Lecture 2: Public Housing
Matching and Market Design 395
Esteban Peralta
University of Michigan
1/23
Today
I Today we start analyzing assignment markets;
I i.e., markets in which objects are assigned to individuals.
I We’ll start with the role of endowments and property rights.
I We’ll then look at the public allocation problem;
I a one-to-one, centralized assignment problem with public
endowments.
I We will start looking at some desirable properties we would
like (or need) assignments to have.
2/23
Endowments and Property Rights
I A basic feature to be determined is whether some individuals
own some of the objects to be assigned or not.
3/23
Endowments and Property Rights
I A basic feature to be determined is whether some individuals
own some of the objects to be assigned or not.
I Public endowment
None of the objects belong to anyone; i.e., all of the objects
belong to the whole society.
Example: Assignment of public housing, students to schools,
parking lots to employees, etc.
3/23
Endowments and Property Rights
I A basic feature to be determined is whether some individuals
own some of the objects to be assigned or not.
I Public endowment
None of the objects belong to anyone; i.e., all of the objects
belong to the whole society.
Example: Assignment of public housing, students to schools,
parking lots to employees, etc.
I Private endowment
Each individual own some object(s) and each object is owned
by some individual(s).
Example: barter economy, with individuals trading goods
without monetary transactions; e.g., homeowners exchanging
houses.
3/23
Endowments and Property Rights
I Mixed endowment
Some objects, but not all, are owned by some individual(s),
but not every individual owns an object.
Example: universities’ dorms:
I public endowment: rooms left by recent graduates.
I private endowment: rooms occupied by sophomores, juniors
and seniors.
4/23
Public Endowment Model
I We’ll start with a one-to-one assignment market with public
endowments.
I The so called public endowment problem.
5/23
Public Endowment Model
I We’ll start with a one-to-one assignment market with public
endowments.
I The so called public endowment problem.
I a set of individuals: I = {i1 , i2 , . . . , in }.
5/23
Public Endowment Model
I We’ll start with a one-to-one assignment market with public
endowments.
I The so called public endowment problem.
I a set of individuals: I = {i1 , i2 , . . . , in }.
I a set of objects: K = {k1 , k2 , . . . , km }.
5/23
Public Endowment Model
I We’ll start with a one-to-one assignment market with public
endowments.
I The so called public endowment problem.
I a set of individuals: I = {i1 , i2 , . . . , in }.
I a set of objects: K = {k1 , k2 , . . . , km }.
I each i has a strict preference ranking Pi over K ∪ {∅}.
I such that k Pi k 0 reads “i prefers k over k 0 .”
5/23
Public Endowment Model
I We’ll start with a one-to-one assignment market with public
endowments.
I The so called public endowment problem.
I a set of individuals: I = {i1 , i2 , . . . , in }.
I a set of objects: K = {k1 , k2 , . . . , km }.
I each i has a strict preference ranking Pi over K ∪ {∅}.
I such that k Pi k 0 reads “i prefers k over k 0 .”
• What’s the ∅ set for?
5/23
Examples
I The previous description can be used to think about different
examples, besides public housing.
I Organ allocation;
I Allocation of parking spaces;
I Allocation of visas;
I etc...
⇒ We use k to denote objects in general; we could use h for
houses, s for schools, k for kidneys, etc.
I we will think about public housing, to fix ideas.
6/23
Why do we care?
I In some of the previous problems, the stakes can be high.
7/23
Why do we care?
I In some of the previous problems, the stakes can be high.
I The US Department of Housing and Urban Development has
shown that re-allocating low-income families from public
housing complexes to mixed-income neighborhoods improved
their mental health, and reduced their obesity and diabetes.
I e.g., Pollack et. al., “The Impact of Public Housing on Social
Networks: A Natural Experiment.”
7/23
Why do we care?
I In some of the previous problems, the stakes can be high.
I The US Department of Housing and Urban Development has
shown that re-allocating low-income families from public
housing complexes to mixed-income neighborhoods improved
their mental health, and reduced their obesity and diabetes.
I e.g., Pollack et. al., “The Impact of Public Housing on Social
Networks: A Natural Experiment.”
I The allocation of temporary-visas to work typically prevents a
beneficiary worker from working in a firm different from her
assignment, thus creating the potential for inefficiencies.
I e.g., Casella and Cox, “A Property Rights Approach to
Temporary Work Visas.”
7/23
Rankings
Consider the following market:
I There are three individuals: Alice, Bob and Carol.
I There are three objects: A, B and C .
I The ranking of each individual is as follows:
PAlice
A
C
B
∅
PBob
C
A
B
∅
PCarol
B
C
A
∅
8/23
Assignments
An assignment specifies who gets what.
9/23
Assignments
An assignment specifies who gets what.
Definition
An assignment is a map µ : I ∪ K → I ∪ K such that:
1. for each individual i ∈ I , µ(i) ∈ K ∪ {∅};
2. for each object k ∈ K , µ(k) ∈ I ∪ {∅};
3. µ(i) = k if, and only if µ(k) = i.
9/23
Assignments
An assignment specifies who gets what.
Definition
An assignment is a map µ : I ∪ K → I ∪ K such that:
1. for each individual i ∈ I , µ(i) ∈ K ∪ {∅};
2. for each object k ∈ K , µ(k) ∈ I ∪ {∅};
3. µ(i) = k if, and only if µ(k) = i.
I Notation: I will use both µ(i) and µi to denote the
assignment of individual i (and similarly for objects).
9/23
Assignments: Examples
µ(Alice) = A, µ(Bob) = B and µ(Carol) = C .
10/23
Assignments: Examples
µ(Alice) = A, µ(Bob) = B and µ(Carol) = C .
µ(Alice) = C , µ(Bob) = A and µ(Carol) = B.
10/23
Assignments: Examples
µ(Alice) = A, µ(Bob) = B and µ(Carol) = C .
µ(Alice) = C , µ(Bob) = A and µ(Carol) = B.
µ(Alice) = A, µ(Bob) = B and µ(Carol) = ∅.
10/23
Assignments: Examples
µ(Alice) = A, µ(Bob) = B and µ(Carol) = C .
µ(Alice) = C , µ(Bob) = A and µ(Carol) = B.
µ(Alice) = A, µ(Bob) = B and µ(Carol) = ∅.
µ(Alice) = µ(Bob) = µ(Carol) = ∅.
10/23
Assignments: Examples
µ(Alice) = A, µ(Bob) = B and µ(Carol) = C .
µ(Alice) = C , µ(Bob) = A and µ(Carol) = B.
µ(Alice) = A, µ(Bob) = B and µ(Carol) = ∅.
µ(Alice) = µ(Bob) = µ(Carol) = ∅.
I What would be an example of a non-assignment?
10/23
The two roles of ∅...
I Notice that ∅ is used to denote the possibility that some
individuals might be assigned to no object, and viceversa.
⇒ µi = ∅ reads “i is unassigned.”
⇒ ∅ is an “object” that can be assigned to multiple individuals.
11/23
The two roles of ∅...
I Notice that ∅ is used to denote the possibility that some
individuals might be assigned to no object, and viceversa.
⇒ µi = ∅ reads “i is unassigned.”
⇒ ∅ is an “object” that can be assigned to multiple individuals.
I Yet ∅ is also part of the individuals’ rankings.
11/23
The two roles of ∅...
I Notice that ∅ is used to denote the possibility that some
individuals might be assigned to no object, and viceversa.
⇒ µi = ∅ reads “i is unassigned.”
⇒ ∅ is an “object” that can be assigned to multiple individuals.
I Yet ∅ is also part of the individuals’ rankings.
I to allow individuals to state whether they would rather have no
object at all than being assigned to certain objects.
I When ∅ is preferred to (i.e., it is higher in the ranking than)
some object, the object is called unacceptable.
11/23
Unacceptable objects
Consider the following preferences:
PAlice
A
C
B
∅
PBob
C
A
B
∅
PCarol
B
C
∅
A
I I won’t typically list ∅ or any unacceptable object:
12/23
Unacceptable objects
Consider the following preferences:
PAlice
A
C
B
∅
PBob
C
A
B
∅
PCarol
B
C
∅
A
I I won’t typically list ∅ or any unacceptable object:
PAlice
A
C
B
PBob
C
A
B
PCarol
B
C
12/23
How should we assign?
I Consider the following preferences
PAlice
A
C
B
PBob
C
A
B
PCarol
B
A
and the following assignment:
µ(Alice) = A, µ(Bob) = B and µ(Carol) = C .
13/23
How should we assign?
I Consider the following preferences
PAlice
A
C
B
PBob
C
A
B
PCarol
B
A
and the following assignment:
µ(Alice) = A, µ(Bob) = B and µ(Carol) = C .
13/23
How should we assign?
I Consider the following preferences
PAlice
A
C
B
PBob
C
A
B
PCarol
B
A
and the following assignment:
µ(Alice) = A, µ(Bob) = B and µ(Carol) = C .
I Is this assignment desirable? Why?
13/23
Individual Rationality
I The problem with the previous assignment is that it is
“forcing” an individual to accept an unacceptable object.
Definition
An assignment µ is individually rational (IR) if for every
individual i, eiher µi = ∅ or µi Pi ∅.
14/23
Individual Rationality
I The problem with the previous assignment is that it is
“forcing” an individual to accept an unacceptable object.
Definition
An assignment µ is individually rational (IR) if for every
individual i, eiher µi = ∅ or µi Pi ∅.
I The following assignment µ0 is individually rational:
µ0 (Alice) = A, µ0 (Bob) = B and µ0 (Carol) = ∅.
14/23
IR: normative and positive considerations
I IR interprets ∅ as an outside option.
⇒ ∅ not only defines what is acceptable.
⇒ IR captures an important normative consideration;
I it takes into account “property rights.”
15/23
IR: normative and positive considerations
I IR interprets ∅ as an outside option.
⇒ ∅ not only defines what is acceptable.
⇒ IR captures an important normative consideration;
I it takes into account “property rights.”
I What property rights?
15/23
IR: normative and positive considerations
I IR interprets ∅ as an outside option.
⇒ ∅ not only defines what is acceptable.
⇒ IR captures an important normative consideration;
I it takes into account “property rights.”
I What property rights?
I ...the right to remain unassigned.
15/23
IR: normative and positive considerations
I IR interprets ∅ as an outside option.
⇒ ∅ not only defines what is acceptable.
⇒ IR captures an important normative consideration;
I it takes into account “property rights.”
I What property rights?
I ...the right to remain unassigned.
⇒ ∅ can be thought of as an object, that belongs to everyone.
15/23
IR: normative and positive considerations
I IR presumes that no individual should be forced to accept an
unacceptable assignment.
⇒ IR captures a basic notion of “fairness.”
I Yet IR is also important from a positive viewpoint;
I in many real-life problems, no individual can actually be forced
to accept an unacceptable assignment.
⇒ A failure of IR can affect participation in centralized markets.
I we will see this later on...
16/23
How should we assign?
Consider the following market:
PAlice
A
C
B
PBob
C
A
B
PCarol
B
C
A
and the following assignment:
µ(Alice) = A, µ(Bob) = B and µ(Carol) = ∅.
I Is this assignment desirable? Why?
17/23
Wastefulness
I The problem with the previous assignment is that it is
“wasting resources.”
18/23
Wastefulness
I The problem with the previous assignment is that it is
“wasting resources.”
Definition
An individually rational assignment µ is wasteful if there is some
object k and some individual i such that:
1. k Pi µi ,
2. µk = ∅.
18/23
Wastefulness
I The problem with the previous assignment is that it is
“wasting resources.”
Definition
An individually rational assignment µ is wasteful if there is some
object k and some individual i such that:
1. k Pi µi ,
2. µk = ∅.
⇒ The following assignment µ0 is nonwasteful:
µ0 (Alice) = A, µ0 (Bob) = B and µ0 (Carol) = C .
18/23
The role of algorithms
I Having a systematic way of finding assignments would be
useful in dealing with “large” centralized markets.
I Indeed, we would have a “step-by-step” procedure, a.k.a, an
algorithm that, no matter what preferences we give, it returns
an assignment.
19/23
The role of algorithms
I Having a systematic way of finding assignments would be
useful in dealing with “large” centralized markets.
I Indeed, we would have a “step-by-step” procedure, a.k.a, an
algorithm that, no matter what preferences we give, it returns
an assignment.
I So, can we find an algorithm that produces IR assignments
that are not wasteful?
I i.e., how could a clearinghouse guarantee, if at all possible, that
the assignment it produces satisfies both of these properties?
19/23
Producing Nonwasteful and IR assignments...
I Suppose that the clearinghouse starts by producing a
“tentative” assignment; let’s call it µ.
I If µ happens to be both NW and IR, we are done.
I What could the clearinghouse do if either property fails?
20/23
A simple algorithm
I The clearinghouse could use the following three steps:
1. Make every individual that finds her object unacceptable drop
her object. Call the new assignment µ0 .
21/23
A simple algorithm
I The clearinghouse could use the following three steps:
1. Make every individual that finds her object unacceptable drop
her object. Call the new assignment µ0 .
I Notice that µ0 is IR; why?
21/23
A simple algorithm
I The clearinghouse could use the following three steps:
1. Make every individual that finds her object unacceptable drop
her object. Call the new assignment µ0 .
I Notice that µ0 is IR; why?
2. For every unassigned object k, match k with any i that prefers
k to µ0i . Call the new assignment µ00 .
I notice that µ0 is IR; why?
21/23
A simple algorithm
I The clearinghouse could use the following three steps:
1. Make every individual that finds her object unacceptable drop
her object. Call the new assignment µ0 .
I Notice that µ0 is IR; why?
2. For every unassigned object k, match k with any i that prefers
k to µ0i . Call the new assignment µ00 .
I notice that µ0 is IR; why?
3. If µ00 is wasteful, repeat 2.
I The final assignment must satisfy both NW and IR. Why?
I Why we might need more than two steps?
21/23
Are we done?
I The previous algorithm produces an assignment that is IR and
NW, but we might be interested in more properties.
1. Is the assignment produced by the previous algorithm
“efficient”?
2. Does it prevent individuals from lying?
22/23
Readings
I Haeringer, chapter 11, sections 11.1.1 and 11.1.2.
23/23