Round-Robin Beyond Additive Agents: Existence and Fairness of Approximate Equilibria

Abstract

Fair allocation of indivisible goods has attracted extensive attention over the last two decades, yielding numerous elegant algorithmic results and producing challenging open questions. The problem becomes much harder in the presence of strategic agents. Ideally, one would want to design truthful mechanisms that produce allocations with fairness guarantees. However, in the standard setting without monetary transfers, it is generally impossible to have truthful mechanisms that provide non-trivial fairness guarantees. Recently, Amanatidis et al. [2021] suggested the study of mechanisms that produce fair allocations in their equilibria. Specifically, when the agents have additive valuation functions, the simple Round-Robin algorithm always has pure Nash equilibria and the corresponding allocations are envy-free up to one good (EF1) with respect to the agents' true valuation functions. Following this agenda, we show that this outstanding property of the Round-Robin mechanism extends much beyond the above default assumption of additivity. In particular, we prove that for agents with cancelable valuation functions (a natural class that contains, e.g., additive and budget-additive functions), this simple mechanism always has equilibria and even its approximate equilibria correspond to approximately EF1 allocations with respect to the agents' true valuation functions. Further, we show that the approximate EF1 fairness of approximate equilibria surprisingly holds for the important class of submodular valuation functions as well, even though exact equilibria fail to exist!

Full text

arXiv:2301.13652v1 [cs.GT] 31 Jan 2023
Round-Robin Beyond Additive Agents:
Existence and Fairness of Approximate Equilibria∗
Georgios Amanatidis1, Georgios Birmpas2, Philip Lazos3,
Stefano Leonardi2, and Rebecca Reiffenhäuser4
1Department of Mathematical Sciences
University of Essex; Colchester, UK
georgios.amanatidis@essex.ac.uk
2Department of Computer, Control, and Management Engineering
Sapienza University of Rome; Rome, Italy
{birbas, leonardi}@diag.uniroma1.it
3Input Output; London, UK
philip.lazos@iohk.io
4Institute for Logic, Language and Computation
University of Amsterdam; Amsterdam, The Netherlands
r.e.m.reiffenhauser@uva.nl
Abstract
Fair allocation of indivisible goods has attracted extensive attention over the last two decades, yield-
ing numerous elegant algorithmic results and producing challenging open questions. The problem
becomes much harder in the presence of strategic agents. Ideally, one would want to design truthful
mechanisms that produce allocations with fairness guarantees. However, in the standard setting with-
out monetary transfers, it is generally impossible to have truthful mechanisms that provide non-trivial
fairness guarantees. Recently, Amanatidis et al. [5] suggested the study of mechanisms that produce
fair allocations in their equilibria. Specifically, when the agents have additive valuation functions,
the simple Round-Robin algorithm always has pure Nash equilibria and the corresponding allocations
are envy-free up to one good (EF1) with respect to the agents’ true valuation functions. Following this
agenda, we show that this outstanding property of the Round-Robin mechanism extends much beyond
the above default assumption of additivity. In particular, we prove that for agents with cancelable valu-
ation functions (a natural class that contains, e.g., additive and budget-additive functions), this simple
mechanism always has equilibria and even its approximate equilibria correspond to approximately EF1
allocations with respect to the agents’ true valuation functions. Further, we show that the approxi-
mate EF1 fairness of approximate equilibria surprisingly holds for the important class of submodular
valuation functions as well, even though exact equilibria fail to exist!
∗This work was supported by the ERC Advanced Grant 788893 AMDROMA “Algorithmic and Mechanism Design Research
in Online Markets”, the MIUR PRIN project ALGADIMAR “Algorithms, Games, and Digital Markets”, and the NWO Veni project
No. VI.Veni.192.153.
1

1 Introduction
Fair division refers to the problem of dividing a set of resources among a group of agents in a way that
every agent feels they have received a “fair” share. The mathematical study of (a continuous version
of) the problem dates back to the work of Banach, Knaster, and Steinhaus [36], who, in a first attempt
to formalize fairness, introduced the notion of proportionality, i.e., each of the 𝑛agents receives at least
1/𝑛-th of the total value from fer perspective. Since then, different variants of the problem have been
studied in mathematics, economics, political science, and computer science, and various fairness notions
have been defined. The most prominent fairness notion is envy-freeness [22,21,37], where each agent
values her set of resources at least as much as the set of any other agent. When the available resources are
indivisible items, i.e., items that cannot be split among agents, notions introduced for infinitely divisible
resources, like proportionality and envy-freeness are impossible to satisfy, even approximately. In the
last two decades fair allocation of indivisible items has attracted extensive attention, especially within the
theoretical computer science community, yielding numerous elegant algorithmic results for various new
fairness notions tailored to this discrete version of the problem, such as envy-freeness up to one good (EF1)
[28,16], envy-freeness up to any good (EFX) [18], and maximin share fairness (MMS) [16]. We refer the
interested reader to the surveys of Procaccia [34], Bouveret et al. [15], Amanatidis et al. [6].
In this work, we study the problem of fairly allocating indivisible goods, i.e., items of non-negative
value, to strategic agents, i.e., agents who might misreport their private information if they have an incen-
tive to do so. Incentivising strategic agents to truthfully report their valuations is a central goal—and often
a notorious challenge—in mechanism design, in general. Specifically in fair division, this seems particu-
larly necessary, since any fairness guarantee on the outcome of a mechanism typically holds with respect
to its input, namely the reported preferences of the agents rather than their true, private preferences
which they may have chosen not to reveal. Without truthfulness, fairness guarantees seem to become
meaningless. Unfortunately, when monetary transfers are not allowed, as is the standard assumption in
fair division, such truthful mechanisms fail to exist for any meaningful notion of fairness, even for simple
settings with two agents who have additive valuation functions [2].
As an alternative, Amanatidis et al. [5] initiated the study of equilibrium fairness: when a mechanism
always exhibits stable (i.e., pure Nash equilibrium) states, each of which corresponds to a fair allocation
with respect to the true valuation functions, the need for extracting agents’ true preferences is mitigated.
Surprisingly, they show that for the standard case of additive valuation functions, the simple Round-Robin
routine is such a mechanism with respect to EF1 fairness. Round-Robin takes as input an ordering of the
goods for each agent, and then cycles through the agents and allocates the goods one by one, giving to
each agent their most preferred available good. For agents with additive valuation functions, Round-Robin
is known to produce EF1 allocations (see, e.g., [30]). Note that, without monetary transfers, what distin-
guishes a mechanism from an algorithm is that its input is the, possibly misreported, agents’ preferences.
To further explore the interplay between incentives and fairness, we take a step back and focus solely
on this very simple, yet fundamental, allocation protocol. It should be noted that the Round-Robin al-
gorithm is one of the very few fundamental procedures one can encounter throughout the discrete fair
division literature. Its central role is illustrated by various prominent results, besides producing EF1 alloca-
tions: it can be modified to produce approximate MMS allocations [3], as well as EF1 allocations for mixed
goods and chores (i.e., items with negative value) [9]. It produces envy-free allocations with high proba-
bility when the values are drawn from distributions [29], it is used to produce a “nice” initial allocation
as a subroutine in the state-of-the-art approximation algorithms for pairwise maximin share fair (PMMS)
allocations [25] and EFX allocations [4], it has the lowest communication complexity of any known fair
division algorithm, and, most relevant to this work, it is the only algorithm for producing fair allocations
for more than two agents that, when viewed as a mechanism, is known to even have equilibria [8].
2

We investigate the existence and the EF1 guarantees of approximate pure Nash equilibria of the Round-
Robin mechanism beyond additive valuation functions, i.e., when the goods already assigned to an agent
potentially change how they value the remaining goods. In particular, we are interested in whether any-
thing can be said about classes that largely generalize additive functions, like cancelable functions, i.e.,
functions where the marginal values with respect to any subset maintain the relative ordering of the
goods, and submodular functions, i.e., functions capturing the notion of diminishing returns. Although
the stability and equilibrium fairness properties of Round-Robin have been visited before [8,5], to the best
of our knowledge, we are the first to study the problem for non-additive valuation functions and go be-
yond exact pure Nash equilibria. Cancelable functions also generalize budget-additive, unit-demand, and
multiplicative valuation functions [12], and recently have been of interest in the fair division literature as
several results can be extended to this class [12,1,19]. For similar reasons, cancelable functions seem to
be a good pairing with Round-Robin as well, at least in the algorithmic setting (see, e.g., Proposition 2.5).
Nevertheless, non-additive functions seem to be massively harder to analyze in our setting and come
with various obstacles. First, it is immediately clear that, even without strategic agents, the input of an
ordinal mechanism implemented as a simultaneous-move one-shot game, like the Round-Robin mecha-
nism we study here, can no longer capture the complexity of a submodular function (see also the relevant
discussion in Our Contributions). As a result, translating this sequential assignment to an estimate on the
value of each agent’s bundle of goods, is not obvious. Lastly, and this applies to cancelable functions as
well, assuming equilibria do exist and enough can be shown about the value of the assigned bundles to
establish fairness, there is no reason to expect that any fairness guarantee will hold with respect to the
true valuation functions, as the agents may misreport their preferences in an arbitrary fashion.
1.1 Contribution and Technical Considerations
We study the well-known Round-Robin mechanism (Mechanism 1) for the problem of fairly allocating a set
of indivisible goods to a set of strategic agents. We explore the existence of approximate equilibria, along
with the fairness guarantees that the corresponding allocations provide with respect to the agents’ true
valuation functions. Qualitatively, we generalize the surprising connection between the stable states of
this simple mechanism and its fairness properties to all approximate equilibria equilibria and for valuation
functions as general as subadditive cancelable and submodular. In more detail, our main contributions can
be summarized as follows:
• We show that the natural generalization of the bluff profile of Aziz et al. [8] is an exact PNE that
always corresponds to an EF1 allocation, when agents have cancelable valuation functions (Theorem
3.2 along with Proposition 2.5). Our proof is simple and intuitive and generalizes the results of Aziz
et al. [8] and Amanatidis et al. [5].
• For agents with submodular valuation functions, we show that there are instances where no (3/4+
𝜀)-approximate PNE exists (Proposition 3.4), thus creating a separation between the cancelable and
the submodular cases. Nevertheless, we prove that an appropriate generalization of the bluff profile
is a 1/2-approximate PNE (Theorem 3.7) that also produces an 1/2-EF1 allocation with respect to
the true valuation functions (Theorem 3.8).
• We provide a unified proof that connects the factor of an approximate PNE with the fairness ap-
proximation factor of the respective allocation. In particular, any 𝛼-approximate PNE results in a
𝛼/2-EF1 allocation for subadditive cancelable agents (Theorem 4.5), and in a 𝛼/3-EF1 allocation for
submodular agents (Theorem 4.4). We complete the picture by providing lower bounds in both cases
(Theorem 4.3 and Proposition 4.8), which demonstrate that our results are almost tight.
3

While this is not the first time Round-Robin is considered for non-additive agents, see, e.g., [13], to the
best of our knowledge, we are the first to study its fairness guarantees for cancelable and submodular
valuation functions, independently of incentives. As a minor byproduct of our work, Theorem 3.8 and
the definition of the bluff profile imply that, given value oracles for the submodular functions, we can use
Round-Robin as a subroutine to produce 1/2-EF1 allocations.
This also raises the question of whether one should allow a more expressive bid, e.g., a value oracle.
While, of course, this is a viable direction, we avoid it here as it comes with a number of issues. Allowing
the input to be exponential in the number of goods is already problematic, especially when simplicity and
low communication complexity are two appealing traits of the original mechanism. Moreover, extracting
orderings from value oracles would essentially result in a mechanism equivalent to ours (if the ordering
of an agent depended only on her function) or to a sequential game (if the orderings depended on all
the functions) which is not what we want to explore here. Note that less information is not necessarily
an advantage towards our goal. While this results in a richer space of equilibria, fairness guarantees are
increasingly harder to achieve.
As a final remark, all the algorithmic procedures we consider run in polynomial time, occasionally
assuming access to value oracles, e.g., Algorithms 2,3,4. Although we do not consider computational
complexity questions here, like how do agents compute best responses or how do they reach approximate
equilibria, we do consider such questions interesting directions for future work.
1.2 Further Related Work
The problem of fairly allocating indivisible goods to additive agents in the non-strategic setting has been
extensively studied; for a recent survey, see Amanatidis et al. [6]. Although the additivity of the valuation
functions is considered a standard assumption, there are many works that explore richer classes of val-
uation functions. Some prominent examples include the computation of EF1 allocations for agents with
general non-decreasing valuation functions [28], EFX allocations (or relaxations of EFX) under agents
with cancelable valuation functions [12,1,19] and subaditive valuation functions [33,20], respectively,
and approximate MMS allocations for submodular, XOS, and subadditive agents [11,23].
Moving to the strategic setting, Caragiannis et al. [17] and Markakis and Psomas [31] were the first
to consider the question of whether it is possible to have mechanisms that are truthful and fair at the
same time, again assuming additive agents. Amanatidis et al. [2] resolved this question for two agents,
showing there is no truthful mechanism with fairness guarantees under any meaningful fairness notion.
As a result, subsequent papers considered truthful mechanism design under restricted valuation function
classes [24,10].
The stability of Round-Robin was first studied by Aziz et al. [8], who proved that it always has PNE by
using a special case of retracted result of Bouveret and Lang [13] (this did not affect the former though;
see [7]). Finally, besides the work of Amanatidis et al. [5] mentioned earlier, the fairness properties of
Round-Robin under strategic agents have recently been studied by Psomas and Verma [35]. Therein it
is shown that Round-Robin, despite being non-truthful, satisfies a relaxation of truthfulness, as it is not
obviously manipulable.
2 Preliminaries
For 𝑎∈N, let [𝑎]denote the set {1,2, . . . ,𝑎}. We will use 𝑁=[𝑛]to denote the set of agents and
𝑀={𝑔1, . . . , 𝑔𝑚}to denote the set of goods. Each agent 𝑖∈𝑁has a valuation function 𝑣𝑖: 2𝑀→R≥0
over the subsets of goods. We assume that all 𝑣𝑖are normalized, i.e., 𝑣𝑖( ∅) =0. We also adopt the shortcut
4

𝑣𝑖(𝑇|𝑆)for the marginal value of a set 𝑇with respect to a set 𝑆, i.e., 𝑣𝑖(𝑇|𝑆)=𝑣𝑖(𝑇∪𝑆) −𝑣(𝑆). If 𝑇={𝑔},
we write 𝑣𝑖(𝑔|𝑆)instead of 𝑣( {𝑔} | 𝑆). For each agent 𝑖∈𝑁, we say that 𝑣𝑖is
•non-decreasing (often referred to as monotone), if 𝑣𝑖(𝑆) ≤ 𝑣𝑖(𝑇)for any 𝑆⊆𝑇⊆𝑀.
•submodular, if 𝑣𝑖(𝑔|𝑆) ≥ 𝑣𝑖(𝑔|𝑇)for any 𝑆⊆𝑇⊆𝑀and 𝑔∉𝑇.
•cancelable, if 𝑣𝑖(𝑆∪ {𝑔}) >𝑣𝑖(𝑇∪ {𝑔}) ⇒ 𝑣𝑖(𝑆)>𝑣𝑖(𝑇)for any 𝑆, 𝑇 ⊆𝑀and 𝑔∈𝑀\ (𝑆∪𝑇).
•additive, if 𝑣𝑖(𝑆∪𝑇)=𝑣𝑖(𝑆) + 𝑣𝑖(𝑇)for every 𝑆, 𝑇 ⊆𝑀with 𝑆∩𝑇=∅.
•subadditive, if 𝑣𝑖(𝑆∪𝑇) ≤ 𝑣𝑖(𝑆) + 𝑣𝑖(𝑇)for every 𝑆, 𝑇 ⊆𝑀.
Throughout this work, we only consider non-decreasing valuation functions, e.g., when we refer to sub-
modular functions, we mean non-decreasing submodular functions. Note that although both submodular
and (subadditive) cancelable functions are strict superclasses of additive functions, neither one is a super-
class of the other.
We will occasionally need an alternative characterization of submodular functions due to Nemhauser
et al. [32].
Theorem 2.1 (Nemhauser et al. [32]).A function 𝑣: 2𝑀→R≥0is (non-decreasing) submodular if and only
if we have 𝑣(𝑇) ≤ 𝑣(𝑆) + Í𝑖∈𝑇\𝑆𝑣(𝑖|𝑆), for all 𝑆,𝑇 ⊆𝑀.
Also, the following lemma summarizes some easy observations about cancelable functions.
Lemma 2.2. If 𝑣: 2𝑀→R≥0is cancelable, then 𝑣𝑖(𝑆∪𝑅)>𝑣𝑖(𝑇∪𝑅) ⇒ 𝑣𝑖(𝑆)>𝑣𝑖(𝑇), implying that
𝑣𝑖(𝑆) ≥ 𝑣𝑖(𝑇) ⇒ 𝑣𝑖(𝑆∪𝑅) ≥ 𝑣𝑖(𝑇∪𝑅), for any 𝑆 ,𝑇 , 𝑅 ⊆𝑀, such that 𝑅⊆𝑀\𝑆∪𝑇. In particular,
𝑣𝑖(𝑆)=𝑣𝑖(𝑇) ⇒ 𝑣𝑖(𝑆∪𝑅)=𝑣𝑖(𝑇∪𝑅).
Note that, for 𝑆, 𝑇 ⊆𝑀, Lemma 2.2 directly implies that arg max𝑔∈𝑇𝑣(𝑔) ⊆ arg max𝑔∈𝑇𝑣(𝑔|𝑆).
Despite the fact that the agents have valuation functions, the mechanism we study (Mechanism 1) is
ordinal, i.e., it only takes as input a preference ranking from each agent. Formally, the preference ranking
≻𝑖, which agent 𝑖reports, defines a total order on 𝑀, i.e., 𝑔≻𝑖𝑔′implies that good 𝑔precedes good 𝑔′in
agent 𝑖’ declared preference ranking.1We call the vector of the agents’ declared preference rankings, ≻=
(≻1, . . . , ≻𝑛), the reported profile for the instance. So, while an instance to our problem is an ordered triple
(𝑁 , 𝑀, v), where v=(𝑣1, . . . , 𝑣𝑛)is a vector of the agents’ valuation functions, the input to Mechanism 1
is (𝑁 , 𝑀, ≻)instead.
Note that ≻𝑖may not reflect the actual underlying values, i.e., 𝑔≻𝑖𝑔′does not necessarily mean that
𝑣𝑖(𝑔)>𝑣𝑖(𝑔′)or, more generally, 𝑣𝑖(𝑔|𝑆)>𝑣𝑖(𝑔′|𝑆)for a given 𝑆⊆𝑀. This might be due to agent 𝑖
misreporting her preference ranking, or due to the fact that any single preference ranking is not expressive
enough to fully capture all the partial orders induced by a submodular function. Nevertheless, a valuation
function 𝑣𝑖does induce a true preference ranking <∗
𝑖|𝑆for each set 𝑆⊆𝑀, which is a partial order, i.e.,
𝑔<∗
𝑖|𝑆𝑔′⇔𝑣𝑖(𝑔|𝑆) ≥ 𝑣𝑖(𝑔′|𝑆)for all 𝑔, 𝑔′∈𝑀. We use ≻∗
𝑖|𝑆if the corresponding preference ranking is
strict, i.e., when 𝑔<∗
𝑖|𝑆𝑔′∧𝑔′<∗
𝑖|𝑆𝑔⇒𝑔=𝑔′, for all 𝑔, 𝑔′∈𝑀\𝑆. For additive (and more generally, for
cancelable) valuations, we drop 𝑆for the notation and simply write <∗
𝑖or ≻∗
𝑖. Finally, for a total order ≻
on 𝑀and a set 𝑇⊆𝑀, we use top(≻, 𝑇 )to denote the “largest” element of 𝑇with respect to ≻.
1See the discussion after the statement of Mechanism 1about why assuming that the reported preference rankings are total
(rather than partial) orders is without loss of generality.
5

2.1 Fairness Notions
A fair division mechanism produces an allocation (𝐴1, . . . , 𝐴𝑛), where 𝐴𝑖is the bundle of agent 𝑖, which
is a partition of 𝑀. The latter corresponds to assuming no free disposal, namely all the goods must be
allocated.
There are several different notions which attempt to capture which allocations are “fair”. The most
prominent such notion in the fair division literature has been envy-freeness (EF) [22,21,37], which has
been the starting point for other relaxed notions, more appropriate for the indivisible goods setting we
study here, as envy-freeness up to one good (EF1) [28,16] and envy-freeness up to any good (EFX) [18]. Here
we focus on EF1.
Definition 2.3. An allocation (𝐴1, . . . , 𝐴𝑛)is
•𝛼-envy-free (𝛼-EF), if for every 𝑖, 𝑗 ∈𝑁,𝑣𝑖(𝐴𝑖) ≥ 𝛼·𝑣𝑖(𝐴𝑗).
•𝛼-envy-free up to one good (𝛼-EF1), if for every pair of agents 𝑖, 𝑗 ∈𝑁, with 𝐴𝑗≠∅, there exists a
good 𝑔∈𝐴𝑗, such that 𝑣𝑖(𝐴𝑖) ≥ 𝛼·𝑣𝑖(𝐴𝑗\ {𝑔}).
When for every agent 𝑗∈𝑁with 𝐴𝑗≠∅, we have 𝑣𝑖(𝐴𝑖) ≥ 𝛼·𝑣𝑖(𝐴𝑗\ {𝑔}) for some good 𝑔∈𝐴𝑗, we
say that (𝐴1, . . . , 𝐴𝑛)is 𝛼-EF1 from agent 𝑖’s perspective, even when the allocation is not 𝛼-EF1!
2.2 Mechanisms and Equilibria
We are interested in mechanisms that produce allocations with EF1 guarantees. When no payments are
allowed, like in our setting, an allocation mechanism Mis just an allocation algorithm that takes as input
the agents’ reported preferences. In particular, Round-Robin, the mechanism of interest here, takes as
input the reported profile ≻and produces an allocation of all the goods. This distinction in terminology
is necessary as the reported input may not be consistent with the actual valuation functions due to the
agents’ incentives. When the allocation returned by M (≻)has some fairness guarantee, e.g., it is 0.5-EF1,
we will attribute the same guarantee to the reported profile itself, i.e., we will say that ≻is 0.5-EF1.
We study the fairness guarantees of the (approximate) pure Nash equilibria of Round-Robin. Given a
preference profile ≻=(≻1, . . . , ≻𝑛), we write ≻−𝑖to denote (≻1, . . . , ≻𝑖−1,≻𝑖+1, . . . , ≻𝑛)and given a pref-
erence ranking ≻′
𝑖we use (≻′
𝑖,≻−𝑖)to denote the profile (≻1,...,≻𝑖−1,≻′
𝑖,≻𝑖+1, . . . , ≻𝑛). For the next def-
inition we abuse the notation slightly: given an allocation (𝐴1, . . . , 𝐴𝑛)produced by M(≻), we write
𝑣𝑖(M (≻)) to denote 𝑣𝑖(𝐴𝑖); similarly for M (≻′
𝑖,≻−𝑖).
Definition 2.4. Let Mbe an allocation mechanism and consider a preference profile ≻=(≻1, . . . , ≻𝑛). We
say that the total order ≻𝑖is an 𝛼-approximate best response to ≻−𝑖if for every total order, i.e., permutation
≻′
𝑖of 𝑀, we have 𝛼·𝑣𝑖(M(≻′
𝑖,≻−𝑖)) ≤ 𝑣𝑖(M ( ≻)). The profile ≻is an 𝛼-approximate pure Nash equilibrium
(PNE) if, for each 𝑖∈𝑁,≻𝑖is an 𝛼-approximate best response to ≻−𝑖.
When 𝛼=1, we simply refer to best responses and exact PNE.
2.3 The Round-Robin Mechanism
We state Round-Robin as a mechanism (Mechanism 1) that takes as input a reported profile (≻1,...,≻𝑛).
For the sake of presentation, we assume that the agents in each round (lines 3–6) are always considered
according to their “name”, i.e., agent 1 is considered first, agent 2 second, and so on, instead of having
a permutation determining the priority of the agents as an extra argument of the input. This is without
loss of generality, as it only requires renaming the agents accordingly. We often refer to the process of
allocating a good to an agent (lines 4–6) as a step of the mechanism.
6

Mechanism 1 Round-Robin( ≻1, . . . , ≻𝑛)// For 𝑖∈𝑁,≻𝑖is the reported preference ranking of agent 𝑖.
1: 𝑆=𝑀;(𝐴1, . . . ,𝐴𝑛)=(∅, . . . , ∅);𝑘=⌈𝑚/𝑛⌉
2: for 𝑟=1, . . . , 𝑘 do // Each value of 𝑟determines the corresponding round.
3: for 𝑖=1, . . . , 𝑛 do // The combination of 𝑟and 𝑖determines the corresponding step.
4: 𝑔=top(≻𝑖, 𝑆 )
5: 𝐴𝑖=𝐴𝑖∪ {𝑔}// The current agent receives (what appears to be) her favorite available good.
6: 𝑆=𝑆\ {𝑔}// The good is no longer available.
7: return (𝐴1, . . . ,𝐴𝑛)
Note that there is no need for a tie-breaking rule here, as the reported preference rankings are assumed
to be total orders. Equivalently, one could allow for partial orders (either directly or via cardinal bids as
it is done in [5]) paired with a deterministic tie-breaking rule, e.g., lexicographic tie-breaking, a priori
known to the agents.
In the rest of the paper, we will assume that 𝑚=𝑘𝑛 for some 𝑘∈N, for simplicity. Note that this is
without loss of generality, as we may introduce at most 𝑛−1 dummy goods that have marginal value of
0 with respect to any set for everyone and append them at the end of the reported preference rankings to
be allocated during the last steps of the mechanism.
We have already mentioned that Round-Robin as an algorithm produces EF1 allocations for additive
agents, where the input is assumed to be any strict variant ≻∗=( ≻∗
1| ∅,≻∗
2| ∅, . . . , ≻∗
𝑛| ∅)of the truthful profile
(<∗
1| ∅,<∗
2| ∅,...,<∗
𝑛| ∅), i.e., the profile where each agent ranks the goods according to their singleton value.
This property fully extends to cancelable valuation functions as well. The proof of Proposition 2.5 is
rather simple, but not as straightforward as the additive case; note that it requires Lemma 3.3 from the
next section.
Proposition 2.5. Let be ≻∗be as described above. When all agents have cancelable valuation functions, the
allocation returned by Round-Robin(≻∗)is EF1.
Proof. Let (𝐴1, . . . ,𝐴𝑛)be the allocation returned by Round-Robin(≻∗). Fix two agents, 𝑖and 𝑗, and let
𝐴𝑖={𝑥1, 𝑥2, . . . ,𝑥𝑘}and 𝐴𝑗={𝑦1, 𝑦2, . . . ,𝑦𝑘}, where the goods in both sets are indexed according to the
round in which they were allocated to 𝑖and 𝑗, respectively. By the way Mechanism 1is defined, we have
𝑥𝑟≻∗
𝑖| ∅ 𝑦𝑟+1, for all 𝑟∈ [𝑘−1]. Therefore, 𝑥𝑟<∗
𝑖| ∅ 𝑦𝑟+1, or equivalently, 𝑣𝑖(𝑥𝑟) ≥ 𝑣𝑖(𝑦𝑟+1), for all 𝑟∈ [𝑘−1].
Thus, by Lemma 3.3, we get 𝑣𝑖(𝐴𝑖\ {𝑥𝑘}) ≥ 𝑣𝑖(𝐴𝑗\ {𝑦1}), and using the fact that 𝑣𝑖is non-decreasing,
𝑣𝑖(𝐴𝑖) ≥ 𝑣𝑖(𝐴𝑗\ {𝑦1}).
3 Existence of approximate PNE
At first glance, it is not clear why Mechanism 1has any pure Nash equilibria, even approximate ones
for a constant approximation factor. For additive valuation functions, however, it is known that for any
instance we can construct a simple preference profile, called the bluff profile, which is an exact PNE. While
the proof of this fact, in its full generality, is fragmented over three papers [8,14,5], we give here a simple
proof that generalizes the existence of exact PNE to cancelable valuation functions. As we shall see later,
extending this result to submodular functions is not possible and even defining a generalization of the
bluff profile which is a 0.5-approximate PNE is not straightforward.
3.1 Cancelable valuations
Defining the bluff profile for cancelable agents, we will start from a strict variant of the truthful profile
(<∗
1| ∅,<∗
2| ∅,...,<∗
𝑛| ∅), i.e., the profile where each agent ranks the goods according to their value (as single-
7

tons) in descending order, as we did for Proposition 2.5. Assume that any ties are broken deterministically
to get the strict version ≻∗=(≻∗
1| ∅,≻∗
2| ∅, . . . , ≻∗
𝑛| ∅). Now, consider Round-Robin(≻∗)and let ℎ1, ℎ 2, . . . , ℎ𝑚
be a renaming of the goods according to the order in which they were allocated and ≻bbe the correspond-
ing total order (i.e., ℎ1≻bℎ2≻b... ≻bℎ𝑚). The bluff profile is the preference profile ≻b=(≻b,≻b,...,≻b),
where everyone ranks the goods in the order they were allocated in Round-Robin(≻∗). The following fact
follows directly from the definition of the bluff profile and the description of Round-Robin.
Fact 3.1. If (≻∗)is a strict version of the truthful preference profile and (≻b)is the corresponding bluff profile,
then Round-Robin(≻b)and Round-Robin(≻∗)both return the same allocation.
An interesting observation about this fact is that, combined with Proposition 2.5 and Theorem 3.2, it
implies that there is at least one PNE of Mechanism 1which is EF1! Of course, it is now known that all
exact PNE of Round-Robin are EF1 for agents with additive valuation functions and, as we will see later
on, even approximate PNE have (approximate) EF1 guarantees for much more general instances, including
the case of subadditive cancelable valuation functions.
Theorem 3.2. When all agents have cancelable valuation functions, the bluff profile is an exact PNE of
Mechanism 1.
We first need to prove the following lemma that generalizes a straightforward property of additive
functions for cancelable functions.
Lemma 3.3. Suppose that 𝑣(·) is a cancelable valuation function. Consider sets 𝑋={𝑥1, 𝑥2, . . . , 𝑥𝑘}and
𝑌={𝑦1, 𝑦2, . . . , 𝑦𝑘}. If for every 𝑗∈ [𝑘], we have that 𝑣(𝑥𝑗) ≥ 𝑣(𝑦𝑗), then 𝑣(𝑋) ≥ 𝑣(𝑌).
Proof. We begin by arguing that it is without loss of generality to first assume that the elements of 𝑋are
ordered by non-increasing value with respect to 𝑣and then also assume that 𝑦𝑗∉{𝑥1, 𝑥2, . . . , 𝑥𝑗−1}, for
any 𝑗∈ [𝑘]. The former is indeed a matter of reindexing, if necessary, the elements of 𝑋and consistently
reindexing the corresponding elements of 𝑌. For the latter, suppose that there exist 𝑗such that 𝑦𝑗=𝑥𝑡
for 𝑡≤𝑗−1 and consider the smallest 𝑡for which this happens. We have 𝑣(𝑥𝑡) ≥ 𝑣(𝑥𝑡+1) ≥ ... ≥𝑣(𝑥𝑗)
by the assumption on the ordering of the elements of 𝑋,𝑣(𝑥𝑗) ≥ 𝑣(𝑦𝑗)by hypothesis, and 𝑣(𝑦𝑗)=𝑣(𝑥𝑡).
Thus, 𝑣(𝑥𝑡)=𝑣(𝑥𝑡+1)=... =𝑣(𝑥𝑗). Now we may rename the elements of 𝑌to {𝑦′
1, . . . ,𝑦′
𝑘}by inserting
𝑦𝑗to the 𝑡-th position, i.e., 𝑦′
𝑡=𝑦𝑗,𝑦′
𝑠=𝑦𝑠−1, for 𝑡+1≤𝑠≤𝑗, and 𝑦′
𝑠=𝑦𝑠, for 𝑠<𝑡or 𝑠>𝑗. Since only
𝑦𝑡, 𝑦𝑡+1, . . . ,𝑦𝑗changed indices but 𝑣(𝑥𝑡)=𝑣(𝑥𝑡+1)=... =𝑣(𝑥𝑗), we again have that 𝑣(𝑥𝑗) ≥ 𝑣(𝑦′
𝑗)for
every 𝑗∈ [𝑘]. Moreover, now the smallest ℓfor which there exist 𝑗>ℓsuch that 𝑦𝑗=𝑥ℓis strictly larger
than 𝑡. By repeating this renaming of the elements of 𝑌we end up with a renaming {𝑦∗
1, . . . ,𝑦∗
𝑘}such that
for every 𝑗∈ [𝑘],𝑣(𝑥𝑗) ≥ 𝑣(𝑦∗
𝑗)and 𝑦∗
𝑗∉{𝑥1, 𝑥2, . . . , 𝑥 𝑗−1}.
So, assuming that the elements of 𝑋are ordered in non-increasing value with respect to 𝑣and that
𝑦𝑗∉{𝑥1, 𝑥2, . . . , 𝑥 𝑗−1}, for any 𝑗∈ [𝑘], suppose towards a contradiction that 𝑣(𝑋)<𝑣(𝑌). That is,
𝑣({𝑥1, 𝑥 2, . . ., 𝑥𝑘}) <𝑣({𝑦1, 𝑦2, . . . ,𝑦𝑘}). Observe that if 𝑣({𝑥1, 𝑥2, . . . , 𝑥𝑘−1}) ≥ 𝑣({𝑦1, 𝑦2, . . . ,𝑦𝑘−1}), this
would imply that 𝑣({𝑥1, . . . , 𝑥𝑘−1, 𝑦𝑘}) ≥ 𝑣({𝑦1, . . . , 𝑦𝑘−1, 𝑦𝑘}), by the definition of cancelable valuations
and the fact that 𝑦𝑘∉{𝑥1, . . . , 𝑥𝑘−1} ∪ {𝑦1, . . . , 𝑦𝑘−1}. This leads to
𝑣({𝑥1, . . . , 𝑥𝑘−1, 𝑥𝑘}) ≥ 𝑣( {𝑥1, . . . , 𝑥𝑘−1, 𝑦𝑘}) ≥ 𝑣({𝑦1, . . . ,𝑦𝑘−1, 𝑦𝑘}) ,
where the first inequality follows from 𝑣(𝑥𝑘) ≥ 𝑣(𝑦𝑘)and Fact 2.2, contradicting our initial assumption.
Therefore, 𝑣({𝑥1, . . . , 𝑥𝑘−1}) <𝑣({𝑦1, . . . , 𝑦𝑘−1}). By repeating the same argument 𝑘−2 more times, we
end up with 𝑣(𝑥1)<𝑣(𝑦1), a contradiction. 
Proof of Theorem 3.2.Now we show that the bluff profile for cancelable valuations is an exact PNE. Con-
sider the goods named ℎ1, . . . ,ℎ𝑚as in the bluff profile, i.e., by the order in which they are picked when
8

each agent reports their preference order to be the one induced by all singleton good values. Consider
agent 𝑖. Her assigned set of goods under the bluff profile is 𝐴b
𝑖={ℎ𝑖, ℎ𝑛+𝑖, . . ., ℎ(𝑘−1)𝑛+𝑖}, where 𝑘=𝑚/𝑛.
Assume now that she deviates from ≻bto ≻𝑖, resulting in some allocated set 𝐴𝑖={𝑦1, 𝑦2, . . . , 𝑦𝑘}, where
we assume 𝑦𝑟to be allocated in round 𝑟. We need to show 𝑣𝑖(𝐴b
𝑖) ≥ 𝑣𝑖(𝐴𝑖).
To this end, we compare the goods allocated to agent 𝑖in both reports, one by one. If 𝑣𝑖(𝑦𝑟) ≤
𝑣𝑖(ℎ(𝑟−1)𝑛+𝑖)for every 𝑟∈ [𝑘], then we are done by applying Lemma 3.3 with 𝐴b
𝑖and 𝐴𝑖. If some of
these inequalities fail, let 𝑟denote the latest round such that 𝑣𝑖(𝑦𝑟)>𝑣𝑖(ℎ(𝑟−1)𝑛+𝑖. Therefore, in the exe-
cution of Mechanism 1with the bluff profile as input, 𝑦𝑟was no longer available in round 𝑟. However, 𝑦𝑟
becomes available in round 𝑟once agent 𝑖deviates. This can only stem from the fact that at some point
before round 𝑟, a good ℎ𝑡with 𝑡>(𝑟−1)𝑛+𝑖was picked (since the overall number of goods picked per
round always stays the same). Clearly, the only agent who could have done so (since she is the only one
deviating from the common bluff order) is agent 𝑖. Therefore, it holds that ℎ𝑡=𝑦𝑗for some 𝑗<𝑟. Now,
we replace the ordered set 𝑌=(𝑦1,𝑦 2, . . ., 𝑦𝑘)by 𝑌′=(𝑦1, . . ., 𝑦𝑗−1, 𝑦𝑟, 𝑦𝑗+1, . . . , 𝑦𝑟−1, 𝑦𝑗, 𝑦𝑟+1, . . . , 𝑦𝑘), i.e.,
we simply exchange 𝑦𝑟and 𝑦𝑗. It will be convenient to rename 𝑦1, . . . , 𝑦𝑘so that 𝑌′=(𝑦′
1, 𝑦 ′
2, . . . ,𝑦′
𝑘)
We claim that it if agent 𝑖reports a preference ranking ≻′
𝑖that starts with all goods in 𝑌′, in that
specific order, followed by everything else, in any order, she still gets 𝐴𝑖but the goods are allocated in the
order suggested by 𝑌′. Indeed, first notice that the first 𝑗−1 rounds of Round-Robin will be the same as
in the run with the original deviation ≻𝑖. Further, 𝑦′
𝑗=𝑦𝑟is allocated earlier under ≻′
𝑖than under ≻𝑖, and
thus it surely is available at the time. After that, rounds 𝑗−1 to 𝑟−1 will be the same as in the run with
the deviation ≻𝑖. Now 𝑦′
𝑟=𝑦𝑗is allocated later than before, namely in round 𝑟, but it is not among the
first (𝑟−1)𝑛+𝑖goods in the bluff order, as noted above, which means it is not allocated to any other agent
in any round before the 𝑟-th under ≻′
𝑖. Finally, rounds 𝑟+1 to 𝑘will be the same as in the run with ≻𝑖.
Although agent 𝑖still is assigned the same set 𝐴𝑖by deviating to ≻′
𝑖, we now have 𝑣𝑖(𝑦′
𝑟)=𝑣𝑖(𝑦𝑗) ≤
𝑣𝑖(ℎ(𝑟−1)𝑛+𝑖, where the inequality holds because both goods are available in round 𝑟of the bluff run, and
agent one prefers ℎ(𝑟−1)𝑛+𝑖. Also, all later goods in 𝑌′remain unchanged, i.e., 𝑦′
𝑠=𝑦𝑠for 𝑠>𝑟. Therefore,
the latest occurrence of some 𝑦′
ℓ>ℎ(ℓ−1)𝑛+𝑖now happens at an earlier point in the sequence, if at all.
Repeating this process until no such occurrence is left yields an ordering 𝑌∗=(𝑦∗
1, 𝑦∗
2, . . . ,𝑦∗
𝑘)of 𝐴𝑖such
that for all 𝑟∈ [𝑘],𝑣𝑖(𝑦∗
𝑟) ≤ 𝑣𝑖(ℎ(𝑟−1)𝑛+𝑖). Now using Lemma 3.3 completes the proof. 
3.2 Submodular valuations
We move on to the much more general class of submodular valuations. In order to define the bluff profile
in this case, we again would like to start from the truthful profile. However, recall that Round-Robin
restricts each agent’s report to specifying an ordering on the good set 𝑀and these preference rankings
are not expressive enough to fully capture submodular valuation functions. In fact, it is not obvious what
‘truthful’ means here without further assumptions on what information is known by the agents. Still, we
define a truthfully greedy allocation and use this as our starting point.
Imagine that, instead of having a full preference profile from the beginning, we only ask the active
agent 𝑖(i.e., the agent to which we are about to allocate a new good) for the good with the largest marginal
value with respect to her current set of goods 𝐴𝑖and give this to her. Let ℎ1, ℎ2, . . . , ℎ𝑚be a renaming of
the goods according to the order in which they would be allocated in this hypothetical truthfully greedy
scenario and ≻bbe the corresponding total order. Like in the cancelable case, the bluff profile is the
preference profile ≻b=(≻b,≻b, . . . , ≻b).
Formally, the renaming of the goods is performed as described in Algorithm 2below. It should be
noted that this definition of the bluff profile is consistent with the definition for cancelable functions,
assuming that all ties are resolved lexicographically.
Also notice that the allocation Round-Robin(≻b)produced under the bluff profile is exactly (𝑋1, 𝑋 2,
. . . ,𝑋𝑛), as described in Algorithm 2, i.e., 𝑋𝑖=𝐴b
𝑖={ℎ𝑖, ℎ𝑛+𝑖, . . . , ℎ(𝑘−1)𝑛+𝑖}, where recall that 𝑘=𝑚/𝑛.
9

Algorithm 2 Greedy renaming of goods for defining the bluff profile
Input: 𝑁,𝑀, value oracles for 𝑣1(·), . . . , 𝑣𝑛(·)
1: 𝑋𝑖=∅for 𝑖∈ [𝑛]
2: for 𝑗=1, . . . , 𝑚 do
3: 𝑖=(𝑗−1) (mod 𝑛) + 1
4: ℎ𝑗=arg max
𝑔∈𝑀\Ðℓ𝑋ℓ
𝑣𝑖(𝑔|𝑋𝑖)// Ties are broken lexicographically.
5: 𝑋𝑖=𝑋𝑖∪ {ℎ𝑗}
6: return (ℎ1, ℎ2, . . . ,ℎ𝑚)
The main result of this section is Theorem 3.7 stating that the bluff profile is a 1
2-approximate PNE
when agents have submodular valuation functions. While this sounds weaker than Theorem 3.2, it should
be noted that for submodular agents Mechanism 1does not have PNE in general, even for relatively simple
instances, as stated in Proposition 3.4. In fact, even the existence of approximate equilibria can be seen as
rather surprising, given the generality of the underlying valuation functions.
Proposition 3.4. There exists an instance where all agents have submodular valuation functions such that
Mechanism 1has no (3
4+𝜀)-approximate PNE.
Proof. Consider an instance with 2 agents and 4 goods 𝑀={𝑔1, 𝑔2, 𝑔3, 𝑔4}, with the following valuation
for all possible 2-sets:
𝑣1({𝑔1, 𝑔2}) =3
𝑣1({𝑔1, 𝑔3}) =3
𝑣1({𝑔1, 𝑔4}) =4
𝑣1({𝑔2, 𝑔3}) =4
𝑣1({𝑔2, 𝑔4}) =3
𝑣1({𝑔3, 𝑔4}) =3
𝑣2({𝑔1, 𝑔2}) =4
𝑣2({𝑔1, 𝑔3}) =4
𝑣2({𝑔1, 𝑔4}) =3
𝑣2({𝑔2, 𝑔3}) =3
𝑣2({𝑔2, 𝑔4}) =4
𝑣2({𝑔3, 𝑔4}) =4
In addition, all individual goods have the same value: 𝑣1(𝑥)=𝑣2(𝑥)=2 for 𝑥∈𝑀, while all 3-sets and
4-sets have value 4, for both agents.
We begin by establishing that this valuation function is indeed submodular for both agents. Observe
for any set 𝑆⊆𝑀and 𝑖∈ [2], 𝑗 ∈ [4]we have:
|𝑆|=0⇒𝑣𝑖(𝑔𝑗|𝑆) ∈ {2}
|𝑆|=1⇒𝑣𝑖(𝑔𝑗|𝑆) ∈ {1,2}
|𝑆|=2⇒𝑣𝑖(𝑔𝑗|𝑆) ∈ {0,1}
|𝑆|=3⇒𝑣𝑖(𝑔𝑗|𝑆)=0,
which immediately implies that both valuation functions are indeed submodular.
Notice that for any reported preferences ≻1,≻2, one of the two agents will receive goods leading to a
value of 3. If this is the agent 1, she can easily deviate and get 4 instead. In particular, if agent 2 has good
𝑔2or 𝑔3first in their preferences then agent 1 can get {𝑔1, 𝑔4}, and if agent 2 has good 𝑔1or 𝑔4as first then
agent 1 can get {𝑔2, 𝑔3}instead. On the other hand, if agent 2 received a value of 3 they can also always
deviate to 4. Notice that for any 𝑔𝑎, agent 2 always has two sets different sets {𝑔𝑎, 𝑔𝑏},{𝑔𝑎, 𝑔𝑐}with value
4 and one {𝑔𝑎, 𝑔𝑑}with value 3. Thus, for any preference of agent 1 with 𝑔ˆ𝑎≻1𝑔ˆ
𝑏≻1𝑔ˆ𝑐≻1𝑔ˆ
𝑑, agent 2 can
10

deviate and get either {𝑔ˆ
𝑏, 𝑔 ˆ
𝑑}or {𝑔ˆ𝑐, 𝑔 ˆ
𝑑}, one of which must have value 4. Therefore, in every outcome
there exists an agent that can deviate to improve their value from 3 to 4. 
Moving towards the proof of Theorem 3.7 for the submodular case, we note that although it is very
different from that of Theorem 3.2, we will still need an analog of the main property therein, i.e., the
existence of a good-wise comparison between the goods an agent gets under the bluff profile and the ones
she gets by deviating. As expected, the corresponding property here (see Lemma 3.5) is more nuanced and
does not immediately imply Theorem 3.7 as we are now missing the analog of Lemma 3.3.
Throughout this section, we are going to argue about an arbitrary agent 𝑖. To simplify the notation,
let us rename 𝑋𝑖=𝐴b
𝑖={ℎ𝑖, ℎ𝑛+𝑖, . . . , ℎ(𝑘−1)𝑛+𝑖}to simply 𝑋={𝑥1, 𝑥2, . . . , 𝑥𝑘}, where we have kept the
order of indices the same, i.e., 𝑥𝑗=ℎ(𝑗−1)𝑛+𝑖. This way, the goods in 𝑋are ordered according to how they
were allocated to agent 𝑖in the run of Mechanism 1with the bluff profile as input.
We also need to define the ordering of the goods agent 𝑖gets when she deviates from the bluff bid ≻b
to another preference ranking ≻𝑖. Let 𝐴𝑖=𝑌={𝑦1, 𝑦2, . . . ,𝑦𝑘}be this set of goods. Instead of renaming
the elements of 𝑌in a generic fashion like in the proof of Theorem 3.2, doing so becomes significantly
more complicated, and we need to do it in a more systematic way, see Algorithm 3.
Algorithm 3 Greedy renaming of goods for the deviating agent 𝑖
Input: 𝑋={𝑥1, 𝑥2, . . . , 𝑥𝑘},𝑌, and a value oracle for 𝑣𝑖( ·)
1: 𝑍=𝑌
2: for 𝑗=|𝑌|, . . . , 1do
3: 𝑦′
𝑗=arg min
𝑔∈𝑍
𝑣𝑖(𝑔| {𝑥1, . . . , 𝑥𝑗−1}) // Ties are broken lexicographically.
4: 𝑍=𝑍\ {𝑦′
𝑗}
5: return (𝑦′
1, 𝑦 ′
2, . . . ,𝑦′
|𝑌|)
In what follows, we assume that the indexing 𝑦1, 𝑦2, . . . ,𝑦𝑘is already the result of Algorithm 3. This
renaming is crucial and it will be used repeatedly. In particular, we need this particular ordering in order to
prove that 𝑣𝑖(𝑥𝑗| {𝑥1, . . . ,𝑥 𝑗−1}) ≥ 𝑣𝑖(𝑦𝑗| {𝑥1, . . . ,𝑥𝑗−1}), for all 𝑗∈ [𝑘], in Lemma 3.5 below. Towards that,
we need to fix some notation for the sake of readability. For 𝑗∈ [𝑘], we use 𝑋𝑗
−and 𝑋𝑗
+to denote the sets
{𝑥1, 𝑥2, . . . , 𝑥 𝑗}and {𝑥𝑗, 𝑥𝑗+1, . . . , 𝑥𝑘}, respectively. The sets 𝑌𝑗
−and 𝑌𝑗
+, for 𝑗∈ [𝑘], are defined analogously.
We also use 𝑋0
−=𝑌0
−=∅. The main high-level idea of the proof is that if 𝑣𝑖(𝑦ℓ|𝑋ℓ−1
−)>𝑣𝑖(𝑥ℓ|𝑋ℓ−1
−)
for some ℓ, then it must be the case that during the execution of Round-Robin(≻b)every good in 𝑌ℓ
−=
{𝑦1, . . ., 𝑦ℓ}is allocated before the turn of agent 𝑖in round ℓ. Then, using a simple counting argument, we
show that agent 𝑖cannot receive all the goods in 𝑌ℓ
−when deviating, leading to a contradiction.
Lemma 3.5. Let 𝑋={𝑥1, 𝑥2, . . . , 𝑥𝑘}be agent 𝑖’s bundle in Round-Robin(≻b), where goods are indexed in
the order they were allocated, and 𝑌={𝑦1, 𝑦2, . . . ,𝑦𝑘}be 𝑖’s bundle in Round-Robin(≻𝑖,≻b
−𝑖), where goods
are indexed by Algorithm 3. Then, for every 𝑗∈ [𝑘], we have 𝑣𝑖(𝑥𝑗|𝑋𝑗−1
−) ≥ 𝑣𝑖(𝑦𝑗|𝑋𝑗−1
−).
Proof. The way goods in 𝑋are indexed, we have that 𝑥𝑗is the good allocated to agent 𝑖in round 𝑗of
Round-Robin(≻b). Suppose, towards a contradiction, that there is some ℓ∈ [𝑘], for which we have
𝑣𝑖(𝑦ℓ|𝑋ℓ−1
−)>𝑣𝑖(𝑥ℓ|𝑋ℓ−1
−). First notice that ℓ≠1, as 𝑥1is, by the definition of the bluff profile, a singleton
of maximum value for agent 𝑖excluding the goods allocated to agents 1 through 𝑖−1 in round 1, regardless
of agent 𝑖’s bid. Thus, ℓ≥2.
Let 𝐵⊆𝑀and 𝐷⊆𝑀be the sets of goods allocated (to any agent) up to right before a good is
allocated to agent 𝑖in round ℓin Round-Robin(≻b)and Round-Robin(≻𝑖,≻b
−𝑖), respectively. Clearly, |𝐵|=
|𝐷|=(ℓ−1)𝑛+𝑖−1. In fact, we claim that in this case the two sets are equal.
11

Claim 3.6. It holds that 𝐵=𝐷. Moreover, {𝑦1, . . . ,𝑦ℓ} ⊆ 𝐵.
Proof of the claim. We first observe that 𝑣𝑖(𝑦𝑗|𝑋ℓ−1
−) ≥ 𝑣𝑖(𝑦ℓ|𝑋ℓ−1
−)>𝑣𝑖(𝑥ℓ|𝑋ℓ−1
−), for every 𝑗∈ [ℓ−1],
where the first inequality follows from way Algorithm 3ordered the elements of 𝑌. Now consider the
execution of Round-Robin(≻b). Since 𝑥ℓwas the good allocated to agent 𝑖in round ℓ,𝑥ℓhad maximum
marginal value for agent 𝑖with respect to 𝑋ℓ−1
−among the available goods. Thus, none of the goods
𝑦1, . . . ,𝑦ℓwere available at the time. That is, 𝑦1, . . . ,𝑦ℓwere all already allocated to some of the agents
(possibly including agent 𝑖herself ). We conclude that {𝑦1, . . . , 𝑦𝑙} ⊆ 𝐵.
Now suppose for a contradiction that 𝐷≠𝐵and consider the execution of Round-Robin(≻𝑖,≻b
−𝑖).
Recall that the goods in 𝐵are still the (ℓ−1)𝑛+𝑖−1 most preferable goods for every agent in 𝑁\ {𝑖}
according to the profile (≻𝑖,≻b
−𝑖). Therefore, all agents in 𝑁\ {𝑖}will get goods from 𝐵allocated to them
up to the point when a good is allocated to agent 𝑖in round ℓ, regardless of what ≻𝑖is. If agent 𝑖also
got only goods from 𝐵allocated to her in the first ℓ−1 rounds of Round-Robin(≻𝑖,≻b
−𝑖), then 𝐷would
be equal to 𝐵. Thus, at least one good which is not in 𝐵(and thus, not in {𝑦1, . . . , 𝑦ℓ}) must have been
allocated to agent 𝑖in the first ℓ−1 rounds. As a result, at the end of round ℓ−1, there are at least two
goods in {𝑦1, . . . ,𝑦ℓ}that have not yet been allocated to 𝑖.
However, we claim that up to right before a good is allocated to agent 𝑖in round ℓ+1, all goods
in 𝐵(and thus in {𝑦1, . . . ,𝑦ℓ}as well) will have been allocated, leaving 𝑖with at most ℓ−1 goods from
{𝑦1, . . ., 𝑦ℓ}in her final bundle and leading to a contradiction. Indeed, this follows from a simple counting
argument. Right before a good is allocated to agent 𝑖in round ℓ+1, the goods allocated to agents in 𝑁\ {𝑖}
are exactly ℓ(𝑛−1) + 𝑖−1≥ (ℓ−1)𝑛+𝑖−1=|𝐵|. As noted above, agents in 𝑁\ {𝑖}will get goods from 𝐵
allocated to them as long as they are available. Thus, no goods from 𝐵, or from {𝑦1, . . . , 𝑦ℓ}in particular,
remain unallocated right before a good is allocated to agent 𝑖in round ℓ+1. Therefore, agent 𝑖may get at
most ℓ−1 goods from {𝑦1, . . . ,𝑦ℓ}(at most ℓ−2 in the first ℓ−1 rounds and one in round ℓ), contradicting
the definition of the set 𝑌. We conclude that 𝐷=𝐵.⊡
Given the claim, it is now easy to complete the proof. Clearly, in the first ℓ−1 rounds of Round-
Robin( ≻𝑖,≻b
−𝑖)at most ℓ−1 goods from {𝑦1, . . . ,𝑦ℓ}have been allocated to agent 𝑖. However, when it
is 𝑖’s turn in round ℓ, only goods in 𝑀\𝐷are available, by the definition of 𝐷. By Claim 3.6, we have
{𝑦1, . . ., 𝑦𝑙} ⊆ 𝐷, and thus there is at least one good {𝑦1, . . . ,𝑦ℓ}that is allocated to another agent, which
contradicts the definition of 𝑌.
We are now ready to state and prove the main result of this section.
Theorem 3.7. When all agents have submodular valuation functions, the bluff profile is a 1
2-approximate
PNE of Mechanism 1. Moreover, this is tight, i.e., for any 𝜀>0, there are instances where the bluff profile is
not a 1
2+𝜀-approximate PNE.
Proof. We are going to use the notation used so far in the section and consider the possible deviation of
an arbitrary agent 𝑖. Like in the statement of Lemma 3.5,𝑋={𝑥1, . . . , 𝑥𝑘}is agent 𝑖’s bundle in Round-
Robin(≻b), with goods indexed in the order they were allocated, and 𝑌={𝑦1, 𝑦2, . . ., 𝑦𝑘}is 𝑖’s bundle
in Round-Robin( ≻𝑖,≻b
−𝑖), with goods indexed by Algorithm 3. Also, recall that 𝑋𝑗
−={𝑥1, . . . , 𝑥 𝑗}and
𝑋𝑗
+={𝑥𝑗, . . . ,𝑥𝑘}(and similarly for 𝑌𝑗
−and 𝑌𝑗
+). We also use the convention that 𝑌𝑘+1
+=∅. For any 𝑗∈ [𝑘],
we have
𝑣𝑖(𝑋𝑗
−) − 𝑣𝑖(𝑋𝑗−1
−)=𝑣𝑖(𝑥𝑗|𝑋𝑗−1
−)
≥𝑣𝑖(𝑦𝑗|𝑋𝑗−1
−)
≥𝑣𝑖(𝑦𝑗|𝑋𝑗−1
−∪𝑌𝑗+1
+)
12

=𝑣𝑖(𝑋𝑗−1
−∪𝑌𝑗+1
+∪ {𝑦𝑗}) − 𝑣𝑖(𝑋𝑗−1
−∪𝑌𝑗+1
+)
=𝑣𝑖(𝑋𝑗−1
−∪𝑌𝑗
+) − 𝑣𝑖(𝑋𝑗−1
−∪𝑌𝑗+1
+)
≥𝑣𝑖(𝑋𝑗−1
−∪𝑌𝑗
+) − 𝑣𝑖(𝑋𝑗
−∪𝑌𝑗+1
+).
The first inequality holds because Lemma 3.5 applies on 𝑋and 𝑌, whereas the second inequality holds
because of submodularity. Finally, the last inequality holds since 𝑋𝑗−1
−⊆𝑋𝑗
−and 𝑣𝑖(·) is non-decreasing,
for every 𝑖∈𝑁. Using these inequalities along with a standard expression of the value of a set as a sum
of marginals, we have
𝑣𝑖(𝑋)=𝑣𝑖(𝑋𝑘
−) − 𝑣𝑖(𝑋0
−)
=
𝑘
Õ
𝑗=1𝑣𝑖(𝑋𝑗
−) − 𝑣𝑖(𝑋𝑗−1
−)
≥
𝑘
Õ
𝑗=1𝑣𝑖(𝑋𝑗−1
−∪𝑌𝑗
+) − 𝑣𝑖(𝑋𝑗
−∪𝑌𝑗+1
+)
=𝑣𝑖(𝑋0
−∪𝑌1
+) − 𝑣𝑖(𝑋𝑘
−∪𝑌𝑘+1
+)
=𝑣𝑖(𝑌) − 𝑣𝑖(𝑋).
Thus, we have 𝑣𝑖(𝑋) ≥ 1
2·𝑣𝑖(𝑌), and we conclude that ≻bis a 1
2-approximate PNE of Mechanism 1.
To show that the result is tight, consider an example with two agents and five goods. The valuation
function of agent 1 is additive and defined as follows on the singletons:
𝑣1(𝑔1)=2𝑣1(𝑔2)=1𝑣1(𝑔3)=1−𝜀1𝑣1(𝑔2)=1−𝜀2𝑣1(𝑔5)=1−𝜀3,
where 1 ≫𝜀3>𝜀2>𝜀1>0.
The valuation function of agent 2 is OXS2and defined by the maximum matchings in the bipartite
graph below, e.g., 𝑣2({𝑔1, 𝑔2}) =2+1=3 and 𝑣2( {𝑔1, 𝑔4, 𝑔5}) =2+1−𝜀2=3−𝜀2.
𝑔1
𝑔2
𝑔3
𝑔4
𝑔5
2
1
1−𝜀1
1−𝜀2
1−𝜀3
It is not hard to see that the bluff profile for this instance consists of the following declared ordering
by both agents: 𝑔1>𝑔2>𝑔3>𝑔4>𝑔5. The allocation produced by Mechanism 1for the bluff profile
is then 𝐴=(𝐴1, 𝐴2), where 𝐴1={𝑔1, 𝑔3, 𝑔5}, and 𝐴2={𝑔2, 𝑔4}. Observe that 𝑣1(𝐴1)=4−𝜀1−𝜀3and
𝑣2(𝐴2)=1. It is easy to see that there is no profitable deviation for agent 1, while the maximum value that
2Roughly speaking, OXS functions generalize unit-demand functions. The set of OXS functions is a strict superset of additive
functions and a strict subset of submodular functions. See, [26,27].
13

agent 2 can attain by deviating is 2 −𝜀1−𝜀2. Agent 2 achieves this by reporting the preference ranking:
𝑔3>𝑔4>𝑔1>𝑔2>𝑔5and getting goods {𝑔3, 𝑔4}. This implies that for any 𝜀>0 one can chose
appropriately small 𝜀1, 𝜀2, 𝜀 3so that the bluff profile is not a 1
2+𝜀-approximate PNE. 
In Section 4, we show that every approximate PNE of Mechanism 1results in an approximately EF1
allocation. Here, as a warm-up, we start this endeavor with an easy result which holds specifically for
the bluff profile (and can be extended to approximate PNE where all agents submit the same preference
ranking) but shows a better fairness guarantee than our general Theorem 4.4.
Theorem 3.8. When all agents have submodular valuation functions 𝑣1, . . . , 𝑣𝑛, the allocation returned by
Round-Robin(≻b)is 1
2-EF1 with respect to 𝑣1, . . . , 𝑣𝑛. Moreover, this is tight, i.e., for any 𝜀>0, there are
instances where this allocation is not 1
2+𝜀-EF1.
Proof. In order to obtain a contradiction, suppose that the allocation (𝐴b
1, 𝐴b
2, . . . ,𝐴b
𝑛)returned by Round-
Robin(≻b)is not 1
2-EF1. That is, there exist agents 𝑖and 𝑗such that 𝑣𝑖(𝐴b
𝑖)<0.5·𝑣𝑖(𝐴b
𝑗\{𝑔}), for all 𝑔∈𝐴b
𝑗.
We are going to show that this allows us to construct a deviation for agent 𝑖where she gets value more than
2𝑣𝑖(𝐴b
𝑖), contradicting the fact that ≻bis a 1
2-approximate PNE. Recall that using the renaming ℎ1, ℎ2, . . .
produced by Algorithm 2, we have 𝐴b
𝑖={ℎ𝑖, ℎ𝑛+𝑖, . . ., ℎ(𝑘−1)𝑛+𝑖}and 𝐴b
𝑗={ℎ𝑗, ℎ𝑛+𝑗, . . . , ℎ(𝑘−1)𝑛+𝑗}.
Let 𝛿be the indicator variable of the event 𝑗<𝑖, i.e., 𝛿is 1 if 𝑗<𝑖and 0 otherwise. We will show
that it is possible for agent 𝑖to get the set {ℎ𝛿𝑛+𝑗, ℎ (1+𝛿)𝑛+𝑗, ℎ (2+𝛿)𝑛+𝑗, . . . ,ℎ(𝑘−1)𝑛+𝑗}, which is either the
entire 𝐴b
𝑗(when 𝑖<𝑗) or 𝐴b
𝑗\ {ℎ𝑗}(when 𝑗<𝑖). In particular, let ≻𝑖be a preference ranking that starts
with all goods in 𝐴b
𝑗in the same order as they were allocated to agent 𝑗in Round-Robin(≻b), followed by
everything else, in any order.
Consider the execution of Round-Robin(≻𝑖,≻b
−𝑖). The crucial, yet simple, observation (that makes
an inductive argument work) is that the first 𝑖−1 goods ℎ1, . . . , ℎ𝑖−1are allocated as before, then good
ℎ𝛿𝑛+𝑗(rather than ℎ𝑖) is allocated to agent 𝑖, and after that the 𝑛−1 top goods for all agents in 𝑁\ {𝑖}
according to ≻b
−𝑖are ℎ𝑖, ℎ𝑖+1, . . ., ℎ𝛿𝑛+𝑗−1, ℎ𝛿𝑛+𝑗+1, . . . ,ℎ𝑛+𝑖−1, and these are allocated in the next 𝑛−1 steps
of the algorithm. As a result, right before a second good is allocated to agent 𝑖, the available goods are
ℎ𝑛+𝑖, ℎ𝑛+𝑖+1, . . . ,ℎ𝑚exactly as in the execution of Round-Robin(≻b).
More generally, right before an 𝑟-th good is allocated to 𝑖, her bundle is {ℎ𝛿𝑛+𝑗, ℎ (1+𝛿)𝑛+𝑗, ℎ(2+𝛿)𝑛+𝑗,
. . . ,ℎ(𝑟−2+𝛿)𝑛+𝑗}, and the available goods are ℎ(𝑟−1)𝑛+𝑖, ℎ (𝑟−1)𝑛+𝑖+1, . . ., ℎ𝑚(as they were in the execution of
Round-Robin(≻b)). Then good ℎ(𝑟−1+𝛿)𝑛+𝑗(rather than ℎ(𝑟−1)𝑛+𝑖) is allocated to agent 𝑖, and after that the
𝑛−1 top goods for all agents according to ≻b
−𝑖are
ℎ(𝑟−1)𝑛+𝑖, ℎ(𝑟−1)𝑛+𝑖+1, . . . ,ℎ(𝑟−1+𝛿)𝑛+𝑗−1, ℎ (𝑟−1+𝛿)𝑛+𝑗+1, . . . , ℎ𝑟𝑛+𝑖−1,
and they are allocated in the next 𝑛−1 steps of the algorithm. At the end, agent 𝑖gets the entire 𝐴b
𝑗or
𝐴b
𝑗\ {ℎ𝑗}plus some arbitrary good, depending on whether 𝑖<𝑗or 𝑗<𝑖. In either case, by monotonicity,
agent 𝑖’s value for her bundle is at least 𝑣𝑖(𝐴b
𝑗\ {ℎ𝑗}) >2𝑣𝑖(𝐴b
𝑖), where the last inequality follows from
our assumption that (𝐴b
1, 𝐴b
2, . . . ,𝐴b
𝑛)is not 1
2-EF1. Therefore, by deviating from ≻bto ≻𝑖, agent 𝑖increases
her value by a factor strictly grater than 2, contradicting Theorem 3.7.
To show that this factor is tight, we again turn to the example given within the proof of Theorem 3.7.
Recall the allocation produced by Mechanism 1for the bluff profile is 𝐴=(𝐴1, 𝐴 2), with 𝐴1={𝑔1, 𝑔3,𝑔 5}
and 𝐴2={𝑔2, 𝑔4}. Observe that agent 1 is envy-free towards agent 2 as 𝑣1(𝐴1)=4−𝜀1−𝜀3>2−𝜀2=𝑣1(𝐴2).
On the other hand, 𝑣2(𝐴2)=1, whereas 𝑣2(𝐴1)=4−𝜀1−𝜀3and 𝑣2(𝐴1\ {𝑔1}) =2−𝜀1−𝜀3. The latter
implies that for any 𝜀>0 one can chose appropriately small 𝜀1, 𝜀2, 𝜀3so that the bluff profile does not
result in a 1
2+𝜀-EF1 allocation with respect to the true valuation functions of the agents. 
14

4 Fairness properties of PNE
In Section 2.3, Proposition 2.5, we state the fairness guarantees of Round-Robin—viewed as an algorithm—
when all agents have cancelable valuation functions. So far, we have not discussed this matter for the
submodular case. It is not hard to see, however, that Theorem 3.8 and the definition of the bluff profile
via Algorithm 2imply that when we have (value oracles for) the valuation functions, then we can use
Round-Robin to algorithmically produce 1
2-EF1 allocations. Using similar arguments, we show next that
for any preference profile ≻=(≻1,...,≻𝑛)and any 𝑖∈𝑁, there is always a response ≻′
𝑖of agent 𝑖to ≻−𝑖,
such that the allocation returned by Round-Robin(≻′
𝑖,≻−𝑖)is 1
2-EF1 from agent 𝑖’s perspective.
Towards this, we first need a variant of Algorithm 2that considers everyone in 𝑁\ {𝑖}fixed to their
report in ≻−𝑖and greedily determines a “good” response for agent 𝑖. An intuitive interpretation of what
Algorithm 4below is doing, can be given if one sees Mechanism 1as a sequential game. Then, given that
everyone else stays consistent with ≻−𝑖, agent 𝑖picks a good of maximum marginal value every time her
turn is up.
Algorithm 4 Greedy response of agent 𝑖to ≻−𝑖
Input: 𝑁,𝑀,≻−𝑖, value oracle for 𝑣𝑖
1: 𝑆=𝑀;𝑋=∅
2: for 𝑗=1, . . . , 𝑚 do
3: ℓ=(𝑗−1) (mod 𝑛) + 1
4: if ℓ=𝑖then
5: 𝑥⌈𝑗/𝑛⌉=arg max
𝑔∈𝑆
𝑣𝑖(𝑔|𝑋)// Ties are broken lexicographically.
6: 𝑋=𝑋∪ {𝑥⌈𝑗/𝑛⌉}
7: 𝑆=𝑆\ {𝑥⌈𝑗/𝑛⌉}
8: else
9: 𝑔=top(≻ℓ, 𝑆 )
10: 𝑆=𝑆\ {𝑔}
11: return 𝑥1≻′
𝑖𝑥2≻′
𝑖... ≻′
𝑖𝑥𝑘≻′
𝑖... // Arbitrarily complete ≻′
𝑖with goods in 𝑀\𝑋.
Proving the next lemma closely follows the proof of Theorem 3.7 but without the need of an analog
of Lemma 3.5, as we get this for free from the way the greedy preference profile ≻′
𝑖is constructed.
Lemma 4.1. Assume that agent 𝑖has a submodular valuation function 𝑣𝑖. If ≻′
𝑖is the ranking returned by
Algorithm 4when given 𝑁,𝑀,≻−𝑖,𝑣𝑖, then the allocation (𝐴′
1, 𝐴′
2, . . . ,𝐴′
𝑛)returned by Round-Robin(≻′
𝑖,≻−𝑖)
is such that for every 𝑗∈𝑁, with 𝐴′
𝑗≠∅, there exists a good 𝑔∈𝐴′
𝑗, so that 𝑣𝑖(𝐴′
𝑖) ≥ 1
2·𝑣𝑖(𝐴′
𝑗\ {𝑔}).
Proof. First, it is straightforward to see that 𝐴′
𝑖=𝑋, as computed in Algorithm