Shapley and Scarf housing markets with consumption externalities

• Date: 01 September 2020
• DOI: http://doi.org/10.1111/jpet.12470
• Author: Claudia Meo,Maria Gabriella Graziano,Nicholas C. Yannelis
• Published date: 01 September 2020

J Public Econ Theory. 2020;22:1481–1514. wileyonlinelibrary.com/journal/jpet © 2020 Wiley Periodicals LLC

|

1481

Received: 13 February 2020

|

Accepted: 25 July 2020

DOI: 10.1111/jpet.12470

ORIGINAL ARTICLE

Shapley and Scarf housing markets with

consumption externalities

Maria Gabriella Graziano

1

|Claudia Meo

2

|Nicholas C. Yannelis

3

1

Dipartimento di Scienze Economiche e

Statistiche and CSEF, Università di Napoli

Federico II, Napoli, Italy

2

Dipartimento di Scienze Economiche e

Statistiche, Università di Napoli Federico II,

Napoli, Italy

3

Department of Economics, Henry B.

Tippie College of Business, University of

Iowa, Iowa City, Iowa

Correspondence

Nicholas C. Yannelis, Department of

Economics, Henry B. Tippie College of

Business, University of Iowa, Pappajohn

Business Building, 21 East Market Street,

Iowa City, IA 52242.

Email: nicholas-yannelis@uiowa.edu and

nicholasyannelis@gmail.com

Abstract

We introduce externalities into the classical model by

Shapley and Scarf; that is, agents care about others

and their preferences are defined over allocations

rather than over single indivisible goods. After col-

lecting some results about the nonexistence of several

cooperative solutions, we focus on stable allocations

and propose domains of preferences that can guar-

antee that they both exist and form a stable set à la

von Neumann and Morgenstern.

1|INTRODUCTION

In this paper we consider the problem of allocating a number of differentiated indivisible

objects to individuals in an efficient and stable manner, when there is an externality in con-

sumption. In particular, we assume that traders may care not only about the good that they

receive, but also about the goods that are received by the others.

The classical version of the model without externalities is due to Shapley and Scarf (1974)

and is known in the literature as the housing market model. It considers an exchange economy

where nagents trade in indivisible objects, say houses, with no transfers of money. Each agent

owns one distinct house when entering the market and desires exactly one house. Agents are

allowed to swap their houses among themselves without, however, any money transfers. Tra-

ders have complete, reflexive, and transitive preference relations over all existing houses and

exchange their houses to make a mutually beneficial trade. An outcome in this market is an

allocation of houses among individuals such that each individual holds exactly one house, that

is, a permutation of the initial endowment.

To determine the outcome, Shapley and Scarf (1974) use the core as the solution concept.

An allocation is in the core if there is no group of individuals that could make every member

strictly better off by reallocating the houses owned by the group overall among the members of

the group itself. In their original model, they prove that the core always exists; however, when

the weak domination replaces the strong one, it may become empty. Few years later, Roth and

Postlewaite (1977) showed that even the core defined by the weak domination is nonempty, and

in fact unique, when no individual is indifferent between any indivisible goods. This unique

core allocation can be determined by using a constructive procedure, called the top trading

cycles (TTC) algorithm, which they attribute to David Gale. They also pointed out another

element for a framework where there is no indifference: allocations

a

exist which are in the core

of a given market but not in the core of the market where

a

itself is the initial endowment. They

defined an allocation

a

to be stable if and only if it is in the core of the market where

a

itself is

the initial endowment. Thus, in contrast with the core notion, they consider a dynamic per-

spective: an allocation is defined to be stable if no coalition of traders can benefit by further

reallocating the items after they have traded. By using the TTC algorithm, they proved that

there are always stable points in the core.

Most of the literature on housing market models, both theoretical and applied, maintains

the assumption adopted by the two aforementioned papers: that is, agents are self‐interested

and have preference relations over the differentiated items to be distributed. However, in highly

diverse range of real‐world problems, such as bequests and inter vivos transfers, this view is too

narrow in that it neglects possible altruistic or envious behaviors as opposed to pure selfish

ones

1

: in terms of preferences, traders may care about both the items they receive and the items

that are allocated to the other agents. Apart from altruism or envy issues, this dependency of the

individuals' preferences on the commodities allocated to the others is central in many matching

problems. The next two examples aim to illustrate this point. Consider the actual problem of

allocating houses to people. It is undeniable that the desirability of a house may depend on the

peers in the nearby houses; moreover, social connections among traders that are involved—

friendship or, even more, kinships—influence the assignment process. It can be the case that

some or all traders may prefer to live close to their relatives/social friends rather than to people

that they do not know at all, and this sort of preference can be even more relevant than the

physical characteristics of the house itself such as size, view or floor. This specific issue has been

explored in Baccara, Imrohoroğlu, Wilson, and Yariv (2012); their paper analyzes, both from an

empirical and theoretical perspective, the impact of network externalities on an assignment

problem of faculty members to newly renovated offices varying in physical characteristics, such

as floor, geographical exposure and size. It is shown that peer connections, divided into three

layers (institutional links, past and current coauthorship links, and friendship links) have a

relevant impact on the final outcomes in the assignment process.

2

The same issue, that is

central in all the public housing sector, concretely emerged in Italy after the earthquake that

heavily struck its central regions in 2016. For the first time, in fact, the wood‐frame houses for

the resident population left homeless have been distributed not by drawing, as it was in the past,

but keeping into consideration the demands by people, trying to recover the same social and

urban structure as in the destroyed villages. As a second example, consider the case of six

1

In this paper, we are mainly interested in other‐regarding preferences as an expression of altruism, envy or, more in

general, affinity among traders; we do not consider several other aspects that have been analyzed in the literature, such

as “peer influences,”“neighborhood effects,”or “bandwagon effect.”

2

In a slightly different framework, Echenique and Yenmez (2007) analyze a many‐to‐one matching where students have

to be assigned to colleges; contrary to the mainstream model, the authors assume that students have preferences over

the other students who would attend the same college and propose an algorithm that finds the solutions for this

assignment problem, if they exist.

1482

|

GRAZIANO ET AL.

colleagues that have to fly to the same conference, but seats are only available in three different

flights (3, 2, and 1 seats, respectively). Besides the personal preferences for each flight, based on

the departure time or the number of stopovers, the choice of each individual can be somehow

influenced by his preferences over the peers. This kind of preferences can be modeled by

ranking, for each trader, not just the six seats available, but all the

6

!

possible permutations

of them.

Starting from these insights into real‐life situations, in this paper, we propose a variant of

the original model by Shapley and Scarf where an externality in consumption is introduced in

the model. In particular, we assume that agents are not purely self‐interested but have other‐

regarding preferences that may depend on what items are allocated to other people in the

market. Formally, each trader has a preference relation over the set of all the allocations rather

than over the set of the indivisible items.

Recently, a strand of literature on one side matching theory and mechanism design has

presented models that manage to focus on preference profiles with externalities. Among them,

Sonmez (1999) studies a general class of allocation problems that include housing markets as a

subclass, and proves that there exists an efficient, individually rational and strategy‐proof so-

lution only if all allocations in the core are Pareto indifferent and that any such solution selects

a core allocation whenever the core is nonempty. Ehlers (2018) obtains the same result by

considering a different notion of core which contains the one considered by Sönmez and allows

blocking for coalitions with some allocations where the non‐blocking agents receive their en-

dowments. Mumcu and Saglam (2007) prove that the core may be empty in housing markets

with externalities. In a recent paper, Hong and Park (2017) consider a market model with

consumption externalities and analyze two solution concepts based on the core. In particular,

they show that the allocation derived from the TTC algorithm is stable and belongs to both

these two solution concepts; moreover, under a further preference restriction, it is the unique

stable allocation in either of these two cores.

3

Our contribution to the literature on housing markets with externalities is twofold. In the

first part of the paper, we illustrate some difficulties that may arise when externalities are

incorporated into housing market models. We present a set of nonexistence results for several

cooperative solutions; more precisely, we show that: (a) the core may often be empty not just in

a general framework as it has been already proved by Mumcu and Saglam (2007), but even

when the class of preferences is restricted to more special settings; (b) the set of stable allo-

cations can also be empty

4

, in contrast with what has been proved for the classical model with

selfish preferences by Roth and Postlewaite (1977); (c) there exist dominance relations over

allocations such that the housing market does not admit stable sets. These negative results lead

into the second, constructive part of the paper where we focus on the notion of stable alloca-

tions and explore two special classes of other‐regarding preferences that guarantee their ex-

istence and their stability à‐la von Neumann and Morgenstern. The first class is formed by

preference profiles exhibiting the following property: for each trader

i

, given two allocations

a

and

b

assigning him different items ai()and

b

i

a

(), is preferred to

b

if and only if any other

allocation allocating ai()to him is preferred to all the allocations that give him

b

i()

. This class

3

The core notion in the presence of externalities has been recently adopted also by Takarada, Kawabata, Yanase, and

Kurata (2020) to analyze what standards regime emerges as an equilibrium in a three‐country trade model. In their

paper, policy standards are primarily intended to control (negative) externalities (e.g., vehicle emissions or the use of

pesticides for agricultural goods) that emerge both within and across countries.

4

We remark that the set of stable allocations can be interpreted as the core of the housing market based on a different

blocking mechanism, where the status quo allocation in place of the initial endowment is considered.

GRAZIANO ET AL.

|

1483