EFX Exists for Four Agents with Three Types of Valuations

Abstract

In this paper, we address the problem of determining an envy-free allocation of indivisible goods among multiple agents. EFX, which stands for envy-free up to any good, is a well-studied problem that has been shown to exist for specific scenarios, such as when there are only three agents with MMS valuations, as demonstrated by Chaudhury et al(2020), and for any number of agents when there are only two types of valuations as shown by Mahara(2020). Our contribution is to extend these results by showing that EFX exists for four agents with three distinct valuations. We further generalize this to show the existance of EFX allocations for n agents when n-2 of them have identical valuations.

Full text

arXiv:2301.10632v1 [cs.GT] 25 Jan 2023
EFX Exists for Four Agents with Three Types of Valuations
Pratik Ghosal∗Vishwa Prakash H.V.†Prajakta Nimbhorkar‡Nithin Varma§
Abstract
In this paper, we address the problem of determining an envy-free allocation of indivisible goods
among multiple agents. EFX, which stands for envy-free up to any good, is a well-studied problem
that has been shown to exist for specific scenarios, such as when there are only three agents with MMS
valuations, as demonstrated by Chaudhury et al. (2020), and for any number of agents when there are
only two types of valuations as shown by Mahara (2020). Our contribution is to extend these results
by showing that EFX exists for four agents with three distinct valuations. We further generalize this to
show the existance of EFX allocations for nagents when n−2 of them have identical valuations.
1 Introduction
Fair division of indivisible goods is a fundamental problem in the field of multiagent systems. The problem
is to allocate a set G={g1,...gm}of mgoods to a group A={a1,...an}of nagents such that each agent
thinks of the allocation as being fair. One of the most well-studied fairness notions is that of envy-freeness.
To quantify this notion, we model each agent ai,i∈[n], as having a valuation function vi: 2G→R≥0on
bundles of goods. An allocation (X1, X2,...Xn)1is said to be envy-free (EF) if all agents value their own
bundle at least as much as that of any other agent, i.e., vi(Xi)≥vi(Xj) for all i, j ∈[n]. It is well-known
that EF allocations may not exist in general and various relaxations of such allocations have been proposed.
Budish (2011) proposed the concept of envy-freeness up to one good (EF1), where the goal is to find an
allocation such that, for each agent ai, there exists some good gin each bundle Xjsuch that aivalues Xi
at least as much as Xj\ {g}. It is known that EF1 allocations always exist and can be found in polynomial
time Lipton et al. (2004). In between the notions of EF and EF1 allocations, lie envy-freeness up to any
good (EFX), which was introduced by Caragiannis et al. (2019b). Given an allocation, an agent aistrongly
envies another agent ajif there exists g∈Xjsuch that aivalues Xj\ {g}over their own bundle Xi. An
allocation is EFX if no agent strongly envies another agent. In other words, each agent aivalues Xiat least
as much as Xj\ {g}for any good gpresent in any Xj.
Contrary to both EF and EF1, the question of whether EFX allocations always exist or not is far from set-
tled and is one of the important questions in contemporary research on fair allocation. Plaut and Roughgarden
(2020) show that when all agents have the same valuation function on bundles, then EFX always exists. They
also showed that when there are only two agents, EFX always exists. Mahara (2020,2021) improved upon
this result and showed the existence of EFX for multiple agents when there are only two valuation functions.
In a recent breakthrough, Chaudhury et al. (2020) showed that EFX always exists for 3 agents when the
valuation functions of agents are additive.2In addition to improving the state of the art, their contributions
also include several new technical ideas to reason about EFX allocations.
In this work, we show the following improvement over the state of the art.
∗Chennai Mathematical Institute. Email: pratik@cmi.ac.in.
†Chennai Mathematical Institute. Email: vishwa@cmi.ac.in.
‡Chennai Mathematical Institute. Email: prajakta@cmi.ac.in.
§Chennai Mathematical Institute. Email: nithinvarma@cmi.ac.in.
1Our convention is to allocate the bundle Xito agent aifor all i∈[n]. Additionally, we only consider allocations that are
complete, i.e., where Si∈[n]Xi=G.
2A valuation v: 2G→R≥0is additive if, for each bundle S⊆ G of goods, v(S) = Pg∈Sv({g}). The result of Akrami et al.
(2022) holds for slightly more general valuation functions, which they call MMS-feasible valuations (see Definition 2.3).
1

Theorem 1.1. Consider a set of nagents with additive valuations where at least n−2agents have identical
valuations. Then, for any set of goods, an EFX allocation always exists. Moreover, this holds even when all
the agents have more general MMS-feasible valuations.
When n= 4, the above theorem implies the following corollary
Corollary 1.2. Consider a set of 4agents with at most 3distinct additive valuations Then, for any set of
goods, an EFX allocation always exists. Moreover, this holds even when all the agents have more general
MMS-feasible valuations.
Theorem 1.1 is the first result for the existence of EFX for an arbitrary number of agents with more than
two distinct valuations and is, in this sense, an improvement over the work of Mahara (2020,2021).
1.1 Overview of Our Techniques
Several of the ideas that we use in our proofs of Theorem 1.1 are attributed to the work of Akrami et al. who
give a simplified proof for the existence of EFX allocations for three agents. Our proof of Theorem 1.1 begins
by considering, what we refer to as an almost feasible EFX allocation. An almost EFX feasible allocation
ensures that the first n−1 bundles are EFX feasible for the first n−2 agents with identical valuations
and that the last bundle is EFX feasible for one of the remaining two agents. Our procedure modifies such
an allocation carefully to get to an EFX allocation, in which case we are done, or to another almost EFX
feasible allocation. The termination of our procedure is ensured by the fact that the resulting almost EFX
feasible allocation is strictly better than the previous one in a concrete sense. The novel challenge arising
in our case is the fact that maintaining the above mentioned invariant and arguing about the increase in
potential is more involved due to a higher number of dependencies caused by a larger number of agents.
1.2 Related Work
The notion of envy-free allocations was introduced by Gamow and Stern and Foley. For indivisible goods,
Lipton et al. and Budish consider a relaxed notion of envy-freeness known as envy-freeness up to one good
(EF1). The notion of envy-freeness up to any good (EFX) was introduced by Caragiannis et al.. The
existence of EFX allocations has been shown in various restricted settings like 2 agents with arbitrary
valuations and any number of agents with identical valuations Plaut and Roughgarden (2020), for additive
valuations with 3 agents Chaudhury et al. (2020), at most two valuations for an arbitrary number of agents
Mahara (2020,2021), for the case when each value of each agent can take one of the two possible values
Amanatidis et al. (2021), etc. EFX allocations for the case when some goods can be left unallocated have
been considered in several papers Brams et al. (2022); Cole et al. (2013); Caragiannis et al. (2019a) etc.
Caragiannis et al. (2019a) show that discarding some items can achieve at least half of the maximum Nash
Welfare whereas Chaudhury et al. show that an EFX allocation always exists for nagents with arbitrary
valuations with at most n−1 unallocated items, Berger et al. improve this to EFX for 4 agents with at most
one unallocated item.
2 Preliminaries
Let A={a1, a2,··· , an}be a set of nagents and let G={g1, g2,··· , gm}be a set of mindivisible goods.
An instance of discrete fair division is specified by the tuple hA,G,Vi, where V={v1(·), v2(·),··· , vn(·)}is
such that for i∈[n], the function vi: 2G→R≥0denotes the valuation of agent aion subsets of goods.
Let a∈ A, g ∈G, S, T ⊆ G, v : 2G→R≥0. To simplify notation, we write v(g) to denote v({g}) and use
S\g,S∪gto denote S\ {g},S∪ {g}, respectively. We also write S >aTto denote va(S)> va(T) and
similarly for <a,≥a,≤aand =a. We use mina(S, T ) and maxa(S, T ) to denote arg minY∈{S,T }va(Y) and
arg maxY∈{S,T }va(Y).
We often use the term bundle to denote a subset of goods. An allocation is a tuple X=hX1, X2,...,Xni
of nbundles such that bundle Xiis assigned to agent aifor all i∈[n] and Si∈[n]Xi=G. Given an allocation
2

X=hX1, X2,··· , Xni, we say that agent aienvies another agent ajif vi(Xj)> vi(Xi). As a shorthand, we
sometimes simply say that agent aienvies the bundle Xj.
Definition 2.1 (Strong Envy).Given an allocation X=hX1, X2,··· , Xni, an agent aistrongly envies an
agent ajif vi(Xi)< vi(Xj\g)for some g∈Xj.
An allocation is EFX if there is no strong envy between any pair of agents.
Definition 2.2 (EFX-Feasibility).A bundle S⊆ G is said to be EFX-feasible w.r.t. a disjoint bundle T
according to valuation v, if for all h∈T,v(T\h)< v(S). Given an allocation X=hX1, X2,··· , Xni,
bundle Xiis EFX-feasible for an agent ajif Xiis EFX-feasible w.r.t. all other bundles in Xaccording to
valuation vj.
An allocation X=hX1, X2,··· , Xniis said to be EFX if for all i∈[n], the bundle Xiis EFX-feasible
for agent ai.
Minimally Envied Subset. If agent aiwith bundle Xienvies an agent ajwith bundle Xj, we call a
subset S⊆Xjaminimally envied subset of Xjfor agent aiif both the following conditions hold.
1. vi(Xi)< vi(S)
2. vi(Xi)≥vi(S\h)∀h∈S
Non-Degenerate Instances Chaudhury et al. (2020); Akrami et al. (2022)An instance I=hA,G,Vi
is said to be non-degenerate if and only if no agent values two different bundles equally. That is, ∀ai∈ A
we have vi(S)6=vi(T) for all S6=T, where S, T ⊆ G.Akrami et al. (2022) showed that it suffices to
deal with non-degenerate instances when there are nagents with general valuation functions, i.e., if each
non-degenerate instance has an EFX allocation, each general instance has an EFX allocation.
In the rest of the paper, we only consider non-degenerate instances. This implies that all goods are
positively valued by all agents as value of the empty bundle is assumed to be zero.
Properties of Valuation Functions A valuation vis said to be monotone if S⊆Timplies v(S)≤
vi(T) for all S, T ⊆ G. Monotonicity is a natural restriction on valuation functions and occurs frequently
in real-world instances of fair division. A valuation vis additive if v(S) = Pg∈Sv({g}) for all S⊆ G.
Additive valuation functions are, by definition, also monotone. Akrami et al. (2022) introduced a new class
of valuation functions called MMS-feasible valuations which are natural extensions of additive valuations.
Definition 2.3. A valuation v: 2G→R≥0is MMS-feasible if for every subset of goods S⊆ G and every
partitions A= (A1, A2)and B= (B1, B2)of S, we have
max(v(B1), v(B2)) >min(v(A1), v(A2))
Plaut and Roughgarden Algorithm In 2020,Plaut and Roughgarden (2020) gave an algorithm to
compute an EFX-allocation when all agents have the same valuation v(·), where the only assumption on v(·)
is that it is monotone. Throughout this paper, we refer to this algorithm as the PR algorithm. Let M⊆ G
be a subset of goods and let abe an agent with valuation v. Let X={X1, X2,··· , Xk}be a k-partition of
M. In its most general form, the PR algorithm takes (X, v, k) as input and outputs a (possibly different)
k-partition Y={Y1, Y2,··· , Yk}. We crucially use the following properties Plaut and Roughgarden (2020)
of the output of the PR algorithm.
1. If Yiis allocated to agent athen agent adoes not strongly envy any other bundle in Y.
2. The value of the least valued bundle does not decrease, i.e.,
min(v(Y1), v(Y2),··· , v(Yk)) ≥min(v(X1), v(X2),··· , v (Xk)).
3

3 EFX for four agents with three valuations
In this section, we show that EFX allocation always exists for nagents when n−2 of the agents have identical
valuations thus prove Theorem 1.1.
Consider a set of nagents A={a1, a2,··· , an−2b1, c1}, a set of mgoods G={g1, g2,··· , gm}and a set
of three valuation functions V={va, vb, vc}such that agents a1, a2,··· , an−2have valuation vaand agents
b1and c1have valuations vband vcrespectively. The valuations vaand vbare assumed to be monotone and
vcis assumed to be MMS-feasible.
Definition 3.1. We say that an allocation X=hX1, X2,··· , Xniis almost EFX-feasible if it satisfies the
following conditions:
1. The first n−1bundles X1, X2,··· , Xn−1are EFX-feasible for agents a1, a2,··· , an−2.
2. Xnis EFX-feasible for either agent b1or agent c1.
We define a potential function φwhich assigns a real value for each allocation X=hX1, X2,··· , Xnias
follows:
φ(X) = min{va(X1), va(X2),··· , va(Xn−1)}.
To prove Theorem 1.1, we first show that almost EFX-feasible allocations always exist. Then we show
that, if an allocation Xis almost EFX-feasible, then either Xis an EFX allocation or there exists another
almost EFX-feasible allocation X′with a strictly higher potential value, i.e., φ(X′)> φ(X). Since φ(X)
cannot grow arbitrarily as φ(X)< va(G), there must exist an almost EFX-feasible allocation which is also
an EFX allocation.
Proof of Theorem 1.1:For any given instance with nagents such that n−2 agents have identical valuations,
an almost EFX-feasible allocation always exists. This can be obtained by running the PR algorithm on G
with the valuation vafor all nagents. Lets call this initial allocation X=hX1, X2,··· , Xni. From the
property 1 of the PR algorithm, all the bundles are EFX-feasible for agents a1, a2,··· , an−2. Let agent c1
pick the most valued bundle from Xaccording their valuation vc. Without loss of generality, we can assume
that the bundle picked by agent c1is Xn. Its is clear that Xnis EFX-feasible for c1. Hence Xis almost
EFX-feasible.
If either one among the agents b1or c1has at least one EFX-feasible bundle other than Xn, say Xk, then
we are done. We allocate hXn, Xkito agent c1and b1respectively, and the remaining bundles to agents
a1, a2,··· , an−2arbitrarily. The resulting allocation is EFX.
In the remainder of the proof, we consider the case that Xnis the only EFX-feasible bundle for both b1
and c1.
Let gband gcbe the least valuable good(s) in Xnaccording to agents b1and c1, respectively. Since Xn
is the most valued bundle and also the only EFX-feasible bundle in Xfor agent b1(or c1), even if we give
the maximum valued bundle from {X1, X2,··· , Xn−1}according to vb(vc, respectively) to agent b1(c1,
respectively), they would strongly envy the bundle Xn. That is
Xn\gb>bmax
b(X1, X2,··· , Xn−1) (1)
Xn\gc>cmax
c(X1, X2,··· , Xn) (2)
Without loss of generality, assume
X1<aX2<a···<aXn−1(3)
Now, we consider the cases which arise when we move the least valued good from Xn(according to b1or c1)
and add it to the bundle X1.
4

Case 1: The bundle Xn\gbremains to be the most favorite bundle for agent b1or the bundle Xn\gc
remains to be the most favorite bundle for agent c1. That is,
Xn\gb>bX1∪gb,or
Xn\gc>cX1∪gc
Here we assume that Xn\gb>bX1∪gb. The procedure is analogous if we consider Xn\gc>cX1∪gc
as we are only using the monotonicity of the valuation functions for Case 1. The new allocation is
X′=hX1∪gb, X2,··· , Xn\gbi. Combining Xn\gb>bX1∪gbwith (1), we get that the bundle
Xn\gbis the most valuable according to vband hence EFX-feasible for agent b1in the new allocation.
Case 1.1: X1∪gb<aX2.
Combining X1∪gb>aX1and (3), we can see that
φ(X′) = va(X1∪gb)> va(X1) = φ(X).
Thus there is an increase in the potential. For agents a1, a2,··· , an−2, the bundle X1∪gbremains
EFX-feasible as no other bundle has increased in value. Furthermore, For agents a1, a2,··· , an−2,
the bundles X2, X3,··· , Xn−1are EFX-feasible when compared to X1∪gbas they are more valu-
able than X1∪gbaccording to va. They are also EFX-feasible when compared to Xn\gbbecause
they were EFX-feasible against a higher valued bundle Xn. Thus, bundles X1∪gb, X2,··· , Xn−1
are EFX-feasible for agents a1, a2,··· , an−2. Therefore, the new allocation is almost EFX-feasible
and has an increased potential.
Case 1.23:X1∪gb>aX2.
Let (X1∪gb)\Zbe a minimally envied subset with respect to X2under valuation va. That is,
(X1∪gb)\Z >aX2, and
((X1∪gb)\Z)\h <aX2∀h∈(X1∪gb)\Z(4)
Now, let the new allocation be
X′=hX′
1, X′
2,··· , X′
ni
=h(X1∪gb)\Z, X2,··· ,(Xn\ {gb})∪Zi
Since (X1∪gb)\Z >aX2, it holds that φ(X′) = va(X2)> va(X1) = φ(X). Thus the potential
has strictly increased.
From (1), we have Xn\gb>bmaxb(X1, X2,··· , Xn−1). From the Case 1 assumption, we also
have Xn\gb>bX1∪gb. Therefore,
X′
n= (Xn\gb)∪Z >bmax
b(X′
1, X′
2, X′
n−1)
Thus X′
nis EFX-feasible for agent b1.
Next, we show that the bundles X′
1, X′
2,··· , X′
n−1are EFX-feasible among themselves (i.e, not
compared with X′
n) to agents a1, a2,··· , an−2.
3Note that we do not have to consider the case that X1∪gb=aX2since the instance is assumed to be non-degenerate.
5

The bundle X1was EFX-feasible w.r.t. X2,··· , Xn−1in X. Therefore, X′
1>aX1is also EFX-
feasible w.r.t. X′
2,· · · , X ′
n−1.
Bundles X′
2,··· , X′
n−1are EFX-feasible w.r.t. each other as they remain unchanged. From (4)
we know that X′
1\h= ((X1∪gb)\Z)\h <aX2∀h∈((X1∪gb)\Z), and from (3) we
have X2<a··· <aXn−1. Therefore, both X′
2,··· , X′
n−1are EFX-feasible w.r.t. X′
1for agents
a1, a2,··· , an−2. Therefore, the bundles X′
1, X′
2,··· , X′
n−1are EFX-feasible among themselves to
agents a1, a2,··· , an−2.
All that remains is to check the EFX-feasibility of bundles X′
1, X′
2,··· , X′
n−1w.r.t. X′
n. If the
bundles X′
1, X′
2,··· , X′
n−1are EFX-feasible w.r.t. X′
n, then we meet all the conditions of the
invariant and hence X′is almost EFX-feasible. Since φ(X′)> φ(X), we have an almost EFX-
feasible solution with increased potential and we are done.
Now, consider the case that one of the bundles in {X′
1, X′
2,··· , X′
n−1}is not EFX-feasible w.r.t.
X′
n. That is,
∃h∈X′
nsuch that X′
n\h >amin
a(X′
1, X′
2,··· , X′
n−1)
=⇒X′
n>amin
a(X′
1, X′
2,··· , X′
n−1)
=⇒min
a(X′
1, X′
2,··· , X′
n) = min
a(X′
1, X′
2,··· , X′
n−1) = X2>aX1
Now, we apply the PR algorithm on X′under the valuation vato get a new allocation X′′. We
can see that X′′ is almost EFX-feasible by relabeling the bundles appropriately if needed. From
the property 2 of the PR algorithm, we also know that mina(X′′)>amina(X′)>aX1. Therefore,
φ(X′′)> va(X1) = φ(X). Thus we obtain a new almost EFX-feasible allocation with increased
potential.
Case 2: The bundle Xn\gbis not the most favorite bundle of agent b1and bundle Xn\gcis not
the most favorite bundle of agent c1. That is,
Xn\ {gb}<bX1∪ {gb},and
Xn\ {gc}<cX1∪ {gc}
In this case, we run the PR algorithm on hX1∪gb, Xn\gbiunder valuation vbto get bundles Yn−1, Yn.
Now the new allocation is X′=hX′
1, X′
2,··· , X′
n−1, X′
ni=hX2, X3,··· , Yn−1, Yni.
We first show that bundles Yn−1and Ynare EFX-feasible for agents b1and c1respectively.
min
b(Yn−1, Yn)>bmin
b((X1∪gb),(Xn\gb))
=Xn\ {gb}(Case 2 assumption)
>bmax
b(X2,··· , Xn−1) ( (1))
Therefore, the bundles Yn−1and Ynare both EFX-feasible for agent b1.
We let agent c1choose their favorite bundle among Yn−1and Yn.w.l.o.g let Yn>cYn−1. From the
6

maximin property of vc, we know the following:
Yn= max
c(Yn−1, Yn) (∵Yn>cYn−1)
≥cmin
c(X1∪ {gc}, Xn\ {gc}) (vcis MMS-feasible)
=Xn\ {gc}(Case 2 assumption)
>cmax
c(X2,··· , Xn−1) (From (2))
Therefore, the bundle Ynis EFX-feasible for agent c1.
Now, recall that the current allocation is X′=hX2, X3,··· , Yn−1, Yni. Depending on the envy from agent
a1, we have the following three cases:
Case 2.1: Agent a1does not strongly envy Yn−1or Yn. Since X2<a··· <aXn−1, agents
a2, a3,··· , an−2also does not strongly envy Yn−1or Yn. Thus, X′is an EFX allocation.
Case 2.2: Agent a1strongly envies both Yn−1and Yn. Then,
Yn>aX2
Yn−1>aX2
X3>aX2From (3)
Therefore, mina(X′) = X2>aX1=φ(X). That is, the minimum has strictly increased. Now we run
the PR algorithm on X′with the valuation vato get an almost EFX-feasible allocation X′′ with a
potential value φ(X′′)> φ(X).
Case 2.3: Agent a1strongly envies Yn−1but not Yn. The other case is similar.4
Let Y′
n−1⊆Yn−1be such that Y′
n−1>aX2but Y′
n−1\h <aX2∀h∈Y′
n−1.
Now consider the new allocation X′′ =hX′′
1,··· , X′′
n−1, X′′
ni=hX2,··· , Y ′
n−1, Yn∪(Yn−1\Y′
n−1)i.
Previously, Ynwas EFX-feasible for agent c1. Now, the value of this bundle has increased and values
of other bundles have not increased. Therefore, the new bundle X′′
nis EFX-feasible for agent c1.
The potential of the new allocation X′′ is φ(X′′ ) = mina(X′′
1, X′′
2,··· , Y ′
n−1) = X2>aX1=φ(X).
That is, the potential value has increased. Now, if the bundles X′′
1, X′′
2,··· , X′′
n−1are EFX-feasible for
agents a1, a2,··· , an−2, we are done.
We know that bundles X′′
1,··· , X′′
n−2are EFX-feasible among themselves for agents a1, a2,··· , an−2.
By the construction of Y′
n−1, it is clear that X′′
1, X′′
2,··· , X′′
n−1=Y′
n−1are EFX-feasible among
themselves for agents a1, a2,··· , an−2. Now, if X′′
1, X′′
2,··· , X′′
n−1are EFX-feasible with respect to
X′′
n, then all the invariant constraints are met and X′′ is a new almost EFX-feasible allocation with a
higher potential value. Otherwise, if one of X′′
1, X′′
2,··· , X′′
n−1is not EFX-feasible w.r.t. X′′
naccording
to valuation va, then we have:
∃h∈X′′
nsuch that X′′
n\h >amin
a(X′′
1, X′′
2,··· , X′′
n−1)
=X2
>amina(X1, X2,··· , Xn)
4If agent a1strongly envies Yn, then give Ynto agent b1and Yn−1to agent c1. We know both Ynand Yn−1are EFX-feasible
for agent b1. Thus we meet the invariant by making X′′
nEFX-feasible for agent b1instead of agent c1.
7

That is, the overall minimum has increased. Now, we run the PR algorithm on X′′ with the valuation
vato get a new allocation Z. Let agent c1pick their favorite bundle. From the property of the PR
algorithm, we know that φ(Z)> φ(X). Thus, we have a new almost EFX-feasible allocation with
higher potential. This concludes the proof.

References
Hannaneh Akrami, Bhaskar Ray Chaudhury, Jugal Garg, Kurt Mehlhorn, and Ruta Mehta. 2022. EFX
Allocations: Simplifications and Improvements. https://doi.org/10.48550/ARXIV.2205.07638
Georgios Amanatidis, Georgios Birmpas, Aris Filos-Ratsikas, Alexandros Hollender, and Alexandros A.
Voudouris. 2021. Maximum Nash welfare and other stories about EFX. Theor. Comput. Sci. 863 (2021),
69–85. https://doi.org/10.1016/j.tcs.2021.02.020
Ben Berger, Avi Cohen, Michal Feldman, and Amos Fiat. 2022. Almost Full EFX Exists for Four Agents.
In Thirty-Sixth AAAI Conference on Artificial Intelligence, AAAI 2022, Thirty-Fourth Conference on
Innovative Applications of Artificial Intelligence, IAAI 2022, The Twelveth Symposium on Educational
Advances in Artificial Intelligence, EAAI 2022 Virtual Event, February 22 - March 1, 2022. AAAI Press,
Virtual Event, 4826–4833. https://ojs.aaai.org/index.php/AAAI/article/view/20410
Steven J. Brams, D. Marc Kilgour, and Christian Klamler. 2022. Two-Person Fair Divi-
sion of Indivisible Items when Envy-Freeness is Impossible. Oper. Res. Forum 3, 2 (2022).
https://doi.org/10.1007/s43069-021-00115-7
Eric Budish. 2011. The Combinatorial Assignment Problem: Approximate Competitive Equi-
librium from Equal Incomes. Journal of Political Economy 119, 6 (2011), 1061–1103.
https://doi.org/10.1086/664613 arXiv:https://doi.org/10.1086/664613
Ioannis Caragiannis, Nick Gravin, and Xin Huang. 2019a. Envy-Freeness Up to Any Item with High Nash
Welfare: The Virtue of Donating Items. In Proceedings of the 2019 ACM Conference on Economics
and Computation, EC 2019, Phoenix, AZ, USA, June 24-28, 2019, Anna Karlin, Nicole Immorlica, and
Ramesh Johari (Eds.). ACM, 527–545. https://doi.org/10.1145/3328526.3329574
Ioannis Caragiannis, David Kurokawa, Herv´e Moulin, Ariel D. Procaccia, Nisarg Shah, and Junxing Wang.
2019b. The Unreasonable Fairness of Maximum Nash Welfare. ACM Trans. Econ. Comput. 7, 3, Article
12 (sep 2019), 32 pages. https://doi.org/10.1145/3355902
Bhaskar Ray Chaudhury, Jugal Garg, and Kurt Mehlhorn. 2020. EFX Exists for Three
Agents. In Proceedings of the 21st ACM Conference on Economics and Computation (Virtual
Event, Hungary) (EC ’20). Association for Computing Machinery, New York, NY, USA, 1–19.
https://doi.org/10.1145/3391403.3399511
Bhaskar Ray Chaudhury, Telikepalli Kavitha, Kurt Mehlhorn, and Alkmini Sgouritsa. 2021. A
Little Charity Guarantees Almost Envy-Freeness. SIAM J. Comput. 50, 4 (2021), 1336–1358.
https://doi.org/10.1137/20M1359134
Richard Cole, Vasilis Gkatzelis, and Gagan Goel. 2013. Mechanism design for fair division: allocating divisible
items without payments. In Proceedings of the fourteenth ACM Conference on Electronic Commerce, EC
2013, Philadelphia, PA, USA, June 16-20, 2013, Michael J. Kearns, R. Preston McAfee, and ´
Eva Tardos
(Eds.). ACM, 251–268. https://doi.org/10.1145/2492002.2482582
D Foley. 1967. Resource allocation and the public sector. Yale Economic Essays 7 (1967), 45–98.
8

G Gamow and M Stern. 1958. Puzzle-math, edn. Viking Adult, URL http://www. worldcat.
org/isbn/0670583359 (1958).
R. J. Lipton, E. Markakis, E. Mossel, and A. Saberi. 2004. On Approximately Fair Allocations
of Indivisible Goods. In Proceedings of the 5th ACM Conference on Electronic Commerce (New
York, NY, USA) (EC ’04). Association for Computing Machinery, New York, NY, USA, 125–131.
https://doi.org/10.1145/988772.988792
Ryoga Mahara. 2020. Existence of EFX for Two Additive Valuations. CoRR abs/2008.08798 (2020).
arXiv:2008.08798 https://arxiv.org/abs/2008.08798
Ryoga Mahara. 2021. Extension of Additive Valuations to General Valuations on the Exis-
tence of EFX. In 29th Annual European Symposium on Algorithms, ESA 2021, September 6-
8, 2021, Lisbon, Portugal (Virtual Conference) (LIPIcs, Vol. 204), Petra Mutzel, Rasmus Pagh,
and Grzegorz Herman (Eds.). Schloss Dagstuhl - Leibniz-Zentrum f¨ur Informatik, 66:1–66:15.
https://doi.org/10.4230/LIPIcs.ESA.2021.66
Benjamin Plaut and Tim Roughgarden. 2020. Almost Envy-Freeness with General Valuations. SIAM
Journal on Discrete Mathematics 34, 2 (2020), 1039–1068. https://doi.org/10.1137/19M124397X
arXiv:https://doi.org/10.1137/19M124397X
9