# Not obviously manipulable allotment rules††thanks: We thank Alejandro Neme
for his helpful comments. We acknowledge financial support from UNSL through
grants 032016, 030120, and 030320, from Consejo Nacional de Investigaciones
Científicas y Técnicas (CONICET) through grant PIP 112-200801-00655, and from
Agencia Nacional de Promoción Científica y Tecnológica through grant PICT
2017-2355.

R. Pablo Arribillaga Instituto de Matemática Aplicada San Luis (UNSL-CONICET)
and Departamento de Matemática, Universidad Nacional de San Luis, San Luis,
Argentina. Emails<EMAIL_ADDRESS>(R. P. Arribillaga) and
<EMAIL_ADDRESS>(A. G. Bonifacio).  Agustín G. Bonifacio22footnotemark:
2

###### Abstract

In the problem of allocating a single non-disposable commodity among agents
whose preferences are single-peaked, we study a weakening of strategy-
proofness called not obvious manipulability (NOM). If agents are cognitively
limited, then NOM is sufficient to describe their strategic behavior. We
characterize a large family of own-peak-only rules that satisfy efficiency,
NOM, and a minimal fairness condition. We call these rules "simple". In
economies with excess demand, simple rules fully satiate agents whose peak
amount is less than or equal to equal division and assign, to each remaining
agent, an amount between equal division and his peak. In economies with excess
supply, simple rules are defined symmetrically. We also show that the single-
plateaued domain is maximal for the characterizing properties of simple rules.
Therefore, even though replacing strategy-proofness with NOM greatly expands
the family of admissible rules, the maximal domain of preferences involved
remains basically unaltered.

_JEL classification:_ D71, D72.

_Keywords:_ obvious manipulations, allotment rules, maximal domain, single-
peaked preferences, single-plateaued preferences.

##  1 Introduction

Consider the problem of allocating a single non-disposable commodity among a
group of agents with single-peaked preferences: up to some critical level,
called the peak, an increase in an agent’s consumption raises his welfare;
beyond that level, the opposite holds. In this context, an allotment rule is a
systematic procedure that allows agents to select an allotment, among many,
according to their preferences. An allotment rule is strategy-proof if
misreporting preferences is never better than truth-telling. In this paper, we
study a weakening of strategy-proofness called not obvious manipulability.

Criteria of incentive compatibility, efficiency, and fairness are considered
the three perennial goals of mechanism design. When those three criteria are
materialized in the properties of strategy-proofness, efficiency, and
symmetry, they single out the “uniform” rule (Sprumont, 1991; Ching, 1994);
whereas relaxing symmetry by permitting an asymmetric treatment of agents
expands the options to the versatile family of “sequential” allotment rules
(Barberà et al., 1997). Insisting on efficiency and on a minimal fairness
condition, and in line with a current strand in the literature, we investigate
what happens when strategy-proofness is weakened. The idea underlying this
approach is that even though manipulations are pervasive, agents may not
realize they can manipulate a rule because they lack information about others’
behavior or they are cognitively limited. A misreport that in a specific
situation (a specific preference profile of the other agents) is a profitable
manipulation may actually be worse than truth-telling in another situation.
Therefore, if we consider an agent with limited information on other agents’
preferences, it may be very unclear whether such manipulation will be
profitable in practice. Assuming that the agent knows all possible outcomes
under any misreport and under truth-telling, there are several different ways
to determine when a misreport is recognized or identified as a profitable
manipulation. Troyan and Morrill (2020), in the context of two-sided matching,
propose a simple and tractable way to do this by considering best/worst-case
scenarios to formalize the notion of obvious manipulation. A manipulation is
obvious if it either makes the agent better off than truth-telling in the
worst case or makes the agent better off than truth-telling in the best case.
An allotment rule is not obviously manipulable (NOM) if it has no obvious
manipulation.

If agents are cognitively limited, then NOM is sufficient to describe their
strategic behavior. Therefore, the question arises to what extent NOM rules
enrich the landscape of strategy-proof rules. We will focus on own-peak-only
rules. This means that the sole information collected by these rules from an
agent’s preference to determine his allotment is his peak amount. Because of
their simplicity, own-peak-only rules are important rules in their own right
and are both useful in practice and extensively studied in the literature.
Furthermore, the own-peak-only property follows from efficiency and strategy-
proofness (see Sprumont, 1991; Ching, 1994). However, since we do not impose
strategy-proofness, we explicitly invoke this property.

Our goal is to introduce and characterize a large family of own-peak-only
rules, which we call "simple". Their definition is as follows. In economies
with excess demand, simple rules fully satiate agents whose peak amount is
less than or equal to equal division and assign to each remaining agent an
amount between equal division and his peak. Symmetrically, in economies with
excess supply, simple rules fully satiate agents whose peak is greater than or
equal to equal division and assign to each remaining agent an amount between
his peak and equal division. We believe that the full family of simple rules
has been dormant in the literature because many of the rules (and families of
rules) identified in the literature belong to this large family, even though
their definition needs more specifications and details. This is the case, for
example, of the family of sequential rules in Barberà et al. (1997) that are
defined throughout an intricated process needed to cope with strategy-
proofness.

In our main result, Theorem 2, we show that an allotment rule is simple if and
only if it is own-peak-only, efficient, satisfies NOM, and meets the equal
division guarantee. This last property is a minimal fairness requirement that
states that whenever an agent demands equal division, the rule must guarantee
him that amount. If we narrow the picture to symmetric rules, i.e., rules that
assign indifferent allotments to agents with the same preferences, then we
also obtain a characterization of a big subfamily of simple rules that
satisfies all five properties (Corollary 1).111Note that although symmetry and
the equal division guarantee can both be considered as fairness requirements,
they are logically independent properties. As we previously said, in contrast,
when we replace NOM with strategy-proofness the only efficient and symmetric
rule that remains is the uniform rule (Ching, 1994). We also provide three
variants of our main characterization: (i) adapting the result to economies
with individual endowments (Theorem 3), (ii) invoking a peak monotonicity
property that also encompasses symmetry and the equal division guarantee
(Theorem 4), and (iii) getting rid of efficiency (Theorem 5).

Next, we analyze the maximality of the domain of preferences (including the
domain of single-peaked preferences) for which a rule satisfying own-peak-
onliness, efficiency, the equal division guarantee, and NOM exists. For the
properties of efficiency, strategy-proofness, and symmetry, the single-
plateaued domain is maximal (Ching and Serizawa, 1998; Massó and Neme, 2001).
In Theorem 6, we show that the single-plateaued domain is maximal for our
properties as well. Therefore, even though replacing strategy-proofness with
NOM greatly expands the family of admissible rules, the maximal domain of
preferences involved remains basically unaltered.

To the best of our knowledge, our paper is the first one that applies Troyan
and Morrill (2020) notion of obvious manipulations to the allocation of a non-
disposable commodity among agents with single-peaked preferences. In the
context of voting, Aziz and Lam (2021) and Arribillaga and Bonifacio (2022)
study non-peaks-only and peaks-only rules, respectively. Other recent papers
that study obvious manipulations, in other situations, are Ortega and Segal-
Halevi (2022), Psomas and Verma (2022), and Arribillaga and Risma (2023).

The rest of the paper is organized as follows. The model and the concept of
obvious manipulations are introduced in Section 2. In Section 3, we present
the main characterization of simple rules. Section 4 analyses some related
characterizations. The maximal domain result is presented in Section 5. To
conclude, some final remarks are gathered in Section 6.

##  2 Preliminaries

###  2.1 Model

A social endowment $\Omega\in\mathbb{R}_{++}$ is an amount of a perfectly
divisible commodity to be distributed among a set of agents
$N=\\{1,2,\ldots,n\\}$. Each $i\in N$ is equipped with a continuous preference
relation $R_{i}$ defined over $\mathbb{R}_{+}\cup\\{\infty\\}$. Call $P_{i}$
and $I_{i}$ to the strict preference and indifference relations associated
with $R_{i},$ respectively. Denote by $\mathcal{U}$ the domain of all such
preferences. Given $R_{i}\in\mathcal{U}$, let
$p(R_{i})=\\{x\in\mathbb{R}_{+}\cup\\{\infty\\}:xR_{i}y\text{ for each
}y\in\mathbb{R}_{+}\cup\\{\infty\\}\\}$ be the set of preferred consumptions
according to $R_{i}$, called the peak of $R_{i}$. When $p(R_{i})$ is a
singleton, we slightly abuse notation and use $p(R_{i})$ to denote its single
element. Agents $i$’s preference $R_{i}\in\mathcal{U}$ is single-peaked if
$p(R_{i})$ is a singleton and, for each pair
$\\{x_{i},x_{i}^{\prime}\\}\subseteq\mathbb{R}_{+}$, we have
$x_{i}P_{i}x_{i}^{\prime}$ as long as either $x_{i}^{\prime}<x_{i}\leq
p(R_{i})$ or $p(R_{i})\leq x_{i}<x_{i}^{\prime}$ holds. Denote by
$\mathcal{SP}$ the domain of all such preferences.

A (generic) domain of preferences $\mathcal{D}$ is a subset of $\mathcal{U}.$
Given a domain of preferences $\mathcal{D}\subseteq\mathcal{U}$, an economy in
$\mathcal{D}$ consists of a profile of preferences $R=(R_{j})_{j\in
N}\in\mathcal{D}^{N}$ and a social endowment $\Omega\in\mathbb{R}_{++}$ and is
denoted by $(R,\Omega)$. Let $\mathcal{E}_{\mathcal{D}}$ be the domain of all
such economies. Given a social endowment $\Omega\in\mathbb{R}_{++}$, the set
of (feasible) allotments of $\Omega$ is
$X(\Omega)=\\{x\in\mathbb{R}^{N}_{+}:\sum_{j\in N}x_{j}=\Omega\\}$. An
(allotment) rule on $\mathcal{E}_{\mathcal{D}}$ is a function
$\varphi:\mathcal{E}_{\mathcal{D}}\longrightarrow\mathbb{R}^{N}_{+}$ such that
$\varphi(R,\Omega)\in X(\Omega)$ for each
$(R,\Omega)\in\mathcal{E}_{\mathcal{D}}.$

Given a rule $\varphi$ defined on a generic domain
$\mathcal{E}_{\mathcal{D}}$, some desirable properties we consider are listed
next.

Efficiency: For each $(R,\Omega)\in\mathcal{E}_{\mathcal{D}}$, there is no
$x\in X(\Omega)$ such that $x_{i}R_{i}\varphi_{i}(R,\Omega)$ for each $i\in N$
and $x_{i}P_{i}\varphi_{i}(R,\Omega)$ for some $i\in N.$

Efficiency is the usual Pareto optimality criterion. Under this condition, for
each economy, the allocation selected by the rule should be such that there is
no other allocation that all agents find at least as desirable and at least
one agent (strictly) prefers.

Next, we introduce a useful property that is equivalent to efficiency on
$\mathcal{E}_{\mathcal{SP}}.$

Same-sidedness: For each $(R,\Omega)\in\mathcal{E}_{\mathcal{SP}}$,

  1. (i)

$\sum_{j\in N}p(R_{j})\geq\Omega$ implies $\varphi_{i}(R)\leq p(R_{i})$ for
each $i\in N,$ and

  2. (ii)

$\sum_{j\in N}p(R_{j})\leq\Omega$ implies $\varphi_{i}(R)\geq p(R_{i})$ for
each $i\in N$.

###### Remark 1

Let $\varphi$ be a rule defined on $\mathcal{E}_{\mathcal{SP}}$. Then,
$\varphi$ is efficient if and only if it is same-sided.

A rule is strategy-proof if, for each agent,truth-telling is always optimal,
regardless of the preferences declared by the other agents. To define it
formally, let $R_{i},R_{i}^{\prime}\in\mathcal{D}$, and
$\Omega\in\mathbb{R}_{++}.$ Preference $R_{i}^{\prime}$ is a manipulation of
$\boldsymbol{\varphi$at$(R_{i},\Omega)}$ if there is
$R_{-i}\in\mathcal{D}^{n-1}$ such that
$\varphi_{i}(R_{i}^{\prime},R_{-i},\Omega)P_{i}\varphi_{i}(R_{i},R_{-i},\Omega).$

Strategy-proofness: For each $i\in N$ and each
$(R_{i},\Omega)\in\mathcal{D}\times\mathbb{R}_{++},$ there is no manipulation
of $\varphi$ at $(R_{i},\Omega)$.

The following is an informational simplicity property stating that if an agent
unilaterally changes his preference for another one with the same peak, then
his allotment remains unchanged.222This property is weaker than the “peak-
only” property, that has been imposed in a number of axiomatic studies. See
Section 6 for more details.

Own-peak-onliness: For each $(R,\Omega)\in\mathcal{E}_{\mathcal{D}},$ each
$i\in N,$ and each $R_{i}^{\prime}\in\mathcal{D}$ such that
$p(R_{i}^{\prime})=p(R_{i}),$ we have
$\varphi_{i}(R,\Omega)=\varphi_{i}(R_{i}^{\prime},R_{-i},\Omega).$

Analyzing the uniform rule, Sprumont (1991) derives the own-peak-only property
from efficiency and strategy-proofness (see also Ching, 1994). Since we do not
impose strategy-proofness, we explicitly invoke own-peak-onliness.

Next, we present a minimal fairness condition. It states that whenever an
agent has equal division as his preference’s peak, the rule must assign it to
him.

Equal division guarantee: For each $(R,\Omega)\in\mathcal{D}$ and each $i\in
N$ such that $\frac{\Omega}{n}\in p(R_{i})$, we have
$\varphi_{i}(R)I_{i}\frac{\Omega}{n}$.

Another well-known fairness property states that agents with the same
preferences should be assigned indifferent allocations.

Symmetry: For each $(R,\Omega)\in\mathcal{D}$ and each $\\{i,j\\}\subseteq N$
such that $R_{i}=R_{j}$, $\varphi_{i}(R,\Omega)I_{i}\varphi_{j}(R,\Omega)$.

###  2.2 Obvious manipulations

The notion of obvious manipulation is introduced by Troyan and Morrill (2020).
They try to single out those manipulations that are easily identifiable by the
agents. A manipulation is obvious if the worst possible outcome under the
manipulation is strictly better than the worst possible outcome under truth-
telling.333Troyan and Morrill (2020) also consider that a manipulation is
obvious when the best possible outcome under the manipulation is strictly
better than the best possible outcome under truth-telling. However, under EFF,
the best possible outcome under truth-telling for an agent is always a peak
alternative for that agent in our model. Therefore, no manipulation becomes
obvious by considering best possible outcomes and so we omit this in our
definition.  When the set of alternatives is infinite and preferences allow
indifferences, worst possible outcomes may not be well-defined and a more
general definition is necessary in this case. We say that a manipulation is
obvious if each possible outcome under the manipulation is strictly better
than some possible outcome under truth-telling.

Before we present the formal definition, we introduce some notation. Given a
rule $\varphi$ defined on $\mathcal{E}_{\mathcal{D}}$ and
$(R_{i},\Omega)\in\mathcal{D}\times\mathbb{R}_{++}$, the option set attainable
with $(R_{i},\Omega)$ at $\varphi$ is

$O^{\varphi}(R_{i},\Omega)=\left\\{\varphi_{i}(R_{i},R_{-i})\in[0,\Omega]\text{
such that }R_{-i}\in\mathcal{D}^{n-1}\right\\}.$

###### Definition 1

Let $\varphi$ be a rule defined on $\mathcal{E}_{\mathcal{D}}$ and let
$R_{i},R_{i}^{\prime}\in\mathcal{D}$ and $\Omega\in\mathbb{R}_{++}$ be such
that $R_{i}^{\prime}$ is a manipulation of $\varphi$ at $(R_{i},\Omega)$.
Then, $R_{i}^{\prime}$ is an obvious manipulation if for each $x^{\prime}\in
O^{\varphi}(R_{i}^{\prime},\Omega)$ there is $x\in O^{\varphi}(R_{i},\Omega)$
such that $x^{\prime}P_{i}x.$ The rule $\varphi$ is not obviously manipulable
(NOM) if it does not admit any obvious manipulation.

The next proposition shows that our definition generalizes Troyan and Morrill
(2020)’s definition. In fact, both definitions are equivalent whenever worst
possible outcomes exist. Given $R_{i}$ and $Y\subseteq[0,\Omega]$, denote by
$W(R_{i},Y)$ the element in $Y$ (if any) such that $xP_{i}W(R_{i},Y)$ for each
$x\in Y\setminus\\{W(R_{i},Y)\\}.$

###### Proposition 1

Let $\varphi$ be a rule defined on $\mathcal{E}_{\mathcal{D}}$ and let
$R_{i},R_{i}^{\prime}\in\mathcal{D}$ and $\Omega\in\mathbb{R}_{++}$ be such
that $R_{i}^{\prime}$ is a manipulation of $\varphi$ at $(R_{i},\Omega)$.
Assume that $W(R_{i},O^{\varphi}(R_{i},\Omega))$ and
$W(R_{i},O^{\varphi}(R^{\prime}_{i},\Omega))$ exist. Then, $R^{\prime}_{i}$ is
an obvious manipulation if and only if
$W(R_{i},O^{\varphi}(R_{i}^{\prime},\Omega)P_{i}W(R_{i},O^{\varphi}(R_{i},\Omega)$.

_Proof._ ($\Longrightarrow$) Let $R^{\prime}_{i}$ be an obvious manipulation
of $\varphi$ at $(R_{i},\Omega)$. As
$W(R_{i},O^{\varphi}(R_{i}^{\prime},\Omega))\in
O^{\varphi}(R_{i}^{\prime},\Omega)$, there is $x\in
O^{\varphi}(R_{i},\Omega))$ such that
$W(R_{i},O^{\varphi}(R_{i},\Omega))P_{i}x$. Then, by definition of
$W(R_{i},O^{\varphi}(R_{i},\Omega))$,

$W(R_{i},O^{\varphi}(R_{i}^{\prime},\Omega))P_{i}W(R_{i},O^{\varphi}(R_{i},\Omega))$
(1)

($\Longleftarrow$) Assume that (1) holds. We will see that $R_{i}^{\prime}$ is
an obvious manipulation of $\varphi$ at $(R_{i},\Omega)$. Let $x^{\prime}\in
O^{\varphi}(R_{i}^{\prime},\Omega).$ Then, by definition of
$W(R_{i},O^{\varphi}(R_{i}^{\prime},\Omega))$ and (1),

$x^{\prime}R_{i}W(R_{i},O^{\varphi}(R_{i}^{\prime},\Omega))P_{i}W(R_{i},O^{\varphi}(R_{i},\Omega)).$

As $W(R_{i},O^{\varphi}(R_{i},\Omega))\in O^{\varphi}(R_{i},\Omega)$, the
proof is complete. $\square$

##  3 Simple rules

In subsection 3.1, the formal definition of simple rule and our two main
characterizations are presented. In subsection 3.2, the independence of the
axioms involved in the characterizations is discussed.

###  3.1 Definition and characterizations

Given $(R,\Omega)\in\mathcal{E}_{\mathcal{SP}},$ let $z(R,\Omega)=\sum_{j\in
N}p(R_{j})-\Omega.$ If $z(R,\Omega)\geq 0$ we say that economy $(R,\Omega)$
has excess demand whereas if $z(R,\Omega)<0$ we say that economy $(R,\Omega)$
has excess supply.555We deliberately include the balanced case where
$z(R,\Omega)=0$ in the excess demand case. No confusion will arise. Given
economy $(R,\Omega)\in\mathcal{E}_{\mathcal{SP}},$ agent $i\in N$ is simple if
either $z(R,\Omega)\geq 0$ and $p(R_{i})<\frac{\Omega}{n}$ or $z(R,\Omega)\leq
0$ and $p(R_{i})>\frac{\Omega}{n}$. Let $N^{\star}(R,\Omega)$ denote the set
of simple agents of economy $(R,\Omega)$.

Given $x,y,z\in\mathbb{R}$, we say that $\boldsymbol{x$isbetween$y$and$z}$ if
$y\leq z$ and $x\in[y,z]$ or $z\leq y$ and $x\in[z,y].$

In economies with excess demand, simple rules fully satiate agents whose peak
amount is less than or equal to equal division and assign to each remaining
agent an amount between equal division and his peak. Symmetrically, in
economies with excess supply, simple rules fully satiate agents whose peak is
greater than or equal to equal division and assign to each remaining agent an
amount between his peak and equal division. Formally,

###### Definition 2

An own-peak-only rule $\varphi$ defined on $\mathcal{E}_{\mathcal{SP}}$ is
simple if for each $(R,\Omega)\in\mathcal{E}_{\mathcal{SP}}$,

  1. (i)

$\varphi_{i}(R,\Omega)=p(R_{i})$ if $i\in N^{\star}(R,\Omega),$ and

  2. (ii)

$\varphi_{i}(R,\Omega)$ is between $\frac{\Omega}{n}$ and $p(R_{i})$ for each
$i\in N\setminus N^{\star}(R,\Omega)$.

An algorithmic procedure to construct a simple rule is provided in Appendix A.
A prominent member of the family of simple rules is the uniform rule, first
characterized by Sprumont (1991).

Uniform rule, $\boldsymbol{u}$: For each $(R,\Omega)\in\mathcal{E}^{N}$ and
each $i\in N,$

$u_{i}(R,\Omega)=\left\\{\begin{array}[]{l l
}\min\\{p(R_{i}),\lambda\\}&\text{if }z(R,\Omega)\geq 0\\\
\max\\{p(R_{i}),\lambda\\}&\text{if }z(R,\Omega)<0\\\ \end{array}\right.$

where $\lambda\geq 0$ and solves $\sum_{j\in N}u_{j}(R,\Omega)=\Omega.$

The following result shows that simple rules cannot be obviously manipulated.

###### Theorem 1

A simple rule defined on $\mathcal{E}_{\mathcal{SP}}$ satisfies NOM.

_Proof._ Let $\varphi$ be a simple rule defined on
$\mathcal{E}_{\mathcal{SP}}$, and let $i\in N$ and
$(R_{i},\Omega)\in\mathcal{SP}\times\mathbb{R}_{++}$.

Claim:
$\boldsymbol{O^{\varphi}(R_{i},\Omega)=\begin{cases}\left[\frac{\Omega}{n},p(R_{i})\right]&\text{
if }p(R_{i})>\frac{\Omega}{n}\\\ \left[p(R_{i}),\frac{\Omega}{n}\right]&\text{
if }p(R_{i})\leq\frac{\Omega}{n}\\\ \end{cases}}$

Assume that $\frac{\Omega}{n}<p(R_{i})$ (the other case is symmetric). That
$O^{\varphi}(R_{i},\Omega)\subseteq[\frac{\Omega}{n},p(R_{i})]$ is clear by
the definition of the rule. Now, let $x\in[\frac{\Omega}{n},p(R_{i})]$. We
will prove that $x\in O^{\varphi}(R_{i},\Omega)$. Consider
$R_{-i}\in\mathcal{SP}^{n-1}$ such that
$p(R_{j})=\frac{\Omega}{n}-\frac{x-\frac{\Omega}{n}}{n-1}$ for each $j\in
N\setminus\\{i\\}$. Then, as $\frac{\Omega}{n}\leq x$,
$p(R_{j})\leq\frac{\Omega}{n}\leq x$ for all $j\in N\setminus\\{i\\}$ and
$\sum_{j\in N\setminus\\{i\\}}p(R_{j})+x=\Omega$. By definition of the rule,
$\varphi_{j}(R,\Omega)=p(R_{j})$ for all $j\in N\setminus\\{i\\}$. Then, by
feasibility, $\varphi_{i}(R,\Omega)=x$. Therefore,
$[\frac{\Omega}{n},p(R_{i})]\subseteq O^{\varphi}(R_{i},\Omega)$. This
completes the proof of the Claim.

To see that $\varphi$ satisfies NOM, assume that $R_{i}^{\prime}$ is a
manipulation of $\varphi$ at $(R_{i},\Omega)$. By the Claim,
$\frac{\Omega}{n}\in O^{\varphi}(R_{i}^{\prime})$. Furthermore, by the Claim
and single-peakedness, $xR_{i}\frac{\Omega}{n}$ for each $x\in
O^{\varphi}(R_{i})$. Then, $R_{i}^{\prime}$ is not an obvious manipulation and
so $\varphi$ is NOM. $\square$

Next, we present our first characterization result.

###### Theorem 2

An own-peak-only rule defined on $\mathcal{E}_{\mathcal{SP}}$ satisfies
efficiency, the equal division guarantee, and NOM if and only if it is a
simple rule.

_Proof._ $(\Longleftarrow)$ Let $\varphi$ be a simple rule on
$\mathcal{E}_{\mathcal{SP}}$. Then, $\varphi$ is trivially own-peak-only and
satisfies the equal division guarantee. By definition, $\varphi$ satisfies
same-sidedness. Therefore, by Remark 1, $\varphi$ is efficient. Lastly, by
Theorem 1, $\varphi$ satisfies NOM.

$(\Longrightarrow)$ Let $\varphi$ be an own-peak-only rule defined on
$\mathcal{E}_{\mathcal{SP}}$ that satisfies efficiency, the equal division
guarantee, and NOM. We now prove that $\varphi$ is simple. Let
$(R,\Omega)\in\mathcal{E}_{\mathcal{SP}}$ and assume that $z(R,\Omega)\geq 0$.
Let $i\in N$. By efficiency, $\varphi_{i}(R,\Omega)\leq p(R_{i})$. There are
two cases to consider:

  1. $\boldsymbol{1}.$

$\boldsymbol{p(R_{i})\leq\frac{\Omega}{n}}$. We need to show that
$\varphi_{i}(R,\Omega)\geq p(R_{i})$ also holds. Assume, by way of
contradiction, that $\varphi_{i}(R,\Omega)<p(R_{i}).$ By own-peak-onliness, we
can assume that $R_{i}$ is such that $\frac{\Omega}{n}\ P_{i}\
\varphi_{i}(R,\Omega).$ Let $R_{i}^{\prime}\in\mathcal{SP}$ be such that
$p(R_{i}^{\prime})=\frac{\Omega}{n}$. By the equal division guarantee,
$\varphi_{i}(R_{i}^{\prime},R_{-i},\Omega)=\frac{\Omega}{n}$. Hence,

$\varphi_{i}(R_{i}^{\prime},R_{-i},\Omega)=\frac{\Omega}{n}\ P_{i}\
\varphi_{i}(R,\Omega)$ (2)

and $R_{i}^{\prime}$ is a manipulation of $\varphi$ at $(R_{i},\Omega)$.
Furthermore, by the equal division guarantee,
$O^{\varphi}(R_{i}^{\prime},\Omega)=\left\\{\frac{\Omega}{n}\right\\}$. As
$\varphi_{i}(R,\Omega)\in O^{\varphi}(R_{i},\Omega),$ by (2), $R_{i}^{\prime}$
is an obvious manipulation of $\varphi$ at $(R_{i},\Omega)$, contradicting
that $\varphi$ is NOM. Thus, $\varphi_{i}(R,\Omega)\geq p(R_{i})$ and, since
by hypothesis $\varphi_{i}(R,\Omega)\leq p(R_{i})$, we have
$\varphi_{i}(R,\Omega)=p(R_{i})$.

  2. $\boldsymbol{2}.$

$\boldsymbol{p(R_{i})>\frac{\Omega}{n}}$. We need to show that
$\varphi_{i}(R,\Omega)\geq\frac{\Omega}{n}$. Assume, by way of contradiction,
that $\varphi_{i}(R,\Omega)<\frac{\Omega}{n}.$ By single-peakedness and own-
peak-onliness, $\frac{\Omega}{n}\ P_{i}\ \varphi_{i}(R,\Omega)$ also holds
here, and the proof follows an argument similar to that of the previous case.

We have proved that if $p(R_{i})\leq\frac{\Omega}{n}$,
$\varphi_{i}(R,\Omega)=p(R_{i})$; whereas if $p(R_{i})>\frac{\Omega}{n},$
$\varphi_{i}(R,\Omega)\in[\frac{\Omega}{n},p(R_{i})]$. To complete the proof
that $\varphi$ is a simple rule, a symmetric argument can be used for the case
$z(R,\Omega)<0$. $\square$

Simple rules only meet a minimal fairness requirement: whenever an agent has
equal division as his peak consumption, the rule must assign it to him.
Despite this requirement, simple rules can fail symmetry. If we want to avoid
this adding symmetry to the picture, the following corollary follows.666As we
will see in Subsection 3.2, the equal division guarantee and symmetry are
independent axioms.

###### Corollary 1

An own-peak-only rule defined on $\mathcal{E}_{\mathcal{SP}}$ satisfies
efficiency, the equal division guarantee, symmetry, and NOM if and only if it
is a symmetric and simple rule.

Ching (1994) shows that if we require strategy-proofness instead of NOM in
Corollary 1, the unique rule that prevails is the uniform rule. Therefore,
Corollary 1 evidences how the family of characterizing rules enriches when NOM
is sufficient to describe the strategic behavior of the agents. An interesting
member of this family is the simple rule that applies the constrained equal
distance criterion to non-simple agents. If we use the distance between
assignments and peak amounts as a measure of the sacrifice made by the agents
at an allocation, this criterion tries to equate that sacrifice among agents
as much as possible. It is easy to see that this criterion is incompatible
with the definition of simple rule when all agents are considered. However, it
is still relevant when applied only to non-simple agents.

CED simple rule, $\boldsymbol{\varphi^{CED}}$: For each
$(R,\Omega)\in\mathcal{E}^{N}$ and each $i\in N,$

  1. (i)

if $i\in N^{\star}(R,\Omega)$, $\varphi^{CED}_{i}(R,\Omega)=p(R_{i})$,

  2. (ii)

if $i\in N\setminus N^{\star}(R,\Omega)$,

$\varphi^{CED}_{i}(R,\Omega)=\left\\{\begin{array}[]{l l
}\max\\{\frac{\Omega}{n},p(R_{i})-d\\}&\text{if }z(R,\Omega)\geq 0\\\
\min\\{\frac{\Omega}{n},p(R_{i})+d\\}&\text{if }z(R,\Omega)<0\\\
\end{array}\right.$

where $d\geq 0$ and solves $\sum_{j\in N}\varphi^{CED}_{j}(R,\Omega)=\Omega.$

This rule is a compelling example of a rule that can be relevant in some
situations but is completely overlooked when the strong incentive
compatibility requirement of strategy-proofness is imposed. We can recover
this rule, however, if we only demand the NOM property.

###  3.2 Independence of axioms

Next, we present some examples that prove the independence of the axioms in
Theorem 2 and Corollary 1.

  * •

Let
$\widetilde{\varphi}:\mathcal{E}_{\mathcal{SP}}\longrightarrow\mathbb{R}_{+}^{N}$
be such that, for each $(R,\Omega)\in\mathcal{E}_{\mathcal{SP}}$ and each
$i\in N$, $\varphi_{i}(R,\Omega)=\frac{\Omega}{n}$ . Then
$\widetilde{\varphi}$ satisfies all properties but efficiency.

  * •

Let
$\varphi^{\star}:\mathcal{E}_{\mathcal{SP}}\longrightarrow\mathbb{R}_{+}^{N}$
be such that, for each $(R,\Omega)\in\mathcal{E}_{\mathcal{SP}}$ and each
$i\in N$,

$\varphi^{\star}(R,\Omega)=\left\\{\begin{array}[]{l l
}(p(R_{1}),p(R_{2}),0,\ldots,0))&\text{if }p(R_{1})+p(R_{2})=\Omega\text{ and
}\\\ &p(R_{j})\notin\\{p(R_{1}),p(R_{2})\\}\text{ for each }j\in
N\setminus\\{1,2\\}\\\ u(R,\Omega)&\text{otherwise}\\\ \end{array}\right.$

Observe that this rule does meet the equal division guarantee. It is clear
that $\varphi^{\star}$ is own-peak-only, efficient, and symmetric. Now, we
prove that $\varphi^{\star}$ also satisfies NOM. Let $i\in N$ and
$(R_{i},\Omega)\in\mathcal{SP}\times\mathbb{R}_{+}$. Notice that
$O^{\varphi^{\star}}(R_{i},\Omega)=O^{u}(R_{i},\Omega)\cup\\{0\\}$ if $i\in
N\setminus\\{1,2\\}$ and
$O^{\varphi^{\star}}(R_{i},\Omega)=O^{u}(R_{i},\Omega)$ if $i\in\\{1,2\\}$.
Then, the fact that $\varphi^{\star}$ does not have obvious manipulations for
agents in $N\setminus\\{1,2\\}$ follows a similar argument as in Theorem 1
using the fact that $u$ is a simple rule. Then, $\varphi^{\star}$ satisfies
all properties but the equal division guarantee.

  * •

Let
$\overline{\varphi}:\mathcal{E}_{\mathcal{SP}}\longrightarrow\mathbb{R}_{+}^{N}$
be such that, for each $(R,\Omega)\in\mathcal{E}_{\mathcal{SP}}$,

$\overline{\varphi}(R,\Omega)=\left\\{\begin{array}[]{l l
}(\frac{1}{3}\Omega,\frac{2}{3}\Omega,0,\ldots,0)&\text{if
}p(R_{1})=p(R_{2})=\Omega\text{ and }\\\ &p(R_{j})=0\text{ for each }j\in
N\setminus\\{1,2\\}\\\ u(R,\Omega)&\text{otherwise}\\\ \end{array}\right.$

Observe that $\overline{\varphi}$ is simple. Thus, it satisfies own-peak-
onliness, efficiency, the equal division guarantee, and NOM. However, it does
not satisfy symmetry.

  * •

Let
$\widehat{\varphi}:\mathcal{E}_{\mathcal{SP}}\longrightarrow\mathbb{R}_{+}^{N}$
be such that, for each $(R,\Omega)\in\mathcal{E}_{\mathcal{SP}}$,

    1. (i)

if $z(R,\Omega)\geq 0,\widehat{\varphi}(R,\Omega)=u(R,\Omega)$.

    2. (ii)

if $z(R,\Omega)<0$, let $\widehat{N}=\\{i\in N:p(R_{i})=\min_{j\in
N}\\{p(R_{j})\\}\\}$. Then, for each $i\in N$,

$\widehat{\varphi}_{i}(R,\Omega)=\left\\{\begin{array}[]{l l
}p(R_{i})&\text{if }i\in N\setminus\widehat{N}\\\ \lambda&\text{if
}i\in\widehat{N}\\\ \end{array}\right.$

where $\lambda\geq 0$ and solves $\sum_{j\in
N}\widehat{\varphi}_{j}(R,\Omega)=\Omega$.

Observe that $\widehat{\varphi}$ is not simple because it could be that
$p(R_{i})<\frac{\Omega}{n}<\widehat{\varphi}_{i}(R)$ for some
$i\in\widehat{N}$. It is easy to see that $\widehat{\varphi}$ is own-peak-
only, efficient, and symmetric. If $z(R,\Omega)<0$ and
$p(R_{i})=\frac{\Omega}{n}$, then $i\in N\setminus\widehat{N}$. Therefore,
$\widehat{\varphi}$ satisfies the equal division guarantee. This implies, by
Theorem 2, that $\widehat{\varphi}$ satisfies all properties but NOM.

  * •

Let
$\underline{\varphi}:\mathcal{E}_{\mathcal{SP}}\longrightarrow\mathbb{R}_{+}^{N}$
be such that, for each $(R,\Omega)\in\mathcal{E}_{\mathcal{SP}}$ and each
$i\in N$,

$\underline{\varphi}_{i}(R,\Omega)=\left\\{\begin{array}[]{l l
}u_{i}(R^{0}_{1},R_{-1},\Omega)&\text{if }z(R_{-1},\Omega)\geq 0,\
0P_{1}\frac{\Omega}{n},\text{ and }\\\ &p(R_{1})<p(R_{j})\text{ for each }j\in
N\setminus\\{1\\}\\\ u_{i}(R,\Omega)&\text{otherwise}\\\ \end{array}\right.$

where $R^{0}_{1}\in\mathcal{SP}$ is such that $p(R^{0}_{1})=0$.

Observe that $\underline{\varphi}$ is not simple because it is not own-peak-
only and it could be that $\varphi_{1}(R,\Omega)<p(R_{1})<\frac{\Omega}{n}$.
It is clear that $\underline{\varphi}$ satisfies the equal division guarantee
and symmetry. To see that $\underline{\varphi}$ is efficient, let
$(R,\Omega)\in\mathcal{E}_{\mathcal{SP}}$ be such that $z(R_{-1},\Omega)\geq
0$, $0P_{1}\frac{\Omega}{n}$, and $p(R_{1})<p(R_{j})\text{ for each }j\in
N\setminus\\{1\\}$. Then, by the definition of $u$ and the fact that $u$
satisfies same-sideness, $\underline{\varphi}_{1}(R,\Omega)=0\leq p(R_{1})$
and $\varphi_{i}(R,\Omega)\leq p(R_{i})$ for each $i\in N\setminus\\{1\\}$.
Then, $\underline{\varphi}_{i}(R,\Omega)\leq p(R_{i})$ for each $i\in N$.
Since in any other case $\underline{\varphi}$ operates as $u$ and $u$
satisfies same-sideness, $\underline{\varphi}$ also satisfies this property.
Thus, by Remark 1, $\underline{\varphi}$ is efficient. Now, we prove that
$\underline{\varphi}$ satisfies NOM. For each $i\in N\setminus\\{1\\}$ and
each $(R_{i},\Omega)\in\mathcal{SP}\times\mathbb{R}_{+}$,
$O^{\underline{\varphi}}(R_{i},\Omega)=O^{u}(R_{i},\Omega)$. Then, the fact
that $\underline{\varphi}$ does not have obvious manipulations for agents in
$N\setminus\\{1\\}$ follows the same arguments used in the proof of Theorem 1
and the fact that $u$ is a simple rule. Let us next consider agent $1$. First,
let $(R_{1},\Omega)\in\mathcal{SP}\times\mathbb{R}_{+}$ be such that
$\frac{\Omega}{n}R_{1}0$. Then,
$O^{\underline{\varphi}}(R_{1},\Omega)=O^{u}(R_{1},\Omega)$. Again, the fact
that $\underline{\varphi}$ does not have obvious manipulations for agent $1$
at $(R_{1},\Omega)$ follows the same arguments used in the proof of Theorem 1
together with the facts that $u$ is a simple rule and that
$\frac{\Omega}{n}\in O^{\varphi}(R^{\prime}_{1},\Omega)$ for each
$R^{\prime}_{1}\in\mathcal{SP}$. Second, let
$(R_{1},\Omega)\in\mathcal{SP}\times\mathbb{R}_{+}$ be such that
$0P_{1}\frac{\omega}{n}$. Then, by single-peakedness of $R_{1}$,
$p(R_{1})<\frac{\Omega}{n}.$ By definition of $\underline{\varphi}$,
$O^{\underline{\varphi}}(R_{1},\Omega)=O^{u}(R^{0}_{1},\Omega)=[0,\frac{\Omega}{n}]$.
Using single-peakedness again and the fact that $0P_{1}\frac{\Omega}{n}$, it
follows that $xR_{1}\frac{\Omega}{n}$ for each $x\in
O^{\varphi}(R_{1},\Omega)$. Furthermore, as $\frac{\Omega}{n}\in
O^{\varphi}(R^{\prime}_{1},\Omega)$ for each $R^{\prime}_{1}\in\mathcal{SP}$,
agent $1$ does not have obvious manipulations of $\underline{\varphi}$ at
$(R_{1},\Omega).$ Therefore, $\underline{\varphi}$ is NOM. We conclude that
$\underline{\varphi}$ satisfies all properties but own-peak-onliness.

##  4 Further characterizations involving NOM

Insisting on the NOM property, in this section we provide new
characterizations of rules in which NOM together with other relevant
properties are invoked. These considerations lead to three interesting
variants of Theorem 2. Insisting on efficiency as well, the first one is an
adaptation of Theorem 2 to economies with individual endowments and the second
one involves a monotonicity condition concerning the peaks of the preference
profile. Lastly, the third one gets rid of efficiency.

###  4.1 Endowments guarantee

In some situations, it is more appropriate to assume that instead of a social
endowment $\Omega\in\mathbb{R}_{+}$, there is a profile
$\omega=(\omega_{i})_{i\in N}\in\mathbb{R}_{+}^{N}$ where, for each $i\in N$,
$\omega_{i}$ denotes agent $i$’s individual endowment of the non-disposable
commodity and $\sum_{j\in N}\omega_{j}=\Omega$. In such cases, it is natural
to replace the equal division guarantee with the following property.

Endowments guarantee: For each $(R,\omega)\in\mathcal{E}_{\mathcal{D}}$ and
each $i\in N$ such that $\omega_{i}\in p(R_{i})$, we have
$\varphi_{i}(R)I_{i}\omega_{i}$.

If, for each $i\in N$, we replace $\frac{\Omega}{n}$ with $\omega_{i}$ in the
definition of simple agent, we can define a simple reallocation rule as one
that assigns: (i) to each simple agent, his peak amount and, (ii) to each
remaining agent, an amount between his endowment and his peak. Then, for
economies with individual endowments, the following variant of Theorem 2 is
obtained.

###### Theorem 3

An own-peak-only rule defined on $\mathcal{E}_{\mathcal{SP}}$ satisfies
efficiency, the endowments guarantee, and NOM if and only if it is a simple
reallocation rule.

The analysis of the independence of axioms involved is similar to that of
Theorem 2 (see Subsection 3.2).

###  4.2 Peak monotonicity

The following monotonicity property says that if the peak of one agent is
greater than or equal to the peak of another agent, then the first agent must
be assigned an amount greater than or equal to that of the second agent.

Peak monotonicity: For each $(R,\Omega)\in\mathcal{E}_{\mathcal{SP}}$ and each
$\\{i,j\\}\subseteq N$ with $i\neq j$, $p(R_{i})\leq p(R_{j})$ implies
$\varphi_{i}(R,\Omega)\leq\varphi_{j}(R,\Omega).$

It is clear that peak monotonicity implies symmetry. Furthermore, we next
prove that peak monotonicity together with NOM imply the equal division
guarantee for an own-peak-only rule defined on $\mathcal{E}_{\mathcal{SP}}$.

###### Lemma 1

Any own-peak-only rule defined on $\mathcal{E}_{\mathcal{SP}}$ that satisfies
peak monotonicity and NOM meets the equal division guarantee.

_Proof._ Let $\varphi$ be an own-peak-only rule defined on
$\mathcal{E}_{\mathcal{SP}}$ that satisfies peak monotonicity and NOM. Assume,
on contradiction, that there are $(R,\Omega)\in\mathcal{E}_{\mathcal{SP}}$ and
$i\in N$ such that $p(R_{i})=\frac{\Omega}{n}$ and
$\varphi_{i}(R,\Omega)\neq\frac{\Omega}{n}$. Consider the case
$\varphi_{i}(R,\Omega)<\frac{\Omega}{n}$ (the other one is symmetric). Let
$R_{i}^{\prime}\in\mathcal{SP}$ be such that $p(R^{\prime}_{i})=\infty$. By
peak monotonicity,
$\varphi_{i}(R^{\prime}_{i},R_{-i},\Omega)\geq\varphi_{j}(R^{\prime}_{i},R_{-i},\Omega)$
for each $j\in N\setminus\\{i\\}$. Then, by feasibility,
$\varphi_{i}(R^{\prime}_{i},R_{-i},\Omega)\geq\frac{\Omega}{n}$. By own-peak-
onliness, we can assume that $R_{i}$ is such that
$\varphi_{i}(R^{\prime}_{i},R_{-i},\Omega)P_{i}\varphi_{i}(R,\Omega)$. Then,
$R^{\prime}_{i}$ is a manipulation of $\varphi$ at $(R_{i},\Omega)$.
Furthermore, if $x\in O^{\varphi}(R^{\prime}_{i},\Omega)$, then
$x=\varphi_{i}(R^{\prime}_{i},R^{\prime}_{-i})$ for some
$R^{\prime}_{-i}\in\mathcal{SP}^{n-1}$. Then, by peak monotonicity and
feasibility, $x\geq\frac{\Omega}{n}$. Therefore, by own-peak-onliness, we can
assume that $xR_{i}\varphi_{i}(R)$. Therefore, $R^{\prime}_{i}$ is an obvious
manipulation of $\varphi$ at $(R,\Omega)$, which is a contradiction. $\square$

Hence, under own-peak-onliness and NOM, peak monotonicity implies the fairness
properties of symmetry and the equal division guarantee.777Remember that, as
we saw in Section 3, symmetry and the equal division guarantee are independent
properties. By using Lemma 1 and a proof similar to that of Theorem 2, the
following characterization is obtained.

###### Theorem 4

An own-peak-only rule defined on $\mathcal{E}_{\mathcal{SP}}$ satisfies
efficiency, peak monotonicity, and NOM if and only if it is a peak monotonic
and simple rule.

Let us check the independence of axioms involved in Theorem 4. Considering the
rules defined in Subsection 3.2, the equal division rule $\widetilde{\varphi}$
satisfies all properties but efficiency; by Lemma 1, rule $\varphi^{\star}$
satisfies all properties but peak monotonicity; and rule $\underline{\varphi}$
satisfies all properties but own-peak-onliness. Lastly, consider the following
rule:

Constrained equal-distance rule, $\boldsymbol{c}$: For each
$(R,\Omega)\in\mathcal{E}_{\mathcal{SP}}$ and each $i\in N,$

$c_{i}(R,\Omega)=\left\\{\begin{array}[]{l l }\max\\{p(R_{i})-d,0\\}&\text{if
}z(R,\Omega)\geq 0\\\ p(R_{i})+d&\text{if }z(R,\Omega)<0\\\
\end{array}\right.$

where $d\geq 0$ and solves $\sum_{j\in N}c_{j}(R,\Omega)=\Omega.$

This rule is own-peak-only by definition; it is same-sided, and therefore
efficient; and it is also peak monotonic. However, since it may not assign the
peak amount to some simple agent, it is not NOM.

###  4.3 Dropping efficiency

Getting rid of efficiency leads us to a straightforward relaxation of simple
rules. A generalized simple rule is an own-peak-only rule that assigns to each
agent an amount between his peak and equal division. It turns out that such
rules are characterized by the equal division guarantee and the NOM property.

###### Theorem 5

An own-peak-only rule defined on $\mathcal{E}_{\mathcal{SP}}$ satisfies the
equal division guarantee and NOM if and only if it is a generalized simple
rule.

To see that a generalized simple rule satisfies the equal division guarantee
and NOM just note that, by the definition of the rule, option sets of an agent
are always included in the interval between the peak of that agent and equal
division. This fact immediately implies the equal division guarantee, whereas
the NOM property follows as in the proof of Theorem 1. Conversely, assume that
an own-peak-only rule satisfies the equal division guarantee and NOM and that
one agent does not get an allotment between his peak and equal division.
Further, by own-peak-onliness, assume w.l.o.g. that whatever he gets is worse
than equal division. Then, reporting a preference with equal division as its
peak amount entails an obvious manipulation by the equal division guarantee,
contradicting NOM. Therefore, this rule must be generalized simple.

Finally, Table 1 summarizes the different characterizations involving NOM
presented in this study.

|  Th. 2 |  Cor. 1 |  Th. 3 |  Th. 4 |  Th. 5
---|---|---|---|---|---
Own-peak-onliness | $+$ | $+$ | $+$ | $+$ | $+$
Efficiency | $+$ | $+$ | $+$ | $+$ |
Equal division guarantee | $+$ | $+$ |  |  | $+$
NOM | $+$ | $+$ | $+$ | $+$ | $+$
Symmetry |  | $+$ |  |  |
Endowments guarantee |  |  | $+$ |  |
Peak monotonicity |  |  |  | $+$ |
Table 1: _Characterizations of NOM rules._

##  5 Maximal domain for NOM

Now, a relevant question that arises is whether the domain of preferences can
be enlarged maintaining the compatibility of the properties. Ching and
Serizawa (1998) show that the single-plateaued domain is maximal for
efficiency, symmetry, and strategy-proofness. As we have seen, when we weaken
strategy-proofness to NOM (and explicitly invoke own-peak-onliness), the class
of rules that also meet the equal division guarantee enlarges considerably.
One may suspect that the maximal domain of preferences for these new
properties enlarges as well. However, as we will see in this section, the
single-plateaued domain is still maximal for these properties.888Ching and
Serizawa (1998) show that the single-plateaued domain is _the_ unique maximal
domain for their properties. We show that this domain is _one_ maximal domain
for our properties. There could exist “pathological” maximal domains
containing just a portion of the single-plateaued domain. Nevertheless, all of
them have to be contained in the domain of convex preferences.

We start with the definition of maximality for a list of properties.

###### Definition 3

A domain $\mathcal{D}^{\star}\subseteq\mathcal{U}$ is maximal for a list of
properties if (i) $\mathcal{SP}\subseteq\mathcal{D}^{\star}$, (ii) there is a
rule defined on $\mathcal{E}_{\mathcal{D}^{\star}}$ satisfying the properties,
and (iii) for each $\mathcal{E}_{\overline{\mathcal{D}}}$ with
$\mathcal{D}^{\star}\subsetneq\overline{\mathcal{D}}\subseteq\mathcal{U}$
there is no rule defined on $\mathcal{E}_{\overline{\mathcal{D}}}$ satisfying
the same properties.

Given $i\in N$ and $R_{i}\in\mathcal{U}$, let $\underline{p}(R_{i})=\inf
p(R_{i})$ and $\overline{p}(R_{i})=\sup p(R_{i}).$ Agent $i$’s preference
$R_{i}\in\mathcal{U}$ is single-plateaued if $p(R_{i})$ is the interval
$[\underline{p}(R_{i}),\overline{p}(R_{i})]$ and, for each pair
$\\{x_{i},x_{i}^{\prime}\\}\subseteq\mathbb{R}_{+}$, we have
$x_{i}P_{i}x_{i}^{\prime}$ as long as either
$x_{i}^{\prime}<x_{i}\leq\underline{p}(R_{i})$ or $\overline{p}(R_{i})\leq
x_{i}<x_{i}^{\prime}$ holds. Denote by $\mathcal{SPL}$ the domain of all such
preferences. Agent $i$’s preference $R_{i}\in\mathcal{U}$ is convex if
$p(R_{i})$ is the interval $[\underline{p}(R_{i}),\overline{p}(R_{i})]$ and,
for each pair $\\{x_{i},x_{i}^{\prime}\\}\subseteq\mathbb{R}_{+}$, we have
$x_{i}R_{i}x_{i}^{\prime}$ as long as either
$x_{i}^{\prime}<x_{i}\leq\underline{p}(R_{i})$ or $\overline{p}(R_{i})\leq
x_{i}<x_{i}^{\prime}$ holds. Denote by $\mathcal{C}$ the domain of all such
preferences.

###### Lemma 2

Let $\mathcal{D}$ be such that
$\mathcal{SP}\subseteq\mathcal{D}\subseteq\mathcal{U}$ and let $\varphi$ be an
own-peak-only rule defined on $\mathcal{E}_{\mathcal{D}}$ that satisfies the
equal division guarantee and NOM. Let $i\in N$ and
$(R,\Omega)\in\mathcal{E}_{\mathcal{D}}$. If $p(R_{i})$ is a singleton, then
$\varphi_{i}(R,\Omega)$ is between $p(R_{i})$ and $\frac{\Omega}{n}$.

_Proof._ Let $\varphi:\mathcal{E}_{\mathcal{D}}\longrightarrow\mathbb{R}_{+}$
be an own-peak-only rule that satisfies the equal division guarantee and NOM.
Let $(R,\Omega)\in\mathcal{E}_{\mathcal{D}}$ and let $i\in N$ be such that
$p(R_{i})\leq\frac{\Omega}{n}$. We will prove that
$\varphi_{i}(R,\Omega)\in[p(R_{i}),\frac{\Omega}{n}].$ By own-peak-onliness
and the fact that $\mathcal{SP}\subseteq\mathcal{D}$, the proof that
$\varphi_{i}(R,\Omega)\geq p(R_{i})$ follows the same lines as Case 1 in part
($\Longrightarrow$) of the proof of Theorem 2. Next, we need to show that
$\varphi_{i}(R,\Omega)\leq\frac{\Omega}{n}$. Assume, by way of contradiction,
that $\frac{\Omega}{n}<\varphi_{i}(R,\Omega)$. By own-peak-onliness and the
fact that $\mathcal{SP}\subseteq\mathcal{D}$, we can assume w.l.o.g. that
$\frac{\Omega}{n}\ P_{i}\ \varphi_{i}(R,\Omega)$. Let
$R^{\prime}_{i}\in\mathcal{SP}$ be such that
$p(R^{\prime}_{i})=\frac{\Omega}{n}.$ By the equal division guarantee,
$R^{\prime}_{i}$ is an obvious manipulation of $\varphi$ at $(R_{i},\Omega)$.
This contradicts the fact that $\varphi$ satisfies NOM. Therefore,
$\varphi_{i}(R,\Omega)\in[p(R_{i}),\frac{\Omega}{n}].$

The case $\frac{\Omega}{n}>p(R_{i})$ is symmetric, so we omit it. $\square$

###### Lemma 3

Let $\mathcal{D}$ be such that
$\mathcal{SP}\subseteq\mathcal{D}\subseteq\mathcal{U}$. If there is an own
peak-only rule defined on $\mathcal{E}_{\mathcal{D}}$ that satisfies
efficiency, the equal division guarantee, and NOM, then
$\mathcal{D}\subseteq\mathcal{C}$.

_Proof._ Let $\mathcal{D}$ be such that
$\mathcal{SP}\subseteq\mathcal{D}\subseteq\mathcal{U}$ and let $\varphi$ be an
own-peak-only rule defined on $\mathcal{E}_{\mathcal{D}}$ that is efficient,
NOM, and meets the equal division guarantee. Assume, by way of contradiction,
that there is $R_{0}\in\mathcal{D}\setminus\mathcal{C}$. Then, there are
$x,y,z\in\mathbb{R}_{+}$ such that $x<y<z$, $xP_{0}y$ and $zP_{0}y$. Without
loss of generality, we can assume that $zR_{0}x$ (the case $xR_{0}z$ is
symmetric).

Let $x^{\prime}=\max\\{w\in[x,y]\text{ such that }wI_{0}x\\}$ and
$y^{\prime}=\min\\{w\in[y,z]\text{ such that }wI_{0}x\\}$. As $R_{0}$ is
continuous, both $x^{\prime}$ and $y^{\prime}$ are well defined,
$x^{\prime}<y<y^{\prime}$, and $y^{\prime}P_{0}w$ for each
$w\in(x^{\prime},y^{\prime})$. Let $(R,\Omega)\in\mathcal{E}_{\mathcal{D}}$ be
such that $\Omega=ny^{\prime}$, $R_{1}=R_{0}$, and
$R_{-1}\in\mathcal{SP}^{n-1}$ is such that
$p(R_{j})\in(y^{\prime},\frac{\Omega-x^{\prime}}{n-1})$ and $\Omega
R_{j}y^{\prime}$ for each $j\in N\setminus\\{1\\}$. By Lemma 2,
$\varphi_{j}(R,\Omega)\in[y^{\prime},p(R_{j})]$ for each $j\in
N\setminus\\{1\\}$. Then, by feasibility and the fact that
$p(R_{j})<\frac{\Omega-x^{\prime}}{n-1}$, it follows that
$\varphi_{1}(R,\Omega)\in(x^{\prime},y^{\prime}]$. If
$\varphi_{1}(R,\Omega)\in(x^{\prime},y^{\prime})$, by the definitions of
$x^{\prime}$ and $y^{\prime}$, $y^{\prime}P_{1}\varphi_{i}(R,\Omega)$. If we
consider $R^{\prime}_{1}\in\mathcal{SP}$ be such that
$p(R^{\prime}_{1})=\frac{\Omega}{n}=y^{\prime}$, by the equal division
guarantee $R^{\prime}_{1}$ is an obvious manipulation of $\varphi$ at
$(R_{1},\Omega)$, contradicting that $\varphi$ satisfies NOM. Therefore,
$\varphi_{1}(R,\Omega)=y^{\prime}$. Hence, again by Lemma 2 and feasibility,
$\varphi_{j}(R,\Omega)=y^{\prime}$ for each $j\in N$. Now, consider the
feasible allocation in which agent $1$ gets $x^{\prime}$ and the remaining
agents get $\frac{\Omega-x^{\prime}}{n-1}$. Since $x^{\prime}I_{1}y^{\prime}$
and $\frac{\Omega-x^{\prime}}{n-1}P_{j}y^{\prime}$ for each $j\in
N\setminus\\{1\\}$, this new allocation Pareto improves upon
$\varphi(R,\Omega),$ contradicting efficiency. We conclude that
$\mathcal{D}\subseteq\mathcal{C}$. $\square$

###### Theorem 6

Domain $\mathcal{SPL}$ is a maximal domain for own-peak-onliness, efficiency,
the equal division guarantee, and NOM.

_Proof._ Let $\mathcal{D}$ be a domain for the properties listed in the
theorem such that $\mathcal{SPL}\subseteq\mathcal{D}$. By Lemma 3,
$\mathcal{D}\subseteq\mathcal{C}$. Assume, by way of contradiction, that there
is $R_{0}\in\mathcal{D}\setminus\mathcal{SPL}$. Without loss of generality, we
can assume that there are $x,y\in\mathbb{R}_{+}$ such that
$x<y<\underline{p}(R_{0})$ and $xI_{0}y$.

Assume further that there is a rule $\varphi$ defined on
$\mathcal{E}_{\mathcal{D}}$ that satisfies the properties listed in the
Theorem. Consider $(R,\Omega)\in\mathcal{E}_{\mathcal{D}}$ with $\Omega=ny$,
$R_{1}=R_{0}$, and $R_{-1}\in\mathcal{SP}^{n-1}$ such that
$p(R_{j})=y+\frac{y-x}{n-1}$ for each $j\in N\setminus\\{1\\}$. By Lemma 2,
$\varphi_{j}(R,\Omega)\in[y,p(R_{j})]$ for each $j\in N\setminus\\{1\\}$.
Then, by feasibility, $\varphi_{1}(R,\Omega)\leq y.$ If
$\varphi_{1}(R,\Omega)<y$ we can assume, by own-peak-onliness and the fact
that $\mathcal{SPL}\subseteq\mathcal{D}$, that $yP_{1}\varphi_{1}(R,\Omega)$.
Next, let us consider $R^{\prime}_{1}\in\mathcal{SP}$ such that
$p(R^{\prime}_{1})=\frac{\Omega}{n}=y.$ By the equal division guarantee,
$R^{\prime}_{1}$ is an obvious manipulation of $\varphi$ at $R_{i}$,
contradicting the fact that $\varphi$ satisfies NOM. Therefore,
$\varphi_{1}(R,\Omega)=y$. Hence, again by Lemma 2 and feasibility,
$\varphi_{j}(R,\Omega)=y$ for each $j\in N$. Now, consider the feasible
allocation in which agent $1$ gets $x$ and the remaining agents get
$y+\frac{y-x}{n-1}$. Since $xI_{1}y$ and $y+\frac{y-x}{n-1}P_{j}y$ for each
$j\in N\setminus\\{1\\}$, this new allocation Pareto improves upon
$\varphi(R,\Omega),$ contradicting efficiency. We conclude that
$\mathcal{D}=\mathcal{SPL}$.

The proof is completed by considering the extension of the uniform rule to
domain $\mathcal{SPL}$ defined in the proof of the Theorem in Ching and
Serizawa (1998). As they observe, this rule satisfies efficiency and strategy-
proofness on $\mathcal{SPL}$. As NOM is a weakening of strategy-proofness,
this rule satisfies NOM as well. Furthermore, it is easy to see that this rule
is own-peak-only and meets the equal division guarantee.999We can also
consider extensions of simple rules defined on $\mathcal{E}_{\mathcal{SP}}$ to
domain $\mathcal{E}_{\mathcal{SPL}}$. Before defining them, we need some
notation. Given $(R,\Omega)\in\mathcal{E}_{\mathcal{SPL}},$ let
$\underline{z}(R,\Omega)=\sum_{j\in N}\underline{p}(R_{j})-\Omega$ and
$\overline{z}(R,\Omega)=\sum_{j\in N}\overline{p}(R_{j})-\Omega.$ For $i\in N$
and $R_{i}\in\mathcal{SPL}$, let $\underline{R}_{i}\in\mathcal{SP}$ be such
that $p(\underline{R}_{i})=\underline{p}(R_{i})$ and let
$\overline{R}_{i}\in\mathcal{SP}$ be such that
$p(\overline{R}_{i})=\overline{p}(R_{i})$. An own-peak-only rule
$\varphi^{\star}$ defined on $\mathcal{E}_{\mathcal{SPL}}$ is an extension of
simple rule $\varphi$ defined on $\mathcal{E}_{\mathcal{SP}}$ if, for each
$(R,\Omega)\in\mathcal{E}_{\mathcal{SPL}},$ (i)
$\varphi^{\star}_{i}(R,\Omega)=\varphi_{i}(\underline{R},\Omega)$ if
$\underline{z}(R,\Omega)\geq 0$, (ii)
$\varphi^{\star}_{i}(R,\Omega)=\varphi_{i}(\overline{R},\Omega)$ if
$\overline{z}(R,\Omega)\leq 0$, (iii)
$\varphi^{\star}_{i}(R,\Omega)\in[\underline{p}(R_{i}),\overline{p}(R_{i})]$
if $\underline{z}(R,\Omega)<0$ and $\overline{z}(R,\Omega)>0$. With similar
arguments to those used in the proof of Theorem 2, it follows that these rules
also are efficient, meet the equal division guarantee, and are NOM.  $\square$

Finally, it is worth mentioning that the single-plateaued domain remains
maximal for the properties involved in the characterization of Corollary 1,
i.e., when symmetry is also considered.

##  6 Final Remarks

Some final remarks are in order. The well-known property of peaks-onliness is
a strong form of the own-peak-only property and requires that if two profiles
of preferences have the same peaks, then the allocations recommended by the
rule in both profiles are the same. Although own-peak-onliness follows from
efficiency and strategy-proofness, peak-onliness does not. If we replace own-
peak-onliness with peak-onliness in the definition of essential rules, in
Section 3, then all our results remain valid replacing own-peak-onliness by
peak-onliness.

In this paper, we introduced the equal division guarantee as a minimal
fairness requirement. We showed that it is not implied by symmetry. However,
two well-known fairness properties, envy-freeness and the equal division lower
bound, imply it for own-peak-only rules. Remember that a rule is envy-free if
no agent prefers some other agent’s allotment to his own; and that a rule
meets the equal division lower bound if, for each agent, the allotment
recommended by the rule is at least as good as equal division. Formally,

Envy-free: For each $(R,\Omega)\in\mathcal{E}_{\mathcal{SP}}$ and each
$\\{i,j\\}\subseteq N$ with $i\neq j$,
$\varphi_{i}(R,\Omega)R_{i}\varphi_{j}(R,\Omega)$.

Equal division lower bound:101010This property is also known as _individual
rationality from equal division._ For each
$(R,\Omega)\in\mathcal{E}_{\mathcal{SP}}$ and each $i\in N$,
$\varphi_{i}(R,\Omega)R_{i}\frac{\Omega}{n}.$

###### Lemma 4

For any own-peak-only rule defined on $\mathcal{E}_{\mathcal{SP}}$, envy-
freeness or the equal division lower bound imply the equal division guarantee.

_Proof._ Let $\varphi$ be an own-peak-only rule defined on
$\mathcal{E}_{\mathcal{SP}}$. First, assume that $\varphi$ meets the equal
division lower bound. Let $(R,\Omega)\in\mathcal{E}_{\mathcal{SP}}$ and $i\in
N$ be such that $p(R_{i})=\frac{\Omega}{n}$. Then,
$\varphi_{i}(R,\Omega)R_{i}\frac{\Omega}{n}=p(R_{i})$ and thus
$\varphi_{i}(R,\Omega)=p(R_{i})$, implying that $\varphi$ meets the equal
division guarantee. Second, assume that $\varphi$ is envy-free but does not
meet the equal division guarantee. Then, there are
$(R,\Omega)\in\mathcal{E}_{\mathcal{SP}}$ and $i\in N$ with
$p(R_{i})=\frac{\Omega}{n}$ such that $\varphi_{i}(R)\neq\frac{\Omega}{n}$.
Consider the case $\varphi_{i}(R)>\frac{\Omega}{n}$ (the other one is
symmetric). By feasibility, there is $j\in N\setminus\\{i\\}$ such that
$\varphi_{j}(R)<\frac{\Omega}{n}$. As $p(R_{i})=\frac{\Omega}{n}$, by own-
peak-onliness, we can assume that $R_{i}$ is such that
$\varphi_{j}(R)P_{i}\varphi_{i}(R)$. Then, $\varphi$ does not satisfy envy-
freeness. This contradiction shows that $\varphi$ meets the equal division
guarantee. $\square$

## References

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##  Appendix A Constructing a simple rule

Even though simple rules are easy to define and we can think of several
different ways to construct them, in this appendix we provide a particular
algorithmic approach.111111Of course, this process will be nothing more than a
way to resolve the system of inequalities posed by the definition of simple
rule once an economy is fixed.

Once we specify an economy $(R,\Omega)\in\mathcal{E}_{\mathcal{SP}}$, the
procedure in Figure 1 shows how we can construct the outcome of a simple rule
in such an economy.

Algorithm:
---
Input |  An economy $(R,\Omega)\in\mathcal{E}_{\mathcal{SP}}$
Initialization |
|  $\alpha_{i}:=\begin{cases}p(R_{i})&\text{ if }i\in N^{\star}(R,\Omega)\\\
\frac{\Omega}{n}&\text{ if }i\in N\setminus N^{\star}(R,\Omega)\\\
\end{cases}$ (initial allotment)
Adjustment |
|  $N\setminus N^{\star}(R,\Omega):=\\{i_{1},i_{2},\ldots,i_{k}\\}$ (order
$N\setminus N^{\star}(R,\Omega)$)
| $t:=1$
|  $\overline{\Omega}_{1}:=\Omega-\sum_{j\in N}\alpha_{j}$ (amount still left
to allocate)
|  $\underline{\Omega}_{1}:=\Omega-\sum_{j\in N}p(R_{j})$ (amount to control
for type of case)
|  IF $z(R,\Omega)\geq 0$ (excess demand case)
|  WHILE $t<k$ DO
|  Choose $\lambda_{t}$ such that
|
$\max\\{0,\underline{\Omega}_{t}+p(R_{i_{t}})-\frac{\Omega}{n}\\}\leq\lambda_{t}\leq\min\\{p(R_{i_{t}})-\frac{\Omega}{n},\overline{\Omega}_{t}\\}$
|  $\alpha_{i_{t}}=\alpha_{i_{t}}+\lambda_{t}$
|  $\overline{\Omega}_{t+1}=\overline{\Omega}_{t}-\lambda_{t}$
|
$\underline{\Omega}_{t+1}=\underline{\Omega}_{t}-\alpha_{i_{t}}+p(R_{i_{t}})$
|  $t=t+1$
|  ELSE (excess supply case: $z(R,\Omega)<0$)
|  WHILE $t<k$ DO
|  Choose $\lambda_{t}$ such that
|
$\max\\{0,\frac{\Omega}{n}-p(R_{i_{t}})-\underline{\Omega}_{t}\\}\leq\lambda_{t}\leq\min\\{\frac{\Omega}{n}-p(R_{i_{t}}),-\overline{\Omega}_{t}\\}$
|  $\alpha_{i_{t}}=\alpha_{i_{t}}-\lambda_{t}$
|
$\underline{\Omega}_{t+1}=\underline{\Omega}_{t}-\alpha_{i_{t}}+p(R_{i_{t}})$
|  $t=t+1$
Output |
|  $\varphi_{i}(R,\Omega)=\begin{cases}\alpha_{i}&\text{ if }i\in
N\setminus\\{i_{k}\\}\\\ \ \Omega-\sum_{j\in
N\setminus\\{i_{k}\\}}\alpha_{j}&\text{ if }i=i_{k}\\\ \end{cases}$ (final
allotment)
Figure 1: _Algorithm to construct a simple rule._

Assume that $z(R,\Omega)\geq 0$ (the other case is similar). To begin, an
initial allotment is performed. Each simple agent is assigned his peak amount
and each non-simple agent is assigned equal division. This frees up an amount
$\overline{\Omega}_{1}\geq 0$ that still has to be divided among non-simple
agents. To do this, we consider a sequential adjustment process. First, we
order non-simple agents arbitrarily. Following this order, in step $t$ of the
adjustment process non-simple agent $i_{t}$’s surplus, $\lambda_{t}$, is
determined fulfilling three requirements: (i) agent $i_{t}$’s final allotment
$\frac{\Omega}{n}+\lambda_{t}$ should be between $\frac{\Omega}{n}$ and
$p(R_{i_{t}})$ to comply with the definition of simple rule; (ii) surplus
$\lambda_{t}$ should be small enough to achieve feasibility, i.e.,
$\lambda_{t}\leq\overline{\Omega}_{t}$; and (iii) surplus $\lambda_{t}$ should
be large enough to maintain the economy in excess demand for not yet adjusted
agents, i.e., ${\Omega}-[\sum_{j\in
N^{\star}(R,\Omega)\cup\\{i_{1},\cdots,i_{t-1}\\}}\alpha_{j}+(\frac{\Omega}{n}+\lambda_{t})]\leq\sum_{j=t+1}^{k}p(R_{i_{j}})$
or, equivalently,
$\underline{\Omega}_{t}+p(R_{i_{t}})-\frac{\Omega}{n}\leq\lambda_{t}$. To
finish step $t$, amounts $\overline{\Omega}_{t}$ and $\underline{\Omega}_{t}$
are updated accordingly. Finally, each simple agent gets as final allotment
his peak, each non-simple agent $i_{t}$ different from $i_{k}$ gets as final
allotment $\frac{\Omega}{n}+\lambda_{t}$ and the allotment of agent $i_{k}$ is
determined by feasibility.

Notice that the algorithm is correct. To see this, observe that in each step
$t$ of the algorithm $\alpha_{i}\leq p(R_{i})$ for all $i\in
N\setminus\\{i_{t}\\}$ and at the beginning of each step $t$,
$\alpha_{i_{t}}=\frac{\Omega}{n}\leq p(R_{i_{t}})$ . Therefore,
$\underline{\Omega}_{t}+p(R_{i_{t}})-\frac{\Omega}{n}\leq\overline{\Omega}_{t}$.
Furthermore, $\underline{\Omega}_{1}\leq 0$ in the initialization and
$\underline{\Omega}_{2}=\underline{\Omega}_{1}-\alpha_{i_{1}}+p(R_{i_{1}})=\underline{\Omega}_{1}-(\frac{\Omega}{n}+\lambda_{1})+p(R_{i_{1}}).$
Therefore, by the choice of $\lambda_{1}$, $\underline{\Omega}_{2}\leq 0$.
Following a similar argument we get that $\underline{\Omega}_{t}\leq 0$ at
each step of the algorithm. Therefore,
$\underline{\Omega}_{t}+p(R_{i_{t}})-\frac{\Omega}{n}\leq
p(R_{i_{t}})-\frac{\Omega}{n}$ holds. Then, the election of $\lambda_{t}$ is
always possible.