Since the core of a housing market with externalities may be empty, we propose as a solution concept the set of allocations that satisfy two basic properties: Pareto efficiency and the No-regret condition. Our main result shows that for any instance of the housing market problem, there always exists at least one allocation that satisfies both properties.Housing markets, the core, externalities, Pareto efficiency, No-regret condition

Alfredo Salgado-Torres

English

Research Papers in Economics

     
 
 
  
  
Volume 31, Issue 1 
  
A solution concept for housing market problems with externalities 
  
 
 
Alfredo Salgado-Torres  
Universidad Carlos III de Madrid 
Abstract 
Since the core of a housing market with externalities may be empty, we propose as a solution concept the set of 
allocations that satisfy two basic properties: Pareto efficiency and the No-regret condition. Our main result shows that 
for any instance of the housing market problem, there always exists at least one allocation that satisfies both 
properties. 
The author is grateful to an anonymous referee and the editor of the Economics Bulletin for helpful comments. The comments from Cecilia 
Vives Coscojuela and the FPI grant of the Spanish Ministry of Science and Innovation are gratefully acknowledged. 
Citation: Alfredo Salgado-Torres, (2011) ''A solution concept for housing market problems with externalities'', Economics Bulletin, Vol. 31 
no.1 pp. 623-630. 
Submitted: Dec 24 2010.   Published: February 17, 2011. 
 
     1 Introduction
An allocation problem consists in assigning a set of objects to a set of agents, that is
the case of two-sided matching, housing market problems, etc. In standard allocation
problems, agents have complete, transitive and strict preferences over the set of objects
and it is assumed that agents’ valuations are independent, i.e. there are no externalities
in the problem.
It is not di￿cult to ￿nd applications of the allocation problem where the absence of
externalities seems unreasonable. For instance, in the case of professional sports leagues,
where clubs compete to win a tournament, it is very reasonable to think that the chance
of winning a league may depend on the complete matching between sportsmen and clubs.
Another example regards housing markets, a house may be more valuable if very impor-
tant people live in the same neighborhood (the case of Bervelly hills), this implies that
the value of a house may depend on who live next door.
Recently, important contributions have emerged in the study of cooperative games
with externalities. Clippel and Serrano (2008) and Macho-Stadler, et. al. (2007) charac-
terize the Shapley Value in the presence of external e￿ects; Hafalir (2009) and Sasaki and
Toda (1996) analyze the stability of the matching in the marriage problem with exter-
nalities; Salgado-Torres (2010) analyzes the existence of stable matchings in many to one
matching problems with externalities; Hafalir (2007) analyzes the problem of e￿ciency in
coalition form games with externalities; Mumcu and Saglam (2007) analyze the existence
of the core in housing market problems with externalities.
One of the main problems in cooperative setups with externalities is the existence
of equilibrium allocations. Following that line, this note deals with the existence of a
reasonable solution concept for housing market problems with externalities.
It is well known that the core of any standard housing market problem is a singleton
and always exists. In addition, it is always possible to ￿nd the core of any housing
market by means of the Top Trading Cycle Algorithm (TTC) (Roth and Postlewaite,
1977; Roth, 1982). In contrast, when we consider the presence of externalities, the core
may contain more than one allocation and it may be empty for some instances of the
housing market (Mumcu and Saglam, 2007).
Given this problem, there are two possible approaches. Either, it is possible to seek
reasonable constrains over the preferences which guarantee the existence of the core or
we could ￿nd an alternative solution concept that satis￿es a set of minimum reasonable
properties. In the ￿rst case, it may be impossible to show how large the constrained
domain of preferences is, hence it is possible that the core only exists in a very small
domain of preferences. In the second approach, the solution concept is weakened trying
to keep as many desirable properties as possible. However, in this case we would be able
to ensure that the solution concept is general enough, since it would be applicable to
any instance of the problem. Since the core is a strong solution concept and we want
to maintain the general conditions of the problem, in this note we follow the second
approach.
First, we analyze the bargaining set already de￿ned in the one to one matching
problem (Klijn and Mass￿ o, 2003). Even when this solution concept is weaker than the
core, we show that the bargaining set may be empty for some instances of the housing
market with externalities.
After that, we consider an alternative solution concept that takes into account two
basic properties of the TTC. First of all, any allocation attained by the TTC is Pareto
1e￿cient, this is a minimum desirable property of a solution concept. Secondly, one of the
main properties of the TTC is that exchange has to be pro￿table, i.e. no agent regrets
having exchanged his initial allocation at the end of the algorithm.
Pareto e￿ciency and the No-regret condition are properties that a reasonable
solution concept has to satisfy. The ￿rst property implies that the economy is in the
frontier of consumption possibilities. The second one is a minimum requirement of non-
myopic behavior of agents. We show that any allocation in the core always satis￿es both
properties.
Our main result shows that for any housing market problem with externalities, there
always exists at least one allocation that satis￿es both properties. Hence, the solution
that associates to any problem the set of allocations that satisfy Pareto e￿ciency and
the No-regret condition is always nonempty.
The rest of the paper is organized as follows. In section 2, we introduce the model; in
section 3, we present our results; in section 4, we conclude.
2 The model
Let A = fa1;:::;ang be a ￿nite set of agents and let H = fh1;:::;hmg be a ￿nite set of
houses and assume that jAj = jHj ￿ 2. An allocation ￿ is a bijection from A to H. Let
M be the set of all possible allocations given the sets of agents and houses. We denote an
allocation by a vector of n entries ￿ = (h;h0;:::;h00) such that, ￿ (a1) = h, ￿ (a2) = h0,...,
￿ (an) = h00.
Each agent a 2 A has a transitive, strict and complete preference relation Pa over
the set of allocations M. For each a 2 A, let Ra denote the weak preference relation
associated to Pa, so for all v;v0 2 M, vRav0 means either vPav0 or v = v0. Note that no
agent is indi￿erent between two di￿erent allocations, even when these allocations assign
the same house to that agent. This implies that the value of a house depends on the
complete allocation. A housing market with externalities or simply a market is a four-
tuple (A;H;P;￿), where P = fPaga2A is the pro￿le of preferences and ￿ 2 M is an
exogenous initial endowment. In what follows, we ￿x the sets of agents and houses A
and H, hence a market is completely described by a preference pro￿le and an initial
endowment, i.e. a tuple (P;￿).
With some abuse of notation, Pa denotes a preference list from the best to the worst
possible allocation for each a 2 A. For instance, Pa = ￿;￿0;:::;￿00 means that ￿ and ￿00
are, respectively, the most and the least preferred allocations for a, ￿Pa￿0 and so on. Let
￿(A0) = fh 2 H : ￿(a) = h some a 2 A0g denote the set of houses initially assigned to
agents A0 ￿ A.
In the standard housing market problem, agents have rational and strict preferences
over the set of houses and it is assumed that preferences are independent each other.
This means that each agent is able to order houses from the best one to the worst one,
and this ranking is independent of the complete allocation between agents and houses.
In the presence of externalities, preferences are interdependent, i.e. there is no a
unique way to order the set of houses, since agents’ values depend on the complete
allocation. For instance, for some agents, a house may be the most preferred if their
parents live next door, but the worst one if their neighbors are unfriendly and unknown
people.
2Example 1 Consider a 3x3 housing market with A = fa1;a2;a3g, H = fh1;h2;h3g, the
set of all feasible allocations is,
￿1 = (h1;h2;h3);
￿2 = (h1;h3;h2);
￿3 = (h2;h1;h3);
￿4 = (h2;h3;h1);
￿5 = (h3;h1;h2);
￿6 = (h3;h2;h1).
Consider the next preferences:
Pa1 = ￿2;￿6;￿3;￿4;￿5;￿1:
Pa2 = ￿2;￿3;￿4;￿1;￿6;￿5:
Pa3 = ￿1;￿2;￿3;￿4;￿5;￿6:
The initial endowment is ￿ = ￿5.
Clearly, under ￿2 the agent a1 considers that h1 is his top choice. However, under ￿1
where a2 and a3 exchange their houses, a1 considers that h1 is his last choice. Hence, the
a1’s value of h1 depends on the complete allocation.
The usual solution concept in standard housing market problems is the core, which is
a single value correspondence that always exists (Roth and Postlewaite, 1977; Roth, 1982).
It is possible to extend this notion of equilibrium to environments with externalities.
Formally,
De￿nition 1 Given any housing market (P;￿), a coalition A0 ￿ A blocks the allocation
￿ via another allocation ￿0 6= ￿ if:
1. ￿0 (a) 6= ￿ (a) for all a 2 A0;
2. ￿0 (a) 2 ￿(A0) for all a 2 A0; and
3. ￿0Ra￿ for all a 2 A0 and ￿0Pa￿ for some a 2 A0.
An allocation ￿ is in the core of a market (P;￿), if it is not blocked by any coalition
via another allocation. Let C (P;￿) denote the core of a market (P;￿). Unlike the
standard problem, in the presence of externalities the core of a housing market may be
empty and it may not be a singleton (Mumcu and Saglam, 2007).
3 Results
In the presence of external e￿ects, the core of housing markets is a too demanding solution
concept, since it requires the existence of an allocation not blocked by any coalition. We
relax this solution and we consider the notion of the bargaining set, already applied in
one to one matching problems (Klijn and Mass￿ o, 2003). The notion of the bargaining
set only considers credible deviations. In order to de￿ne this solution concept in our
setup, we require some additional de￿nitions.
De￿nition 2 Given a housing market (P;￿), a coalition A0 ￿ A is able to enforce an
allocation ￿ if, for all a 2 A0, if ￿ (a) 6= ￿(a) implies that ￿ (a) 2 ￿(A0￿fag).
3Note that the previous de￿nition always takes into account the initial endowment of
the market and it is independent of the preference pro￿le. Further any agent is always
able to enforce the initial endowment for any market (P;￿).
De￿nition 3 An objection against an allocation ￿ is a pair (A0;￿0) such that ￿0 2 M
can be enforced by A0 ￿ A, and ￿0Pa￿ for all a 2 A0.
De￿nition 4 A counter-objection against an objection (A0;￿0), is a pair (T;￿00) where
￿00 2 M can be enforced by T ￿ A and it is satis￿ed:
1. T￿A0 6= ￿, A0￿T 6= ￿ and T \ A0 6= ￿; and
2. ￿00Pa￿ for all a 2 T￿A0 and ￿00Pa￿0 for all a 2 T \ A0.
If there is a counter-objection against an objection (A0;￿0), implies that the objection
of agents T \A0 is not credible since ￿00Pa￿0 for all a 2 T \A0. Further, no agent in T￿A0
su￿ers when T enforces ￿00, since ￿00Pa￿ for all a 2 T￿A0.
An objection (A0;￿0) against an allocation ￿ is justi￿ed if there is no counter-objection
against (A0;￿0), i.e. only credible objections are justi￿ed. The bargaining set is the
set of allocations that have no justi￿ed objections. Let B (P;￿) denote the bargaining
set of a market (P;￿). Note that an objection made by either one agent or the grand
coalition is always justi￿ed.
It is possible to show that the bargaining set may be empty for some instances of
problem, as we show in the next result.
Proposition 1 There exists a housing market with externalities (P;￿), such that B (P;￿)
is empty.
Proof. Consider a 3x3 housing market with A = fa1;a2;a3g and H = fh1;h2;h3g.
Agents’ preferences are,
Pa1 = ￿6;￿3;￿2;￿1;￿4;￿5:
Pa2 = ￿3;￿2;￿6;￿1;￿4;￿5:
Pa3 = ￿2;￿1;￿6;￿3;￿4;￿5:
The initial endowment is ￿ = ￿1. We show that there is a justi￿ed objection for each
allocation:
￿ ￿1 has the justi￿ed objection (fa1;a2;a3g;￿2);
￿ ￿2 has the justi￿ed objection (fa1;a2g;￿3), since only (fa1;a3g;￿6) may form a
counter-objection but ￿2Pa3￿6 with fa3g = fa1;a3g￿fa1;a2g;
￿ ￿3 has the justi￿ed objection (fa1;a3g;￿6), since only (fa2;a3g;￿2) may form a
counter-objection but ￿3Pa2￿2 with fa2g = fa2;a3g￿fa1;a1g;
￿ ￿4 has the justi￿ed objection (fa1;a2;a3g;￿6);
￿ ￿5 has the justi￿ed objection (fa1;a2;a3g;￿2); and
￿ ￿6 has the justi￿ed objection (fa2;a3g;￿2), since only (fa1;a3g;￿3) may form a
counter-objection but ￿6Pa1￿3 with fa1g = fa1;a3g￿fa2;a3g.
Then the set of allocations B (P;￿) is empty.
4Since neither the core nor the bargaining set always exist, we relax these solution
concepts by taking into account two basic properties of the Top Trading Cycle Algorithm
(TTC). First of all, any allocation attained by the TTC is Pareto e￿cient, which is
a very desirable property since it implies that it is not possible to improve one agent
without harming another one. Formally,
De￿nition 5 Given any market (P;￿) an allocation ￿ dominates another one ￿0, if ￿Ra￿0
for all a 2 A and ￿Pa￿0 for some a.
An allocation ￿ is Pareto e￿cient if it is not dominated by any other allocation.
Let PE (P;￿) denote the set of Pareto e￿cient allocations.
Secondly, at each step of the TTC agents point to the owner of his most favorite house
and agents (in a cycle) exchange their initial endowments, i.e. exchange is pro￿table. At
the ￿nal step of the TTC no agent regrets having exchanged his initial endowment.
Hence, a second natural property of any allocation in the solution of the problem is the
No-regret condition. Formally,
De￿nition 6 Given any market (P;￿), an allocation ￿ satis￿es the No-regret condi-
tion if ￿0Ra￿ for all a 2 A such that ￿0 (a) 6= ￿(a).
Let NR(P;￿) denote the set of allocations that satisfy the No-regret condition
given a market (P;￿). De￿ne S (P;￿) = PE (P;￿) \ NR(P;￿).
In the next result, we show that any allocation in the core of a housing market with
externalities satis￿es both properties.
Claim 1 Let (P;￿) be any instance of the housing market with externalities, then C (P;￿) ￿
S (P;￿).
Proof. If C (P;￿) = ￿ the proposition is trivially satis￿ed, hence assume that the core
of the market is not empty. Assume that there is some allocation ￿ 2 C (P;￿) but
￿ = 2 PE (P;￿) \ NR(P;￿), by de￿nition ￿ is a Pareto e￿cient allocation. Hence, by
assumption ￿ = 2 NR(P;￿). This implies that there exists an agent a 2 A such that
￿ (a) 6= ￿(a) and ￿Pa￿. Set S = fag and ￿0 = ￿, then ￿ is blocked by the coalition fag
via ￿0, a contradiction.
Example 2 Consider a 3x3 housing market with A = fa1;a2;a3g, H = fh1;h2;h3g and
the next preferences:
Pa1 = ￿2;￿1;￿3;￿4;￿5;￿6:
Pa2 = ￿2;￿3;￿4;￿1;￿6;￿5:
Pa3 = ￿1;￿2;￿3;￿4;￿5;￿6:
The initial endowment is ￿ = ￿4.
It is possible to show that, PE (P;￿) = f￿1;￿2g and NR(P;￿) = f￿2;￿3;￿4g, whereas
S (P;￿) = C (P;￿) = f￿2g.
In Example 2, we show that neither PE (P;￿) * NR(P;￿) nor NR(P;￿) * PE (P;￿).
In the next result, we show that for any instance of the problem it is always possible to
￿nd at least one allocation that satis￿es both Pareto e￿ciency and the No-regret
condition.
5Proposition 2 For any instance of the housing market with externalities (P;￿), the set
of allocation S (P;￿) is always nonempty.
Proof. Let (P;￿) be any housing market, if ￿ is Pareto e￿cient the proposition is
trivially satis￿ed, since ￿ 2 NR(P;￿) by de￿nition.
Assume that ￿ is not Pareto e￿cient, hence there exists an allocation ￿ 6= ￿ that
dominates ￿. If ￿ is Pareto e￿cient the proposition is proven, since ￿ 2 NR(P;￿) given
￿, otherwise there exist another ￿0 6= ￿ that dominates ￿, and so on. De￿ne the sequence
of allocations f￿;￿;￿0;:::g, such that ￿ dominates ￿, ￿0 dominates to ￿, and so on.
Let ￿ denote a partial order over the set M, such that ￿ ￿ ￿0 if and only if ￿Ra￿0
for all a 2 A. Under the order ￿, the sequence f￿;￿;￿0:::g is increasing in M. Hence,
f￿;￿;￿0:::g has to converge to a Pareto e￿cient allocation ￿￿, since M is a ￿nite set. By
construction ￿￿ 6= ￿ and ￿￿Ra￿ for all a 2 A, hence ￿￿ 2 NR(P;￿). This completes the
proof.
The previous result has two main implications. First of all, the converse of Claim
1 does not hold. Secondly, unlike the core the set of allocations that satisfy Pareto
e￿ciency and the No-regret condition is always nonempty. Hence, we are able to
ensure that the solution that associates the set of allocations S (P;￿) to any housing
market problem with externalities (P;￿) is always nonempty.
4 Conclusions
Mumcu and Saglam (2007) show that the core of a housing market with externalities
may not exist. We show that a weaker solution concept, i.e. the bargaining set has
the same problem, it is possible to ￿nd at least one instance of the problem where the
bargaining set is empty.
We propose as a solution concept the set of allocations that satisfy two desirable
properties: Pareto e￿ciency and the No-regret condition. The ￿rst property does
not need to be justi￿ed, this is a minimum requirement for any economic problem. The
second one implies that agents are not myopic, in the sense that no agent regrets having
exchanged his initial endowment. We argue that both properties are desirable for a
reasonable solution concept for housing markets in the presence of externalities. Our
main result shows that for any instance of the problem, there always exists at least one
allocation that satis￿es Pareto e￿ciency and the No-regret condition.
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A solution concept for housing market problems with externalities

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