A Solution To The Random Assignment Problem On The Full Preference Domain

A solution to the random assignment problem on the full
preference domain∗
Akshay-Kumar Katta†
Jay Sethuraman ‡
January 2004; Revised February 2005
Abstract
We consider the problem of allocating a set of indivisible objects to agents in a fair and
efficient manner. In a recent paper, Bogomolnaia and Moulin consider the case in which all
agents have strict preferences, and propose the probabilistic serial (PS) mechanism; they
define a new notion of efficiency, called ordinal efficiency, and prove that the probabilistic
serial mechanism finds an envy-free ordinally efficient assignment. However, the restrictive
assumption of strict preferences is critical to their algorithm. Our main contribution is
an analogous algorithm for the full preference domain in which agents are allowed to be
indifferent between objects. Our algorithm is based on a reinterpretation of the PS mech-
anism as an iterative algorithm to compute a “flow” in an associated network. In addition
we show that on the full preference domain it is impossible for even a weak strategyproof
mechanism to find a random assignment that is both ordinally efficient and envy-free.
∗
This research was supported by an NSF grant DMI-0093981 and an IBM partnership award.
†
Department of Industrial Engineering and Operations Research, Columbia University, New York, NY; email:
ark2001@columbia.edu
‡
Department of Industrial Engineering and Operations Research, Columbia University, New York, NY; email:
jay@ieor.columbia.edu
1

1 Introduction
The problem of allocating a number of indivisible objects to a number of agents, each desiring
at most one object, in a fair and efficient manner is fundamental in many applications. If
only one indivisible object must be allocated to n > 1 agents, there is really only one fair and
efficient solution1: assigning the object to an agent chosen uniformly at random among the
n agents. When there are many objects the problem becomes substantially more interesting
for a number of reasons: (i) many definitions of fairness and efficiency are possible; (ii) richer
mechanisms emerge; and (iii) since preferences of the agents over the objects have to be solicited
from the agents, truthfulness of the allocation mechanism becomes important.
A natural approach to the problem of allocating multiple objects among many agents is
to generalize the simple “lottery” mechanism: order the agents uniformly at random and let
them successively choose an available object according to this (random) order; thus the first
agent picks his favorite object, the second agent picks his favorite object among the remaining
objects, etc. This is the random priority mechanism, and has a number of attractive features:
it is ex post efficient and truthful (or strategyproof: revealing true preferences is a dominant
strategy for all the agents); it is fair in the weak sense of equal treatment of equals (agents with
identical preferences are treated in an identical manner, a priori); however, it is not efficient
when agents are endowed with utility functions consistent with their preferences, that is, it is
not ex ante efficient. Moreover, it is not fair in the stronger sense of envy-freeness (described
later).
A second solution to this problem adapts the competitive equilibrium with equal incomes
(CEEI) solution for the fair division of unproduced commodities. This solution is envy-free
and ex ante efficient, but is not strategyproof. Moreover, the computational and informational
requirements of implementing this mechanism are prohibitive: it requires the solution of a
fixed-point problem, and requires a complete knowledge of the utility functions of the agents.
In contrast, the random priority mechanism is very simple to implement, and only requires the
agents to report their “preferences” over objects.
Our work is inspired by the remarkable paper of Bogomolnaia and Moulin [5], who proposed
a new mechanism (PS) for the random assignment problem. Their mechanism combines the
attractive features of the random priority mechanism and the CEEI solution: it requires the
agents to report their preferences over objects, not the complete utility functions, and yet
computes a random assignment that is ordinally efficient and envy-free. Ordinal efficiency is
stronger than ex post efficiency but weaker than ex ante efficiency; given the “ordinal” nature of
the input to the mechanism (only preference rankings are used, not complete utility functions),
this is perhaps the most meaningful notion of efficiency for an ordinal mechanism. Finally, the
mechanism proposed by Bogomolnaia and Moulin is not truthful, it satisfies a weaker version
of that property.
Unfortunately, the work of Bogomolnaia and Moulin [5] (and most of the existing work
on this problem so far) assumes that all preferences are strict, which is a fairly restrictive
1
We assume that monetary compensations are not allowed.
2

requirement in many practical settings. A natural question then is to ask if these results have
analogs in the more general case of a full preference domain. As discussed by Bogomolnaia, Deb,
and Ehlers [4], there are both practical and technical reasons for studying the full preference
domain. Moreover, it has generally been recognized that the natural preference domain for
many allocation problems in economic settings is the full preference domain.
Contributions. Our main contribution in this paper is a solution to the random assignment
problem in which indifferences are permitted. The solution we propose can be viewed as a
natural extension of the PS mechanism to the full domain. Our proposed solution is ordinally
efficient and envy-free. Furthermore we show that for this richer preference domain, it is
not possible for any mechanism to find an envy-free, ordinally efficient assignment and still
satisfy the weaker version of strategyproofness satisfied by the PS mechanism in the strict
preference domain. This observation indeed reveals that the full preference domain is subtler
and fundamentally different from the strict preference domain for the problems studied here.
Our techniques rely on standard tools from network flow theory: the mechanism we propose
can be viewed as an iterative application of an algorithm to compute the maximum flow in a
parametric network. We first consider an extreme special case in which each agent partitions
the available objects into acceptable and unacceptable objects, but is indifferent between the
objects in each of these classes. Our algorithm for the full preference domain is an iterative
application of the algorithm for this special case. This special case is an interesting problem in
its own right and has been independently studied by Bogomolnaia and Moulin [7] recently. We
show that the algorithm proposed for this special case is identical to an algorithm to compute
a lexicographically optimal flow in a network; this establishes a connection to the sharing
problem and its variants, which have been studied by operations researchers.
The remainder of this paper is organized as follows. In §2 we describe the problem, define
various notions of fairness and efficiency, and describe the notation. A description of relevant
past work appears in §3 followed by the impossibility result in §4. We consider the special
case of two indifference classes in §5, and provide an algorithm to solve it by posing it as a
parametric maximum flow problem. This algorithm serves as the basis for the main result,
described in §6, which also contains a characterization of ordinally efficient assignments as well
as a method to compute them.
2 Problem Description and Definitions
Problem Description. We consider the problem of allocating indivisible objects to agents
so that each agent receives at most one object. Agents have preferences over the objects, and
the allocation mechanism takes this profile of preferences as input. The preference ordering of
each agent is assumed to be transitive and complete (every pair of objects is comparable). Thus,
given a pair of objects, an agent strictly prefers one to the other or is indifferent between them;
both the indifference relation and the strict preference relation are transitive. The preference
ordering of agent i is denoted by ≥i, with >i and =i denoting the strict preference relation and
3

the indifference relation of agent i respectively. The set of all preference orderings is called the
preference domain, and denoted by A. Throughout the paper we let N be the set of agents
and A be the set of objects. Without loss of generality, we assume that |N| = |A|, and we
denote this common cardinality by n; we also assume that no agent prefers being unassigned to
receiving an object. These assumptions can be relaxed easily by introducing dummy objects
or dummy agents and appropriately modifying the preferences of the existing agents or by
introducing arbitrary preferences for the dummy agents.
A deterministic assignment is an allocation in which each agent receives exactly one object
and each object is allocated to exactly one agent. More formally, it is a one-to-one corre-
spondence between the set of agents, N, and the set of objects, A. Often, it is convenient
to think of a deterministic assignment as a 0 − 1 matrix, with rows indexed by agents and
columns indexed by objects: a 0−1 matrix represents a deterministic assignment if and only if
it contains exactly one 1 in each row and each column. (Such matrices are called permutation
matrices.) We let D be the set of all deterministic assignments.
A random assignment is a probability distribution over deterministic assignments; the
corresponding convex combination of permutation matrices is a stochastic matrix, whose (i, j)th
entry represents the probability with which agent i receives object j. For our purposes, different
random assignments that give rise to the same stochastic matrix are equivalent, so we do not
distinguish a random assignment from its associated matrix. Given a random assignment
matrix P, we let Pi be the ith row, which represents the allocation for agent i in this random
assignment. (An allocation is simply a probability distribution over the set of objects A.) We
let R be the set of all random assignments.
A random assignment mechanism is simply a mapping from An to R. Our objective
is to find a random assignment mechanism satisfying some desirable efficiency and fairness
properties. To describe these properties formally, we extend the agents’ preferences over objects
to preferences over random assignments. Given two random assignments P and Q, an agent
i prefers P to Q (denoted P i Q) if and only if the allocation Pi stochastically dominates
the allocation Qi, where the stochastic dominance is with respect to ≥i, agent i’s preference
ordering over sure objects. Formally,
P i Q ⇐⇒
X
k:k≥ij
pik ≥
X
k:k≥ij
qik, ∀j ∈ A. (1)
If, in addition,
P
k:k≥ij pik
P
k:k≥ij qik for some j ∈ A, then agent i strictly prefers P to Q,
and this is denoted by P ≻i Q; on the other hand, if
P
k:k≥ij pik =
P
k:k≥ij qik for all j ∈ A,
then agent i is indifferent between P and Q, and this is denoted by P ∼i Q. We emphasize
that this relation i is only a partial order on the set of all random assignments, and so not
every pair of random assignments is comparable. Finally, a random assignment P dominates
a random assignment Q (denoted P Q), if every agent prefers P to Q, that is, if P i Q for
all i ∈ N. Even if P and Q are comparable for every agent, it may be the case that P 6≻ Q
and Q 6≻ P; this is the case if some agent prefers P to Q and another prefers Q to P. We are
now ready to describe the notions of efficiency and fairness adopted here.
4

Efficiency. A random assignment P is ordinally efficient if it is not dominated by any other
random assignment Q. If we apply this definition to a deterministic assignment—a random
assignment in which the entries are 0 or 1—then we recover the familiar definition of Pareto
efficiency. Thus, ordinal efficiency can be viewed as a natural extension of Pareto efficiency
to the random assignment setting. Other extensions are possible and have been considered in
the literature; we briefly discuss two such extensions, one weaker and one stronger. A random
assignment is ex post efficient if it can be represented as a probability distribution over Pareto
efficient deterministic assignments. A random assignment is ex ante efficient if for any profile
of utility functions consistent with the preference profile of the agents, the resulting expected
utility vector is Pareto efficient: for any random assignment in which the expected utility of
some agent is strictly greater, there must be another agent whose expected utility is strictly
lower. It is easy to show that ex ante efficiency implies ordinal efficiency, which implies ex
post efficiency. The relationship between the various notions of efficiency are explored further
by Abdulkadiroglu and Sonmez [2] and McLennan [17]. The attractive feature of ordinal
efficiency is that its informational requirements are low: it relies only on agents’ preferences
over objects, not on their preferences over random allocations; yet, it is stronger than ex post
efficiency (which also only requires preferences over objects). For this reason we shall restrict
ourselves to ordinally efficient assignments in the rest of this paper.
Fairness. A random assignment P is envy-free if each agent prefers her allocation to that
of any other agent. Formally, P is envy-free if Pi i Pi′ for all i, i′ ∈ N. A weaker version of
envy-freeness can also be defined: P is weakly envy-free if no agent strictly prefers someone
else’s allocation to hers, that is, if Pi′ 6≻i Pi for all i, i′ ∈ N. A random assignment P satisfies
equal treatment of equals if agents with identical preferences get identical allocations (Pi = Pi′
if ≥i≡≥i′ ). Another notion of fairness is anonymity: a random assignment mechanism is
anonymous if its outcome depends only on the profile of preferences and does not depend on
the identity of the agents. Most of the mechanisms we consider will be anonymous mechanisms.
Again, it is easy to show that envy-freeness implies weak envy-freeness and equal treatment
of equals. (It is easy to see that weak envy-freeness and equal treatment of equals are not
comparable: it is easy to construct assignments that satisfy one property but not the other.)
In most of this paper, we shall restrict ourselves to envy-free assignments.
Incentives. A random assignment mechanism is said to be strategyproof if revealing the
true preference ordering is a dominant strategy for each agent. Again a weaker notion of
strategyproofness can be defined: A mechanism is weakly strategyproof if by falsifying her
preference list an agent cannot obtain an allocation that she strictly prefers to her true alloca-
tion. A mechanism is group strategyproof if it is not possible for any coalition S to improve its
allocation (meaning, each agent in S is atleast as well off, and at least one of them is strictly
better off) by falsifying their preferences.
5

3 Related Work
Strict Preferences. Several solutions have been proposed to the random assignment prob-
lem. The earliest work on this problem is due to Hylland and Zeckhauser [14], who adapt the
competitive equilibrium with equal incomes (CEEI) solution. The mechanism is “expensive”
both in terms of its informational requirements (needs utility functions of all individuals) and
in terms of its computational requirements (requires the solution of a fixed-point problem).
The resulting solution is envy-free and Pareto efficient with respect to the utility functions,
but the mechanism is not strategyproof. Gale [10] conjectured that this is the best possible,
specifically, that no strategyproof mechanism that elicits utility functions and achieves Pareto
efficiency with respect to these utility functions can find a “fair” solution, even in the weaker
sense of equal treatment of equals. This conjecture was proved by Zhou [26].
The prohibitive cost of CEEI motivated research on finding simpler mechanisms. A natural
candidate was the random priority (RP) mechanism: order the agents uniformly at random,
and let them successively choose an object in that order. This was analyzed by Abdulkadiroglu
and Sonmez [1] who showed that RP is equivalent to the unique core allocation of the Shapley-
Scarf housing market [23] in which each agent is endowed with an object chosen uniformly at
random; Roth and Postlewaite [22] had earlier shown that any Shapley-Scarf housing market
with strict preferences has a unique core, and Roth [21] proved that the core (from random
endowments) is strategyproof, anonymous (does not depend on the labels of the agents), and
ex-post Pareto efficient. The equivalence of RP with the core from random endowments implies
that RP itself is strategyproof, anonymous, and always an ex-post Pareto efficient solution; in
fact, Abdulkadiroglu and Sonmez showed [1] that the only Pareto efficient matching mecha-
nisms are serial dictatorships, which are like RP, except that the initial ordering of the agents
is chosen in a deterministic fashion. Svensson [25] showed that serial dictatorships are the only
rules satisfying satisfying strategyproofness, neutrality and nonbossiness. Zhou [26] showed that
the solution computed by RP may not be efficient if agents are endowed with utility functions
consistent with their preferences.
Bogomolnaia and Moulin [5] considered mechanisms that combined the “best” of CEEI
and RP: the CEEI solution has strong efficiency and fairness properties, but is not strate-
gyproof; the RP solution is strategyproof but has weaker efficiency and fairness properties.
The key contribution of Bogomolnaia and Moulin [5] is the definition of an intermediate no-
tion of efficiency—ordinal efficiency–and natural mechanisms to find all ordinally efficient solu-
tions. They also showed that one such mechanism—the probabilistic serial (PS) mechanism—is
weakly strategyproof, and achieves an envy-free, ordinally efficient solution. In a result parallel
to Zhou’s impossibility theorem, they show that no strategyproof mechanism can achieve both
ordinal efficiency and fairness, even in the weak sense of equal treatment of equals. All of their
results were derived in the setting of a strict preference domain; the extension to the “full
domain” (that allows for “subjective indifferences”) was left as a challenging open problem at
the end of their work, and is addressed here.
The PS mechanism was introduced by Cres and Moulin [9] in the context of a scheduling
6

problem with opting out. In that model, every agent has a job that needs unit time on a
machine by a certain deadline; thus the objects are really possible time-slots to which jobs can
be assigned. Agents have strictly decreasing, positive, utilities for the time-slots to which their
jobs can be feasibly assigned, and a utility of zero for not being assigned a time-slot before their
deadline. (Thus agents’ preference orderings are identical; only their deadlines and utilities may
vary.) Cres and Moulin [9] show that in their model the PS solution stochastically dominates
the RP solution. Bogomolnaia and Moulin [6] introduced the concept of ordinal efficiency in
that model and provided two different characterizations of the PS mechanism: it is the only
mechanism that is ordinally efficient, strategyproof, and satisfies equal treatment of equals;
and it is the only mechanism that is ordinally efficient and envy-free.
Full domain. As mentioned earlier, most of the work dealing with the assignment problem
is in the setting of the strict preference domain. Important exceptions include the work of
Svensson [24] and the recent work of Bogomolnaia et al. [4]. Svensson [24] introduces a
class of rules—Serially Dictatorial Rules—that satisfy a number of desirable properties like ef-
ficiency, neutrality, and strategyproofness. This class of rules was generalized by Bogomolnaia
et al. [4] to the class of Bi-Polar Serially Dictatorial Rules, who, more importantly, charac-
terized these rules by essential single-valuedness, nonbossiness, strategyproofness, and Pareto
indifference. Further, they show that an assignment rule satisfies single-valuedness, efficiency,
strategyproofness and weak non-bossiness if and only if it is a selection from a Bi-Polar Seri-
ally Dictatorial Rule. They compare and contrast these rules to allocation rules arising from
traditional exchange-based approaches, and also include an illuminating discussion of the dif-
ficulties and surprises one encounters when studying the full preference domain. In particular,
they show that results that hold for the full preference domain may not hold for the strict
preference domain, and vice-versa. While these two papers study the full preference domain,
they consider only deterministic mechanisms, and so cannot deal with fairness issues. (Recall
that monetary compensations are not permitted.) In contrast, we are concerned with finding
a random assignment that is efficient and envy-free.
4 An Impossibility Result
Our first result establishes that on the full preference domain, ordinal efficiency and envy-
freeness are incompatible with strategyproofness even in the weak sense. This is in contrast to
the result of Bogomolnaia and Moulin [5], who showed that on the strict preference domain,
the PS mechanism is weakly strategyproof and finds an envy-free, ordinally efficient solution.
Example 1. Let N = {1, 2, 3}, A = {a, b, c}, and consider the following preference profile:
1 {a, b} c
2 a b c
3 a c b
7

Agent 1 is indifferent between a and b, but prefers both of these to c; agent 2 prefers a to b to
c, and agent 3 prefers a to c to b. It is easy to verify that
a b c
1 0 3/4 1/4
2 1/2 1/4 1/4
3 1/2 0 1/2
(2)
is ordinally efficient and envy-free. We first argue that this is the only assignment that is both
ordinally efficient and envy-free. To that end, we make a few observations. For agents 2 and
3 to not envy each other, we must have
p2a = p3a;
similarly, for agents 1 and 2 to not envy each other, we must have
p1a + p1b = p2a + p2b,
which also implies
p1c = p2c.
Also, ordinal efficiency implies that p1a = 0 and p3b = 0: if p1a 0, then agent 1 can give up
her share of a in exchange for an equal amount of b from agents 2 or 3; if p3b 0, agent 3
can give up her share of b in exchange for an equal amount of c from agents 1 or 2. In either
case, the resulting assignment dominates the original one. Taking all of these observations
together, we see that p2a = p3a = 1/2; as p3b = 0, p3c = 1/2 (because agent 3’s allocation
should sum to 1). Since p1c = p2c, we must have p1c = p2c = 1/4. This means that p1b = 3/4
and p2b = 1/4. Thus, the only envy-free, ordinally efficient assignment for this instance is the
one shown in (2).
Now suppose agent 1 reports that she prefers a to b to c. Then, envy-freeness implies
p1a = p2a = p3a = 1/3, and that agents 1 and 2 must receive identical allocations. As described
earlier, p3b must be 0, otherwise the assignment cannot be ordinally efficient. Thus, p3c = 2/3,
which implies p1c = p2c = 1/6, which implies p1b = p2b = 1/2. Thus, agent 1’s allocation is
(1/3, 1/2, 1/6) if she submits a preference list a b c; whereas her allocation is (0, 3/4, 1/4) if
she submits her true preference list {a, b} c. Both these allocations are uniquely determined
by ordinal efficiency and envy-freeness, and the former stochastically dominates the latter. So
it is impossible to find an envy-free, ordinally efficient allocation even if the mechanism is only
required to be weakly strategyproof.
This also formally verifies our intuition that the strict preference domain is indeed special
for the random assignment problem, and allowing for indifferences changes the character of
the problem.
8

5 Random Assignment with Dichotomous Preferences
In designing a mechanism for the random assignment problem on the full preference domain,
it seems natural to first understand some extreme special cases. The case of no indifferences
(i.e., strict preferences) is one extreme; the case of dichotomous preferences, discussed in this
section, is the other. Specifically, we assume that each agent partitions the set of objects into
two classes: the set of acceptable objects, and the set of unacceptable objects. Three reasons
underlie our focus on this special case: first, our solution to the overall problem is essentially an
iterative application of the ideas that help solve this special case; second, stronger results can
be proved for this special case, which may be of independent interest; and third, the algorithm
used to solve this special case is identical to the algorithm for finding a lexicographically optimal
(lex-opt) flow in a network, first discovered by Megiddo [19]. We note that, independently of
our work, Bogomolnaia and Moulin [7] have studied the same problem, using somewhat similar
methods, and have obtained the same results. In particular, the impossibility result of §4 does
not hold, and in fact, the proposed mechanism is group strategyproof and finds an ordinally
efficient, envy-free assignment. All the results of this section were originally proved in [7], so
we simply state them without detailed proofs. However, we describe the mechanism in detail
because our point of view is algorithmic, whereas the description in [7] is structural. Our main
contribution in this section is simply the observation that the random assignment problem with
dichotomous preferences is closely related to a classical “sharing” problem considered earlier
in the operations research literature. We briefly discuss this relationship at the end of this
section.
In an instance of the random assignment problem with dichotomous preferences, each agent
indicates only the subset of objects that she finds acceptable, each of which gives her unit
utility; the other objects are unacceptable, yielding zero utility. We assume that each agent
finds at least one object acceptable; agents who do not meet this assumption are irrelevant
to the problem as they will necessarily have zero utility in all solutions. Let Li ⊆ A be the
set of all acceptable objects for agent i; recall that she is indifferent between any two objects
in Li. Thus, for a random assignment P, the utility (or welfare-level) ui of agent i is simply
P
j:j∈Li
pij, the probability that she is assigned an acceptable object.
Remark. Note that the random assignment problem with dichotomous preferences can be
solved by adapting the random priority algorithm as well; we do not describe this adaptation
because (a) the results for the natural adaptation of the random priority mechanism are weaker;
and (b) our goal is to use this mechanism to eventually find an ordinally efficient assignment
on the full preference domain, and we know from Bogomolnaia and Moulin [5] that the random
priority mechanism does not find an ordinally efficient assignment.
5.1 Algorithm and Analysis
Overview. Suppose the preference profile in a given instance of the random assignment
problem is such that some set of three agents have only two acceptable objects among them.
9

Then, it is obvious that the combined utilities of these three agents cannot exceed 2. The
general algorithm for solving the problem builds on this trivial observation: it consists of
locating such a “bottleneck” subset of agents, and allocating their acceptable objects amongst
them in a “fair” way, eliminating these agents and their acceptable objects, and recursively
applying the idea on the remaining set of agents and objects. To formalize this, let Γ(Y )
denote the set of objects acceptable to at least one agent in Y , for any Y ⊆ N, and let
v = min
Y ⊆N
|Γ(Y )|
|Y |
,
with X denoting the largest cardinality set Y ⊆ N with |Γ(Y )|/|Y | = v. If v ≥ 1, there are
more objects than agents competing for it, so each agent can be assigned a distinct object,
and there is nothing more to do. If v 1, the algorithm allocates the objects in Γ(X) among
the agents in X such that the utility of any agent i ∈ X is exactly v; such an assignment
would completely use up the objects in Γ(X), so none of these objects can even be fractionally
assigned to the other agents; moreover, the agents in X cannot be assigned any other object.
Thus, the agents in X and the objects in Γ(X) can be removed from the problem as they play
no further role. The same calculations are carried out in the residual problem with agents
N X, objects A Γ(X), and the preference profiles of the agents restricted to the these
“currently available” objects. Note that this is exactly the algorithm proposed in [7].
The only remaining details are the identification of the “bottleneck” subset X of agents,
and the precise allocation of the “bottleneck” set of objects, Γ(X), to the agents in X. It is
an elementary exercise to show both of these problems can be solved simultaneously as the
problem of finding a maximum flow in an appropriately defined network. We provide the
argument for the sake of completeness.
Flows and cuts. Network flow models arise naturally in the design and analysis of
communication, logistics, and transportation networks, as well as in many other contexts.
These models are among the best-understood and most used optimization models in practice.
(For background on network flow problems and techniques, we recommend [3].)
A directed graph G = (V, A) consists of a set of nodes V and a set of directed arcs A, which
are simply ordered pairs of distinct nodes. We will use the terms arcs and edges interchangeably.
A network is simply a directed graph with some additional data associated with the nodes and
arcs such as capacities and costs. We can view the nodes of a network as representing demand
or supply points of a commodity that is transported via the arcs. Given any network with a
source node s ∈ V , a sink node t ∈ V , and capacities uij on arc (i, j) ∈ A, the maximum flow
problem is to find the maximum amount of flow that can be sent from s to t without exceeding
arc capacities. A classical result in network flow theory connects the maximum flow problem
to a related problem called the minimum cut problem. Before describing this result, we need
to define the notion of a cut and its capacity.
Cut : A cut is any partition of V into S and S̄ := V S such that s ∈ S and t ∈ S̄; it is
sometimes called an s − t cut. Since S̄ = V S, we often refer to the cut simply by S instead
of the pair (S, S̄).
10

Figure 1: A network
Capacity of a cut: The capacity of the cut (S, S̄) is the sum of the capacities of all the arcs
that are directed from a node in S to a node in S̄ (i.e.
P
(i,j)∈A:i∈S,j∈S̄ uij).
Obviously, the maximum flow from s to t can be no more than the capacity of any s − t
cut; in particular, the maximum flow from s to t is at most the minimum capacity s − t cut.
The max-flow min-cut theorem asserts that the maximum s − t flow is equal to the capacity
of minimum s − t cut. This is a classical result in network flow theory, and can be proved
easily from linear programming duality. Hence, to show that a given flow on a network is
optimal, it is enough to demonstrate that there is a cut in the network with the same capacity
as the value of the flow. For example, consider the network shown in Figure 1, where the
label uij, fij on arc (i, j) indicates its capacity and current flow respectively. The value of the
flow in the network ( i.e the amount of flow out of s or into t ) is 15. Now consider the cut
(S, S̄) where S = {s, 1} and S̄ = {2, t}. The set of arcs which go from a node in S into a
node in S̄ is {((s, 2), (1, 2), (1, t)} and hence the capacity of this cut is 6+1+8=15. Therefore,
we can conclude that the given flow is a maximum flow. The relevance to the problem under
consideration should be clear by now: we plan to identify the bottleneck subset as an s − t cut
in a suitably defined graph; the precise manner by which the bottleneck objects are distributed
among the bottleneck agents will be given by a maximum s − t flow.
Consider the bipartite graph with nodes corresponding to N and A, and directed edges
(i, j) for i ∈ N, j ∈ Li. Let these edges have infinite capacity. Augment the network by adding
a source node s, a sink node t; edges with capacity λ ≥ 0 going from s to each node in N,
and edges with unit capacity going from each node in A to t. Let V = N ∪ {s} ∪ {t} denote
the set of vertices in the augmented network. We view λ ≥ 0 as a parameter, and study the
minimum-capacity s − t cuts (or simply minimum cuts) in the network as λ is varied. For the
problem under consideration, it is clear that λ will be varied only in the interval [0, 1]. Fix any
such λ, and consider any s − t cut (S, S̄). It is clear that S must be of the form s ∪ X ∪ W for
some X ⊆ N and W ⊆ A. Furthermore since the “original” edges (i.e. those from the agents
to their acceptable objects) have infinite capacity, W ⊇ Γ(X) in any minimum cut (otherwise,
we will have an edge of infinite capacity in the cut). The capacity of such a cut is then
λ(|N| − |X|) + |W|,
11

which shows that W = Γ(X) in any minimum cut. Thus, any minimum cut (S, S̄) is completely
determined by the set X, and and is of the form S = s ∪ X ∪ Γ(X) with capacity
λ(|N| − |X|) + |Γ(X)|.
When several minimum cuts exist, we pick the minimum-cut with the largest |X|. (This is
unique.)
For λ sufficiently close to 0, it is clear that X = ∅ gives the minimum cut, whose capacity
grows linearly in λ. We consider two possibilities, depending on whether or not X = ∅ is a
minimum cut for λ = 1. If it is, it will be clear from the following argument that each agent can
be assigned an acceptable object with probability 1, and there is nothing more to do. If X = ∅
is not a minimum cut for λ = 1, let λ∗ ∈ (0, 1) be the smallest value of λ for which X = ∅ is
not the only minimum cut, that is, some set X with |X| 0 is also a minimum cut. (By our
convention, X = ∅ could be a minimum cut, but is not the only one.) Let X∗ be the minimum
cut for λ = λ∗. By definition, for any λ λ∗, X = ∅ was the minimum-cut, with capacity
λ|N|. It is clear (by continuity) that the capacity of the minimum-cut for λ = λ∗ must be
exactly λ∗|N|; since X∗ is a minimum-cut for λ = λ∗ and has capacity λ∗(|N|−|X∗|)+|Γ(X∗)|,
we have
λ∗
|N| = λ∗
(|N| − |X∗
|) + |Γ(X∗
)|,
from which we get
λ∗
=
|Γ(X∗)|
|X∗|
. (3)
Let λ = λ∗, and consider any Y ⊆ N; the capacity of the cut s ∪ Y ∪ Γ(Y ) is simply
λ∗(|N| − |Y |) + |Γ(Y )|, which must be at least as large as the minimum-cut capacity λ∗|N|.
This, combined with Eq. (3), establishes
λ∗
= min
Y ⊆N
|Γ(Y )|
|Y |
; X∗
= arg min
Y ⊆N
|Γ(Y )|
|Y |
.
By our convention, X∗ is the largest cardinality subset of agents with this property.
Consider the network with λ = λ∗. By the max-flow min-cut theorem, the maximum flow
from s to t in this network matches the capacity of the minimum cut, which equals λ∗|N|.
However, every edge from s has capacity exactly λ∗ (and there are exactly |N| such edges), so
all of these edges carry a flow of λ∗ in a maximum flow; in particular, λ∗ units reach the sink
from each i ∈ X∗, which gives us the required random assignment.
The quantity λ∗ is the (smallest) breakpoint of the min-cut capacity function of the para-
metric network. The breakpoints of the min-cut capacity function of a parametric network are
well-understood [11]. Efficient algorithms to compute these breakpoints have been discovered
by several researchers, see [3]; the fastest of these methods, discovered by Gallo, Grigoriadis
and Tarjan [11], is based on the preflow-push algorithm for the maximum flow problem due to
Goldberg and Tarjan [12]. Gallo et al. [11] propose algorithms to find maximum flows in an
n-node, m-arc network for O(n) values of the parameter in O(nm log(n2/m)) time. In addi-
tion, they study the min-cut capacity as a function of a parameter λ and propose algorithms
12

Figure 2: A network
to compute its smallest (or largest) breakpoint in O(nm log(n2/m)) time; a more complicated
algorithm but with essentially the same running time in fact finds all the breakpoints of the
min-cut capacity function. Since these algorithms are technical and since they are fairly well-
understood, we do not describe it in detail, but refer the interested reader to the original
papers. We next illustrate these ideas by means of an example.
Illustrative example. Let N = {1, 2, 3, 4, 5, 6}, A = {a, b, c}, and consider the following
preference profile:
1 {a, b}
2 b
3 {a, b, c}
4, 5, 6 c
The initial augmented network is shown in Figure 2. Let λ be a small enough number,
say, 1/4. Consider the cut (S1, S̄1) where S1 = {s}. By definition, the capacity of this cut
is precisely the sum of the capacities on the edges {(s, 1), (s, 2), . . . , (s, 6)}, each of which has
capacity 1/4. Therefore, the cut S1 has capacity 3/2. It is easy to verify that when λ = 1/4,
this is the unique s − t min-cut. (For instance, the cut S2 = {s, 4, 5, 6, c} has capacity 7/4.)
As λ is gradually increased from 0, S1 remains the unique min-cut until λ = 1/3. At this
point, the cut S1 has capacity 2, but so does S2. By our convention, we shall choose S2 as
our cut, so λ∗ = 1/3 is the smallest breakpoint and X∗ = {4, 5, 6} is the bottleneck set. The
corresponding maximum flow gives agents 4, 5 and 6 1/3 of c.
Next, we remove the nodes {4, 5, 6, c} from the network of Figure 2 to get the network
shown in Figure 3. In this network it is clear that for any λ ≤ 1/3, S1 will be the unique min-
cut. As λ is increased further, at λ = 2/3, both S1 and S3 = {s, 1, 2, 3, a, b} have capacity 2 and
are min-cuts. Therefore, λ∗ = 2/3 is the smallest breakpoint for network, and X∗ = {1, 2, 3}
13

Figure 3: A network
is the bottleneck set. We need to distribute the objects {a, b} among the agents {1, 2, 3} such
that each agent gets 2/3 units of an acceptable object, and we can do so by finding a maximum
flow in this network with λ set to 2/3. But observe that there are multiple ways of achieving
this. For example we could give 1/3 of a and 1/3 of b to 1, 2/3 of b to 2, 2/3 of a to 3; (or)
we could give 2/3 of a to 1, 2/3 of b to 2, 1/3 of a and 1/3 of b to 3. Of course, each agent’s
welfare remains the same in any such solution, so any random assignment is acceptable. One
such solution is:
a b c
1 1/3 1/3 0
2 0 2/3 0
3 2/3 0 0
4 0 0 1/3
5 0 0 1/3
6 0 0 1/3
(4)
At this stage, all the objects have been distributed among the agents and so the algorithm
ends.
Properties. We now state some attractive properties of the random assignment found by the
parametric max-flow algorithm. Recall that an n-vector x Lorenz dominates an n-vector y if
and only if upon rearranging their coordinates in increasing order, we have
Pk
i=1 xi ≥
Pk
i=1 yi,
for all k = 1, 2, . . . , n. An n-vector x is lexicographically greater than an n-vector y if and
only if upon rearranging their coordinates in increasing order, xi is larger than yi in the first
coordinate in which they differ. Note that if x Lorenz dominates y, then x is lexicographically
greater than y, but the converse is not true.
Theorem 1 1. The parametric max-flow algorithm finds a random assignment whose utility
vector is Lorenz dominant. In particular, the random assignment is ordinally efficient.
2. The random assignment P computed by the parametric max-flow algorithm is envy-free.
3. The parametric max-flow mechanism is group strategyproof.
14

Theorem 1 is proved formally in [7], so we will not repeat the proofs here. A weaker version
of the first statement of Theorem 1 is due to Megiddo [19], proved in the context of a sharing
problem, described next.
5.2 The Sharing Problem
The random assignment problem with dichotomous preferences is closely related to the sharing
problem, first introduced by Brown [8]. Brown’s work was partly motivated by a coal-strike
problem: during a coal-strike, some “non-union” mines could still be producing. In this case,
how should the limited supply of coal be distributed equitably among the power companies
that need it? Since power companies vary in size, it would not be desirable to give each power
company the same amount of coal. Moreover, even if such an equal sharing was desirable, the
distribution system may not allow for a perfect distribution because of capacity constraints.
Brown [8] models the distribution system by a network in which there are multiple sources
(representing the coal-producers), multiple sinks (representing power companies), and multiple
transshipment nodes; each edge in the network has a capacity which is an upper bound on
the amount of coal that can traverse that edge. Also, each sink node has a positive “weight”
reflecting its relative importance, and the utility of any sink node is the amount of coal it
receives divided by its weight. The goal is to distribute the supply of coal so as to maximize
the utility of the sink that is worst-off. This maximin objective is in the spirit of Rawls [20].
Following Brown [8], a number of authors have explored a variety of related models, arising in
diverse applications. Itai and Rodeh [16] study the minimax sharing problem (minimize the
utility of the sink that is best-off) and the optimal sharing problem (simultaneously maximizes
the utility of the worst-off sink and minimizes the utility of the best-off sink). In contrast
to these papers in which only one or two of the sinks enter the objective function explicitly,
Megiddo [18, 19] considers the lexicographic sharing problem, where the objective is to lexico-
graphically maximize the utility vector of all the sinks, where the kth component of the utility
vector is the kth smallest utility. The algorithm described earlier in §5.1 is precisely the one
that finds a lex-optimal flow for this sharing problem. We note that flow sharing problems arise
in other contexts as well such as network transmission [15, 16] and network vulnerability [13];
we refer the reader to the paper of Gallo et al. [11] for more details.
6 Random Assignment: The full preference domain
We now extend the parametric max-flow algorithm to compute an envy-free, ordinally efficient
assignment on the full preference domain. (In view of the impossibility result of §4, the
algorithm cannot even be weakly strategyproof).
Algorithm for the full preference domain and its analysis. The algorithm we propose
is an iterative application of parametric maximum flow algorithm that terminates in (at most)
n phases. The network structure remains the same as in Section 5, with each agent having
15

EPS
1. Initialize A′ = A; P ≡ 0; c(i, 1) = 0 for all i ∈ N; and k = 1.
2. Construct G with each agent i ∈ N pointing to the objects H(i, A′); every
object in A′ connected to a common sink through an arc of capacity 1 ; and
every agent i ∈ N connected from the source through an arc of capacity
c(i, k) + λ .
3. Solve the corresponding parametric max-flow problem. Let λ∗
k be the smallest
break point and B be the bottleneck set of objects.
4. For an agent i, if H(i, A′) ⊆ B then update c(i, k + 1) = 0 and give her an
amount c(i, k)+λ∗
k from the set H(i, A′) ; if not update c(i, k+1) = c(i, k)+λ∗
k
5. Update A′ = A′ − B and k = k + 1. If A′ 6= φ then go to step 2. Else,
terminate the algorithm.
Figure 4: The EPS algorithm
outgoing arcs to her most preferred objects among the set of objects available at the current
phase. The capacities of the arcs from the source to the agents and the set of available objects
vary from phase to phase. In phase k, the capacity of the arc from the source to agent i ∈ N
is c(i, k) + λ, where c(i, k) is a constant determined at the end of the (k − 1)st phase, and λ
is the parameter that will be increased gradually; as before, infinite capacity arcs connect the
agents to their most preferred available objects, and each arc connecting an object to the sink
has unit capacity. Let λ∗
k be the smallest breakpoint in the associated parametric maximum
flow problem. Note that λ∗
k is the smallest value of λ for which some subset B of the available
objects is completely consumed; this set B can be viewed as the bottleneck set of objects. If
i’s best object in this phase belongs to this set B, then we give her c(i, k) + λ∗
k units of this
object, and set c(i, k + 1) = 0 (this is because i has no share over the other remaining objects
as she has not yet started “eating” them); if not we set c(i, k+1) = c(i, k)+λ∗
k (this is because
i has already “eaten” this amount of her best available object at the end of this phase, and
this object will still be available for consumption, possibly by other agents as well; in a sense,
this is like pledging an amount of c(i, k) + λ∗
k to agent i). The objects in B are removed from
the set of available objects, and we move to the (k + 1)st phase. Since |B| ≥ 1 in each phase,
the procedure ends in at most n phases. We set c(i, 1) = 0 for all i ∈ N, as initially none of the
agents has started eating. We record these observations formally in Figure 4, where H(i, A′)
denotes agent i’s most preferred set of objects in A′, and P is the random assignment matrix
computed by the algorithm.
When specialized to the strict preference domain, it is easy to see that the algorithm
described in Figure 4 is identical to the PS algorithm. Therefore, we call this algorithm
EPS for “extended” PS. Note that in the EPS algorithm, the most expensive computation is
16

solving the parametric maximum flow problem. As noted before, the complexity of parametric
maximum flow algorithm is O(nm log(n2/m)), where m is the number of arcs in the network.
Therefore, the overall complexity of the algorithm EPS is O(
Pr
k=1 n2mk log(n2/mk)), where
mk is the number of arcs in the kth phase and r is the total number of phases.
Properties. We now proceed to prove that the random assignment P generated by the EPS
algorithm is ordinally efficient and envy-free. Let P be a random assignment matrix over the
preference profile ≥. Given this, we define a binary relation τ(P, ≥) over the set of objects, A,
as follows:
∀a, b ∈ A : a τ(P, ≥) b ⇐⇒ { ∃ i ∈ N : a ≥i b and pib 0} (5)
The binary relation is strict if in the above expression a i b, that is, an agent i strictly
prefers a to b and pib 0. To save notation, we denote the above relation in short by aτb
when the random assignment matrix is clear from the context. The relation τ(P, ≥) is cyclic
if there exists a cycle of relations a1τa2, ......, a(r−1)τar, ...., aRτa1 . τ(P, ≥) is strictly cyclic if
it is cyclic and at least one of the binary relations in the cycle is strict. We next characterize
ordinally efficient assignments. This result parallels the characterization of ordinal efficiency
in the strict preference domain due to Bogomolnaia and Moulin [5].
Lemma 2 The random assignment P is ordinally efficient with respect to the profile ≥ if and
only if the relation τ(P, ≥) is not strictly cyclic .
Proof.
Only if. Suppose the relation τ has a strict cycle. Let a1τa2; a2τa3; ...; aR−1τaR; aRτa1
be the cycle; we assume, without loss of generality, that the first relation is a strict one. By
definition of τ, we can construct a sequence i1, ..., iR in N such that: pi1,a2 0 and a1 i1 a2;
pi2,a3 0 and a2 ≥i2 a3; . . . piR,a1 0 and aR ≥iR
a1 (note that the agents ik, k = 1, ..., R,
may not be all different). Let
δ = min
1≤k≤R
pikak+1
with the understanding aR+1 = a1. Note that δ 0. Let Q = P +∆ = (δia) where δikak
= δ ,
δikak+1
= −δ, for k = 1, ..., K − 1 and δia = 0 otherwise. By construction, Q is a bistochastic
matrix. We go from Pik
to Qik
by shifting some probability from object ak+1 to an object ak,
which is at least as good as ak+1 according to ik. Moreover, this preference is strict for k = 1.
Therefore, Q stochastically dominates P in a strict sense (if the same agent appears more than
once, we use the transitivity of stochastic dominance). Hence, P cannot be ordinally efficient.
17

If. Suppose P is not ordinally efficient. Let Q be a matrix than stochastically dominates
P, and let i1 be an agent for whom Qi1 6= Pi1 . By definition of stochastic dominance, there
exist two objects a1, a2 such that
a1 i1 a2; qi1,a1 pi1,a1 and qi1,a2 pi1,a2 (6)
In other words, a1τ(P, ≥)a2 in a strict sense. Next by the feasibility of Q, there exists an agent
i2 such that qi2,a2 pi2a2 . Repeating the argument, we find an a3 such that a2 ≥i2 a3 and
pi2,a3 qi2,a3 ; and hence a2τa3. Due to finiteness of A and N, we will eventually find a cycle
of the relation τ. If this cycle contains a1 (or) if this cycle is strict, then we are done.
Now suppose that the cycle is not strict. Let the cycle be made up of the objects
(b1, b2, . . . , bn) and the corresponding agents be (i1, i2, . . . , in). Note that we will have qi1,b2
1, . . . , qin−1,bn 1, qin,b1 1 and qi1,b1 0, . . . , qin,bn 0. Also note that we will have
b1 =i1 b2, . . . , bn =in b1 (as the cycle is not strict). Now we make the following transformation
: q′
i1,b1
= qi1,b1 − δ, q′
i1,b2
= qi1,b2 + δ, . . . , q′
in,bn
= qin,bn − δ, q′
in,b1
= qin,b1 + δ, and for all other
i ∈ N, j ∈ A; q′i, j = qi,j, where
δ = max{k : qij,aj − k ≥ pij ,aj , qij ,aj+1 + k ≤ pij ,aj+1 ∀i = 1, · · · , n} (7)
Note that the new matrix Q′ still stochastically dominates P in a strict sense. Now if we
repeat the exercise of finding a cycle for this matrix Q′, we will never find the cycle which we
found in Q ( because for one particular (agent,object) pair, say (i,c), involved in the cycle we
have made q′(i, c) = p(i, c) and hence this pair will never enter the cycle in the future). Since
the number of nonstrict cycles is finite, we will eventually find a strict cycle of the relation τ.
We are now ready to prove that the EPS algorithm finds an ordinally efficient assignment.
Theorem 3 The algorithm proposed is ordinally efficient.
Proof. Suppose not. Then, by Lemma 2, we can find a cycle in the relation τ over the
assignment matrix Q: a1τa2, . . . , arτar+1, . . . , aRτa1. Without loss of generality, assume that
the last binary relation is strict. Let ir be an agent such that ar ≥ir ar+1 and pir,ar+1 0
(r ∈ 1, . . . , R, with the convention aR+1 = a1). Let sr be the phase of the EPS algorithm
during which ar was in the bottleneck set. Since ar is at least as good as ar+1 for the agent ir,
and ir gets a positive amount of ar+1 , ar+1 could not have been in the bottleneck set before
ar. Therefore, sr ≤ sr+1. Moreover, if ar is strictly preferred over ar+1 , then sr sr+1.
Note that sR+1 = s1.Therefore, s1 ≤ s2 ≤ . . . ≤ sR s1. But, this is a contradiction because
s0 = sR+1. Therefore, the algorithm is ordinally efficient.
Next, we prove the that assignment P found by the EPS algorithm is envy-free.
Theorem 4 The matrix P given by the EPS algorithm is envy-free.
18

Proof. Let the preference profile be ≥. Consider any two agents i, j ∈ N and an object
a ∈ A. Define the set M(i, a) = {b : b ≥i a}. Let k be the first phase of the algorithm by the
end of which all the objects in M(i, a) are used up. Note that in the first k phases, i will be
allocated objects only from the set M(i, a). Therefore, the number of units allocated to i from
the objects in M(i, a) is
P
l∈M(i,a) pi,l = λ∗
1 + . . . + λ∗
k ≥
P
l∈M(i,a) pj,l (since j could have used
some other objects outside the set M(i, a)). Therefore, i cannot envy j and hence EPS always
finds an envy-free random assignment.
Illustrative example. Let N = {1, 2, 3}, A = {a, b, c}, and consider the following preference
profile:
1 {a, b} c
2 a b c
3 a c b
Agent 1 is indifferent between a and b, but prefers both of these to c; agent 2 prefers a to b to
c, and agent 3 prefers a to c to b. At the beginning of first phase, agent 1 is connected to both
a and b; agents 2 and 3 are connected only to a. The smallest breakpoint in the parametric
maximum flow problem is λ∗
1 = 1
2. The corresponding minimum cut is s ∪ {2, 3} ∪ {a} (
following the notation of section 5). We update p2,a = 1
2; p3,a = 1
2; c(1, 2) = 1
2; c(2, 2) = 0;
c(3, 2) = 0; A′ = {b, c}. At this stage, a is fully consumed. In the second phase agents 1 and
2 are connected to b and agent 3 is connected to c. The smallest breakpoint is λ∗
2 = 1
4 and
the corresponding minimum cut is s ∪ {1, 2} ∪ {b}. We update p1,b = 3
4 ; p2,b = 1
4 ; c(1, 3) = 0;
c(2, 3) = 0; c(3, 3) = 1
4;A′ = {c}. At this stage, both a and b are totally consumed. In the
third phase all the agents are connected to c. The smallest breakpoint is λ∗
3 = 1
4 and the
corresponding minimum cut is s∪{1, 2, 3}∪{c}. We update p1,c = 1
4 ; p2,c = 1
4; p(3, c) = 1
2. At
this stage, all the objects are fully consumed and the algorithm ends. The resulting random
assignment is
a b c
1 0 3/4 1/4
2 1/2 1/4 1/4
3 1/2 0 1/2
(8)
Variable eating rates. In the EPS algorithm, all the agents have same eating rates. Suppose
that each agent i ∈ N has a different eating rate function wi(·), where
R t=1
t=0 wi(t)dt = 1. The
EPS algorithm can be modified in a simple way to accommodate for these different eating
rate functions. The only modification required is that the parametric maximum flow problem
needs to be redefined appropriately, which we do here. We refer to this algorithm as EER.
Let tk be the starting time of the kth phase. The capacity of the arc from the source to
agent i ∈ N is c(i, k) +
R t
tk
wi(u)du , where c(i, k) is a constant determined at the end of the
(k − 1)st phase, and each agent points to her most preferred set of objects. In this problem
the parameter is t ≥ tk. The smallest breakpoint, in this context, is the value of t after which
the maximum flow stops increasing linearly with time. A method for doing this was described
19

in Gallo, Grigoriadis and Tarjan [11]. We denote the smallest breakpoint by tk+1 and the
corresponding bottleneck by B. If i’s preferred set of objects is a subset of B, then we give
her c(i, k) +
R tk+1
tk
wi(u)du amount from these objects (this can be done, as before, based on
the maximum flow solution at the breakpoint tk+1) and also update c(i, k + 1) = 0 ; if not we
update c(i, k + 1) = c(i, k) +
R tk+1
tk
wi(u)du. Since |B| ≥ 1 , this procedure also has to end in
at most n phases. As before, c(i, 1) = 0 and t1 = 0.
Let PEER denote the random assignment obtained from the EER algorithm. Then PEER
is ordinally efficient. This can be seen from the proof of Theorem 3 . Notice that the proof
relies only on the phase at which an object is in the bottleneck set; and not upon the rate at
which it is consumed. So, essentially the same argument goes through. In fact, more is true:
every ordinal efficient can be found in this way; we omit the straightforward proof. (This result
parallels a result of Bogomolnaia and Moulin [5] for the strict preference domain.)
Theorem 5 Fix a preference profile ≥. Then, a random assignment matrix P is ordinally
efficient if and only if there exists a profile of eating speeds w = (wi)i∈N such that P = PEER.
7 Concluding remarks
In this paper we describe the first mechanism to find an ordinally efficient envy-free solution
to the random assignment problem over the full preference domain. The proposed algorithm
has nice, interesting connections to the sharing problem studied in the late 70s and the 80s in
the operations research literature; we discuss these connections as well. We note that the EPS
algorithm is applicable more generally: for instance, the algorithm and all its properties extend
immediately to domains in which agents have partially ordered preferences; the only change
necessary in the description of the algorithm is that at any stage, each agent should point to
her maximal set of objects (i.e. those objects that are not dominated by other objects). This
extension includes the case in which indifferences are allowed, but the indifference relation is
not transitive: for example, an agent may be indifferent between a and b, and between b and
c, but may prefer a to c.
Acknowledgments
We are grateful to two anonymous referees for their constructive comments and for their many
useful suggestions on an earlier version of this paper. Their comments prompted us to revise
some parts of the paper significantly, resulting in a substantial improvements. A version of this
paper was presented at the Fall 2003 Midwest Theory Conference in Bloomington, Indiana.
We gratefully acknowledge the comments of the participants of that conference. Finally, we
thank Anna Bogomolnaia, Herve Moulin, and Tayfun Sonmez for their detailed comments on
this work.
20

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