we are given a
description of consumer preferences, the assets held by the agents in the
economy and the technical processes available for production.
These algorithms are used extensively in discussing the consequences of
changes in economic policies - such as the establishment of NAFTA.
This conceptual apparatus has many flaws:
• It isn’t very good with economies of scale,
• or in dealing with an unforeseen future.
• And we are certainly not in equilibrium at the present time.
Our old equilibrium has been given a swift kick in the rear end,
and we are trying to find our way to a new equilibrium.
But, imperfect as it is, the conceptual framework of economic
theory is indispensible as a tool of analysis and repair.
It ‘s the only game in town!