Essays on Fair, Efficient, and Incentive-Compatible Resource and Cost

Allocation

Stergios Athanassoglou

Submitted in partial fulfillment of the

Requirements for the degree

of Doctor of Philosophy

in the Graduate School of Arts and Sciences

COLUMBIA UNIVERSITY

2008

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ABSTRACT

Essays on Fair, Efficient, and Incentive-Compatible Resource and Cost

Allocation

Stergios Athanassoglou

This dissertation develops and analyzes resource and cost allocation mechanisms,

which, within certain modeling frameworks for an economy, produce outcomes that

meet axiomatic criteria of efficiency and fairness. An equally important goal is to

ensure that these procedures provide incentives for people to reveal private infor-

mation, such as their preferences or valuations, truthfully. The main technical tools

that we employ derive from game theory and discrete optimization, in particular

mechanism design and network flows.

We investigate allocation mechanisms in four distinct economic environments. In

Chapter 2 we study a generalization of the well-known Shapley-Scarf housing mar-

kets and develop an allocation mechanism that satisfies ordinal efficiency, individual

rationality, and no justified envy. In addition, we show that in this more general model strategyproofness is incompatible with ordinal efficiency and individual ra-

tionality. In Chapter 3 we focus on minimum cost spanning tree games and provide

an axiomatic characterization of Bird's rule in terms of efficiency, core-selection,

tree invariance, and merge-proofness. In Chapter 4, we study partnership dissolu-

tion mechanisms under the novel assumption that agents act as maximum-regret

minimizers. We analyze two intuitive classes of mechanisms and derive the regret-

optimizing behavior that they induce. On a more general level, we display the

limitations that ex-post efficiency and ex-post individual rationality impose on our

mechanism design problem. In Chapter 5, we develop a dynamic framework for

the repeated allocation of a scarce resource. We analyze priority-based mecha-

nisms, which aim to mitigate allocation inequities that may build over time, and

we describe the equilibrium behavior that they give rise to.

Contents

List of Tables vi

List of Figures vii

Acknowledgments ix

Chapter 1: Introduction 1

1.1 House Allocation with Fractional Endowments 3

1.2 Minimum Cost Spanning Tree Games and Merging Behavior . . . . 5

1.3 Minimizing Regret when Dissolving Partnerships 7

1.4 Fair Allocation Over Time 9

Chapter 2: House Allocation with Fractional Endowments 12

2.1 Introduction 12

2.1.1 Motivation 12

2.1.2 Contributions 15

2.2 Model Description 16

i

2.2.1 Preliminaries 16

2.2.2 Mechanisms and Properties 17

2.3 Related Work 22

2.4 The Controlled-Consuming (CC) Algorithm 24

2.4.1 An example 25

2.4.2 The Algorithm , 30

2.4.3 Properties 35

2.5 Impossibility Results 40

2.5.1 Strategyproofness 40

2.5.2 Strict and Weak Core Allocations 46

2.6 Extensions 49

2.6.1 Rill Domain 49

2.6.2 Arbitrary Endowment Profiles 50

2.6.3 Arbitrary Fractions of Houses Available in the Market . . . 51

2.7 Conclusion 51

Chapter 3: Minimum Cost Spanning Tree Games and Merging Be-

havior 54

3.1 Introduction 54

3.1.1 Motivation 54

3.1.2 Contributions 56

3.2 Model Description 56

ii

3.2.1 Preliminaries 56

3.2.2 Prim's algorithm 57

3.2.3 Allocation Rules and Properties 58

3.2.4 Merge-proofness 59

3.2.5 Special Allocation Rules 60

3.3 Related Work 61

3.4 Characterization Result 62

Chapter 4: Minimizing Regret When Dissolving Partnerships 74

4.1 Introduction 74

4.1.1 Motivation and Related Work 74

4.1.2 Contributions 77

4.2 Model Description 78

4.2.1 Two Agents 78

4.2.2 Regret 79

4.2.3 Properties of Mechanisms 81

4.3 Linear Mechanisms 82

4.3.1 Linear Mechanisms and Regret 83

4.3.2 Linear Mechanisms under Bayes-Nash 89

4.4 Binary Search Mechanism 91

4.4.1 Binary Search and Regret 92

4.4.2 Uniform distribution and Bayesian-Nash equilibria 95

iii

4.5 General Mechanisms 99

4.5.1 Ex-post Efficiency and Regret 99

4.5.2 Ex-post Individual Rationality and Bayes-Nash 102

4.6 Conclusion 103

4.7 Appendix 104

4.7.1 Multiple Players with General Endowments 105

Chapter 5: Fair Allocation Over Time 108

5.1 Introduction 108

5.1.1 Motivation 108

5.1.2 Contributions 110

5.2 Model Description I l l

5.2.1 Preliminaries I l l

5.2.2 Allocation Mechanisms 112

5.3 Related Work 114

5.4 2-Period Model 116

5.4.1 Priority Mechanism 1 116

5.4.2 Priority Mechanism 2 120

5.4.3 The Uniform [0,1] Nash equilibrium 122

5.4.4 Mechanism Comparisons 124

5.4.5 Monotonicity Comments 130

5.5 N-Period Models 133

iv

5.5.1 General Framework 133

5.5.2 Limiting Equilibrium as x grows 142

5.5.3 Comparison to Canonical Allocations 146

5.6 Extensions 148

5.6.1 Inefficiency of non-MC equilibria 148

5.6.2 A Priority Generalization 149

5.7 Conclusion 151

v

List of Tables

4.1 Actual and Optimal profits: Case analysis 80

4.2 Regret derivation for multiple agent case 105

VI

List of Figures

2.1 The CC Algorithm: Initial network for example 3 26

2.2 The CC Algorithm: Example 3, iteration 1 27

2.3 The CC Algorithm: Example 3, iteration 2 28

2.4 The CC Algorithm: Example 3, iteration 3 29

2.5 The CC Algorithm: Example 3, iteration 4 30

2.6 The CC Algorithm: Example 3, iterations 5 and 6 30

2.7 The CC Algorithm: Example 4 36

3.1 M.c.s.t. before and after insertion of dummy node d 65

3.2 The three merged graphs 67

3.3 The m.c.s.t. and its subtrees 70

3.4 Leaf children 71

4.1 Regret of different linear mechanisms 89

5.1 2-period priority mechanism I 118

vii

5.2 An illustration of priority classes in a multi-period setting 134

viii

IX

Acknowledgments

First and foremost, this work would not have been possible without the support

and encouragement of my adviser, Jay Sethuraman. Jay's razor-sharp intellect,

kind demeanor, and gentle spirit have been an inspiration throughout my time as

a graduate student. I feel so very fortunate to have had him as my mentor.

Special thanks are in order for Professors Steven Brams, Tim Huh, Garud Iyengar

and Nicolas Stier-Moses, who graciously agreed to serve on my committee. I am

extremely grateful for their time and help. I also want to thank Professor Ward

Whitt, whose care and personal warmth have enhanced my time at Columbia.

I would like to recognize my friend Chris Lee, who has supported me in so many

ways over the years and who I consider nothing short of family. Ditto for Constan-

tine Farmakidis, my childhood friend and fellow NFL enthusiast. Jim Tripp, my

wonderful mentor and squash partner extraordinaire, has been a constant reminder

of why you should sometimes go with your gut and reply to that random summer-

internship mass e-mail. Thanks are also due to Rishi Talreja for, among so many other things, the countless hours we have spent in such productive endeavors as

DVD-watching. Chuck Crow has been the living embodiment of our mutual hero's

ever-so-wise insight: "[In the NFL] you play to win the game". Abhinav Verma,

Anuj Manuja, Thiam Hui Lee, Attakrit Asvanunt, and Yori Zwols have been my trusty fellow IEOR warriors.

More and more I come to realize the enduring and central value of family. My

brother, Vassili, has simultaneously and consistently been my best friend, role-

model, caring parent, and pro-bono therapist. Is that the best you can do, Bilako?

Last but certainly not least, my greatest thanks go out to my loving, endlessly

patient, constantly supportive parents, Stathis and Irina. To you, mama and baba,

I, quite literally, owe everything.

This work is dedicated to the loving memory of my grandparents: Pappou Mino and Yiayia Stergia, and Pappou Vassili and Yiayia Vitto.

x

For Yiayia Fani

xi

Chapter 1

Introduction

This dissertation comprises four essays, which address different models of cost and

resource allocation. The first three deal with static environments, whereas the

fourth introduces a dynamic element to the allocation process. The overarching

objective is to design allocation mechanisms that induce self-interested market par-ticipants to act in ways that are desirable from the point of view of a central planner.

In our context, such market behavior involves formal concepts of efficiency, fairness,

and truthfulness.

Our work draws on ideas and insights of mechanism design, a field of economic the-

ory that deals with the design of markets to implement precise economic outcomes.

While one could write tomes of expository notes on the subject, we content our-selves with stating that the field has had a profound effect on both the theory and

practice of economics. Its progenitors were just recognized with the Nobel Prize in Economics and their ideas have been successfully applied to a host of real-world

settings. Two notable applications include the design of FCC spectrum auctions

in the 1990's and the recent redesign of school choice mechanisms in Boston and

2

New York City. In the case of the former, by cleverly setting up the rules of its

spectrum auctions, the federal government was able to generate a huge amount of

revenue from the sale of its licenses (according to some, orders of magnitude more than anyone expected at the time). With regard to the latter, Boston and New York redesigned the way that children get assigned to public schools so that there is

no longer any incentive for parents to game the system by stating untruthful school

preferences (i.e., by lying about their relative ranking of different schools). In both cases, theoretical insights realized significant welfare gains and effected substantial

public policy change.

Despite these success stories, such lofty economic outcomes do not come for free.

Mechanism design problems often involve tradeoffs that stem from fundamental

impossibility results: If we insist on finding a mechanism that "does too many

things at once," then quite simply such a mechanism might not exist. The practi-

cal implications of these ideas are profound as they signify the limitations inherent

in organizing a society based on a set of rigorous principles. On an abstract level,

such results are also quite deep, since they imply that there is no function that can simultaneously satisfy a set of properties. A mechanism designer is thus called

to relax the formal criteria the mechanism is required to meet, or to effectively

decide which properties matter most to the allocation procedure. Since such con-

flicting properties typically involve formal criteria of efficiency and fairness, the

philosophical questions imbedded within these choices are far from trivial.

We now turn to a brief description of the four chapters of this dissertation.

3

1.1 House Allocation with Fractional

Endowments

In our first essay we employ basic network flow theory to develop a practical al-

location mechanism. Our modeling framework generalizes classical economic envi-

ronments such as Shapley-Scarf housing markets (52) and the random assignment problem. We assume a collection of agents each owning arbitrary fractions of differ-ent objects (houses). Each agent has a complete and transitive ordinal preference relation over the set of houses, allowing for indifferences, and wishes to be allocated

the fractional equivalent of one 'full' house. Fractional allocations are partially

ordered via stochastic dominance. Obviously, an agent enters the economy with

the aspiration that she will receive a better allocation than her own endowment.

Since participation in such reallocation procedures is generally voluntary, designing

a mechanism in a way that affords such a guarantee to all agents is important. The

challenge is to set up a systematic procedure that will produce an efficient and fair

allocation, while guaranteeing each agent a (weakly) better allocation than her own endowment and simultaneously protecting against gaming behavior.

We need to be clear about how these properties are defined in the context of our

mechanism design problem. With regard to efficiency, we consider a generalization

of the familiar Pareto criterion to accommodate fractional allocations ("ordinal effi-ciency"), which was introduced by Bogomolnaia and Moulin in a seminal paper (9). Participation constraints are formalized with individual rationality: A mechanism

satisfies this property if all agents are unequivocally no worse off after the mecha-

nism has determined their new allocation, as compared to their initial endowment.

Fairness is captured by the concept of envy-freeness: If no agent prefers another's

allocation to her own then the economy's allocation is envy-free. Clearly, we may

4

encounter cases where envy-freeness conflicts with individual rationality. To recon-

cile the two in a meaningful way, Yilmaz (54) introduced the concept of no-justified envy. Intuitively, justified envy considerations may only arise when two agents have the same initial endowment, or when an agent is in some sense dispropor-

tionately rewarded in comparison to her peers. Finally, the model assumes that

agents' preferences are private and have to be elicited by the mechanism designer.

It is desirable that agents do not find it in their best interest to be untruthful in

the underlying revelation game. In this regard, strategyproofness ensures that be-

ing truthful about one's preferences is a dominant strategy. Its weak counterpart

provides a guarantee that truthful preference revelation not be dominated by any

other strategy.

In our work we develop a mechanism, which we refer to as the Controlled-Consuming

(CC) mechanism, that computes a fractional allocation of houses to agents that sat-isfies ordinal efficiency, individual rationality, and no justified envy. Our mechanism can be efficiently implemented with a simple parametric maximum flow algorithm

whose running time is polynomial in the number of agents, and our analysis extends

to the full preference domain. At the same time, the generality of the model gives

rise to a number of interesting impossibility results. We show that individual ratio-

nality, ordinal efficiency, and no justified envy conflict with weak strategyproofness in the strict preference domain, a result which remains true in less general models.

We also show, somewhat counter-intuitively, that individual rationality, ordinal ef-

ficiency, and strategyproofness are incompatible in the strict preference domain.

This suggests that requiring any mechanism to be strategyproof is fundamentally

at odds with relatively standard criteria of efficiency and individual rationality. Fi-

nally, we show that the strict core of the associated cooperative game is empty and

that the weak core conflicts with individual rationality.

5

1.2 Minimum Cost Spanning Tree Games and

Merging Behavior

Minimum cost spanning tree games, first introduced by Claus and Kleitman (16), assume a collection of agents who jointly wish to invest in the formation of a network, which will connect them to a source (for example, a power plant or a water reservoir). Connecting one agent to another (or the source) involves a certain cost, which reflects the difficulty of establishing a link between the two parties. This

environment can be abstracted by a complete undirected network whose node set

represents the source and agents, and whose link costs capture the cost of connecting

one agent to another or to the source. One obvious way that agents can gain access

to the source is by connecting to it directly, but this may clearly be very inefficient.

Instead, a network designer would wish to identify the minimum cost spanning tree

(m.c.s.t.) of the network, which may or may not be unique. Once a m.c.s.t. has been determined, the question immediately arises: How should agents share the

total cost of the joint investment? Since it is natural to expect that agents will collaborate with one another in the formation of the network and the allocation

of subsequent costs, the tools of cooperative game theory have been extensively

applied to this problem.

One way to perform the allocation would be to simply divide up the total cost

by the number of agents and assign everyone an equal share. Another would be

to pick an agent at random and assign her the full cost of the m.c.s.t. A third

would be to assign each agent the cost of her incoming edge in the m.c.s.t, which

we take to be rooted at the source node. This last rule was introduced by Bird (7) and is referred to as Bird's rule. Indeed, there is no shortage of ways we could

imagine to resolve this problem. At the same time, we would want the method we

6

choose to be, in some sense, stable, fair, and not too complicated. For example, one

can easily imagine a situation where the previous "egalitarian" rule would not be

tenable: Say an agent could connect directly to the source at a cost that is less than

her egalitarian share. Why should that agent not reject the proposed cost-sharing arrangement and go it alone?

Thus, it becomes clear that proposed rules will need to satisfy certain formal cri-

teria of stability and fairness in order for them to be acceptable. A particularly

important one is captured by the concept of core selection, which ensures that the

final allocation is such that no subset of agents will find it in its collective best in-

terest to deviate and form its own sub-network to the source. The core is a central

concept in cooperative game theory and refers to exactly those allocations that are

robust to sub-coalitional deviations. Not all cooperative games have a non-empty

core, but m.c.s.t. games do, as Bird (7) and Granot and Huberman (24) showed. Thus, insisting on a rule that always selects a core allocation becomes compelling.

In addition, we can imagine agents having the legal right to merge into one entity

and demand to be treated as such by the adopted cost sharing rule. Such merging

behavior would involve assigning a cost to the new "mega-agent" while the merged

agents connect to one another at minimum cost. If our allocation rule is such that

this maneuver yields no benefit to any subset of users, then it is merge-proof. This

property, first mentioned by Claus and Kleitman (16), is important to consider, since merging behavior typically leads to inefficient networks. In addition, when

merging is not sanctioned by law, it is still important to ensure that no group of

agents feels unfairly treated in not being allowed to collaborate in this way. Fi-

nally, we note two additional regularity properties that cost-sharing rules generally

adhere to. Efficiency is satisfied if the cost of the m.c.s.t. equals the sum of in-

dividual costs. On the other hand, tree invariance forces the final allocation to

be insensitive to changes in the cost structure of the graph which do not alter the

7

m.c.s.t. In essence, tree invariance stipulates that the cost shares only be a function

of the m.c.s.t that the cost structure of the network will induce. It is important for

computational reasons, as the size of the problem gets bigger.

In our essay we provide an axiomatic characterization of Bird's rule. Assuming a

priority ordering of the agents, we show that Bird's rule is the only rule which simul-

taneously satisfies efficiency, core-selection, tree invariance, and merge-proofness.

Our result is tight, in that one can find rules that satisfy all possible subsets of

three of the above properties while failing the fourth.

1.3 Minimizing Regret when Dissolving

Partnerships

Partnership dissolution models assume a group of agents, each owning an arbitrary

fraction of a partnership and having a private valuation over the sole possession

of the commonly-held good. Valuations are taken to be independent and random,

and their distributions are assumed to be common knowledge. Agents are called to

submit bids, which a mechanism takes as input and determines how the partnership

will be split and at what price.

The canonical example of a partnership dissolution model is the bilateral bargaining

game introduced by Chatterjee and Samuelson (14). The process goes as follows. A buyer and a seller each submit a sealed bid reflecting their valuation of the good.

If the seller's bid is greater than the buyer's then the good is sold at a price that

is a convex combination of the two bids; otherwise, no deal is struck. Chatterjee and Samuelson derive necessary and sufficient conditions for the Bayesian-Nash

equilibrium of this game, and compute it explicitly in some special cases. They

further note that equilibrium bidding behavior may give rise to situations when

8

mutually beneficial trade does not occur. The elusive property here is ex-post

efficiency: We cannot ensure that trade happens if and only if the seller's valuation

is lower than the buyer's. In a seminal paper, Myerson and Satterthwaite (43) show that this situation is, in some sense, unavoidable. They characterize the set of all

Bayesian incentive compatible and interim individually rational mechanisms and

note that none of them yield ex-post efficient equilibrium strategies. The Revelation

Principle (see Myerson (41, 42)) then implies that there is no such mechanism at all. In a later paper, Cramton et al (15) generalize the Chatterjee-Samuelson model to include any kind of multi-agent partnership where the good in question is co-owned

among the partners in an arbitrary way. They extend the results of Myerson and

Satterthwaite and show that the only partnerships that can be efficiently dissolved

(that is, for which an incentive-compatible, interim individually rational and ex-post efficient mechanism exists) are ones which are not too far from the equal endowments vector in a way that depends on the given distributions.

In our work we revisit the partnership dissolution framework under the assumption

that agents act as maximum-regret minimizers as opposed to profit maximizers.

The regret of an agent is defined to be the difference between her optimal payoff

and her actual payoff, where the optimal payoff is calculated by assuming that the

agent knows the other agent's bid. That is, an agent's optimal payoff is the best she

could have done in hindsight. In contrast to the classical profit-maximizing model,

valuations are not assumed to follow a particular distribution and are simply taken

to lie in a certain interval. Given that it is often unrealistic to model underlying

distributions as common knowledge, we believe that this aspect of our model adds

robustness to our analysis. In this context, we examine general partnership profiles

(a special case of which would be traditional buyer-seller auctions) and derive regret-optimizing bidding strategies for certain intuitive mechanisms. These include the

class of linear mechanisms, where the price is set to be a convex combination of

9

agents' bids, and a binary search mechanism, where the price is set at the midpoint

of successively smaller intervals. We further analyze these mechanisms under a

Bayesian-Nash profit-maximizing setting. Focusing on the equal-endowment case,

we exhibit equilibrium strategies for both mechanisms and discuss appealing ex-post

efficiency and rationality properties that emerge. We close by showing that ex-post

efficiency and ex-post individual rationality impose significant restrictions on the

kinds of mechanisms that may be used. These results are, in a way, comparable

to the classical profit-maximizing setting where the only partnerships which can be

dissolved efficiently are ones that are not too far from the equal endowments vector.

1.4 Fair Allocation Over Time

In our last essay we consider the problem of allocating an over-demanded, divisible

resource over a fixed time horizon. We assume a very high number of agents,

competing for the same infinitesimal fraction of the resource in each time period. An

agent derives utility from consuming the resource, and these utilities are randomly

distributed across agents. Real-life examples of this economic environment include

Internet bandwidth: It is a huge, over-demanded resource sought after by a (for all intents and purposes) continuum of users. Individual demand is minuscule compared to total supply, but if it were to be tallied up, it would not be able to be

satisfied in its entirety. Further, we may assume that different people derive different

amounts of utility from the utilization of the Internet and that this variation is also

temporal.

In this economic environment we wish to explore ways of performing an efficient

and fair allocation over time. We could, of course, just allocate randomly at each time period. This arrangement is ex-ante fair in that each agent gets the same

ex-ante utility. It is not, however, fair in an interim sense. In particular, if an agent

10

has received the resource 10 times more than another, should the two of them be

treated the same way as the allocation process moves forward? A random procedure

would do just that, not taking into account the imbalance between agents that has been built up over time.

In order to address this kind of inequity, we introduce a family of priority-based

mechanisms. In these mechanisms, an agent's allocation history is taken into ac-

count in present allocation decisions. In particular, agents are placed within priority

classes in accordance with their past behavior. To agents within a certain class,

the resource is randomly distributed, provided they actually request it. We assume

that an agent makes the decision whether or not to request access to the resource

after observing the (randomly distributed) utility that will be derived if it is ac-tually granted. If that utility is high enough, then it is worthwhile to risk losing

higher priority in future assignments. For example, in a 2-period model one kind of

priority mechanism would grant higher priority in period 2 to all those agents who

did not receive the resource in period 1. Alternatively, another kind of mechanism

would grant this kind of preferential treatment to all agents who did not request the

resource in period 1. In both cases, an agent is called to assess the tradeoff between

requesting access in period 1 and potentially dropping down a priority class, and

retaining her high priority in anticipation of allocation decisions in period 2.

A particularly interesting parallel to this work can be found in the economics litera-

ture on voting procedures and storable votes, which was introduced by Casella (11). In her model, voters were asked to weigh the costs and benefits of casting or storing

individual votes as individual decisions arose over a fixed time horizon. A decision

to abstain from voting on a relatively unimportant issue now would result in the

opportunity to cast more votes on a later issue, which a voter may care more about.

Similarly to our model, Casella showed that storable votes generally result in ex-

ante welfare gains over the status-quo "one issue, one vote" procedure. Inspired in

11

part by her work, Jackson and Sonnenschein (30) studied the problem of linking decisions over time in a more general framework. They introduced a mechanism

that compels agents to declare their types in a way that mirrors the underlying type

distribution and proved that it can achieve full efficiency. Finally, we briefly note

that using priority-based methods in game-theoretic queueing models has been in-

vestigated widely and the reader is referred to Hassin and Haviv (28) for a thorough and incisive survey.

In our work, we focus on two canonical priority mechanisms. We formulate the

Bayesian game that they induce, and investigate Nash equilibrium strategies that

emerge. We provide a thorough treatment of 2-period models and comment on

monotonicity and uniqueness properties that the equilibrium strategy will, or in

some cases will not, satisfy. Focusing on market-clearing equilibria, we compare

ex-ante the outcomes of our priority mechanisms to generic allocations in terms

of efficiency and probability of access. Turning to the multi-period version of the

model, we describe the form of the equilibrium, investigate its structural properties,

and compute it explicitly in some special cases. In particular, when aggregate

demand is high enough, the game admits an equilibrium strategy that is insensitive

to its fluctuation. We provide a simple recursive procedure to efficiently compute

this strategy. We close with a brief comment on a more general priority framework.

12

Chapter 2

House Allocation with Fractional

Endowments

2.1 Introduction

2.1.1 Motivation

In this essay, we consider the problem of allocating a number of objects (say houses) to agents in an efficient and fair manner. Agents have complete and transitive pref-erences over the houses, and each agent wishes to be allocated the equivalent of

at most one house. The distinguishing feature of our model is that agents may be

endowed with fractional amounts of various houses. This model is a common gener-alization of several well-studied models that have received a lot of attention in the

literature. If each agent is endowed with a distinct house, we recover the housing

market model, first considered by Shapley and Scarf (52). For this model, Shapley and Scarf proposed the Top-Trading Cycles mechanism (attributed to Gale) that

13

finds the unique core allocation of the associated cooperative game. The TTC mech-

anism is Pareto efficient, strategyproof, and is individually rational (precise defini-tions are given later). At the other extreme, if agents have no endowments, we re-cover the random assignment problem considered by, among others, Abdulkadiroglu

&; Sonmez (1), and Bogomolnaia & Moulin (9). Abdulkadiroglu and Sonmez (1) study the random priority mechanism: agents are ordered randomly, they choose

houses in this order, and each agent picks her most preferred house among the set

of houses still available. This mechanism is strategyproof, ex-post Pareto efficient, and satisfies equal treatment of equals. An alternative mechanismprobabilistic serialfor the same problem was proposed by Bogomolnaia and Moulin (9): at each point in time, agents consume their best available houses at unit rate. The

resulting assignment can be implemented as a lottery over efficient deterministic

assignments. Bogomolnaia and Moulin (9) showed that the probabilistic serial mechanism finds an allocation that is envy-free and is ordinally efficient (a stronger form of efficiency), but satisfies strategyproofness only in a weaker sense. Finally, Abdulkadiroglu and Sonmez (2) and Yilmaz (54) have considered house allocation problems with existing tenants: in this model, some agents have no endowments

("new tenants") and others are endowed with a distinct house ("existing tenants"). In such models, in addition to fairness and efficiency, it is natural to require in-

dividual rationality: absent such a requirement, agents may not participate in the

mechanism in environments where such participation is voluntary. Abdulkadiroglu

and Sonmez (2) designed a natural mechanism for this problem that specializes to the TTC mechanism when there are no new tenants, and to the RP mechanism

when there are no existing tenants. Recently, Yilmaz (54) proposed a mechanism that specializes to the PS mechanism when there are no existing tenants, but finds

a solution that is different from the TTC mechanism when there are no new ten-

14

ants.1 Note that in models with endowments, expecting envy-free assignments that

are also individually rational may be unreasonable, as these two requirements may

be in obvious conflict with each other. A key contribution of Yilmaz (54) is his definition of justified and unjustified envy, which shows how to interpret the equity requirement when agents have different endowments.

Applications. House allocation and random assignment models have significant

real-life applications. Allocation of scarce on-campus housing is one celebrated

example of the usefulness of the TTC algorithm. Our expanded model addresses

situations in which the goods themselves are indivisible, but there is an initial

lottery over them (that is described by agents' fractional endowments). This lottery, via the Birkhoff-von Neumann theorem (8), can be thought of as a probability distribution over initial deterministic allocations. In keeping with the dorm-room

analogy, fractional endowments could reflect the relative "right" an agent has over

different rooms. In this environment, agents wish to enter a reallocation process

that improves upon their initial lottery.

Another way of addressing fractional individual ownership of a good would be

to assume that goods are divisible entities. This would contrast to the classical

models where goods are assumed to be indivisible and a fractional allocation of

goods to agents is viewed as a lottery assignment. This approach makes particular

sense in markets where time-sharing is an option and where goods are not perfect

complements. If a good may be consumed at different times during the course of a

given time period, then owning a fraction of it simply reflects the amount of time

an agent is entitled to consuming it. 1Sethuraman (51) proposed a solution for this problem that specializes to the PS mechanism

when there are no existing tenants and to the TTC mechanism when there are no new tenants.

15

2.1.2 Contributions

In our model, agents are allowed to have arbitrary endowments over the houses.

Thus, the model we proposeHouse allocation with fractional endowmentsis a common generalization of most of the existing models in the context of house allo-

cation. For the house allocation problem with fractional endowments, we design an

algorithm to find an assignment that is individually rational and ordinally efficient,

and does not have any "justified" envy. The algorithm is efficient (its running-time is polynomial in the size of the input), and works by solving a sequence of maximum flow problems. It is in the spirit of earlier work of Katta and Sethuraman (32) and Yilmaz (54), and can be viewed as a generalization of these results to this sub-stantially more general model. We further show that these propertiesindividual

rationality, ordinal efficiency, and no justified envyare incompatible with even the mildest incentive compatibility conditions: any mechanism which satisfies them is

vulnerable to manipulation by some agent. This result is true even in Yilmaz's sim-

pler model. Somewhat surprisingly, we show that ordinal efficiency and individual

rationality alone are incompatible with strategyproofness in the strict preference

domain, a result which is not true in any of the less general models considered

in the literature. Furthermore, our result holds even if all the goods are equally

shared by all the agents, a restricted class of instances of the general model we

examine. From a cooperative game theory standpoint, we show that in this more

general model the strict core of the associated cooperative game is empty, and that

inclusion in the weak core conflicts with individual rationality.

Organizat ion of t h e Chap te r . We discuss the model more formally in Sec-

tion 2.2 and provide an overview of the related work in Section 2.3. Section 2.4

is the main contribution of the essay and contains a description of the algorithm

to find a solution for any given instance of the house allocation problem with frac-

16

tional endowments; it also proves that the algorithm finds an assignment that is

individually rational, ordinally efficient, and has no justified envy. Section 2.5 con-tains some impossibility results that show, in effect, that strategyproof mechanisms

must violate individual rationality or efficiency. We discuss several extensions in

Chapter 2.6 and we provide concluding thoughts in Chapter 2.7

2.2 Model Description

2.2.1 Preliminaries

Consider a market with n agents / = {1,2,..., n} and n houses H = {hi, h2,..., hn}. Suppose agent % is endowed with e^ units of house hj, with each e^ [0,1]. We assume that each agent owns at most the equivalent of a full house, and that at

most one unit of any house is owned by the agents. In other words, the endowment

matrix can be represented by a doubly sub-stochastic matrix 2

hi h2 . . . hn

1 en ei2 . . . ei

2 e2i e22 . . . enn

fl &nl &n2 "nni

with the rows indexed by the agents and the columns by the houses. Each agent

i has (ordinal) preferences over the set of houses expressed by the complete and transitive relation. >zi 3 If houses hj and hk are such that hj >Zi hk and hk >Zi hj

2A11 of the results extend to the more general case in which an arbitrary amount (instead of 1) of each house is available in the market. See 2.6 for this and other generalizations.

3 All of our results extend in a straightforward manner to the case in which >a is quasi-transitive,

and also to the case in which hi is a partial order. We omit the details.

17

then agent i is indifferent between houses hj and hk, denoted by hj ~j hk- If hj hi hk, but hk )ti hj, then agent i strictly prefers hj to hk, denoted by hj >i hk-It is clear that the relation ~< is symmetric and transitive, and that the relation

>~i is antisymmetric and transitive.

An allocation for agent i is a vector pi (pn, pi2,..., Pin) such that ^ . pij < 1. The interpretation is that in allocation pi, agent i consumes p^ units of house hj. As is clear from the definition of an allocation, we consider environments in which agents

desire at most the equivalent of "one house." The allocations for all the agents

can be described by an assignment matrix, with the rows indexing the agents and

columns indexing the houses; like the endowment matrix, the assignment matrix

will be a doubly sub-stochastic matrix. In this work, we explore mechanisms for

allocating the houses to the agents satisfying some desirable properties, described

next.

2.2.2 Mechanisms and Properties

A mechanism is a function that determines an assignment matrix for every possible

profile of preferences and endowments. The desirable properties of a mechanism

typically stem from efficiency, truthfulness, and equity considerations.

Efficiency. For agent i, an allocation pi dominates qi, denoted pi hi Qi, whenever

5 3 Pa ^ 5Z qik> for a11 h G H-

If at least one of the above inequalities is strict, then pi strictly dominates qi, and is

denoted pi yt qi. (The dominance relation described here is simply the first-order stochastic dominance.) Note that h is a partial order and certain allocations are not comparable, for example (0,1,0) and (1/2,0,1/2). The dominance relation

18

defined on individual allocations extends to assignment matrices in a natural way:

an assignment matrix P dominates an assignment matrix Q if p^ yt qi for every agent i; P strictly dominates Q if P dominates Q, and if Pi ^ , qi for some agent i. An assignment P is said to be ordinally efficient (or simply efficient) if P is not strictly dominated by any assignment Q. A mechanism is efficient if it determines an efficient assignment for every profile of preferences and endowments.

Individual Rationality. Consider an environment in which participation is vol-

untary. If the mechanism finds an allocation pt for agent i such that pi >a et, then

agent i will always participate. Otherwise i may choose not to participate, which

may result in an inefficient assignment. So, an assignment P is said to be individ-

ually rational if Pi >a e^ for each agent i. A mechanism is individually rational if it

always determines an individually rational allocation.

Truthfulness. In many application contexts, preferences of the agents are not

observable, but should be elicited from them. A natural, but fairly strong, re-

quirement then is a mechanism in which it is a (weakly) dominant strategy for agents to reveal their preferences truthfully. As not every pair of allocations can

be compared, there are two versions of this property. A mechanism is said to be

strategyproof if for every agent i, the allocation she obtains by reporting her true preferences (weakly) dominates the allocation she obtains by reporting any other preference, regardless of what the other agents do. A mechanism is said to be

weakly strategyproof if for every agent i, the allocation she obtains by reporting her true preferences is not dominated by the allocation she obtains by reporting any

other preference, regardless of what the other agents do.

Equity. A minimal requirement of fairness is the familiar property of equal treat-

ment of equals (ETE), which states that two agents with identical endowments and preferences should receive identical allocations. Formally a mechanism satis-

fies ETE if it finds an allocation such that pi = p# whenever e = e,' and >^ , = >:#,

19

for any pair of agents i and i'.

A stronger requirement is envy-freeness, which states that each agent's allocation (weakly) dominates every other agent's allocation. That is, for any agent i, pi >Zi py for every agent i'. Indeed this property has been considered in many economic

contexts, notably in the house allocation problem with no endowments. For the

model with endowments, however, envy-freeness is too strong a requirement as it is

in obvious conflict with individual rationality. For instance, suppose agents i and

i' both have house hj as their most preferred choice, but i owns hj. In this case individual rationality dictates that i be allocated hj fully, but i' will necessarily envy i in this allocation. One way to overcome this difficulty is to require only that

agents with the same initial endowments not envy each other. In other words, a

mechanism is envy-free if it finds an allocation such that Pi >zi pi> whenever e^ = e^.

Recently, a stronger notionno justified envyhas been proposed by Yilmaz (54). In the model with endowments, Yilmaz distinguishes between two kinds of envy:

justified and unjustified envy. The difference is explained by the following two examples, both due to Yilmaz (54).

Example 1. Consider the following instance of the house allocation problem with

three agents {1,2,3} and three houses {a, 6, c}. Agent 1 prefers a to 6 and b to c; agents 2 and 3 prefer b to a and a to c. The initial endowments are specified

in braces, next to the preference ordering. Here, agent 1 is endowed with house b,

agent 2 with a, and agent 3 with c.

1 : aybyc {b} 2 : by ay c {a} 3 : by ay c {c}

It is clear that the only individually rational and efficient assignment is

20

a b c

1 1 0 0

2 0 1 0

3 0 0 1

Clearly agent 3 will envy both agents 1 and 2. However, this envy is not justified because it is not possible for agents 1 and 2 to give up any portion of their endow-

ments to agent 3, receive a positive share of house c and still maintain individual

rationality. In contrast, consider the following example:

Example 2. 1 : ay cy b {b} 2 : by cy a {a} 3 : by ay c {c}

The assignment discussed earliergiving a to 1, b to 2, and c to 3is still indi-

vidually rational and efficient. However there are other individually rational and

efficient allocations because agents 1 and 2 are willing to give up some of b and a

respectively for any house in the sets {a, c} and {6, c} respectively. In this context, if all of c is allocated to agent 3, then this agent could justifiably envy agents 1 and 2. This is because instead of giving agents 1 and 2 their best houses, the mechanism

could have found a different allocation in which agents 1 and 2 do a little worse,

still maintain individual rationality, and agent 3 does a little better. In particular,

the assignment

a b c

21

is individually rational, efficient, and is envy-free.

Yilmaz (54) formalizes these observations into the following definition: an agent i justifiably envies an agent i' if i's allocation does not dominate z"s, and if i's allocation is an individually rational allocation for agent i'. Formally, i justifiably envies i' if

Pi >tiPv and pi t.i' ev.

Equivalently, i does not justifiably envy i' if

Pi hi Pi' or pi fa ei>.

We say that a mechanism satisfies no justified envy if in the assignment it deter-mines, no two agents justifiably envy each other.4

Observe that in the presence of individual rationality, no justified envy implies that agents with identical endowments do not envy each other. In the rest of the paper,

we shall work with no justified envy as our equity criterion.

We conclude this section by pointing out that the model we consider generalizes

some of the most prominent models studied in the house allocation literature. In

particular:

If the endowment matrix is a permutation matrix, we recover the classical

house trading model of Shapley and Scarf (52) in which each agent owns a distinct house.

If the endowment matrix is identically zero, we get the random assignment

problem considered by Abdulkadiroglu and Sonmez (1), Bogomolnaia and Moulin (9), Katta and Sethuraman (32), and others.

4We could require further that agent i"s allocation dominate i's endowment. But this makes it more difficult for justified envy to exist, so no justified envy becomes easier to satisfy.

22

If the endowment matrix is {0,1} with each column sum at most 1 and each row sum at most 1, we obtain the house allocation problem with existing

tenants, considered by Abdulkadiroglu and Sonmez (2) and Yilmaz (54).

2.3 Related Work

Housing Markets. Shapley and Scarf first introduced house allocation problems

in a seminal paper (52). In their problem each agent owns an indivisible good (a "house"), but has strict preferences over all the houses, including the house she owns. By means of an algorithm called the Top Trading Cycles algorithm

(attributed to Gale), Shapley and Scarf find a reallocation of the houses that is in the core of the associated cooperative game. Roth and Postlewaite (49) showed that when preferences are strict, the strict core of this cooperative game is unique, and

Roth (48) showed that the TTC mechanism is strategyproof. Furthermore, Ma (34) characterized the TTC mechanism using individual rationality, Pareto efficiency,

and strategyproofness. If preferences are not required to be strict, Shapley and

Scarf (52) showed that the strict core may be empty. Recently, Quint and Wako (47) provided an efficient algorithm for the house trading problem with indifferences that

either finds a strict core allocation or verifies that the strict core is empty. This

algorithm is a natural generalization of the TTC algorithm. The common element

in this line of research is that each agent is assumed to own exactly one house.

Random Assignment. In the random assignment problem individual agents do

not own any houses; instead the houses can be thought of as a collectively-owned

social endowment. This gives rise to a much different problem than the house

trading problem. First, the core is not a relevant concept because individual agents

do not own houses, but Pareto efficiency is. Second, since every agent has, a priori,

the same rights over the houses in the market, designing a fair mechanism becomes

23

an important consideration. A natural solution to the problem is to randomly

order agents and let them choose their most preferred available houses according to

this order. This is the random priority mechanism and it has a number of desirable

properties such as strategyproofness and equal treatment of equals. However, it may

fail to be (ordinally) efficient and agents may envy one another. In a remarkable paper Bogomolnaia and Moulin (9) considered the random assignment problem with strict preferences; they proposed the Probabilistic Serial (PS) mechanism, which computes a random assignment of houses to agents that is ordinally efficient,

envy-free, and weakly strategyproof. They also showed that strategyproofness is

incompatible with ordinal efficiency and the very mild fairness requirement of equal

treatment of equals. Extending their work to the full preference domain, Katta and

Sethuraman (32) proposed an ordinally efficient and envy-free mechanism (EPS), while showing that efficiency and fairness conflict with strategyproofness, even in

the weak sense.

House Allocation with Existing Tenants. Abdulkadiroglu and Sonmez (1) considered a hybrid model that has the features of both the house trading problem

and the random assignment problem. In this model some agents ("existing ten-ants" ) are endowed with distinct houses, whereas other agents have no endowments; each house is either owned by some agent, or is owned by all the agents collectively.

For this model, Abdulkadiroglu and Sonmez (1) proposed a mechanismTTCfrom random orderingsthat is individually rational, efficient in the ex-post sense, and

strategyproof. The mechanism they propose specializes to the TTC mechanism

when all the agents are existing tenants, and to the random priority mechanism

when none of the agents is an existing tenant. Yilmaz (54) observed that this mechanism may privilege existing tenants at the expense of new ones in a way that

may not always be justifiable. Motivated by this observation, Yilmaz proposed "no justified envy" (discussed earlier) as a notion of fairness, and described an alterna-

24

tive mechanism for the house allocation problem with existing tenants. The new

mechanism he proposed (PSe) is individually rational and ordinally efficient, and satisfies no-justified envy. Yilmaz also showed that no justified envy, individual ra-tionality, and strategyproofness are incompatible, even when preferences are strict.

Yilmaz's mechanism specializes to the probabilistic serial mechanism when none of

the agents is an existing tenant.

2.4 The Controlled-Consuming (CC) Algorithm

In this section we design an efficient algorithm to find an allocation satisfying indi-

vidual rationality, ordinal efficiency, and no justified envy. To make the discussion transparent and to keep the notation short, we shall restrict attention to the case

in which the agents have strict preferences and have doubly stochastic endowment

matrices. In Section 2.6, we show how our algorithm can be adapted to deal with

indifferences and more general endowment profiles.

The CC algorithm falls under the general class of simultaneous eating algorithms,

first introduced by Bogomolnaia and Moulin (9). In particular, it allows each agent to "eat" her most preferred available house at rate 1, as long as there is

some way to complete the assignment so that the individual rationality constraints

are not violated; this continues until some house is completely consumed, or some

individual rationality constraint is in danger of being violated. In the latter case the

agents, whose continued consumption of their best available houses would violate

some individual rationality constraint, are forbidden from consuming their most

preferred houses even if they are available, and they move on to their next best

house. Before we describe the algorithm formally, it is useful to consider an example

in detail.

25

2.4.1 An example

Example 3. Consider the following instance:

1 : a>-c)~b {.996, .01c} 2 : b>-ayc {.99a, .01c} 3 : bycya {Ma, .016, .98c}

The algorithm finds the final allocations by solving a sequence of maximum-flow

problems (for background on maximum flows, see Ahuja et al. (4)) on specific networks associated with the given instance. The networks all have the same set of

nodes, but the set of arcs will change during the course of the algorithm. The nodes

of the network are as follows: for each agent i, we introduce 3 nodes i^, i@), i(3), one for each "preference level"; there is a node for each house; and finally, there is

a source node s and a sink node t. The arcs of the initial network are as follows:

the source is connected to each node i^), with the capacity of the arc being eih if h is the fcth most preferred house of agent i; each "house" node is connected to the

sink with an arc of capacity 1; and finally, there are k infinite capacity arcs from

node i(fc) to the house nodes, one to each of agent i's k most preferred houses. The initial network is shown in Figure 2.1.

We first make a few observations:

Any flow from s to t determines an assignment in a natural way: the amount

of house h allocated to agent i, denoted pih, is given by the total amount of

flow in the arcs (i^), h), for k = 1,2,3.

The maximum flow from s to t is 3 (use the endowments as flows).

Any flow of 3 units from s to t determines an individually rational assignment:

the only way to send 3 units of flow from s to t is for each arc from s to i^) to

26

Figure 2.1: Initial network for example 3

carry a flow equal to its capacity, which is equal to z's endowment of her kth.

best house; the only way for this flow to reach the sink is via one of the arcs

leaving i^), and each of these arcs is to a house that i (weakly) prefers to her kth. best house. So the individual rationality constraints will be satisfied for

every agent i.

The flow given by the endowments (i.e., the assignment in which each agent is given her endowment) is individually rational, but may not be ordinally efficient. To find an ordinally efficient assignment, we employ a variation of the "simultaneous

eating" algorithm due to Bogomolnaia and Moulin (9): for each agent i, we consider increasing the capacity on the arc (s,i^) at unit rate; because of this increase, we can decrease the capacity (at unit rate) on the first positive capacity arc in the sequence (s, i(2)), (s, i(z))i > (s> *(n)) j a n d still maintain individual rationality. The algorithm we propose does exactly this with one important exception: if an agent

i has a positive endowment e of her most preferred house, the capacity of the arc

(s, i(i)) is set to this endowment until time e so as to maintain individual rationality.

27

On Example 3, these ideas translate to the network shown in Figure 2.2.

Figure 2.2: Example 3, Iteration 1, Ao = 0

The interval of interest in this case is [0,0.01] because for A > 0.01, arc (s, 1(2)) has negative capacity; also for A > 0.01, the capacity of arc (s, 3(X)) should be A, not 0.01 as dictated by individual rationality. Interpreting A as time, we wish to find

the largest value of A in the interval [0,0.01] for which the maximum flow from s to t is still 3. In this case, the largest such value of A is 0.01, at which point the

network is updated as follows: the capacity of arc (s,3(i)) becomes A, the capacity of arc (s, 3(2)) is set to 0.99 A; moreover, as the capacity of arc (s, 1(2)) has dropped to 0, we set the capacity of arc (s, 1(3)) to 1 A. The updated network is shown in Figure 2.3.

We increase A as long as the maximum s-t flow is 3, and the interval of interest

in this case is [0.01,0.99]. Now, observe that for A > 1/2, the maximum s-t flow

28

is strictly below 3: for A > 1/2, the capacity of arc (s, 2(X)) is above 0.5, as is the capacity of arc (s, 3(i)); as these arcs are only connected to b, and as node b cannot carry a flow of more than 1, the maximum flow will have to be below 3. (Observe also that the amount that can reach node c is below 1.) That the maximum s-t flow is below 3 for A > 1/2 can also be certified by exhibiting a minimum-cut. Consider

the cut defined by the subset of nodes {s, 2(X), 3(i), b}. The arcs that contribute to the capacity of this cut are all the arcs of the form (s, i^)) except for (s, 2(i)) and (s, 3(i)), and the arc (b,t); the total capacity therefore is 3 2A + 1 = 4 2A, which is below 3 when A > 1/2. The minimum-cut also helps us identify some arcs

of the form (i^),h) that cannot carry a positive flow: for example, the arc (1(3), b) cannot carry a positive amount of flow because the minimum-cut exhibited requires

arcs (2(i), 6) and (3(i),6) to each carry 1/2 units of flow. We can thus delete all the other arcs to b in the network, freeze the capacity of arcs (s, 2(i)) and (s, 3(i)) to 1/2, declare b as unavailable, and let each agent consume their best remaining

house at unit rate. The resulting network is shown in Figure 2.4.

29

Figure 2.4: Example 3, Iteration 3, A2 = 1/2

We increase A as long as the maximum s-t flow is 3, and the interval of interest in this

case is [0.5,0.99]. Note that A = 0.51 is a breakpoint: house a becomes a bottleneck as the arcs (l(i), a) and (2(2), o) must carry flows of A and 0.49 respectively in any s-t flow of 3 units; the corresponding minimum-cut is {s, l(i), 2(i), 2(2), 3(i), a, b} with a capacity of 3.51 A. To get the updated network, we delete all the arcs to a other

than those from 1 )^ and 2(2), freeze the capacity of arcs (s, l(x)) and (s, 2(2)) to 0.51 and 0.49 respectively, declare a as unavailable, and let each agent consume their

best remaining house at unit rate. The resulting network is shown in Figure 2.5.

Continuing in this fashion for two more iterations (shown in Figure 2.6, we find the final assignment to be

a b c

1 .51 0 .49

2 .49 .5 .01

3 0 .5 .5

30

Figure 2.5: Example 3, Iteration 4, A3 = .51

Figure 2.6: Example 3, Iteration 5 (A = 0.99) and Iteration 6, (A = 1)

2.4.2 The Algorithm

We now turn to a more formal description of the algorithm. Recall that / =

{1 ,2 , . . . , n} denotes the set of agents and H = {hi, h2, . , hn}, the set of houses.

31

We assume strict preferences and a doubly stochastic endowment matrix. Let hi.k) denote agent z's A;th most preferred house (thus /ij(1) hi /^(2) hi hi ^ ( n ) ) - F r

convenience, denote e ,^ by ei(k). As mentioned earlier, the algorithm finds the

final assignment by solving a sequence of (parametric) maximum-flow problems on specific networks associated with the given instance. The networks all have the

same set of nodes, but the set of arcs changes during the course of the algorithm.

The nodes of the network are as follows: for each agent i, we introduce n nodes

*(i)j *(2)> > *(n)> o n e f r e a c n "preference level"; there is a node hj for each house; and finally, there is a source node s and a sink node t. The algorithm progresses

by examining a sequence of networks at times 0 = Ao < Ai < A2 < . . . < Xz = 1.

At each of these instants, the network is updated (some arcs are deleted, some arc-capacities are changed), as is the auxiliary information maintained by the algorithm, described next.

The algorithm maintains auxiliary information for each agent and for each house.

Each house is either available or unavailable to an agent i; a house that is available

to all agents is said to be publicly available, and a house available to no agent is said

to be unavailable. The structure of the algorithm is such that any house unavailable

to an agent at time Xt will remain unavailable to that agent thereafter. Initially

all the houses are publicly available. For each agent i, the algorithm maintains a

best house and a next house: the best house for agent i is her most preferred house

among the houses available to her; if her best house is hi(k), her next house is the

smallest > k for which the arc (s, i^)) has positive capacity (If there is no such house, there is no next house for agent i). Note that the best and next houses for any agent change over time and will be updated by the algorithm.

The arcs of the initial network are as follows: the source is connected to each node

i(fc), with the capacity of the arc being the corresponding endowment e^ each node hj is connected to the sink with an arc of capacity 1; and finally, there are k

32

infinite capacity arcs from node i^) to the house nodes, one to each of agent z's k most preferred houses. The capacity of arc (i, h) is denoted u^.

To complete the description of the algorithm we specify how, given all the data at

time A4, the algorithm finds Xt+i and updates the network as well as the auxiliary

information. We start with the network at time At and make the following changes

for each agent i:

If agent i's best house is her fcth most preferred house, the capacity of the arc

(s, i(fc)) is set to

max I A - ^2 (2-1) * e=i '

where A is a parameter that is gradually increased from its current value of

At-

If the maximum in Expression (2.1) is achieved by the first term, we say that agent % is consuming his best house; if the maximum in Expression (2.1) is achieved by the second term, we say that agent i is claiming his best house.

Clearly an agent can consume or claim a house only if it is available to her,

and only when it is her best house.

If agent i is consuming her best house, then the capacity of the arc associated

with her next house, if any, is altered. Specifically, if agent z's next house is

her jth most preferred house, the capacity of the arc (s,i(j)) is set to j k

/ j Us,i(e) ~ A = 2_^ Us,iw + U*>*(j) "* (2-2) e=i e=i

The interpretation of these two steps is very straightforward: for each agent, we

increase the capacity of the arc to the best available house at unit rate (the first term in Expression (2.1)), except when individual rationality requires a larger quantity

33

of that house to be set aside for this agent (the second term in Expression (2.1)). In the former case, the increase is accompanied by a corresponding decrease in the

guarantee of the next best house, which explains why the second step applies only

when in the first step the capacity increases at unit rate. Note also that these

updates ensure that

k k

5 3 ".*> ^ 5 3 ev) (2-3) t=i t=\

When A = At, the maximum s-t flow is easily seen to be n. We now solve a

parametric maximum-flow problem on this updated network by gradually increasing

A from its current value of Xt. Define Xt+i as the time at which at least one of the

following two events occurs:

(a) The capacity of some arc becomes zero;

(b) Any further increase of A will cause the maximum s-t flow to be strictly below n.

If Event (a) occurs, the capacities for all the arcs are fixed by setting A = Ai+i; in particular, the arcs whose capacities become zero will have their capacities fixed at

zero, the next houses for the corresponding agents are updated, and the algorithm

continues. If Event (b) occurs, the algorithm proceeds by identifying a minimum s-t cut whose capacity is strictly below n for any larger value of A. When there

are many such minimum cuts, we pick one with the maximum number of nodes on

the source side.5 Such a cut will be of the form s U Xt+i U Vt+i, where Xt+i is a

subset of nodes of the form i^) and lt+i, a subset of the house nodes. As arcs of the form (i(fc), h) have infinite capacity, any node i^ for which there is an arc to a node outside Yt+i cannot belong to Xt+i, because any such cut has infinite capacity.

5Such a cut is unique, see Lovasz and Plummer (33).

34

Once such a cut is identified, the network is updated as follows: the capacities for

all the arcs are fixed by setting A = \t+i. If an agent i has a node i^ Xt+i, and

if her total consumption (including all her copies in Xt+i of the bottleneck set Yt+\ increases with A (this occurs if and only if her next house is outside Yt+i), then the houses in Yt+\ are declared unavailable to her. Note that any agent i who does

not have any nodes i(k) S Xt+i will not receive any portion of the houses in Yt+\. Finally, whenever necessary, the best and the next houses are updated for all the

agents and the algorithm continues.

To analyze the number of iterations needed for the algorithm to terminate, observe

that each iteration ends with the occurrence of Event (a) or Event (b) (or both); Event (a) causes at least one agent's next house to change; Event (b) causes at least one agent's best house to change. As any agent prefers her best house to

her next house, we see that the number of possibilities for each agent is 0(n), so the algorithm terminates in 0(n2) iterations. Each iteration of the algorithm involves finding the smallest breakpoint of a parametric max-flow problem. Even

though some of the capacities are nonlinear because of Expression (2.1), it is clear that we can find each \ t by solving at most (n + 1) maximum-flow problems from s to t. (Each capacity is a piecewise linear function with at most 2 pieces; and once an agent becomes a "consumer," she cannot become a "claimer".) Thus the entire algorithm can be implemented by solving 0{n3) maximum flow problems in a network with 0(n2) nodes and 0(n3) arcs. Our analysis of the running time is very loose, and a more careful implementation will likely be substantially faster,

but we do not investigate this aspect any further as it falls outside the scope of this

paper.

We illustrate the algorithm using the following example.

Example 4. Consider the following instance:

35

1 : aybyc {.996, .01c} 2 : byayc {.99a, .01c} 3 : bycya {.Ola, Mb, .98c}

The only difference from Example 3 is in the preference ordering of agent 1: she

preferred c to b in Example 3, but in this example she prefers b to c. The net-

works examined by the algorithm and the corresponding breakpoints are shown in

Figure 2.7; the final assignment is

a b c

1 .99 0 .01

2 .01 .98 .01

3 0 .02 .98

Comparing the allocations of agent 1 in Examples 3 and 4, we arrive at the following

result.

Proposition 1. The CC mechanism is not weakly strategy proof.

2.4 .3 P r o p e r t i e s

We formally show that the CC mechanism is individually rational, ordinally effi-

cient, and satisfies no justified envy. Let P be the outcome of the CC algorithm.

Proposition 2. The CC mechanism is individually rational.

Proof. Let P be the assignment found by the CC algorithm on an instance. The

CC algorithm satisfies Inequality (2.3) at each point in time for each agent i. But this is the definition of individual rationality for agent i.

36

3~-^

K 1

1 (

1 Z OX-

I^tTa\

-^fbV-S5r-

V 1

1 (

J 1 /

\or \or

-J>2

> . /

-~foV-~^zr

. 1

1 N , / 1 /

37

We turn to ordinal efficiency. The CC algorithm is a "simultaneous eating" algo-

rithm (we can find eating speed functions such that the assignment found by the simultaneous eating algorithm with these eating speed functions is the assignment

P); Bogomolnaia and Moulin (9) showed that any assignment found by such an al-gorithm is ordinally efficient and every ordinally efficient assignment can be found

this way. It follows then that P is ordinally efficient. Nevertheless, we present a

direct proof of this result. The proof uses an alternative characterization of ordinal

efficiency, due to Bogomolnaia and Moulin (9), and extended to the full preference domain by Katta and Sethuraman (32). Given an assignment matrix P and pref-erence relations >zt for each agent i, define the binary relation r{P, >Z) over the set of houses H as follows:6

hrh' { 3i I :h>zih! and piihi > 0}.

In the above context the relation is strict \ih>- h!'. The relation r is cyclic if there

exists a cycle of relations h\Th,2, h2Th%,..., hk-irhk, hkrhi It is strictly cyclic if at

least one of the relations in the cycle is strict.

Proposition 3. Let P be a random assignment matrix for the preference profile y. Then P is ordinally efficient if and only if the relation T(P, >Z) is not strictly cyclic.

Proposition 4. The CC mechanism is ordinally efficient.

Proof. Suppose not. Consider an instance for which the assignment P found by

the CC algorithm is not ordinally efficient. Then there is a set of agents and a set of

houses such that the relation r is strictly cyclic. Suppose without loss of generality

that the set of agents is {1 ,2 , . . . , k}, the set of houses {hi, hi, , hk} and suppose that hi X, hi+i for each agent i (interpreting hk+i as hi), and Pi,hi+i > 0. As agent

6Note that T depends on both the given assignment and the preference relation, but we suppress this dependence because it is usually clear from the context.

38

i prefers hi to /i;+i, and as Pi,hi+i > 0, house hi becomes unavailable for agent i

before house hi+i does. Let A, be the time at which house hi becomes unavailable

for agent i, and suppose Ai = min^{A^}. We claim that the minimum s-t cut S at time Ai contains all the house nodes hi, /12, , hk- Clearly, it contains hi as hi

becomes unavailable to agent 1 at exactly this time. Since agent k is later assigned

a positive amount of hi, the cut S must contain some copy of agent k with an arc

to hi (recall that agents with no copy in the cut will not get assigned any amount of houses in the cut); but this copy of agent k will have an arc to hk as well, because hk >-i hi. Therefore hk S. Applying the same argument, we see that

{hi, hi,. ., hk} C S. However, house hi is declared unavailable to agent 1, which implies her next house at time Ai should be outside of S; it follows then that agent

1 cannot get any more of the houses in S as all such arcs will be deleted, and so

Pi,h2 must be zero, a contradiction.

Finally, we prove that in the assignment P no agent i justifiably envies another agent i'.

Proposition 5. The CC mechanism satisfies no justified envy.

Proof. Let P be the assignment found by the CC algorithm on an instance of the

problem. Consider an agent i and let hi >~i /12 >~i >- hn. We will show that

agent i cannot justifiably envy any agent i'. In fact, we will show that for every house hk and every agent i', either i's partial allocation of her k most preferred

houses dominates z"s or that i's partial allocation fails to be individually rational

for i'.

Let Afe be the time at which house hk becomes unavailable to agent i. From

the structure of the algorithm, it is clear that, at time A ,^ none of the houses

{hi,fi2, ,hk} are available to agent i, and the total amount allocated to her from this subset is precisely Afc. Since houses hi,h2,...,hk are no longer avail-

39

able to i, we know that there exists a minimum s-t cut S = {s,Xh,Yk} where {*(i)> *(2)i > Hk)} Q Xk C /(.) and {hi, h2,..., hk} C Yk C H. In fact, all houses in Yfe are no longer available to i.

Consider another agent i'. Suppose all of the houses in Yk (including {hi, /i2, , hk}) are no longer available for agent i'. In this case, it is clear that

with equality if and only if agent i' has also been claiming or consuming houses in

the set throughout the interval [0, A^]. Thus, i's partial allocation dominates i"s, and we are done.

Now, suppose at least one house in Yk is still available to some agent i'. By the

structure of the algorithm we know im,*^)' *) ^ %.k, for some I. Thus, at the very least agent z"s Tth most preferred house is still available to her -let us denote

it by h^. We know that the flow of all edges crossing the cut must be at capacity

in a max flow, so that ^2h^.,hl Peh Ylh^-,h., ei'h- Hence we may infer that:

2J Vi'h = 22 ^i'h

Recall that all houses in Yk are no longer available to i. However, at least one of

agent i"s I most preferred houses (which are all in the set Yk) is still available to i'. So, since she was being allocated a share of houses in the set {h : h >ai h^} continuously up to time Xk, and will continue to do so for a positive amount of

time, we must have 52hhi,h Wh > Afc = Y,hyihkVih > Eh^h,, Pih-

Thus we can conclude:

2_j tvh = z2 Pi>h > z l Pih hhi'K^ hhiiK\ ht-i'h^

40

Hence i's allocation of agent i"s /'th most preferred houses violates individual ra-

tionality for agent i', and agent i cannot justifiably envy agent i'. The above establishes that either i's allocation dominates i"s, or it fails to be individually

rational for i'.

Since the mechanism is individually rational, we note the following corollary of no

justified envy.

Corollary 1. In the assignment P found by the CC algorithm, no two agents with identical endowments envy each other. That is, for any two agents i and i', we have

2.5 Impossibility Results

2.5.1 Strategyproofness

As noted before the CC mechanism is not strategyproof, even in the weak sense.

The following result rules out the existence of an alternative mechanism satisfying

individual rationality, efficiency, no justified envy, and weak strategyproofness.

Theo rem 1. Consider the strict preference domain and fix \I\ > 3. Any mech-anism satisfying individual rationality, efficiency, and no justified envy cannot be strategyproof, even in the weak sense.

We note that to prove such an impossibility result for | / | > k, it is enough to consider the case \I\ = k as long as individual rationality is required. Any instance

with k agents can be extended to one with a greater number of agents by letting

agents k + 1 , . . . , n own a distinct house, which they prefer to any other house in

the market.

41

Proof. Let / = {1,2,3}, H = {a,b,c}, and consider the following preference and endowment profile:

1 : aybyc {c} 2 : O- a >- b {a} 3 : ay cyb {b}

Individual rationality dictates that p2b = 0. Furthermore, ordinal efficiency implies

that p2a = 0; otherwise agent 2 can exchange a part of her share of a for house c

from agents 1 or 2, resulting in a Pareto improving assignment for all agents. From

these two observations, we get p2c = 1, and p\c = pzc = 0. Since any allocation

that 3 obtains will be individually rational for 1 and vice versa, no justified envy implies that 1 and 3 need to receive identical allocations. Thus we obtain

a b c

1 i i 0 x 2 2 u

2 0 0 1

3 I | 0 Now consider what happens if agent 1 changes her preferences to a y c >- b.

Applying individual rationality for agents 1 and 2, we get pn = p2b = 0, by which

Psb = 1. Then, from ordinal efficiency we get pia = p2c 1. For agent 1, this

allocation dominates the original one, so weak strategyproofness is violated.

The impossibility result can be strengthened if we insist on strategyproofness in the

strong sense: the following result shows that (strong) strategyproofness is incom-patible with individual rationality and efficiency. This is somewhat surprising as

typically individual rationality and efficiency are viewed as fairly mild requirements.

The proof relies on an elaborate construction of Bogomolnaia and Moulin (9) and the elegant reasoning they use in their proof of a related (but different) impossibility

42

result.7

Theorem 2. Consider the strict preference domain and fix \I\ > 4. There is no mechanism that satisfies individual rationality, ordinal efficiency, and strategyproof-ness.

Proof. We start with the following fact about strategyproof mechanisms (we omit the easy proof, see Bogomolnaia and Moulin (9)):

Fact 1. Consider two orderings of houses Oi = hi X, /12 >~i >~i hn and oy = h\ >~i h'2 >~i ... >-{ h'n. Suppose for some k, {hi, ...,hk} = {h[, ,h'k}. Consider a mechanism and suppose pi and ^ are the allocations it finds when agent i reports the preference order o~i and a%> respectively, for some fixed preferences of the other agents. If is strategyproof, thenY$=iPu = YA=IP'U-

Our proof of Theorem 2 proceeds by examining a sequence of profiles, noting down,

in each case, the implications of the various properties; eventually, we shall show

that these implications are inconsistent. In the rest of the proof, we suppress the

>~i notation in describing an agent's preferences so that a X, b >~i c X; d is simply

denoted abed; the identity of the agent is usually clear from the context. We also use

IR for individual rationality, OE for ordinal efficiency, and SP for strategyproofness.

Consider an instance of the problem with | / | = 4 and suppose each agent owns 1/4 of each house.

Profile 1: In this profile agents 1, 2, and 3 have the preference order abed

and agent 4's preference order is adbc. Here IR implies that pia = 1/4 for all

i. Also, without loss of generality, let agent 1 have the smallest amount of b

among the agents 1,2, and 3. It is clear that pn, a < | . 7Bogomolnaia and Moulin (9) show that strategyproofness is incompatible with efficiency and

equal treatment equals for the random assignment problem with strict preferences.

43

a b e d

abed

abcd{2) adbc

\ 1 4 1 4

Profile 2: Here agent l 's preference is bade, and the others' preferences are

identically abed. By OE, pla = 0; and by IR, p^ = Pia + Pib = 1/2, for

i 2,3,4. Now, OE implies that pu = 1/2- So we get

a b e d

bade 0 \ 0 | a6cii(3)

Profile 3: Consider the profile in which agent 1 has the preference bdac, and

the others, abed. Here OE implies that p\a = 0. For agent 1, SP and Fact 1

applied to profiles 2 and 3, we get p\b = 1/2 and pid = 1/2. Also, as p\a 0,

we must have max{p2a>P3a)P4a} > 1/3- Assume, without loss of generality, that the maximum is attained by agent 4, so that pa = /3 > 1/3 > 1/4.

These observations are summarized as

a b e d

bdac 0 | 0 | abcd(2)

abed (3

We have isolated agents 1 and 4 as being special: agent 1 has the least amount

of 6 in Profile 1 and agent 4 has the largest amount of a in Profile 3. In the

44

rest of the profiles, agents 2 and 3 always have the preference order abed (like in Profiles 1-3); agents 1 and 4 will have different preference orderings in different profiles and these will be specified in each case.

Profile 4: Here the preference order of agents 1 and 4 is abdc (agents 2 and 3 have the preference order abed). By IR, we have Pia pn, = 1/4 for all i. Then, by OE, we have pu = pu = Pic = Pu = 1/2. So we get

a b e d

abdc

abcd(2) abdc

l 4 1 4 1 4

1 4 1 4 1 4

0 1 2

0

1 2

0 1 2

Profile 5: Now, consider the profile in which agent l's preference is bade and

agent 4's is adbc. By OE, pa = 0. Also, SP and Fact 1 applied to agent 1

and Profiles 1 and 5, we have P26 = 1/4 + ex. So we get

a b c d

bade 0 \ + a

abcd(2) adbc

Profile 6: Now, consider the profile in which l's preference is bdac and 4's

is adbc. Clearly, OE implies pia = 0. Also, SP and Fact 1 applied to agent

1 and Profiles 5 and 6 yields pu = 1/4 + a. Furthermore, applying Fact 1

to agent 4 and Profiles 3 and 6, we get p^a (3 > 1/4. Now let pu = 7 and Pic = e and p^ = S. All these observations are summarized as

45

bdac

a b c d

0 \ + a e 7

abcd{2) adbc (3 5

We shall now argue that 7 > 1/4. If e = 0 then 7 = 1 1/4 a =4> 7 > \

because, by definition, a < 1/3 < 1/2. If, on the other hand, e > 0, then OE

implies that p2d = Psd = 0. As /? > 1/4, we must have 6 < 3/4, which implies

7 = 1 S > 1/4. In either case, we have 7 > 1/4.

Profile 7: Suppose agent l's preference is abdc and agent 4's is adbc. By IR,

Pia = 1/4 for all i. Then SP and Fact 1 applied to agent 1 and Profiles 5 and

7 implies pn, = a. Furthermore SP and Fact 1 applied to agent 1 and Profiles

6 and 7 implies pXc e, and so pu = 7 > 1/4. Consider agent 4. By OE,

P4b = 0; by SP and Fact 1 applied to agent 4 and Profiles 4 and 7, we have

Pia + Pib + Pid = 1, which implies p4d = 3/4. So we get

abdc

abcd{2) adbc

a b e d

I oc e 7 1 4

I 0 2 4 u 4

But since 7 > \ we have 7 + | > 1, a contradiction.

Since the model considered here generalizes several well-studied models in the liter-

ature, the impossibility results of these special cases automatically carry over. The

most prominent of these are stated in the following theorem.

46

Theo rem 3.

(i) (Bogomolnaia and Moulin (9)) Consider the strict preference domain and fix \I\ > 4. There is no mechanism that satisfies ordinal efficiency, equal treatment of equals, and strategyproofness.

(ii) (Yilmaz (54)) Consider the strict preference domain and fix \I\ > 3. There is no mechanism that satisfies individual rationality, no justified envy, and strategyproofness.

(Hi) (Katta and Sethuraman (32)) Consider the full preference domain and fix \I\ > 3. There is no mechanism that satisfies ordinal efficiency, envy-freeness, and weak strategyproofness.

2.5.2 Strict and Weak Core Allocations

The core is a central concept in cooperative game theory. Since our model allows

for the existence of non-zero endowments, it is natural to investigate the core of

the associated cooperative game. Fix an assignment P and consider any nonempty

coalition S of agents. The subproblem associated with the coalition S then consists

of the agents in S and their collective endowments. Let Q be a reallocation of the endowments of S to the agents in S. We say that P is in the strict core if for each

nonempty coalition S of agents, and for every assignment Q satisfying

z2 iih ~ z2eih' ^ e ^ ' ies ies

we have

Pi hi Qi, Vi S.

Any coalition S for which this condition fails is said to block the assignment P.

Note that if an assignment P is not blocked by the grand coalition S = I, then

47

it is clearly ordinally efficient. If it is not blocked by any singleton coalition, it is

individually rational.

As the allocations we deal with are vectors, which are only partially ordered by first-

order stochastic dominance, there is an alternative definition of the core, called the

weak core. An assignment P is in the weak core if for any coalition S, it is not

possible to find an assignment Q satisfying

2_, 9ih ~ z2 eih> ^ ^' ies ies

and

hi Pi, Vz e S,

with at least one of the inequalities strict. Note, however, that an assignment may

be in the weak core, and yet not satisfy our definition of individual rationality.

Shapley and Scarf (52) showed that the strict core may be empty in the housing market model, when indifferences are allowed. The next example shows that, for

the more general model considered here, the strict core may be empty even when

preferences are strict.

Example 5. Let / = {1,2,3}, H = {a, 6, c}, and consider the following preference and endowment profile:

1 : a X 6 >- c {l/3o, 1/36, l/3c} 2 : ayb^c {l/3o, 1/36, l/3c} 3 : aycyb {l/3a, 1/36, l/3c}

Inclusion in the strict core implies individual rationality (simply apply the strict core condition to all singleton coalitions). In this example any individually rational assignment P will have to satisfy:

(l) Pla = P2a = P3a = 1/3

48

(ii) 1/3 < pife < 2/3, 1/3 < p2b < 2/