# Variable population manipulations of reallocation rules in economies with
single-peaked preferences111I thank Pablo Arribillaga, Jordi Massó, William
Thomson, the Associate Editor, and two anonymous referees for very detailed
comments. Of course, all errors are my own. I acknowledge financial support
from Universidad Nacional de San Luis through grants 032016 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. Part of this work was written
during my visit at the Department of Economics, University of Rochester, under
a Fulbright scholarship.

Agustín G. Bonifacio222Instituto de Matemática Aplicada San Luis (UNSL-
CONICET), Av. Italia 1556, 5700, San Luis, Argentina. E-mail:
<EMAIL_ADDRESS>
###### Abstract

In a one-commodity economy with single-peaked preferences and individual
endowments, we study different ways in which reallocation rules can be
strategically distorted by affecting the set of active agents. We introduce
and characterize the family of _iterative_ reallocation rules and show that
each rule in this class is _withdrawal-proof_ and _endowments-merging-proof_ ,
at least one is _endowments-splitting-proof_ and that no such rule is _pre-
delivery-proof._

_Journal of Economic Literature_ Classification Numbers: D63, D71, D82

_Keywords:_ single-peakedness, withdrawal-proofness, endowments-merging-
proofness, endowments-splitting-proofness, pre-delivery-proofness.

##  1 Introduction

This paper studies a variable population model of an economy consisting of one
non-disposable commodity and a set of agents with individual endowments of
that commodity. In this context, we investigate different ways in which
reallocation rules, i.e., systematic ways of selecting reallocations for each
possible configuration of agents’ preferences and endowments, can be
strategically distorted by affecting the set of active agents. We limit our
analysis to the case where agents’ preferences are single-peaked: up to some
critical level, called the peak, an increase in an agent’s consumption raises
her welfare; beyond that level, the opposite holds. This model has been
extensively studied (see, for example, Bonifacio, 2015; Klaus et al., 1997,
1998). We allow for variable population as in Moreno (1996, 2002).333One-
commodity economies with single-peaked preferences and _a social endowment_
are studied, for example, by Sprumont (1991) assuming a fixed population. The
first studies of the extension of this model to allow for variable populations
are Thomson (1994a, 1995b). To illustrate this type of problem, consider the
distribution of a task (e.g., teaching hours) among the members of a group
with concave disutility of labor (which induces single-peaked preferences over
the time they dedicate to work). From one period to the next one, external
factors (e.g., research and administrative duties) might affect preferences
and a reallocation of the time assigned to each agent in the first period
(taken as a benchmark for the second period) could benefit everyone. Another
application of this model is a pollution problem in which countries have
different rates of pollution and could trade via money transfers their
pollution quotas.

Our analysis will be conducted over reallocation rules which are _own-peak-
only_ (the sole information collected by the rule from an agent’s preference
to determine her reallocation is her peak amount) and meet the _endowments
lower bound_ (no agent is made worse off than at her endowment). Two
monotonicity properties are appealing in this model. First, a population
monotonicity. Since variable population is allowed, it is natural to ask for a
monotonicity condition requiring the arrival of new agents to affect all
agents present before the change in the same direction. Adding the proviso
that agents entering the economy do not change the sign of the excess demand
of the economy, we get the property of _one-sided population monotonicity_
(see Thomson, 1995b). Second, a resource monotonicity. If, in case of excess
demand, the individual endowments decrease (or increase in case of excess
supply), then no individual is better off after the change. We call this
property _one-sided endowments monotonicity_ (see Thomson, 1994b).

Our first result is a characterization of the family of reallocation rules
that satisfy the four previously mentioned properties (Theorem 1). This family
of reallocation rules, which we call “iterative”, resembles the family of
weakly sequential reallocation rules (Bonifacio, 2015) in that each rule in
the family can be described by a step-by-step procedure that, at each stage,
guarantees levels of consumption to the agents which are adjusted throughout
the process. Weakly sequential reallocation rules, in turn, follow closely the
definition of the sequential rules presented in Barberà et al. (1997) for
economies with single-peaked preferences and a social endowment to be
allotted.444Ehlers (2002b) extends the rules of Barberà et al. (1997) to the
domain of single-plateaued preferences. Our characterization turns out to be
different from Barberà et al. (1997)’s since their characterization of
sequential rules marries _efficiency_ (allocations chosen by the rule satisfy
the Pareto criterion) with _strategy-proofness_ (no agent ever has incentives
to misrepresent preferences) and _replacement monotonicity_ (if a change in
one agent’s preferences results in that agent receiving at least as much as
before, then all other agents receive at most as much as before). Iterative
reallocation rules are always _efficient_ (Lemma 1) but need not be _strategy-
proof_ nor _replacement monotonic_.555See Appendix B for more details.

Most of the literature on asymmetric rationing with single-peaked preferences
has been focused on a specific property of immunity to manipulation:
_strategy-proofness_. Moulin (1999) characterizes the family of rationing
rules along fixed paths by means of _efficiency_ , _strategy-proofness_ , a
_resource monotonicity_ , and _consistency_ (see also Ehlers, 2002a).666See
also Moulin (2017), and references therein, for a general treatment of
strategy-proof allocation with single-peaked preferences. We deliberately
depart from that stance. Instead, and following Thomson (2014), we examine the
robustness of iterative reallocation rules to various types of manipulations
by affecting the set of active agents. The manipulations we consider are the
following:

  1. (i)

Instead of participating, an agent withdraws with her endowment. The rule is
applied without her. She then trades with one of the agents that did
participate the resources they control between the two of them in such a way
that both end up better off. A rule immune to this type of manipulation is
called _withdrawal-proof_.

  2. (ii)

Two agents can merge their endowments, and one of them withdraw. The rule is
applied without this second agent. The agent who stays may be assigned an
amount that can be divided between the two in such a way that both become at
least as well off as they would have been without the manipulation, and at
least one of them is better off. A rule immune to this type of manipulation is
called _endowments-merging-proof_.

  3. (iii)

An agent may split her endowment with some outsider (an agent with no
endowment). The rule is applied and the guest then transfers her assignment to
the agent who invited her in. The first agent may prefer her final assignment
to what she would have received without the manipulation. A rule immune to
this type of manipulation is called _endowments-splitting-proof_.

  4. (iv)

An agent may make a pre-delivery to some other agent of the trade the latter
would be assigned if she participated. The rule is applied without the second
agent. At her final assignment, the first agent may be better off than she
would have been without the manipulation. A rule immune to this type of
manipulation is called _pre-delivery-proof_.

It turns out that all iterative rules satisfy both _withdrawal-proofness_ and
_endowments-merging-proofness_ (Corollary 1 and Lemma 4). _Endowments-
splitting-proofness_ is satisfied by the proportional reallocation rule
(Remark 2). Finally, no iterative reallocation rule satisfies _pre-delivery-
proofness_ (Corollary 2).

The rest of the paper is organized as follows. In Section 2 the model and some
basic properties of reallocation rules are presented. In Section 3, iterative
reallocation rules are defined and characterized. The different variable
population manipulations are discussed in Section 4. Final comments are
gathered in Section 5.

##  2 Preliminaries

###  2.1 Model

We consider the set of natural numbers $\mathbb{N}$ as the set of potential
agents. Denote by $\mathcal{N}$ the collection of all finite subsets of
$\mathbb{N}.$ Each $i\in\mathbb{N}$ is characterized by an endowment
$\omega_{i}\in\mathbb{R}_{+}$ of the good and a continuous preference relation
$R_{i}$ defined over $\mathbb{R}_{+}$. Call $P_{i}$ and $I_{i}$ to the strict
preference and indifference relations associated with $R_{i},$ respectively.
We assume that agents’ preferences are single-peaked, i.e., each $R_{i}$ has a
unique maximum $p(R_{i})\in\mathbb{R}_{+}$ such that, for each pair
$\\{x_{i},x_{i}^{\prime}\\}\subset\mathbb{R}$, 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
$\mathcal{R}$ the domain of single-peaked preferences defined on
$\mathbb{R}_{+}$. Given $N\in\mathcal{N},$ an economy consists of a profile of
preferences $R\in\mathcal{R}^{N}$ and an individual endowments vector
$\omega=(\omega_{i})_{i\in N}\in\mathbb{R}^{N}_{+}$, and is denoted by
$e=(R,\omega)$. If $S\subset N$ and $R\in\mathcal{R}^{N},$ let
$R_{S}=(R_{j})_{j\in S}$ denote the restriction of $R$ to $S.$ We often write
$N\setminus S$ by $-S$. Using similar notation for the vector of endowments,
$e^{\prime}=(R_{S}^{\prime},R_{-S},\omega_{S}^{\prime},\omega_{-S})$ stands
for the economy where the preference and endowment of agent $i\in S$ are
$R_{i}^{\prime}$ and $\omega_{i}^{\prime},$ and those of agent $i\notin S$ are
$R_{i}$ and $\omega_{i}.$ Let $\mathcal{E}^{N}$ be the domain of economies
with agents in $N$. Given $e=(R,\omega)\in\mathcal{E}^{N},$ let
$z(e)=\sum_{j\in N}(p(R_{j})-\omega_{j}).$ If $z(e)\geq 0$ we say that economy
$e$ has excess demand whereas if $z(e)<0$ we say that economy $e$ has excess
supply. Let $\mathcal{E}=\bigcup_{N\in\mathcal{N}}\mathcal{E}^{N}$ denote the
set of all potential economies. For each $N\in\mathcal{N}$ and each
$e\in\mathcal{E}^{N},$ let $X(e)=\\{x\in\mathbb{R}^{N}_{+}:\sum_{j\in
N}x_{j}=\sum_{j\in N}\omega_{j}\\}$ be the set of reallocations for economy
$e,$ and let $X=\bigcup_{e\in\mathcal{E}}X(e).$ A reallocation rule is a
function $\varphi:\mathcal{E}\to X$ such that $\varphi(e)\in X(e)$ for each
$e\in\mathcal{E}.$ For each $N\in\mathcal{N},$ each $i\in N,$ and each
$e\in\mathcal{E}^{N},$ let $\Delta\varphi_{i}(e)=\varphi_{i}(e)-\omega_{i}$ be
agent $\boldsymbol{i$^{\prime}snettradeat$e}$.

###  2.2 Basic properties

The next informational simplicity property states that if an agent
unilaterally changes her preference for another one with the same peak, then
her allotment remains unchanged.

Own-peak-only: For each $e=(R,\omega)\in\mathcal{E}^{N},$ each $i\in N,$ and
each $R_{i}^{\prime}\in\mathcal{R}$ such that $p(R_{i}^{\prime})=p(R_{i}),$ if
$e^{\prime}=(R_{i}^{\prime},R_{N\setminus\\{i\\}},\omega)$ then
$\varphi_{i}(e^{\prime})=\varphi_{i}(e).$

This property is weaker than the “peak-only” property,777See Appendix B for a
discussion on this fact. that has been imposed in a number of axiomatic
studies. Analyzing the uniform rule, Sprumont (1991) derives the own-peak-only
property from other axioms (see also Ching, 1992, 1994).

The following property requires respecting ownership of the resource, and also
can be seen as giving incentive to participate in the exchange process. It
says that no agent can get a reallocation that she finds worse than her
endowment.

Endowments lower bound: For each $e=(R,\omega)\in\mathcal{E}^{N},$ and each
$i\in N,$ $\varphi_{i}(e)R_{i}\omega_{i}.$888This property is commonly known
in the literature as _individual rationality._

Next, we present our two monotonicity properties. The first one requires that
as population enlarges, and the new resources and preferences considered are
not as disruptive as to modify the status of the economy from excess demand to
excess supply or vice versa, the welfare of each of the initially present
agents should move in the same direction.

One-sided population monotonicity: For each $e=(R,\omega)\in\mathcal{E}^{N},$
each $N^{\prime}\subset N,$ and each
$e^{\prime}=(R_{N^{\prime}},\omega_{N^{\prime}})\in\mathcal{E}^{N^{\prime}},$
$z(e)\ z(e^{\prime})\geq 0$ implies either
$\varphi_{i}(e)R_{i}\varphi_{i}(e^{\prime})$ for each $i\in N^{\prime}$ or
$\varphi_{i}(e^{\prime})R_{i}\varphi_{i}(e)$ for each $i\in N^{\prime}$.

The second one requires all agents to benefit from a favorable change in the
amount to allocate. Given two vectors $x,y\in\mathbb{R}^{N},$ define
$x\geqslant y$ if and only if $x_{i}\geq y_{i}$ for each $i\in N.$

One-sided endowments monotonicity: For each $e=(R,\omega)\in\mathcal{E}^{N},$
and each $\omega^{\prime}\in\mathbb{R}^{N}$ such that
$\omega^{\prime}\geqslant\omega,$ if $e^{\prime}=(R,\omega^{\prime}),$ then
$z(e^{\prime})\geq 0$ implies $\varphi_{i}(e^{\prime})R_{i}\varphi_{i}(e)$ for
each $i\in N,$ and $z(e)\leq 0$ implies
$\varphi_{i}(e)R_{i}\varphi_{i}(e^{\prime})$ for each $i\in N$.

The usual Pareto optimality property states that, for each economy, the
reallocation selected by the rule should be such that there is no other
reallocation that all agents find at least as desirable and at least one agent
prefers. In this model, it is equivalent to the following same-sidedness
condition:

Efficiency: For each $e=(R,\omega)\in\mathcal{E}^{N},$ $z(e)\geq 0$ implies
$\varphi_{i}(e)\leq p(R_{i})$ for each $i\in N,$ and $z(e)\leq 0$ implies
$\varphi_{i}(e)\geq p(R_{i})$ for each $i\in N.$

###### Lemma 1

Each own-peak-only and one-sided endowments monotonic reallocation rule that
meets the endowments lower bound is efficient.

_Proof._ Let $\varphi$ be an _own-peak-only, one-sided endowments monotonic_
rule that meets the _endowments lower bound_ , and assume $\varphi$ is not
_efficient_. Then, there is $e=(R,\omega)\in\mathcal{E}^{N}$, that without
loss of generality we assume $z(e)\geq 0$, and $i\in N$ such that
$\varphi_{i}(e)>p(R_{i}).$ This implies that

$\varphi_{i}(e)\leq\omega_{i}.$ (1)

To see that (1) holds, first assume $p(R_{i})\leq\omega_{i}<\varphi_{i}(e)$.
By single-peakedness, $\omega_{i}P_{i}\varphi_{i}(e)$, contradicting the
_endowments lower bound_. Second, assume $\omega_{i}<p(R_{i})$. Let
$\widetilde{R}_{i}\in\mathcal{R}$ be such that
$p(\widetilde{R}_{i})=p(R_{i})$, and
$\omega_{i}\widetilde{P}_{i}\varphi_{i}(e)$ and let
$\widetilde{e}=(\widetilde{R}_{i},R_{-i},\omega)$. By the _own-peak-only_
property, $\varphi_{i}(\widetilde{e})=\varphi_{i}(e).$ Hence,
$\omega_{i}\widetilde{P}_{i}\varphi_{i}(\widetilde{e})$, contradicting the
_endowments lower bound_. Therefore, (1) holds. Next, let
$\omega_{i}^{\prime}\in\mathbb{R}_{+}$ be such that
$\omega_{i}^{\prime}=p(R_{i})$, and let
$e^{\prime}=(R,\omega_{i}^{\prime},\omega_{-i}).$ By the _endowments lower
bound_ , $\varphi_{i}(e^{\prime})=p(R_{i})$ and therefore
$\varphi_{i}(e^{\prime})P_{i}\varphi_{i}(e)$, contradicting _one-sided
endowments monotonicity_ since $z(e^{\prime})\geq 0$. $\square$

The following result will be useful in the rest of the paper.

###### Lemma 2

Let $\varphi$ be an efficient and own-peak-only reallocation rule that meets
the endowments lower bound. Let $e=(R,\omega)\in\mathcal{E}^{N}$ and $i\in N.$
If either $z(e)\geq 0\text{ and }p(R_{i})\leq\omega_{i}$, or $z(e)\leq 0\text{
and }p(R_{i})\geq\omega_{i}$, then $\varphi_{i}(e)=p(R_{i}).$

_Proof._ Let $\varphi$ satisfy the properties in the lemma and let
$e\in\mathcal{E}^{N}$ and $i\in N$ . Assume $z(e)\geq 0$ and
$p(R_{i})\leq\omega_{i}.$ Since $z(e)\geq 0,$ by _efficiency,_
$\varphi_{i}(e)\leq p(R_{i}).$ If $p(R_{i})=\omega_{i}$,
$\varphi_{i}(e)=p(R_{i})$ by the _endowments lower bound_. Suppose, then, that
$p(R_{i})<\omega_{i}$ and $\varphi_{i}(e)<p(R_{i}).$ Let
$R_{i}^{\prime}\in\mathcal{R}$ and $x_{i}\in\mathbb{R}_{+}$ be such that
$p(R_{i}^{\prime})=p(R_{i}),$ $\varphi_{i}(e)<x_{i}<p(R_{i})$ and
$x_{i}I_{i}^{\prime}\omega_{i}.$ Let
$e^{\prime}=(R_{i}^{\prime},R_{N\setminus\\{i\\}},\omega).$ By the _own-peak-
only_ property, $\varphi_{i}(e^{\prime})=\varphi_{i}(e).$ Then,
$\omega_{i}P_{i}^{\prime}\varphi_{i}(e^{\prime}),$ contradicting the
_endowments lower bound._ A similar reasoning establishes the same conclusion
when $z(e)\leq 0\text{ and }p(R_{i})\geq\omega_{i}.$ $\square$

##  3 Iterative reallocation rules

In this section we present a well-behaved family of reallocation rules. They
resemble the weakly sequential reallocation rules999These rules, in turn,
follow closely the definition of the sequential rules presented in Barberà et
al. (1997) for economies with a social endowment to be allotted. (Bonifacio,
2015) in that both families of rules are defined through an easy step-by-step
procedure. The two families, although overlap, are different. This is
discussed thoroughly in Appendix B.

###  3.1 Definition

For each $N\in\mathcal{N}$ and each $e=(R,\omega)\in\mathcal{E}^{N},$ let
$Q(e)\equiv\\{q\in\mathbb{R}^{N}:\sum_{j\in N}q_{j}=0\text{ and
}\omega+q\geqslant 0\\}$ be the possible net trades of endowments in economy
$e$ and let $Q\equiv\bigcup_{e\in\mathcal{E}}Q(e).$ Next, define
$\mathcal{Q}\equiv\\{(q,e)\in Q\times\mathcal{E}:q\in Q(e)\\}.$ Each element
of $\mathcal{Q}$ specifies a net trade in a particular economy. An iterative
reallocator is a function that, for each $N\in\mathcal{N}$ and each economy
$e=(R,\omega)\in\mathcal{E}^{N}$, starting from the individual endowments of
the agents (i.e., from a net trade $q_{i}^{0}$ equal to zero for each agent
$i\in N$), generates iteratively a sequence of net trades
$q^{0},q^{1},\ldots,q^{|N|-1}.$ Its repeated application is constrained to
follow monotonic features.

###### Definition 1

An iterative reallocator is a function $g:\mathcal{Q}\to\mathcal{Q}$ such that
$g(q,e)\in\mathcal{Q}(e)$ and, for each $N\in\mathcal{N},$ and each
$e=(R,\omega)\in\mathcal{E}^{N},$ if $(q^{t},e)=g(q^{t-1},e)\equiv
g^{t}(0,e),$ then:101010Here $g^{t}$ denotes $g$ compose with itself $t$
times.

  1. (i)

for each $i\in N,$ and each $t\geq 1,$

$q^{t}_{i}=p(R_{i})-\omega_{i}\text{ \ whenever \ }\begin{cases}z(e)\geq
0\text{ \ and \ }p(R_{i})\leq\omega_{i}+q_{i}^{t-1}\\\ z(e)<0\text{ \ and \
}p(R_{i})\geq\omega_{i}+q_{i}^{t-1}.\\\ \end{cases}$

  2. (ii)

for each $i\in N,$ and each $t\geq 1,$

$q^{t}_{i}\geq q^{t-1}_{i}\text{ \ whenever \ }z(e)\geq 0\text{ and
}p(R_{i})>\omega_{i}+q_{i}^{t-1}$ $q^{t}_{i}\leq q^{t-1}_{i}\text{ \ whenever
\ }z(e)<0\text{ and }p(R_{i})<\omega_{i}+q_{i}^{t-1}.$

  3. (iii)

for each $\widetilde{e}=(R,\widetilde{\omega})\in\mathcal{E}^{N}$ such that
$\widetilde{\omega}\geqslant\omega$ and
$(\widetilde{q}^{|N|-1},\widetilde{e})=g^{|N|-1}(0,\widetilde{e}),$

$\widetilde{\omega}+\widetilde{q}^{|N|-1}\geq\omega+q^{|N|-1}\text{ \ whenever
\ }z(e)\leq 0\text{ \ or \ }z(\widetilde{e})\geq 0.$

  4. (iv)

for each $\widetilde{N}\subset N,$ each
$\widetilde{e}=(R_{\widetilde{N}},\omega_{\widetilde{N}})\in\mathcal{E}^{\widetilde{N}},$
and
$(\widetilde{q}^{|\widetilde{N}|-1},\widetilde{e})=g^{|\widetilde{N}|-1}(0,\widetilde{e}),$

$\left[\widetilde{q}_{i}^{{|\widetilde{N}|-1}}-q_{i}^{{|N|-1}}\right]\left[\widetilde{q}_{j}^{{|\widetilde{N}|-1}}-q_{j}^{|N|-1}\right]\geq
0\text{ \ whenever \ }\begin{cases}z(e)\geq 0\text{ \ and \
}z(\widetilde{e})\geq 0\\\ z(e)\leq 0\text{ \ and \ }z(\widetilde{e})\leq 0\\\
\end{cases}$

for each $\\{i,j\\}\subset\widetilde{N}.$

Let us put in words the above definition for the case of excess demand (this
is, when $z(e)\geq 0$). The first two conditions relate to the behavior of the
net trades of an economy throughout the iterations of $g.$ Condition (i) says
that if at stage $t-1$ agent $i$’s peak is not higher than her endowment
_plus_ her net trade, i.e. $p(R_{i})\leq\omega_{i}+q_{i}^{t-1},$ then agent
$i$’s net trade is set at $p(R_{i})-\omega_{i}$ from stage $t$ onward.
Condition (ii) establishes that if at stage $t-1$ agent $i$’s peak is higher
than her endowment _plus_ her net trade, i.e.
$p(R_{i})>\omega_{i}+q_{i}^{t-1},$ then her net trade should not decrease from
stage $t-1$ to stage $t,$ i.e. $q_{i}^{t}\geq q_{i}^{t-1}.$ The last two
conditions relate to the behavior of the iterations of $g$ between two
different economies. Condition (iii) states that, in another economy
$\widetilde{e}$ with the same agents and preferences where no agent has lower
endowment and the increase in the resources is not disruptive, i.e.
$z(\widetilde{e})\geq 0,$ the resources available to each agent in the last
stage of the iterations cannot be smaller than the resources available to each
agent in the last stage of the iterations in the original economy. Finally,
Condition (iv) says that for any subeconomy with excess demand, if the net
trade of one agent in the last stage of the iteration is not smaller (bigger)
than the net trade that same agent gets in the original economy, then the net
trade of each of the other agents in the subeconomy should not be smaller
(bigger) than the net trade that agent gets in the original economy either.

###### Remark 1

_Note that, as there are $|N|$ agents in the economy, at most $|N|-1$
adjustments take place. Therefore, $(q^{|N|+t-1},e)=g^{|N|+t-1}(0,e)$ implies
$q^{|N|+t-1}=q^{|N|-1}$ for each $t\geq 1.$ _

Each iterative reallocator induces a reallocation rule in a straightforward
way:

###### Definition 2

A reallocation rule $\varphi:\mathcal{E}\to X$ is iterative if there is an
iterative reallocator $g:\mathcal{Q}\to\mathcal{Q}$ such that, for each
$N\in\mathcal{N}$ and each $e\in\mathcal{E}^{N},$
$(q^{|N|-1},e)=g^{|N|-1}(0,e)$ implies $\Delta\varphi(e)=q^{|N|-1}.$

A prominent member of the class of iterative reallocation rules is the uniform
reallocation rule, first studied by Thomson (1995a) (see also Klaus et al.,
1997, 1998), that adapts the celebrated uniform rule characterized by Sprumont
(1991) to the model with individual endowments:

Uniform reallocation rule, $\boldsymbol{u}$: for each $N\in\mathcal{N},$ each
$e\in\mathcal{E}^{N},$ and each $i\in N,$

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

where $\lambda(e)\geq 0$ and solves $\sum_{j\in N}u_{j}(e)=\sum_{j\in
N}\omega_{j}.$

Within the class of iterative reallocation rules, the uniform reallocation
rule is the only one that supports _envy-free_ redistributions, meaning by
this that for no $N\in\mathcal{N},$ $e\in\mathcal{E}^{N},$ and pair of agents
$\\{i,j\\}\subset N$ such that $\omega_{i}-\Delta u_{j}(e)\in\mathbb{R}_{+},$
we have $(\omega_{i}-\Delta u_{j}(e))P_{i}u_{i}(e)$ (see Moreno, 2002, Theorem
1).

To see that the uniform reallocation rule is an iterative reallocation rule,
given $e=(R,\omega)\in\mathcal{E}^{N}$ and $q^{0}=(0,\ldots,0),$ consider the
iterative reallocator $g:\mathcal{Q}\to\mathcal{Q}$ defined as follows. If
$(q^{t},e)=g(q^{t-1},e)$ then, for each $i\in N$ and each $t=1,\ldots,|N|-1,$

$q_{i}^{t}=\left\\{\begin{array}[]{l l }p(R_{i})-\omega_{i}&\text{if }i\in
N^{t}\\\ \lambda^{t}&\text{if }i\in N\setminus N^{t}\\\ \end{array}\right.$

where

$\lambda^{t}=\lambda^{t-1}+\frac{\sum_{j\in
N^{t}\setminus(\cup_{s=0}^{t-1}N^{s})}[\omega_{j}+\lambda^{t-1}-p(R_{j})]}{|N\setminus
N^{t}|},$

$\lambda^{0}=0$, $N^{0}=\emptyset$, and

$N^{t}=\left\\{\begin{array}[]{l l }\\{j\in
N:p(R_{j})\leq\omega_{j}+q_{j}^{t-1}\\}&\text{if }z(e)\geq 0\\\ \\{j\in
N:p(R_{j})>\omega_{j}+q_{j}^{t-1}\\}&\text{if }z(e)<0\\\ \end{array}\right.$

It is easy to see that $(q^{|N|-1},e)=g^{|N|-1}(0,e)$ implies
$q^{|N|-1}=\Delta u(e)$ for each $N\in\mathcal{N}$ and each
$e\in\mathcal{E}^{N}.$

###### Example 1

Consider $e=(R,\omega)^{\\{1,2,3,4\\}}$ with $p(R_{1})=0,$ $p(R_{2})=2,$
$p(R_{3})=3.5,$ and $p(R_{4})=10$; and $\omega_{1}=9,$ $\omega_{2}=1,$
$\omega_{3}=0,$ and $\omega_{4}=2.$ Then, as $z(e)=15.5-12>0,$

$N^{1}=\\{1\\},\ \lambda^{1}=0+\frac{9+0-0}{3}=3,\text{ \ and thus \
}q^{1}=(-9,3,3,3)$,

$N^{2}=\\{1,2\\},\ \ \lambda^{2}=3+\frac{1+3-2}{2}=4,\text{ \ and thus \
}q^{2}=(-9,1,4,4),$

$N^{3}=\\{1,2,3\\},\ \ \lambda^{3}=4+\frac{0+4-3.5}{1}=4.5,\text{ \ and thus \
}q^{3}=(-9,1,3.5,4.5)$.

Therefore, $u_{1}(e)=0$, $u_{2}(e)=2$, $u_{3}(e)=3.5$, and $u_{4}(e)=6.5.$
$\Diamond$

Another iterative reallocation rule, that will be analyzed in Section 4.3, is
the proportional reallocation rule that we present next.

Proportional reallocation rule, $\boldsymbol{\varphi^{p}}$: for each
$N\in\mathcal{N},$ each $e\in\mathcal{E}^{N},$ and each $i\in N,$

$\varphi^{p}_{i}(e)=\left\\{\begin{array}[]{l l
}\min\\{p(R_{i}),\lambda(e)\omega_{i}\\}&\text{if }z(e)\geq 0\\\
\max\\{p(R_{i}),\lambda(e)\omega_{i}\\}&\text{if }z(e)\leq 0\\\
\end{array}\right.$

where $\lambda(e)\geq 1$ and solves $\sum_{j\in
N}\varphi_{j}^{p}(e)=\sum_{j\in N}\omega_{j}.$111111For this rule to be well-
defined, we need to constrain the domain of economies to those in which all
individual endowments are always strictly positive.

###  3.2 Characterization

The next result states that the class of iterative rules is characterized by
the _own-peak-only_ property, the _endowments lower bound_ , _one-sided
endowments monotonicity_ and _one-sided population monotonicity:_

###### Theorem 1

A reallocation rule satisfies the own-peak-only property, the endowments lower
bound, one-sided endowments monotonicity, and one-sided population
monotonicity if and only if it is an iterative reallocation rule.

_Proof._ ($\Longrightarrow$) Let $\varphi$ be an _own-peak-only_ , _one-sided
endowments monotonic_ , and _one-sided population monotonic_ reallocation rule
that meets the _endowments lower bound_. By Lemma 1, $\varphi$ is also
_efficient_. The iterative reallocator $g:\mathcal{Q}\to\mathcal{Q}$ is
constructed as follows. Given $(q^{t-1},e)\in\mathcal{Q},$ define $q^{t}$ such
that $(q^{t},e)=g(q^{t-1},e)$ as

$q^{t}=\varphi(e^{t})-\omega$ (2)

where economy $e^{t}=(R,\omega^{t})\in\mathcal{E}^{N}$ is such that, for each
$i\in N,$

$\omega_{i}^{t}=\left\\{\begin{array}[]{l l}p(R_{i})&\text{if
}z(e)[p(R_{i})-\omega_{i}^{t-1}-q_{i}^{t-1}]\leq 0\\\
\omega_{i}^{t-1}+q_{i}^{t-1}&\text{otherwise,}\end{array}\right.$

with $\omega_{i}^{0}=\omega_{i}$ and $q_{i}^{0}=0.$

Let us assume that $e=(R,\omega)\in\mathcal{E}^{N}$ is such that $z(e)\geq 0.$
The other case is similar. We need to see that $\Delta\varphi(e)=q^{|N|-1}$
where $q^{|N|-1}$ is such that $(q^{|N|-1},e)=g^{|N|-1}(0,e)$. In order to do
this, for each $t=1,\ldots,|N|-1,$ let $q^{t}$ be such that
$(q^{t},e)=g(q^{t-1},e)=g^{t}(0,e)$ (notice that $q^{0}=0$).

Claim 1: Let $\boldsymbol{t\in\\{1,\ldots,|N|-1\\}}.$ If
$\boldsymbol{p(R_{i})\geq\omega_{i}^{t-1}+q_{i}^{t-1}}$ for each
$\boldsymbol{i\in N,}$ then $\boldsymbol{q^{t}=\Delta\varphi(e)}$. Consider
first the case $t=1.$ As $q^{0}=0,$ by the hypothesis $p(R_{i})\geq\omega_{i}$
for each $i\in N.$ The _endowments lower bound, efficiency,_ and feasibility
imply $\varphi(e)=\omega,$ and therefore $\Delta\varphi(e)=0.$ Note that,
since in $e^{1}$ no agent’s peak is less than her endowment and $z(e^{1})\geq
0$, by the same reasoning as before $\varphi(e^{1})=\omega.$ Thus,
$q^{1}=\omega-\omega=0.$ Next, assume the claim is true for each $t<T.$ Then
$q_{i}^{T-1}=0$ for each $i\in N$ and, again, since in $e^{T}$ no agent’s peak
is less than her endowment and $z(e^{T})\geq 0,$ we get
$q_{i}^{T}=0=\Delta\varphi(e).$ This proves the claim.

Claim 2: Let $\boldsymbol{t\in\\{1,\ldots,|N|-1\\}}.$ If $\boldsymbol{i\in N}$
is such that $\boldsymbol{p(R_{i})\leq\omega_{i}^{t-1}+q_{i}^{t-1}},$ then
$\boldsymbol{q^{t}_{i}=\Delta\varphi_{i}(e)}$. Let $i\in N$ be such that
$p(R_{i})\leq\omega_{i}^{t-1}+q_{i}^{t-1}.$ First, notice that when $t=1$,
$p(R_{i})\leq\omega_{i}^{0}+q_{i}^{0}$ implies, as $q^{0}_{i}=0$ and
$\omega_{i}^{0}=\omega_{i}$, that $p(R_{i})\leq\omega_{i}.$ Then, by Lemma 2,
$\varphi_{i}(e)=p(R_{i})$ and, therefore,

$\Delta\varphi_{i}(e)=p(R_{i})-\omega_{i}.$ (3)

Next, let $t\in\\{1,\ldots,|N|\\}$. Since $\omega_{i}^{t}=p(R_{i})$ and
$z(e^{t})\geq 0,$ by Lemma 2 applied to economy $e^{t}$, we have
$\varphi_{i}(e^{t})=p(R_{i})$ and then $q^{t}_{i}=p(R_{i})-\omega_{i}.$ Hence,
by (3), $q^{t}_{i}=\Delta\varphi_{i}(e).$ This proves the claim.

Claims 1 and 2 show that if $q^{|N|-1}$ is such that
$(q^{|N|-1},e)=g^{|N|-1}(0,e)$, then $q^{|N|-1}=\Delta\varphi(e).$ It remains
to be checked that function $g$ satisfies conditions (i)-(iv) in Definition 1.
Condition (i) is clear by Claims 1 and 2. Condition (ii) follows from the next
claim.

Claim 3: for each $\boldsymbol{t=1,\ldots,|N|-1,}$ if $\boldsymbol{i\in N}$ is
such that $\boldsymbol{p(R_{i})>\omega_{i}^{t-1}+q_{i}^{t-1}},$ then
$\boldsymbol{q^{t}_{i}\geq q_{i}^{t-1}}$. Let $i\in N$ be such that
$p(R_{i})>\omega_{i}^{t-1}+q_{i}^{t-1}$. Consider first the case $t=1.$ Since
$q^{0}=0$ and $\omega_{i}^{0}=\omega_{i},$ by the hypothesis
$p(R_{i})>\omega_{i}.$ Then, $\omega_{i}^{1}=\omega_{i}.$ This implies, as
$z(e^{1})\geq 0,$ that $\varphi_{i}(e^{1})\geq\omega_{i}^{1}$ by the
_endowments lower bound._ Hence, $q_{i}^{1}=\varphi_{i}(e^{1})-\omega_{i}\geq
0=q^{0}_{i}$ and thus $q_{i}^{1}\geq q^{0}_{i}$. Next, assume the claim is
true for each $t<T.$ Then $q_{i}^{T-1}\geq q_{i}^{T-2}\geq\ldots\geq
q^{0}_{i}=0.$ Since $p(R_{i})>\omega_{i}^{T-1}+q_{i}^{T-1}$ implies
$\omega_{i}^{T}=\omega_{i}^{T-1}+q_{i}^{T-1}$ and $z(e^{T})\geq 0,$ by the
_endowments lower bound_
$\varphi_{i}(e^{T})\geq\omega_{i}^{T}=\omega_{i}+\sum_{k=1}^{T-1}q_{i}^{k}.$
Then,

$q_{i}^{T}=\varphi_{i}(e^{T})-\omega_{i}\geq\sum_{k=1}^{T-1}q_{i}^{k}\geq
q_{i}^{T-1}.$

This proves the claim.

Condition (iii) follows from the definition of $g$ and _one-sided endowments
monotonicity_ of $\varphi,$ whereas condition (iv) is a consequence of the
definition of $g$ and _one-sided population monotonicity_ of $\varphi.$

($\Longleftarrow$) Let $\varphi$ be an iterative reallocation rule. Then there
exists an iterative reallocator $g$ such that, for each
$e=(R,\omega)\in\mathcal{E}^{N},$ if $q^{|N|-1}$ is such that
$(q^{|N|-1},e)=g^{|N|-1}(0,e)$, then $\Delta\varphi(e)=q^{|N|-1}.$ We will
consider only the case $z(e)\geq 0,$ since an analogous argument can be used
in the case $z(e)<0$. Next, we prove that $\varphi$ is _efficient_ ,121212This
is used to prove the two monotonicity properties. _one-sided endowments
monotonic_ , and _one-sided population monotonic_.
Efficiency: We need to show that $\varphi_{i}(e)\leq p(R_{i})$ for each $i\in
N.$ Suppose $\varphi_{i}(e)\neq p(R_{i}).$ Then $\omega_{i}+q_{i}^{|N|-1}\neq
p(R_{i})$. If $q_{i}^{|N|-1}>p(R_{i})-\omega_{i},$ by (i) in Definition 1 we
have $q_{i}^{|N|}=p(R_{i})-\omega_{i}.$ Then, $q_{i}^{|N|-1}\neq q_{i}^{|N|},$
contradicting Remark 1. Thus, $q_{i}^{|N|-1}\leq p(R_{i})-\omega_{i}$ implying
$\Delta\varphi_{i}(e)=q_{i}^{|N|-1}\leq p(R_{i})-\omega_{i}$ and
$\varphi_{i}(e)\leq p(R_{i}).$

One-sided endowments monotonicity: Let
$\widetilde{e}=(R,\widetilde{\omega})\in\mathcal{E}^{N}$ be such that
$\widetilde{\omega}\geqslant\omega$ and $z(\widetilde{e})\geq 0,$ let
$\widetilde{q}^{|N|-1}$ be such that
$(\widetilde{q}^{|N|-1},\widetilde{e})=g^{|N|-1}(0,\widetilde{e})$ and
consider $i\in N$ such that $p(R_{i})>\widetilde{\omega}_{i}.$ By condition
(iii) in Definition 1,
$\widetilde{\omega}_{i}+\widetilde{q}_{i}^{|N|-1}\geq\omega_{i}+q_{i}^{|N|-1}.$
Then, $\varphi_{i}(\widetilde{e})\geq\varphi_{i}(e)$ and, as by _efficiency_
$p(R_{i})\geq\varphi_{i}(\widetilde{e}),$ we have
$\varphi_{i}(\widetilde{e})R_{i}\varphi_{i}(e).$
One-sided population monotonicity: Let $\widetilde{N}\subset N$ and
$\widetilde{e}=(R_{\widetilde{N}},\omega_{\widetilde{N}})\in\mathcal{E}^{\widetilde{N}}$
be such that $z(\widetilde{e})\geq 0.$ Take $\\{i,j\\}\subseteq\widetilde{N}.$
By condition (iv) in Definition 1,
$[\widetilde{q}_{i}^{{|\widetilde{N}|}}-q_{i}^{{|N|}}][\widetilde{q}_{j}^{{|\widetilde{N}|}}-q_{j}^{|N|}]\geq
0.$ Assume, without loss of generality, that
$\widetilde{q}_{i}^{{|\widetilde{N}|}}\geq q_{i}^{{|N|}}.$ Then,
$\widetilde{q}_{j}^{{|\widetilde{N}|}}\geq q_{j}^{{|N|}}.$ This implies
$\varphi_{i}(\widetilde{e})\geq\varphi_{i}(e)$ and
$\varphi_{j}(\widetilde{e})\geq\varphi_{j}(e).$ As $z(\widetilde{e})\geq 0,$
by _efficiency,_ $p(R_{i})\geq\varphi_{i}(\widetilde{e})$ and
$p(R_{j})\geq\varphi_{j}(\widetilde{e}).$ Thus,
$\varphi_{i}(\widetilde{e})R_{i}\varphi_{i}(e)$ and
$\varphi_{j}(\widetilde{e})R_{i}\varphi_{j}(e).$
To complete the proof, notice that $\varphi$ satisfies the _own-peak-only_
property because $g$ does, and meets the _endowments lower bound_ because, for
each agent, the adjustment process at each step guarantees an amount at least
as good as the individual endowment. $\square$

The independence of the axioms involved in the characterization of Theorem 1
is analyzed in Appendix A.

##  4 Variable population manipulations

In this section, we analyze each of the four properties of immunity to
manipulation presented in the introduction and its relations with the family
of iterative reallocation rules.

###  4.1 Withdrawal-proofness

Consider an economy and suppose that an agent withdraws with her endowment and
the reallocation rule is applied without her. It could be the case that the
amount that some other agent received in the reallocation together with the
endowment of the agent that withdrew could be re-divided between the two of
them in such a way that both agents get (strictly) better off with respect to
the assignments they would have obtained if the first agent had not withdrawn.
We require immunity to this sort of behavior:

Withdrawal-proofness: For each $e=(R,\omega)\in\mathcal{E}^{N},$ each
$\\{i,j\\}\subset N$ and each $(x_{i},x_{j})\in\mathbb{R}^{2}_{+}$ such that
$x_{i}+x_{j}=\varphi_{i}(e^{\prime})+\omega_{j},$ where
$e^{\prime}=(R_{N\setminus\\{j\\}},\omega_{N\setminus\\{j\\}}),$ it is not the
case that $x_{k}P_{k}\varphi_{k}(e)$ for each $k\in\\{i,j\\}.$

The next result shows that _withdrawal-proofness_ is implied by some of the
properties discussed in Subsection 2.2.

###### Lemma 3

Each efficient, own-peak-only, and one-sided population monotonic reallocation
rule that meets the endowments lower bound is withdrawal-proof.

_Proof._ Let $\varphi$ satisfy the hypothesis of the Theorem. By Lemma 1,
$\varphi$ is also _efficient_. Assume $\varphi$ is not _withdrawal-proof._
Then, there are $e=(R,\omega)\in\mathcal{E}^{N},$ $\\{i,j\\}\subset N,$ and
$(x_{i},x_{j})\in\mathbb{R}^{2}_{+}$ such that, if
$e^{\prime}=(R_{N\setminus\\{j\\}},\omega_{N\setminus\\{j\\}}),$ then

$x_{i}+x_{j}=\varphi_{i}(e^{\prime})+\omega_{j},$ (4)

and

$x_{k}P_{k}\varphi_{k}(e)\text{ for each }k\in\\{i,j\\}.$ (5)

Assume $z(e)\geq 0.$ The case $z(e)\leq 0$ can be handled similarly. By (5),
$z(e)>0.$ By _efficiency,_ $\varphi_{k}(e)\leq p(R_{k})$ for each $k\in N.$ By
(5), $\varphi_{k}(e)<x_{k}$ for each $k\in\\{i,j\\}$ and therefore, by (4),

$\varphi_{i}(e)+\varphi_{j}(e)<x_{i}+x_{j}=\varphi_{i}(e^{\prime})+\omega_{j}.$
(6)

Claim: there is $\boldsymbol{k^{\star}\in N\setminus\\{i,j\\}}$ such that
$\boldsymbol{\varphi_{k^{\star}}(e^{\prime})<\varphi_{k^{\star}}(e)}$.
Otherwise,

$\sum_{k\in N\setminus\\{i,j\\}}\varphi_{k}(e^{\prime})\geq\sum_{k\in
N\setminus\\{i,j\\}}\varphi_{k}(e)$ (7)

and, since $\sum_{k\in N\setminus\\{j\\}}\omega_{k}=\sum_{k\in
N\setminus\\{j\\}}\varphi_{k}(e^{\prime}),$ by (6) and (7) we have

$\sum_{k\in N}\omega_{k}=\sum_{k\in
N\setminus\\{j\\}}\varphi_{k}(e^{\prime})+\omega_{j}>\sum_{k\in
N}\varphi_{k}(e)=\sum_{k\in N}\omega_{k},$

which is absurd. This proves the Claim.

Now, by the Claim and _efficiency_ ,
$\varphi_{k^{\star}}(e^{\prime})<\varphi_{k^{\star}}(e)\leq p(R_{k^{\star}}).$
This implies

$\varphi_{k^{\star}}(e)P_{k^{\star}}\varphi_{k^{\star}}(e^{\prime})$ (8)

and also, by _efficiency,_ $z(e^{\prime})\geq 0.$ By (5), $\varphi_{j}(e)\neq
p(R_{j})$ holds. Then, by Lemma 2, $\omega_{j}\leq p(R_{j});$ and by the
_endowments lower bound,_ $\varphi_{j}(e)\geq\omega_{j}.$ It follows from this
and (6) that
$0\leq\varphi_{j}(e)-\omega_{j}<\varphi_{i}(e^{\prime})-\varphi_{i}(e).$
Therefore, $\varphi_{i}(e^{\prime})>\varphi_{i}(e).$ As $z(e^{\prime})\geq 0,$
by _efficiency,_ $\varphi_{i}(e^{\prime})\leq p(R_{i}).$ Thus,
$\varphi_{i}(e)<\varphi_{i}(e^{\prime})\leq p(R_{i})$ and

$\varphi_{i}(e^{\prime})P_{i}\varphi_{i}(e).$ (9)

Note that, as $z(e)\geq 0$ and $z(e^{\prime})\geq 0,$ (8) and (9) contradict
_one-sided population monotonicity_. We conclude that $\varphi$ is
_withdrawal-proof._ $\square$

As a consequence of the previous result and Theorem 1, the whole class of
_iterative_ reallocation rules precludes this kind of manipulation.

###### Corollary 1

Each iterative reallocation rule is withdrawal-proof.

###  4.2 Endowments-merging-proofness

Another manipulation involving variable population is the following. Consider
an economy and a pair of agents in that economy. One of those agents gives her
endowment to the other and withdraws. The reallocation rule is applied without
the first agent and with the second agent’s enlarged endowment. The allocation
that the second agent obtains could be divided between the two agents in such
a way that each agent is at least as well off as she would have been if the
merging had not taken place, and at least one of them is better off. We
require immunity to this sort of behavior:

Endowments-merging-proofness: For each $e=(R,\omega)\in\mathcal{E}^{N},$ each
$\\{i,j\\}\subset N$ and each $(x_{i},x_{j})\in\mathbb{R}^{2}_{+}$ such that
$x_{i}+x_{j}=\varphi_{i}(e^{\prime}),$ where
$e^{\prime}=(R_{N\setminus\\{j\\}},\omega_{i}^{\prime},\omega_{N\setminus\\{i,j\\}})$
and $\omega_{i}^{\prime}=\omega_{i}+\omega_{j},$ it is not the case that
$x_{k}R_{k}\varphi_{k}(e)$ for each $k\in\\{i,j\\},$ and
$x_{k}P_{k}\varphi_{k}(e)$ for at least one $k\in\\{i,j\\}.$

Each rule in the class of iterative reallocation rules precludes such
manipulations.

###### Lemma 4

Each iterative reallocation rule is endowments-merging-proof.

_Proof._ Let $\varphi$ be an iterative reallocation rule. By Theorem 1,
$\varphi$ is _one-sided endowments monotonic_. By Lemma 1, $\varphi$ is also
_efficient_. Assume $\varphi$ is not _endowments-merging-proof._ Then, there
are $e=(R,\omega)\in\mathcal{E}^{N},$ $\\{i,j\\}\subset N,$ and
$(x_{i},x_{j})\in\mathbb{R}^{2}_{+}$ such that, if
$e^{\prime}=(R_{N\setminus\\{j\\}},\omega_{i}^{\prime},\omega_{N\setminus\\{i,j\\}}),$
then

$x_{i}+x_{j}=\varphi_{i}(e^{\prime}),$ (10) $x_{k}R_{k}\varphi_{k}(e)\text{
for each }k\in\\{i,j\\},$ (11)

and

$x_{k}P_{k}\varphi_{k}(e)\text{ for at least one }k\in\\{i,j\\}.$ (12)

Assume $z(e)\geq 0.$ By (12), $z(e)>0.$ By _efficiency,_ $\varphi_{k}(e)\leq
p(R_{k})$ for each $k\in N.$ By (11) and (12), $x_{k}\geq\varphi_{k}(e)$ for
each $k\in\\{i,j\\}$ and $x_{k}>\varphi_{k}(e)$ for at least one
$k\in\\{i,j\\}.$ Therefore, by (10),

$\varphi_{i}(e^{\prime})=x_{i}+x_{k}>\varphi_{i}(e)+\varphi_{j}(e).$ (13)

Claim 1: there is $\boldsymbol{k^{\star}\in N\setminus\\{i,j\\}}$ such that
$\boldsymbol{\varphi_{k^{\star}}(e^{\prime})<\varphi_{k^{\star}}(e)}$.
Otherwise,

$\sum_{k\in N\setminus\\{i,j\\}}\varphi_{k}(e^{\prime})\geq\sum_{k\in
N\setminus\\{i,j\\}}\varphi_{k}(e)$ (14)

and, since $\sum_{k\in N\setminus\\{j\\}}\omega_{k}=\sum_{k\in
N\setminus\\{j\\}}\varphi_{k}(e^{\prime}),$ by (13) and (14) we have

$\sum_{k\in N}\omega_{k}=\sum_{k\in
N\setminus\\{j\\}}\varphi_{k}(e^{\prime})>\sum_{k\in
N}\varphi_{k}(e)=\sum_{k\in N}\omega_{k},$

which is absurd.

Now, by Claim 1 and _efficiency_ ,
$\varphi_{k^{\star}}(e^{\prime})<\varphi_{k^{\star}}(e)\leq p(R_{k^{\star}}),$
which implies $z(e^{\prime})\geq 0.$ Let
$e^{\prime\prime}=(R_{N\setminus\\{j\\}},\omega_{N\setminus\\{j\\}}).$ As
$z(e^{\prime})\geq 0,$ it follows that $z(e^{\prime\prime})\geq 0.$ By
Corollary 1, $\varphi$ is _withdrawal-proof,_ which implies that

$\varphi_{i}(e)+\varphi_{j}(e)\geq\varphi_{i}(e^{\prime\prime})+\omega_{j}.$
(15)

Since
$(\omega_{i}^{\prime},\omega_{N\setminus\\{i,j\\}})\geqslant\omega_{N\setminus\\{j\\}},$
by _one-sided endowments monotonicity,_
$\varphi_{k}(e^{\prime})R_{k}\varphi_{k}(e^{\prime\prime})$ for each $k\in
N\setminus\\{j\\}.$ Then _efficiency_ implies

$\varphi_{k}(e^{\prime})\geq\varphi_{k}(e^{\prime\prime})\text{ for each }k\in
N\setminus\\{j\\}.$ (16)

Combining (13) and (15) we obtain

$\varphi_{i}(e^{\prime})>\varphi_{i}(e^{\prime\prime})+\omega_{j}.$ (17)

Claim 2: there is $\boldsymbol{k^{\star\star}\in N\setminus\\{i,j\\}}$ such
that
$\boldsymbol{\varphi_{k^{\star\star}}(e^{\prime\prime})>\varphi_{k^{\star\star}}(e^{\prime})}$.
Otherwise,

$\sum_{k\in N\setminus\\{i,j\\}}\varphi_{k}(e^{\prime\prime})\leq\sum_{k\in
N\setminus\\{i,j\\}}\varphi_{k}(e^{\prime})$ (18)

and, since $\sum_{k\in N\setminus\\{j\\}}\omega_{k}=\sum_{k\in
N\setminus\\{j\\}}\varphi_{k}(e^{\prime\prime}),$ by (17) and (18) we have

$\sum_{k\in N}\omega_{k}=\omega_{j}+\sum_{k\in
N\setminus\\{j\\}}\varphi_{k}(e^{\prime\prime})<\sum_{k\in
N\setminus\\{j\\}}\varphi_{k}(e^{\prime})=\sum_{k\in N}\omega_{k},$

which is absurd.

Therefore, by Claim 2, there is $k^{\star\star}\in N\setminus\\{i,j\\}$ such
that
$\varphi_{k^{\star\star}}(e^{\prime\prime})>\varphi_{k^{\star\star}}(e^{\prime}).$
This contradicts (16). We conclude that $\varphi$ is _endowments-merging-
proof._ $\square$

###  4.3 Endowments-splitting-proofness

Consider an economy and assume that an agent in the economy transfers some of
her endowment to another agent that was not initially present; the rule is
applied, and the guest transfers her assignment to the agent who invited her
in. The first agent could obtain an amount that she prefers to her initial
assignment. We require immunity to this type of behavior:

Endowments-splitting-proofness: For each $e=(R,\omega)\in\mathcal{E}^{N},$
each $i\in N,$ each $j\notin N,$ each $R_{j}\in\mathcal{R},$ and each
$(\omega_{i}^{\prime},\omega_{j}^{\prime})\in\mathbb{R}^{2}_{+}$ such that
$\omega_{i}^{\prime}+\omega_{j}^{\prime}=\omega_{i},$ we have
$\varphi_{i}(e)R_{i}[\varphi_{i}(e^{\prime})+\varphi_{j}(e^{\prime})],$ with
$e^{\prime}=(R,R_{j},\omega_{i}^{\prime},\omega_{N\setminus\\{i\\}},\omega_{j}^{\prime}).$

Not all iterative reallocation rules satisfy this property. The following
example shows that the uniform reallocation rule violates _endowments-
splitting-proofness._

###### Example 2

_Let $e=(R,\omega)\in\mathcal{E}^{\\{1,2,3\\}}$ be such that $p(R_{1})=4,$
$p(R_{2})=0,$ $p(R_{3})=\omega_{1}=\omega_{2}=2$ and $\omega_{3}=1.$ Then,
$u_{1}(e)=3,$ $u_{2}(e)=0,$ and $u_{3}(e)=2.$ Next, let $R_{4}\in\mathcal{R}$
be such that $p(R_{4})=4$ and let $\omega_{4}^{\prime}=1.$ Consider the
economy
$e^{\prime}=(R,R_{4},\omega_{1}^{\prime},\omega_{\\{2,3\\}},\omega_{4}^{\prime})\in\mathcal{E}^{\\{1,2,3,4\\}}$
with $\omega^{\prime}_{1}=1$ (notice that
$\omega_{1}=\omega_{4}^{\prime}+\omega_{1}^{\prime}$). It follows that
$u_{1}(e^{\prime})=u_{3}(e^{\prime})=u_{4}(e^{\prime})=\frac{5}{3},$
$u_{2}(e^{\prime})=0,$ and $u_{1}(e^{\prime})+u_{4}(e^{\prime})=\frac{10}{3}\
P_{1}\ 3=u_{1}(e).$ This implies that $u$ is not _endowments-splitting-proof._
_ $\Diamond$

Priority reallocation rules131313Given a linear $\prec$ order over the set of
potential agents $\mathbb{N}$, the _priority reallocation rule_
$\varphi^{\prec}$ for economies with excess demand (supply) satiates all
suppliers (demanders) and demanders (suppliers) according to order $\prec$,
respecting the endowments lower bound. For economies with excess supply, a
symmetric procedure is performed. See Appendix A for a formal definition. It
is easy to see that such reallocation rules are iterative in our sense.
violate the property as well.

###### Example 3

_Consider $\prec$ as the usual “less than” order in $\mathbb{N}$. Let
$e=(R,\omega)\in\mathcal{E}^{\\{1,3,4\\}}$ be such that $p(R_{1})=0,$
$\omega_{1}=4,$ $p(R_{3})=p(R_{4})=6,$ and $\omega_{3}=\omega_{4}=2.$ Then,
$\varphi^{\prec}_{1}(e)=0,$ $\varphi^{\prec}_{3}(e)=6,$ and
$\varphi^{\prec}_{4}(e)=2.$ Next, let $R_{2}\in\mathcal{R}$ be such that
$p(R_{2})=4$ and let $\omega_{2}=1.$ Consider the economy
$e^{\prime}=(R,R_{2},\omega_{\\{1,3\\}},\omega_{2},\omega_{4}^{\prime})\in\mathcal{E}^{\\{1,2,3,4\\}}$
with $\omega_{4}^{\prime}=1$ (notice that
$\omega_{4}=\omega_{4}^{\prime}+\omega_{2}=2$). It follows that
$\varphi^{\prec}_{1}(e^{\prime})=0,$ $\varphi^{\prec}_{2}(e^{\prime})=4,$
$\varphi^{\prec}_{3}(e^{\prime})=3,$ and $\varphi^{\prec}_{4}(e^{\prime})=1.$
However,
$\varphi^{\prec}_{2}(e^{\prime})+\varphi^{\prec}_{4}(e^{\prime})=4+1=5\ P_{4}\
2=\varphi^{\prec}_{4}(e).$ This implies that $\varphi^{\prec}$ is not
_endowments-splitting-proof._ _ $\Diamond$

However, the proportional reallocation rule is immune to endowments’
splitting:

###### Remark 2

The proportional reallocation rule is endowments-splitting-proof.

_Proof._ Suppose $\varphi^{p}$ is not _endowments-splitting-proof._ Then,
there are $e=(R,\omega)\in\mathcal{E}^{N},$ $i\in N,$ $j\notin N,$
$R_{j}\in\mathcal{R},$ and
$(\omega_{i}^{\prime},\omega_{j}^{\prime})\in\mathbb{R}^{2}_{+}$ with
$\omega_{i}^{\prime}+\omega_{j}^{\prime}=\omega_{i}$ such that, if
$e^{\prime}=(R,R_{j},\omega_{i}^{\prime},\omega_{N\setminus\\{i\\}},\omega_{j}^{\prime}),$
then

$[\varphi_{i}^{p}(e^{\prime})+\varphi_{j}^{p}(e^{\prime})]P_{i}\varphi_{i}^{p}(e).$
(19)

Consider first the case $z(e)\geq 0.$ By (19), $\varphi_{i}^{p}(e)<p(R_{i})$
and therefore

$\lambda(e)\omega_{i}=\varphi_{i}^{p}(e)<\varphi_{i}^{p}(e^{\prime})+\varphi_{j}^{p}(e^{\prime}).$
(20)

Since $z(e^{\prime})\geq z(e)\geq 0,$

$\varphi_{i}^{p}(e^{\prime})+\varphi_{j}^{p}(e^{\prime})\leq\lambda(e^{\prime})\omega_{i}^{\prime}+\lambda(e^{\prime})\omega_{j}^{\prime}=\lambda(e^{\prime})\omega_{i}.$
(21)

By (20) and (21),

$\lambda(e)<\lambda(e^{\prime}).$ (22)

It follows that there is $k\in N\setminus\\{i\\}$ such that
$\varphi^{p}_{k}(e^{\prime})<\varphi^{p}_{k}(e).$ Otherwise, there is a
violation of feasibility by (20). Then,
$\lambda(e^{\prime})\omega_{k}=\varphi^{p}_{k}(e^{\prime})<\varphi^{p}_{k}(e)\leq\lambda(e)\omega_{k},$
which implies $\lambda(e^{\prime})<\lambda(e),$ contradicting (22). If
$z(e)\leq 0$ and $z(e^{\prime})\leq 0,$ the proof is similar to the previous
one. Assume then that $z(e)\leq 0$ and $z(e^{\prime})\geq 0.$ By (19),
$\varphi_{i}^{p}(e)>p(R_{i})\geq\varphi_{i}^{p}(e^{\prime}).$ This implies the
existence of $k\in N\setminus\\{i\\}$ such that
$\varphi_{k}^{p}(e)<\varphi_{k}^{p}(e^{\prime}).$ But then,

$p(R_{k})\leq\varphi_{k}^{p}(e)<\varphi_{k}^{p}(e^{\prime})\leq p(R_{k}),$

which is a contradiction. Therefore, $\varphi^{p}$ is _endowments-splitting-
proof._ $\square$

###  4.4 Pre-delivery-proofness

Consider now the case in which one agent makes a “pre-delivery” to some other
agent of the trade that this second agent would be assigned if she had
participated with everyone else. After the rule is applied, the first agent
may end up with an amount she prefers to his assignment if she had not carried
out the pre-delivery. We require immunity to this sort of behavior.

Pre-delivery-proofness: For each $e=(R,\omega)\in\mathcal{E}^{N}$ and each
$\\{i,j\\}\subset N$ such that $\omega_{i}+\omega_{j}-\varphi_{j}(e)\geq 0,$
$\varphi_{i}(e)R_{i}\varphi_{i}(e^{\prime})$ where
$e^{\prime}=(R_{N\setminus\\{j\\}},\omega_{i}^{\prime},\omega_{N\setminus\\{i,j\\}})$
and $\omega_{i}^{\prime}=\omega_{i}+\omega_{j}-\varphi_{j}(e).$

The endowments rule, that in each economy assigns to agents their own
endowment, is trivially _pre-delivery-proof_. However, as long as we require
the rule to be _efficient_ and _own-peak-only_ , and meet the _endowments
lower bound_ , we reach an impossibility.

###### Theorem 2

No efficient and own-peak-only reallocation rule that meets the endowments
lower bound is pre-delivery-proof.

_Proof._ Let reallocation rule $\varphi$ be efficient, own-peak-only and meet
the endowments lower bound. Let $e=(R,\omega)\in\mathcal{E}^{\\{1,2,3\\}}$ be
such that $0<p(R_{1})=\omega_{2}=\omega_{3}<\omega_{1}<p(R_{2})=p(R_{3}).$
Then $z(e)>0$ and, as $p(R_{1})<\omega_{1},$ by Lemma 2 we have
$\varphi_{1}(e)=p(R_{1}).$ By feasibility, there is $i^{\star}\in\\{2,3\\}$
such that $\varphi_{i^{\star}}(e)<\omega_{1}.$ Assume, without loss of
generality, that $i^{\star}=2.$ Let
$\omega_{2}^{\prime}=\omega_{2}+\omega_{1}-\varphi_{1}(e).$ Then
$\omega_{2}^{\prime}=\omega_{1}.$ Consider now the economy
$e^{\prime}=(R_{\\{2,3\\}},\omega_{2}^{\prime},\omega_{3}).$ It follows that
$z(e^{\prime})>0.$ By _efficiency,_ $\varphi_{2}(e^{\prime})\leq p(R_{2}),$
and since $p(R_{2})>\omega_{2}^{\prime},$ by the _endowments lower bound_ we
have $\varphi_{2}(e^{\prime})\geq\omega_{2}^{\prime}=\omega_{1}.$ By
feasibility then, $\varphi_{2}(e^{\prime})=\omega_{1}.$ Therefore,
$\varphi_{2}(e)<\omega_{1}=\varphi_{2}(e^{\prime})<p(R_{2}),$ which implies
$\varphi_{2}(e^{\prime})P_{2}\varphi_{2}(e).$ Thus, $\varphi$ is not _pre-
delivery-proof._ $\square$

Of course, the previous result extends to the whole class of iterative
reallocation rules.

###### Corollary 2

No iterative reallocation rule is pre-delivery-proof.

##  5 Final comments

We conclude with some remarks. One may ask whether the definition of
_withdrawal-proofness_ can be made with just one of the agents involved in the
manipulation strictly improving. However, not even the uniform reallocation
rule satisfies this variant, as the following example shows:

###### Example 4

Consider $e=(R,\omega)^{\\{1,2,3,4\\}}$ with $p(R_{1})=p(R_{4})=1,$
$p(R_{2})=4,$ $p(R_{3})=3,$ and $\omega_{1}=\omega_{4}=3,$
$\omega_{2}=\omega_{3}=1.$ Then, $z(e)>0$ and $u_{1}(e)=u_{4}(e)=1,$
$u_{2}(e)=u_{3}(e)=3.$ When agent 4 withdraws, if
$e^{\prime}=(R_{\\{1,2,3\\}},\omega_{\\{1,2,3\\}}),$ then
$u_{1}(e^{\prime})=1,$ $u_{2}(e^{\prime})=u_{3}(e^{\prime})=2.$ So $x_{2}=4\
P_{2}\ 3=u_{2}(e),$ $x_{4}=1\ R_{1}\ 1=u_{4}(e),$ and
$x_{2}+x_{4}=5=u_{2}(e^{\prime})+\omega_{4}.$ $\Diamond$

In Thomson (2014), the property of _withdrawal-proofness_ is presented in this
variant: one of the agents involved in the manipulation can be indifferent
between the amount she gets from the rule and the amount she gets after the
manipulation is performed. However, in all the impossibility examples
presented there both agents get strictly better off, so our version does not
hold in those examples (classical multi-commodity exchange model with
homothetic and quasi-linear preferences) either.

The results obtained in our paper are in sharp contrast with the findings in
models with several goods and classical preferences. The Walrasian
reallocation rule is neither _withdrawing-proof,_ nor _endowments-merging-
proof,_ nor _endowments-splitting-proof_. These negative results are obtained
by Thomson (2014) in two classical subdomains: (i) the domain of economies in
which preferences are homothetic and strictly convex, and individual
endowments are proportional, and (ii) the domain of economies in which
preferences are quasi-linear and strictly convex. The Walrasian reallocation
rule, however, is _pre-delivery-proof_ on the classical domain (see Thomson,
2014). “Constrained dictatorial rules”, defined by maximizing the welfare of a
particular agent subject to each of the others finding their assignment at
least as desirable as their endowment, satisfy none of these various
requirements either (see Thomson, 2022).

Several questions are left open for further research. For example, finding out
which properties are required for a rule to be _endowments-merging-proof_ or
_endowments-splitting-proof_ , and knowing whether iterative reallocation
rules exhaust the rules that are _withdrawal-proof_ or _endowments-merging-
proof_. Coalitional versions of the manipulations studied in Section 4 seem
worth studying as well.

##  Appendix A Independence of axioms in Theorem 1

In order to study the independence of axioms in the characterization of
Theorem 1, next we consider several reallocation rules. For each
$N\in\mathcal{N}$ and each $e\in\mathcal{E}^{N}$, let $N^{+}(e)=\\{i\in
N:p(R_{i})>\omega_{i}\\}$ be the set of _demanders_ of $e$. Agents in
$N\setminus N^{+}$ are called _suppliers_. Let $\mathcal{S}(e)\equiv\sum_{j\in
N\setminus N^{+}}(\omega_{j}-p(R_{j}))$.

Given a linear $\preceq$ order over the set of potential agents $\mathbb{N}$,
the _priority reallocation rule_ $\varphi^{\preceq}$ for each economy with
excess demand (supply) satiates all suppliers (demanders) and demanders
(suppliers) according to order $\preceq$, respecting the endowments lower
bound. So if $e=(R,\omega)\in\mathcal{E}^{N}$ is such that $z(e)\geq 0,$

$\varphi^{\preceq}_{i}(e)=\left\\{\begin{array}[]{l l }p(R_{i})&\text{if }i\in
N\setminus N^{+}(e)\\\
\min\left\\{p(R_{i}),\omega_{i}+\mathcal{S}(e)-\sum_{j\in N^{+}:j\prec
i}\Delta\varphi_{j}(e)\right\\}&\text{if }i\in N^{+}(e)\\\ \end{array}\right.$

In case $z(e)<0$, the rule is defined similarly. Priority reallocation rules
allow the definition of a reallocation rule which is not _one-sided population
monotonic_.

Reallocation rule $\boldsymbol{\overline{\varphi}}$: for each
$N\in\mathcal{N},$ each $e\in\mathcal{E}^{N},$ and each $i\in N,$

$\overline{\varphi}_{i}(e)=\left\\{\begin{array}[]{l l
}\varphi^{\preceq}_{i}(e)&\text{if }|N|\text{ is odd}\\\
\varphi^{\succeq}_{i}(e)&\text{if }|N|\text{ is even}\\\ \end{array}\right.$

where $\succeq$ is the dual of $\preceq$.

Next, let us recall the celebrated _uniform_ rule, first characterized by
Sprumont (1991).

Uniform rule, $\boldsymbol{\varphi^{u}}$: for each $N\in\mathcal{N},$ each
$e\in\mathcal{E}^{N},$ and each $i\in N,$

$\varphi^{u}_{i}(e)=\left\\{\begin{array}[]{l l
}\min\\{p(R_{i}),\lambda(e)\\}&\text{if }z(e)\geq 0\\\
\max\\{p(R_{i}),\lambda(e)\\}&\text{if }z(e)<0\\\ \end{array}\right.$

where $\lambda(e)$ and solves $\sum_{j\in N}\varphi^{u}_{j}(e)=\sum_{j\in
N}\omega_{j}.$

Since this rule does not take into account individual endowments, it trivially
does not meet the _endowments lower bound_.

The following rule satiates as many agents as possible. For economies with
excess demand, demanders are satiated according to their claims. First minimal
demands are satiated uniformly. If there is some supply left, then the next
smallest demands are satiated, and so on. This reallocation rule is not _one-
sided endowments monotonic_.

Maximally satiating reallocation rule $\boldsymbol{\varphi^{\text{max}}}$: For
each $N\in\mathcal{N}$ and each $e\in\mathcal{E}^{N}$ such that $z(e)\geq 0,$
partition $N^{+}(e)$ into subsets $N_{1},N_{2},\ldots,N_{s}$ such that (i) for
each $t\in\\{1,\ldots,s\\}$, $p(R_{i})-\omega_{i}=p(R_{j})-\omega_{j}$ for
each $i,j\in N_{t}$, and (ii) $p(R_{i})-\omega_{i}<p(R_{j})-\omega_{j}$ if
$i\in N_{r}$, $j\in N_{s}$, and $r<s.$ Then,

$\varphi^{\text{max}}_{i}(e)=p(R_{i})$

if $i\in N\setminus N^{+}$, and

$\varphi^{\text{max}}_{i}(e)=\min\left\\{p(R_{i}),\omega_{i}+\frac{1}{|N_{t}|}\left(\mathcal{S}(e)-\sum_{j\in\cup_{r=1}^{t-1}N_{r}}\Delta\varphi_{j}(e)\right)\right\\}$

if $i\in N_{t}$ and $t\in\\{1,\ldots,s\\}.$ The formula when $z(e)<0$ is
obtained similarly.

| Own-peak-only | Endow LB | OS endow mon | OS pop mon
---|---|---|---|---
$\overline{\varphi}$ | $+$ | $+$ | $+$ | $-$
$\varphi^{\text{max}}$ | $+$ | $+$ | $-$ | $+$
$\varphi^{u}$ | $+$ | $-$ | $+$ | $+$
? | $-$ | $+$ | $+$ | $+$
Table 1: _Independence of axioms in the characterization of Theorem 1. _

Each one of the previously presented rules satisfies all properties of the
characterization in Theorem 1 except one. This is shown in Table 1. For
example, reallocation rule $\varphi^{u}$ satisfies both monotonicity
properties and the _own-peak-only_ property, but does not meet the _endowments
lower bound_. Whether there is a reallocation rule that satisfies all
properties except the _own-peak-only_ property remains an open question.

##  Appendix B Iterative reallocation rules are neither peak-only nor
strategy-proof

The following strong form of the _own-peak-only_ property requires that if two
profiles of preferences have the same peaks, then the reallocations
recommended by the rule in both profiles are the same.

Peak-only: For each $e=(R,\omega)\in\mathcal{E}^{N}$ and each
$R^{\prime}\in\mathcal{R}$ such that $p(R_{i}^{\prime})=p(R_{i})$ for each
$i\in N$, if $e^{\prime}=(R^{\prime},\omega)$ then
$\varphi(e^{\prime})=\varphi(e).$

Iterative reallocation rules are not in general _peak-only_ , because they can
be _bossy_ : it could be that a change in one agent’s preferences (preserving
the peak amount) only affects other agents’ reallocations (see Example 5).
When a rule is _own-peak-only_ and _non-bossy_ 141414Non-bossiness: For each
$e=(R,\omega)\in\mathcal{E}^{N}$, each $i\in N$, and each
$R^{\prime}_{i}\in\mathcal{R}$, if $e^{\prime}=(R^{\prime}_{i},R_{-i},\omega)$
and $\varphi_{i}(e^{\prime})=\varphi_{i}(e^{\prime})$ then
$\varphi(e^{\prime})=\varphi(e).$, then it is also _peak-only_. Sequential
allocation rules in Barberà et al. (1997) satisfy _replacement monotonicity_ ,
which in turn implies _non-bossiness_. For this reason, _own-peak-onliness_ is
equivalent to _peak-onliness_ for those rules.

A reallocation rule is _strategy-proof_ if, for each agent, truth-telling is
always optimal, regardless of the preferences declared by the other agents.
Formally,

Strategy-proofness: For each $e=(R,\omega)\in\mathcal{E}^{N}$, each $i\in N$,
and each $R_{i}^{\prime}\in\mathcal{R}$, if
$e^{\prime}=(R^{\prime}_{i},R_{-i},\omega)$ then
$\varphi_{i}(e)R_{i}\varphi_{i}(e^{\prime}).$

_Strategy-proofness_ is a strong property. Together with _efficiency_ , it
implies the _own-peak-only_ property for reallocation rules (this is proven,
for example, in Lemma 3 of Klaus et al., 1998). Since we do not impose
_strategy-proofness_ in our paper, we need to explicitly invoke the _own-peak-
only_ property.

To see that iterative reallocation rules need not be _peak-only_ nor
_strategy-proof_ , consider reallocation rule $\varphi^{\star}$ that works as
follows.151515Here we use the terminology presented in Appendix A. For each
$N\in\mathcal{N}$ and each $(R,\omega)\in\mathcal{E}^{N}$ with $z(e)\geq 0$,
let $i^{\star}$ be the agent in $N\setminus N^{+}(e)$ (the set of suppliers)
with the lowest index.161616If $N\setminus N^{+}(e)=\emptyset$, simply
allocate to each agent their endowment. Next, define a linear order $\ll$ over
$N^{+}(e)$ (the set of demanders) saying that $i\ll j$ if and only if either
$p(R_{i})-\omega_{i}<p(R_{j})-\omega_{j}$ or
$p(R_{i})-\omega_{i}=p(R_{j})-\omega_{j}$ and $i<j$. If
$0P_{i^{\star}}\sum_{j\in N}\omega_{j}$, then we apply the priority
reallocation rule according to $\ll$, i.e.,
$\varphi^{\star}(e)=\varphi^{\ll}(e)$. If $\sum_{j\in
N}\omega_{j}R_{i^{\star}}0$, then we apply the uniform reallocation rule,
i.e., $\varphi^{\star}(e)=\varphi^{u}(e)$. The definition of the rule for
economies with excess supply is similar. It is easy to see that reallocation
rule $\varphi^{\star}$ belongs to the class of iterative reallocation rules.
The following example shows that $\varphi^{\star}$ is neither _peak-only_ nor
_strategy-proof_.

###### Example 5

Let $e=(R,\omega)\in\mathcal{E}^{\\{1,2,3\\}}$ be such that $\omega_{1}=9$,
$\omega_{2}=1$, $\omega_{3}=4$, $p(R_{1})=1$ and $0\ P_{1}\ 14$, $p(R_{2})=7$,
and $p(R_{3})=9$. Since $p(R_{3})-\omega_{3}=9-4=5<6=7-1=p(R_{2})-\omega_{2}$,
we have $\varphi^{\star}_{1}(e)=1$, $\varphi^{\star}_{2}(e)=4$, and
$\varphi^{\star}_{3}(e)=9$. Let $\widetilde{R}_{1}\in\mathcal{R}$ be such that
$p(\widetilde{R}_{1})=p(R_{1})$ and $14\ \widetilde{P}_{1}\ 0$, an consider
economy $\widetilde{e}=(\widetilde{R}_{1},R_{-1},\omega)$. Then,
$\varphi_{1}(\widetilde{e})=1$, $\varphi_{2}(\widetilde{e})=5$, and
$\varphi_{3}(\widetilde{e})=8$. Thus, although $\varphi^{\star}$ is _own-peak-
only_ , it is certainly not _peak-only_.

To see that $\varphi^{\star}$ does not satisfy _strategy-proofness_ , let
$R_{2}^{\prime}\in\mathcal{R}$ be such that $p(R_{2}^{\prime})=5.5$ and
consider economy $e^{\prime}=(R_{2}^{\prime},R_{-2},\omega)$. Since
$p(R_{2}^{\prime})-\omega_{2}=4.5<5=9-4=p(R_{3})-\omega_{3}$, we have
$\varphi^{\star}_{2}(e^{\prime})=5.5$. Therefore,
$\varphi^{\star}_{2}(e^{\prime})=5.5\ P_{2}\ 4=\varphi_{2}^{\star}(e).$
$\Diamond$

Finally, it is worth mentioning that the weakly sequential rules in Bonifacio
(2015) are immune to misrepresentation of preferences and endowments in a very
general way: they are _bribe-proof_.171717This concept was introduced by
Schummer (2000) and applied to the model with a social endowment by Massó and
Neme (2007). This property does not allow any group of agents to compensate
one of its subgroups to misrepresent their preferences or endowments in order
that, after an appropriate redistribution of what the rule reallocates to the
group (adjusted by the resource surplus or deficit they all engage in by
misreporting), (i) each agent in the misrepresenting subgroup obtain a
preferred amount, and (ii) the rest of the agents in the group are not made
worse-off.

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