Individually Rational Rules for the Division Problem when the Number of Units to be Allotted is Endogenous

We study individually rational rules to be used to allot, among a group of agents, a perfectly divisible good that is freely available only in whole units. A rule is individually rational (at a preference prole) if each agent nds that her allotment is at least as good as any whole unit of the good. We study and characterize two individually rational and e¢ cient rules, whenever agentspreferences are symmetric single-peaked on the set of possible allotments. The two rules are in addition envy-free, but they di¤er on wether envy-freeness is considered on losses or on awards. Our main result states that (i) the constrained equal losses rule is the unique individually rational and e¢ cient rule that satises justied envy-freeness on losses and (ii) the constrained equal awards rule is the unique individually rational and e¢ cient rule that satises envy-freeness on awards. The work of G. Bergantiños is partially supported by grants ECO2014-52616-R from the Spanish Ministry of Science and Competitiveness, GRC 2015/014 from "Xunta de Galicia", and 19320/PI/14 from Fundación Séneca de la Región de Murcia. J. Massó acknowledges nancial support from the Spanish Ministry of Economy and Competitiveness, through the Severo Ochoa Programme for Centres of Excellence in R&D (SEV-2015-0563) and grant ECO2014-53051, and from the Generalitat de Catalunya, through grant SGR2014-515. The paper was partly written while J. Massó was visiting the Department of Economics at Stanford University and acknowledges its hospitality as well as nancial support received from the Spanish Ministry of Education, Culture and Sport, through project PR2015-00408. The work of A. Neme is partially supported by the Universidad Nacional de San Luis, through grant 319502, and by the Consejo Nacional de Investigaciones Cientícas y Técnicas (CONICET), through grant PIP 112-200801-00655. yFacultade de Económicas, Universidade de Vigo. 36310, Vigo (Pontevedra), Spain. E-mail: gbergant@uvigo.es zUniversitat Autònoma de Barcelona and Barcelona Graduate School of Economics. Departament dEconomia i dHistòria Econòmica, Campus UAB, Edici B. 08193, Bellaterra (Barcelona), Spain. E-mail: jordi.masso@uab.es xInstituto de Matemática Aplicada de San Luis. Universidad Nacional de San Luis and CONICET. Ejército de los Andes 950. 5700, San Luis, Argentina. E-mail: aneme@unsl.edu.ar


Introduction
Consider the allotment problem faced by a group of agents who may share a homogeneous and perfectly divisible good, available only in whole units. Examples of this kind of good are shares representing ownership of a company, bonds issued by a company to …nance its business operations, treasury bills issued by the government to …nance its short term needs, or any type of …nancial assets with (potentially large) face values or tickets of a lottery. The good could also be workers, with a …xed working day schedule, to be shared among departments or divisions of a big institution or company, with a …xed salary budget. Agents' risk attitudes, wealth or labor requirements and salary budgets induce single-peaked preferences on their potential allotments of the good, the set of nonnegative real numbers. A solution of the problem is a rule that selects, for each pro…le (of agents'preferences), a non-negative integer number of units of the good to be allotted and a vector of allotments (a list of non-negative real numbers, one for each agent) whose sum is equal to this integer. Observe that although the good is only available in integer amounts agents'allotments are allowed to take non-integer values; yet, their sum has to be an integer. Namely, in the above examples agents are able to share a …nancial asset or a lottery ticket by getting portions of it or time of a worker. But, for most pro…les, the sum of agents'best allotments will be either larger or smaller than any integer number and hence, an endogenous rationing problem emerges, positive or negative depending on whether the chosen integer is smaller or larger than the sum of agents'best allotments. Sprumont (1991) studied the problem when the amount of the good to be allotted is …xed. He characterized the uniform rule as the unique e¢ cient, strategy-proof and anonymous rule on the domain of single-peaked preferences. The present paper can be seen as an extension of Sprumont (1991)'s paper to a setting where the amount to be allotted of a divisible good has to be an integer, which may depend on agents'preferences.
We are interested in situations where the good is freely available to agents, but only in whole units. Hence, an agent will not accept a proposal of an allotment that is strictly worse than any integer amount of the good. For an agent with a (continuous) singlepeaked preference, the set of allotments that are at least as good as any integer amount of the good (the set of individually rational allotments) is a closed interval that contains the best allotment, that we call the peak, and at least one of the two extremes of the interval is an integer. If preferences are symmetric, the peak is at the midpoint of the interval.
Our main concern then is to identify rules that select, for each pro…le of agents' symmetric single-peaked preferences, a vector of individually rational allotments. We call such rules individually rational. But since the set of individually rational rules is extremely large, and some of them are arbitrary and non-interesting, we would like to focus further on rules that are also e¢ cient, strategy-proof, and satisfy some minimal fairness requirement. A rule is e¢ cient if it selects, at each pro…le, a Pareto optimal vector of allotments: no other choices of (i) integer unit of the good to be allotted or (ii) vector of allotments, or (iii) both, can make all agents better o¤, and at least one of them strictly better o¤. We characterize the class of all individually rational and e¢ cient rules on the domain of symmetric single-peaked preferences by means of three properties. First, the allotted amount of the good is the closest integer to the sum of agents'peaks. Second, all agents are rationed in the same direction: all receive more than their peaks, if the integer to be allotted is larger than the sum of the peaks, or all receive less, otherwise. Third, the rule selects a vector of allotments that belong to the agents' individually rational intervals. A rule is strategy-proof if it induces, at each pro…le, truth-telling as a weakly dominant strategy in its associated direct revelation game. Our fairness requirements will be related to two alternative and well-known notions of envy-freeness, that we will adapt to our setting (justi…ed envy-freeness on losses and envy-freeness on awards). 1 We show that there is no rule that is individually rational, e¢ cient and strategy-proof on the domain of symmetric single-peaked preferences. We then proceed by studying separately two subclasses of rules on the symmetric single-peaked domain; those that are individually rational and e¢ cient and those that are individually rational and strategyproof. For the …rst subclass, we identify the family of the constrained equal losses rules and the family of the constrained equal awards rules as the unique families of rules that, in addition of being individually rational and e¢ cient, satisfy also either justi…ed envyfreeness on losses or envy-freeness on awards, respectively. These rules divide the e¢ cient integer amount of the good in such a way that all agents experience either equal losses or equal gains, subject to the constraint that all allotments have to be individually rational. Speci…cally, a constrained equal losses rule, evaluated at a pro…le, selects …rst the e¢ cient number of integer units (if there are two, it selects one of them). Then, to allot this integer amount it proceeds with the rationing from the vector of peaks, by either reducing or increasing the allotment of each agent (depending on whether the sum of the peaks is larger or smaller than the integer amount to be allotted) until the total amount is allotted. However, it makes sure that the extremes of agents'individually rational intervals are not overcome by excluding any agent from the rationing process as soon as one of the extremes of the agent's individually rational interval is reached, and it continues with the rest. A constrained equal awards rule is de…ned similarly but instead it uses, as the starting vector of the rationing process, either the vector of lower bounds or the vector of upper bounds of the individually rational intervals, depending on whether the sum of the peaks is larger or smaller than the integer amount to be allotted, but makes sure that no agent's peak is overcome by excluding her from the rationing process as soon as her peak is reached, and it continues with the rest.
For the subclass of individually rational and strategy-proof rules, we show in contrast that although there are many rules satisfying the two properties simultaneously, they are not very interesting; for instance, none of them is unanimous. A rule is unanimous if, whenever the sum of the peaks is an integer, the rule selects this integer and it allots it according to the agents'peaks. We show then that individual rationality and strategyproofness are indeed incompatible with unanimity. At the end of the paper we extend some of our general and possibility results to the case where agents'single-peaked preferences are not necessarily symmetric. Moreover, we argue why relevant strategy-proof rules in the classical division problem (i.e., the uniform rule and all sequential dictator rules) are not satisfactory in our setting. In particular, we show …rst that extended uniform rules are e¢ cient on the domain of all single-peaked preference pro…les but they are neither strategy-proof nor individually rational. 2 Finally, we show that all sequential dictator rules are e¢ cient on the domain of all symmetric single-peaked preference pro…les but they are neither individually rational nor strategyproof, even on this subdomain. 3 Before …nishing this Introduction we mention some of the most related papers to ours. As we have already said, Sprumont (1991) proposed the division problem of a …xed amount of a good among a group of agents with single-peaked preferences on their potential allotments and provided two characterizations of the uniform rule, using strategy-proofness, e¢ ciency and either anonymity or envy-freeness. Then, a very large literature followed Sprumont (1991) by taking at least two di¤erent paths. The …rst contains papers providing alternative characterizations of the uniform rule. See for instance Ching (1994), Sönmez (1994) and Thomson (1994aThomson ( , 1994bThomson ( , 1995Thomson ( and 1997, whose characterizations we brie ‡y discuss in the last section of the paper. The second group of papers proposed alternative rules when the problem is modi…ed by introducing additional features or considering alternative domains of agents' preferences, or both. For instance, Ching (1992) extended the characterization of Sprumont (using envy-freeness) to the domain of single-plateaued preference pro…les and Bergantiños, Massó and Neme (2012a, 2012b 2 An extended uniform rule allots, at each pro…le, the e¢ cient integer amount as the uniform rule would do it (if there are two e¢ cient integers, it selects one of them). It is not strategy-proof because an agent may have incentives to misreport his preferences to induce a di¤erent choice of the integer amount, and it is not individually rational because the vector of allotments selected by the uniform rule is not individually rational in general. However, the adapted versions proposed in Bergantiños, Massó and Neme (2015), the constrained extended uniform rules, satisfy individual rationality, e¢ ciency and equal treatment of equals but they remain manipulable. 3 A sequential dictator rule, given a pre-speci…ed order on the set of agents, proceeds by letting agents choose sequentially their peaks, rationing only the last agent whose allotment is the remainder amount that ensures that the sum of the allotments is equal to an e¢ cient integer amount. Sequential dictator rules are not strategy-proof because the agent at the end of the ordering may have incentives to misreport her preference to induce a di¤erent amount to allot. They are not individually rational because the agent at the end of the ordering is rationed independently of her individually rational interval. and 2015), Manjunath (2012) and Kim, Bergantiños and Chun (2015) studied alternative ways of introducing individual rationality in the division problem. But in contrast with the present paper they assume that the quantity of the good to be allotted is …xed. Adachi (2010), Amorós (2002), Anno and Sasaki (2013), Cho and Thomson (2013), Erlanson and Flores-Szwagrzak (2015) and Morimoto, Serizawa and Ching (2013) contain the multi-dimensional analysis of the division problem when several commodities have to be allotted among the same group of agents, but again the quantities of the goods to be allotted are …xed. The paper is organized as follows. The next section presents the problem, preliminary notation and basic de…nitions. Section 3 contains the de…nitions of the properties of the rules that we will be concerned with. Section 4 describes the rules and states a preliminary result. Section 5 contains the main results of the paper for symmetric singlepeaked preferences. Section 6 contains two …nal remarks.

The problem
We study situations where each agent of a …nite set N = f1; : : : ; ng wants an amount of a perfectly divisible good that can only be obtained in integer units, but arbitrary portions of each unit can be freely allotted. We assume that n 2 and denote by x i 0 the total amount of the good allotted to agent i 2 N: Since all units of the good are alike, the amount x i may come from di¤erent units. We assume that there is no limit on the (integer) number of units that can be allotted. Hence, and once N is …xed, the set of feasible (vector of) allotments is where R + = [0; +1) is the set of non-negative real numbers and N 0 = f0; 1; 2; : : :g is the set of non-negative integers. 4 Each agent i has a preference relation i , de…ned on the set of potential allotments, which is a complete and transitive binary relation on R + . That is, for all x i ; y i ; z i 2 R + ; either x i i y i or y i i x i ; and x i i y i and y i i z i imply x i i z i ; note that re ‡exivity (x i i x i for all x i 2 R + ) is implied by completeness. Given i , let i be the antisymmetric binary relation on R + induced by i (i.e., for all x i ; y i 2 R + , x i i y i if and only if y i x i does not hold) and let i be the indi¤erence relation on R + induced by i (i.e., for all x i ; y i 2 R + , x i i y i if and only if x i i y i and y i i x i ). We assume that i is continuous (i.e., for each x i 2 R + the sets fy i 2 R + j y i i x i g and fy i 2 R + j x i i y i g are closed) and that i is single-peaked on R + ; namely, there exists a unique p i 2 R + , the peak of i , such that p i i x i for all x i 2 R + nfp i g and x i i y i holds for any pair of allotments x i ; y i 2 R + such that either y i < x i p i or p i x i < y i . For each i 2 N; let p i i be an agent i's single-peaked preference such that p i 2 R + is the peak of p i i : We say that agent i's single-peaked preference i is symmetric on Notice two things. First, the peak of a symmetric single-peaked preference conveys all information about the whole preference. Thus, we will often identify a symmetric single-peaked preference i with its peak p i . Second, for each x 2 R + , there exists a unique integer k x 2 N 0 such that k x x < k x + 1: Hence, the following notation is well-de…ned: ( k x when x k x + 0:5 k x + 1 when x > k x + 0:5: where N is the set of agents and = ( 1 ; : : : ; n ) is a pro…le of single-peaked preferences on R + , one for each agent in N . Since the set N will remain …xed we often write instead of (N; ) and refer to as a problem and as a pro…le, interchangeably. To emphasize agent i's preference i in the pro…le we often write it as ( i ; i ): We denote by P the set of all problems and by P S the set of all problems where agents' preferences are symmetric single-peaked.
Since preferences are idiosyncratic, they have to be elicited. A rule on P is a function f assigning to each problem 2 P a feasible allotment f ( ) = (f 1 ( ) ; : : : ; f n ( )) 2 F A: We will also consider rules de…ned only on P S : Any rule on P can be restricted to operate only on P S : To study rules on P S selecting individually rational allotments, the following intervals will play a critical role. Fix a problem 2 P S ; with its vector of peaks (p 1 ; : : : ; p n ). For each i 2 N; de…ne the associated closed interval ; dp i e] if p i bp i c + 0:5: When no confusion arises we write l i instead of l i (p i ) and u i instead of u i (p i ) : Allotments outside the interval [l i ; u i ] are strictly worse to some integer allotment (either to bp i c or to dp i e), and they will not be acceptable to i; if agent i has free access to any integer amount of the good. Since each interval [l i ; u i ] depends only on p i ; we call it the individually rational interval of p i (Proposition 2 will show the exact relationship between individually rational rules on P S and the individually rational intervals). Given p i 2 R + ; [l i ; u i ] can be seen as the unique interval with the properties that p i is equidistant to the two extremes (i.e. p i = l i +u i 2 ), at least one of the two extremes is an integer, and its length is at most one. For instance, the individually rational interval of p i = 1:8 is [1:6; 2] and of p i = 2:3 is [2; 2:6]:

Properties of rules
We now describe possible properties that a rule f on P (or on P S ) may satisfy. Again, the properties de…ned on P can be straightforwardly extended to P S by restricting their de…nitions to the set of problems in P S : We start with the property of individual rationality, the one that we found more basic for the class of problems we are interested in, which is the main focus of this paper. Since we are assuming that all integer units of the good are freely available, even for a single agent, a rule is individually rational if each agent considers her allotment at least as good as any integer number of units of the good.
Individual rationality. For all 2 P, i 2 N and k 2 N 0 , f i ( ) i k: The next two properties are also appealing. E¢ ciency says that, for each problem, the vector of allotments selected by the rule is Pareto undominated in the set of feasible allotments, while a rule is strategy-proof if agents can never obtain a strictly better allotment by misrepresenting their preferences.
E¢ ciency. For all 2 P, there does not exist y 2 F A such that y i i f i ( ) for all i 2 N and y j j f j ( ) for at least one j 2 N: Strategy-proofness. For all 2 P; i 2 N and single-peaked preference 0 i , We say that agent i manipulates f at via 0 We will also consider other desirable properties of rules. Participation says that agents will not have interest in obtaining integer units of the good in addition to their received allotments. To de…ne it formally, we need some additional notation. For each k 2 N 0 and i with peak p i such that k p i ; let p i k i be the single-peaked preference on R + obtained from i by shifting it downwards in k units; namely, for each pair x i ; y i 2 R + ; Participation. For all 2 P, i 2 N and k 2 N 0 such that k p i ; Unanimity says that the rule selects the pro…le of peaks whenever it is a feasible vector of allotments. Equal treatment of equals says that agents with the same preferences receive equal allotments.
Unanimity. For all 2 P such that P j2N p j 2 N 0 ; f i ( ) = p i for all i 2 N: Equal treatment of equals. For all 2 P and i; j 2 N such that i = j ; f i ( ) = f j ( ) : Envy-freeness says that the rule selects a vector of allotments with the property that no agent would strictly prefer the allotment of another agent.
Envy-freeness. For all 2 P and i; j 2 N , The next three properties are alternative versions of envy-freeness, adapted to our context when agents have symmetric single-peaked preferences and they have free access to any integer amount of the good. Given that, each agent is willing to accept a noninteger allotment proposed by the rule insofar her participation in the problem helps her to circumvent the integer restriction. Hence, envy-freeness may take as reference, not the absolute amounts received but instead, how other agents are treated with respect to their peaks or to their individually rational intervals. The emphasis is then on the losses or the awards that agents'allotments represent with respect to their peaks or to the extremes of their individually rational intervals, respectively. First, envy-freeness on losses says that each agent prefers her loss (with respect to her peak) to the loss of any other agent.
Envy-freeness on losses. For all 2 P S and i; j 2 N , f i ( ) i max fp i + (f j ( ) p j ); 0g : 5 Second, justi…ed envy-freeness on losses quali…es the previous property by requiring that each agent i prefers her loss (i.e., f i ( ) p i ) to the loss of any other agent j (i.e., f j ( ) p j ), only if j's allotment is strictly preferred by j to any integer. Since agents can obtain freely any integer number of units of the good, it may be understood that it is not legitimate for i to express envy of another agent j who is receiving an allotment that j considers indi¤erent to an integer because it is as if the rule would not allot to j any amount. Hence, i's envy towards j is only justi…ed if j strictly prefers her allotment to any integer amount.
Justi…ed envy-freeness on losses. For all 2 P S and i; j 2 N such that f j ( ) j k for all Envy-freeness on awards roughly says that each agent prefers her award, with respect to her individually rational allotment, to any amount between her award and the award of any other agent. To state it formally, let f be a rule on P S . De…ne, for each 2 P S and i 2 N , the award of i (at ( ; f )) with respect to i's individually rational interval as When no confusion arises we write a i instead of a i ( ; f ) : Envy-freeness on awards. For all 2 P S and i; j 2 N , To see why envy-freeness on awards is a desirable property consider for example the case where a i = f i ( ) l i ; a j = f j ( ) l j and a i < x < a j : If l i + x i f i ( ); i may argue that the non-integer amount received by j was too large and that there is a compromise, x 2 [a i ; a j ], that may be used to solve the integer problem in a more fair way. Example 1 might also help to better understand this property. By setting x = 0:3 we have that f 1 ( ) = 0 1 0:3 = l 1 + x: Nevertheless, by setting x = 0:1 we have that f 1 ( ) = 0 1 0:1 = l 1 + x, and so f would not satisfy envy-freeness on awards. In this case agent 1 can argue that agent 2 is receiving at f ( ) (compared with the individually rational points l 2 = 0:2 and l 1 = 0) more than her (a 2 = 0:3 versus a 1 = 0).
Again, envy-freeness is based on absolute references: it requires comparisons of allotments directly. In contrast, our two notions of envy-freeness are relative: they disregard the integer amounts allotted to the agents and compare (using losses or awards as references) only those fractions received away from the peaks or the relevant extremes of the individually rational intervals.
Finally, group rationality is an extension of individual rationality to groups of agents. It says that each subset of agents receives a total allotment that is (in aggregate terms) "at least as good as"any other total allotment they could receive only by themselves. 6 For all such x; f i ( ) i l i + x is equivalent to f i ( ) i u i x since i is symmetric single-peaked and, by the de…nition of the extremes of the individually rational interval, p i = li+ui 2 : Group rationality. For all 2 P S , S N and k 2 N 0 , The following statements hold. 7 (R1.1) If f is e¢ cient on P, then f is unanimous. (R1. 2) If f is envy-free on losses on P S , then f satis…es justi…ed envy-freeness on losses on P S .
3) If f is group rational on P S , then f is individually rational on P S .

Rules
In this section we adapt, to our setting with endogenous integer amounts, fair and wellknown rules that have already been used to solve the division problem with a …xed amount. Since our main results will be relative to symmetric single-peaked preferences, we already restrict the rules we consider in the next two sections to operate on P S . This is important because the rules will allot the integer amount that is closest to the sum of the peaks, which is always the e¢ cient amount only if single-peaked preferences are symmetric. Since at pro…les where P j2N p j = p + 0:5, p and p + 1 are both at the same distance of 0:5 from P j2N p j ; many rules will share the same principles but they will be di¤erent only to the extend that they select the smaller or the largest closest integer at some at those pro…les. Hence, we will be de…ning classes of rules. Although we will be interested only in their constrained versions (to ensure that they are individually rational) we also present their unconstrained versions for further reference and because they may help the reader to understand the constrained ones. We start with the class of equal losses rules. At any pro…le p, an equal losses rule selects the feasible vector of allotments by the following egalitarian procedure. Start from the vector of peaks p and, if this is an unfeasible vector of allotments, decrease or increase all agents' allotments in the same amount until the closest integer j P respectively is allotted, stopping the decrease (if this is the case) of any agent's allotment, as soon as the zero allotment is reached.

Equal losses.
We say that f is an equal losses rule if, for all 2 P S ; The proofs are immediate.

Fig. 1 An equal losses rule f EL
A constrained equal losses rule proceeds by following the same egalitarian procedure but now the increase or decrease of the allotment of agent i; starting from p i ; stops as soon as i's allotment is equal to the relevant extreme of i's individually rational interval.

Constrained equal losses.
We say that f is a constrained equal losses rule if, for all 2 P S ; where b is the unique real number for which it holds that P j2N (p j min fb ; p j l j g) = p or P j2N (p j + min fb ; u j p j g) = p + 1. Denote by F CEL the set of all constrained equal losses rules. 8 Corollary 1 below (that follows from Proposition 1) will establish the existence of such unique real number , as well as the existence of the real numbers b , , and b , used to de…ne the other three rules below.
Observe that for any pair f; f 0 2 F CEL ; f ( ) = f 0 ( ) for all 2 P S except for those pro…les for which P j2N p j = p +0:5: But in this case, for all Thus, for any pair f; f 0 2 F CEL , any pro…le 2 P S and any i 2 N; Figure 2 represents a rule f CEL 2 F CEL at pro…les and ; where [p 1 + p 2 ] < p 1 +p 2 < p + 0: Fig. 2 A constrained equal losses rule f CEL An equal awards rule follows the same egalitarian procedure used to de…ne equal losses rules, but instead of starting from the vector of peaks, it starts from the vector of relevant extremes of the individually rational intervals and it increases (or decreases) all agents' allotments in the same amount until the integer number of units is allotted, making sure that no agent receives a negative allotment.
Equal awards. We say that f is an equal awards rule if, for all 2 P S , where is the unique real number for which Denote by F EA the set of all equal awards rules. Figure 3 represents a rule f EA 2 F EA at pro…les ; 0 and ; where p +0:  Fig. 3 An equal awards rule f EA A constrained equal awards rule proceeds by following the same egalitarian procedure but now the increase or decrease of the allotment of each agent i; starting from the relevant extreme of i's individually rational interval, stops as soon as i's allotment is equal to p i .

Constrained equal awards.
We say that f is a constrained equal awards rule if, for all 2 P S ; where b is the unique real number for which Denote by F CEA the set of all constrained equal awards rules.
Observe that for any pair f; f 0 2 F CEA ; f ( ) = f 0 ( ) for all 2 P S except for those pro…les for which P j2N p j = p + 0:5: But in this case, for all Thus, for any pair f; f 0 2 F CEA , any pro…le 2 P S and any i 2 N; Figure 4 represents a rule f CEA 2 F CEA at pro…les and ; where [p 1 +p 2 ] < p 1 +p 2 < p + 0:5 and [ p 1 + p 2 ] > p 1 + p 2 > p + 0:5: The existence of the unique numbers , b , and b in each of the above de…nitions is guaranteed by Proposition 1 below.
Proposition 1 For each 2 P S , the appropriate statement below holds.
Notice that if p j k j + 0:5, then l j = k j and u j = p j + (p j k j ) = 2p j k j : Similarly, if p j > k j + 0:5, then l j = p j (k j + 1 p j ) = 2p j (k + 1) and u j = k j + 1: Hence, Since We now show that X holds as well. By (3), Since l j u j for all j 2 N; (6) holds.
To prove (P1.1), assume P j2N p j p + 0:5 holds. We distinguish between two cases, depending on the relationship between t and p .
Case 1: t p . By (5), P j2N l j t and so P j2N l j p : Case 2: t > p : By de…nition of p and (3), p and t are integer numbers. Hence, t p +1, and so (6); P j2N l j p . To prove (P1.2), assume P j2N p j p + 0:5 holds. We distinguish between two cases, depending on the relationship between t and p + 1.
Case 1: p + 1 t. By (5), P j2N u j p + 1: Case 2: p + 1 > t. By de…nition of p and (3), p + 1 and t are integer numbers. Hence, p t, and so P j2N p j p P j2N p j t: Since p + 0:5 P j2N p j p + 1, holds. Thus, p + 1 P j2N p j P j2N p j t; which implies p + 1 2 P j2N p j t: By (6), p + 1 P j2N u j . Proposition 1 implies that the real numbers ; b ; and b used to de…ne the four families of rules do exist and they are unique, and hence the rules are well-de…ned. To see that, observe that any f EL 2 F EL and f CEL 2 F CEL start allotting the good from p in a continuous and egalitarian (or constrained egalitarian) way until the full amount is allotted. On the other hand, any f EA 2 F EA and f CEA 2 F CEA start allotting the good from the vector of relevant extremes of the individually rational intervals in a continuous and egalitarian (or constrained egalitarian) way until the full amount is allotted. Proposition 1 guarantees that the direction of the allotment process goes in the right direction to reach the full amount, from either one of the two starting vectors. So, Corollary 1 holds.

Corollary 1
The real numbers ; b ; and b , used to de…ne respectively the families of rules F EL ; F CEL ; F EA and F CEA do exist and they are unique.
5 Results for symmetric single-peaked preferences

Individual rationality and basic impossibilities
In the next proposition we present some results related with the properties of rules, whenever they operate on problems where agents' preferences are symmetric single-peaked. The …rst result characterizes individually rational rules by stating that a rule is individually rational if and only if, for all pro…les, the rule selects a vector of allotments that belong to the individually rational intervals of their associated peaks. The second result characterizes individually rational and e¢ cient rules. We also show that some basic incompatibilities among properties of rules hold, even when agents' preferences are restricted to be symmetric single-peaked.

Proposition 2
The following statements hold.
There is no rule on P S satisfying group rationality and e¢ ciency.
(P2.4) There is no rule on P S satisfying individual rationality, e¢ ciency and strategyproofness.
(P2.5) There is no rule on P S satisfying individual rationality and envy-freeness on losses.
We now prove that f satis…es (E2.1). We …rst show that for all 2 P S , P j2N f j ( ) 2 fp ; p + 1g : Suppose that P j2N f j ( ) < p : By (E2.2) for all i 2 N; f i ( ) p i and there exists j 2 N such that f j ( ) < p j : Let y 2 F A be such that for all i 2 N; f i ( ) y i p i ; f j ( ) < y j p j and P j2N y j = p : Since by single-peakedness y i i f i ( ) for all i 2 N and y j j f j ( ); y Pareto dominates f ( ); a contradiction with the e¢ ciency of f: If P j2N f j ( ) > p + 1 the proof proceeds similarly. We distinguish among three cases, depending on the relationship between P j2N p j and p + 0:5.  9 There are however rules on P S satisfying simultaneously individual rationality and envy-freeness.
For instance, the rule f that, at each pro…le, assigns to each agent the closest integer to her peak. To see that f is not e¢ cient, consider the problem (N; ) 2 P S where N = f1; there exists j 2 N such that 2p j f j ( ) < y j and so y j j 2p j f j ( ) j f j ( ), a contradiction with the e¢ ciency of f . Case 2: P j2N p j = p + x with x > 0:5: To obtain a contradiction, suppose that P j2N f j ( ) = p : By (E2.2), for all i 2 N; f i ( ) p i . By individual rationality, there exists j 2 N such that 2p j f j ( ) > y j and so y j j 2p j f j ( ) j f j ( ), a contradiction with the e¢ ciency of f .
We now show that f is e¢ cient. By (E2.1), it is enough to consider two cases, depending on whether P j2N f j ( ) is equal to p or to p + 1.
for all i 2 N; and y j 0 2 (f j 0 ( ) ; p j 0 + (p j 0 f j 0 ( ))) for some j 0 2 N: By (E2.1) and our assumption, P j2N f j ( ) = p P j2N p j p + 0:5: Thus, p < P j2N y j < p + 1: Since P j2N y j 2 N 0 , we have a contradiction. Case 2: P j2N f j ( ) = p + 1: By (E2.2), f i ( ) p i for all i 2 N: Suppose f is not e¢ cient. Then, there exists y 2 F A that Pareto dominates f ( ) : Since preferences are symmetric single-peaked, for all i 2 N; and y j 0 2 (p j 0 (f j 0 ( ) p j 0 ) ; f j 0 ( )) for some j 0 2 N: By (E2.1) and our assumption, P j2N f j ( ) = p + 1 P j2N p j p + 0:5: Hence, Thus, p < P j2N y j < p + 1: Since P j2N y j 2 N 0 ; we have a contradiction. To apply the property of group rationality Table 1 indicates for each subset of agents with cardinality two the aggregate loss, assuming the best integer amount is allotted (i.e., for each S N with jSj = 2; min Table 1 Observe that 0:4 = P j2N p j Our main objective in this paper is to identify individually rational rules to be used to solve the division problem when the integer number of units is endogenous and agents' preferences are symmetric single-peaked. Part (P2.1) in Proposition 2 characterizes the class of all individually rational rules. Since this class is large, it is natural to ask whether individual rationality is compatible with other additional properties. E¢ ciency and strategy-proofness emerge as two of the most basic and desirable properties. However, (P2.4) in Proposition 2 says that no rule satis…es individual rationality, e¢ ciency and strategy-proofness simultaneously. In the next two subsections we study rules that are individually rational and e¢ cient (Subsection 5.2) and rules that are individually rational and strategy-proof (Subsection 5.3). For the …rst case, we identify the family of constrained equal losses rules and the family of constrained equal awards rules as the unique ones that in addition of being individually rational and e¢ cient satisfy also either justi…ed envy-freeness on losses or envy-freeness on awards, respectively (Theorem 1). In contrast, in Subsection 5.3 we …rst show that although there are individually rational and strategy-proof rules, they are not very interesting. For instance, we show in Proposition 4 that individually rationality and strategy-proofness are indeed incompatible with unanimity.

Individual rationality and e¢ ciency
Let 2 P S be a problem. Denote by IRE ( ) the set of feasible vectors of allotments satisfying individual rationality and e¢ ciency. It is easy to see that, by using similar arguments to the ones used to check that (P2.1) and (P2.2) in Proposition 2 hold, this set can be written as IRE ( ) = fx 2 R N + j P j2N x j 2 fp ; p + 1g and, for all i 2 N , l i x i p i when P j2N x j = p and p i x i u i when P j2N x j = p + 1g: By Proposition 1, the set IRE( ) is non-empty. Hence, a rule f satis…es individual rationality and e¢ ciency if and only if, for each 2 P S ; f ( ) 2 IRE ( ) : However, individual rationality and e¢ ciency are properties of rules that apply only to each problem separately. They do not impose conditions on how the rule should behave across problems. Thus, and given two di¤erent criteria compatible with individual rationality and e¢ ciency, a rule can choose, in an arbitrary way, at problem an allocation in IRE( ), following one criterion, while choosing at problem 0 an allocation in IRE( 0 ), following the other criterion. For instance the rule f that selects f 2 F CEL ( ) when p is odd and f 2 F CEA ( ) when p is even satis…es individual rationality and e¢ ciency. 10 Thus, it seems appropriate to require that the rule satis…es an additional property in order to eliminate this kind of arbitrariness. We will focus on two alternative properties related to envy-freeness: justi…ed envy-freeness on losses and envy-freeness on awards. But then, the consequence of requiring that rules (in addition of being individually rational and e¢ cient) satisfy either one of these two forms of non-envyness is that only one family of rules is left, either the family of constrained equal losses rules or the family of constrained equal awards rules, respectively. Theorem 1, the main result of the paper, characterizes axiomatically the two families on the domain of symmetric single-peaked preferences. Before proving Theorem 1, we provide in Proposition 3 preliminary results on the two families of rules that will be useful along the proof of Theorem 1 and in the sequel.  Justi…ed envy-freeness on losses. Let j 2 N be such that f j ( ) j k for all k 2 N 0 : We want to show that for all i 2 N; f i ( ) i maxfp i + (f j ( ) p j ); 0g: We distinguish among three cases, depending on the relationship between P j2N p j and p + 0:5: Case 1: P j2N p j < p + 0:5: By de…nition, f j ( ) = p j minfb ; p j l j g for all j 2 N . If p j l j b , then f j ( ) = l j ; which contradicts (8) because f j ( ) j l j u j and either l j or u j is an integer. Hence, Let i 2 N be arbitrary. We distinguish between two cases, depending on the relationship between b and p i l i . First, b p i l i . Then, by (9), Then, by de…nition, f i ( ) = l i : Since, by (9), single-peakedness implies that f i ( ) i maxfp i + (f j ( ) p j ); 0g: Case 2: P j2N p j > p + 0:5: By de…nition, f j ( ) = p j + min fb ; u j p j g for all j 2 N . If u j p j b , then f j ( ) = u j ; which contradicts (8) because f j ( ) j l j u j and either l j or u j is an integer. Hence, Let i 2 N be arbitrary. We distinguish between two cases, depending on the relationship between b and u i p i . First, b u i p i . Then, by (10), which means that f i ( ) = maxfp i + (f j ( ) p j ); 0g: Hence, f i ( ) i maxfp i + (f j ( ) p j ); 0g: Second, b > u i p i . Then, by de…nition, f i ( ) = u i : Since, by (10), single-peakedness implies that f i ( ) i maxfp i + (f j ( ) p j ); 0g: Case 3: P j2N p j = p + 0:5: Two cases are possible, The former is similar to Case 1 and the latter is similar to Case 2.
Participation. Let 2 P S , i 2 N and k 2 N 0 be such that k p i . We want to show We distinguish between two cases, depending on whether P j2N f i ( ) is equal to p or to p + 1.

Case 1:
P j2N f i ( ) = p : Since (as we have already proved) f is individually rational and e¢ cient, we can use (P2.2) and assert that P j2N p j p + 0:5: Then, f i ( ) = p i min fb ; p i l i g where b satis…es P j2N f j ( ) = p : Since p 0 i = p i k and k is an integer, p 0 = p k. We distinguish between two subcases, depending on whether P j2N p 0 j is strictly smaller than or equal to p 0 + 0:5.

Subcase 2:
P j2N p 0 j = p 0 +0:5: Again two subcases are possible. First, P j2N f j ( 0 ) = p 0 : Then, using the same argument to the one used in Subcase 1, and, by an argument similar to the one used in the …rst subcase, we conclude that f i ( ) i k + f i ( 0 ) :

Case 2:
P j2N f i ( ) = p + 1: Since (as we have already proved) f is individually rational and e¢ cient, we can use (P2.2) and assert that P j2N p j p + 0:5: Then, f i ( ) = p i + min fb ; u i p i g where b satis…es P j2N f j ( ) = p + 1: Since p 0 i = p i k and k is an integer, p 0 = p k. We distinguish between two subcases, depending on whether P j2N p 0 j is strictly larger than or equal to p 0 + 0:5. Subcase 1: which implies that f i ( ) i k + f i ( 0 ).

Subcase 2:
P j2N p 0 j = p 0 + 0:5: Again two subcases are possible. First, P j2N f j ( 0 ) = p 0 + 1: Then, using the same argument to the one used in Subcase 1, Envy-freeness on awards. We distinguish among three cases, depending on the relationship between P j2N p j and p + 0:5: Case 1: P j2N p j < p + 0:5: By de…nition, f i ( ) p i for all i 2 N: Suppose that f does not satisfy envy-freeness on awards. Then, there exist i; j 2 N and x 2 [min fa i ; a j g ; max fa i ; a j g] such that Hence, f i ( ) is not the peak of i and so f i ( ) < p i : Since f i ( ) = l i + minf b ; p i l i g; b < p i l i and hence Thus, a i = b : We distinguish between two subcases.
Subcase 2: minf b ; p j l j g = p j l j < b : By de…nition; f j ( ) = p j and a j = f j ( ) l j = p j l j : Thus, x 2 [p j l j ; b ] and where the equality follows from (12). By single-peakedness, f i ( ) i l i +x; a contradiction with (11).
Case 2: P j2N p j > p + 0:5: By de…nition, f i ( ) p i for all i 2 N: Suppose that f does not satisfy envy-freeness on awards. Then, there exist i; j 2 N and Hence, f i ( ) is not the peak of i and so f i ( ) > p i : Thus, a i = b : We distinguish between two subcases.
it must be the case that x = b . Hence, by (13), which is a contradiction.
Subcase 2: minf b ; u j p j g = u j p j < b : By de…nition; f j ( ) = p j and a j = u j f j ( ) = u j p j : Thus, x 2 [u j p j ; b ] and where the equality follows from (14). By single-peakedness, f i ( ) i u i x; a contradiction with (13).
Participation. Let 2 P S , i 2 N and k 2 N 0 be such that k p i . We want to show We distinguish between two cases, depending on whether P j2N f i ( ) is equal to p or to p + 1.

Case 1:
P j2N f i ( ) = p : Since (as we have already proved) f is individually rational and e¢ cient, we can use (P2.2) and assert that P j2N p j p + 0:5: Then, f i ( ) = p i minf b ; p i l i g where b satis…es P j2N f j ( ) = p : Since p 0 i = p i k and k is an integer, p 0 = p k. We distinguish between two subcases, depending on whether P j2N p 0 j is strictly smaller than or equal to p 0 + 0:5.

Subcase 2:
P j2N p 0 j = p 0 + 0:5: Again two cases are possible. First, P j2N f j ( 0 ) = p 0 : Then, using the same argument to the one used in Subcase 1, and, by an argument similar to the one used in the …rst subcase, we conclude that f i ( ) i k + f i ( 0 ).

Case 2:
P j2N f i ( ) = p + 1: Since (as we have already proved) f is individually rational and e¢ cient, we can use (P2.2) and assert that P j2N p j p + 0:5: Since p 0 i = p i k and k is an integer, p 0 = p k. We distinguish between two subcases, depending on whether P j2N p 0 j is strictly larger than or equal to p 0 + 0:5. Subcase 1: which implies that f i ( ) i k + f i ( 0 ).
Subcase 2: P j2N p 0 j = p 0 + 0:5: Again two subcases are possible. First, P j2N f j ( 0 ) = p 0 + 1: Then, the same argument used in Subcase 1 shows that and, by an argument similar to the one used in the …rst subcase, we conclude that f i ( ) i k + f i ( 0 ). Let f be a rule satisfying individual rationality, e¢ ciency, and justi…ed envy-freeness on losses. Let 2 P S be a problem. By (7), it is su¢ cient to distinguish between two cases. Hence, for all i 2 N; f i ( ) = p i min fb ; p i l i g : Thus, at pro…le , f coincides with a constrained equal losses rule: Assume now that x j < x i for some pair i; j 2 N: By single peakedness, p i x j i p i x i : Since holds, by justi…ed envy-freeness on losses, there must exist y j 2 N 0 such that f j ( ) j y j : By individual rationality, f j ( ) = l j : Let S be the set of agents with the largest loss from the peak: Namely, S = fi 0 2 N j x i 0 x j 0 for all j 0 2 N g: Since N is …nite, S 6 = ;: Moreover, our assumption that x j < x i for some pair i; j 2 N implies S ( N: For each b j 2 S; set b = x| and observe that Hence, for all i 2 N; f i ( ) = p i + min fb ; u i p i g : Thus,at pro…le , f coincides with a constrained equal losses rule. Assume now that x j < x i for some pair i; j 2 N: By single peakedness, p i + x j i p i + x i : Since holds, by justi…ed envy-freeness on losses, there must exist y j 2 N 0 such that f j ( ) j y j : By individual rationality, Let S and b be de…ned as in Case 1. Then, for each at pro…le , f coincides with a constrained equal losses rule.
(T1.2) Let f be a constrained equal awards rule. By Proposition 3, f satis…es individual rationality, e¢ ciency and envy-freeness on awards.
Let f be a rule satisfying individual rationality, e¢ ciency, and envy-freeness on awards. Let 2 P S be a problem. By (7), it is su¢ cient to distinguish between two cases. Case 1: P j2N f j ( ) = p : By (15), for each i 2 N , f i ( ) = l i + a i , where 0 a i p i l i . We …rst prove that if a i < a j for some pair i; j 2 N; then a i = p i l i : Assume not; then, there exist i; j 2 N such that a i < a j and a i < p i l i : Let x 2 R + be such that x 2 (a i ; min fa j ; p i l i g] : Since f i ( ) = l i + a i < l i + x p i , single-peakedness implies that l i + x i f i ( ) where x 2 (a i ; a j ], contradicting envy-freeness on awards. Let S be the set of agents with the largest award from the peak: Namely, S = fi 0 2 N j a i 0 a j 0 for all j 0 2 N g: Since N is …nite, S 6 = ;: We consider two subcases. Assume not; then, there exist i; j 2 N such that a i < a j and a i < u i p i : Let x 2 R + be such that x 2 (a i ; min fa j ; u i p i g] : Since p i u i x < u i a i = f i ( ), singlepeakedness implies that u i x i f i ( ) where x 2 (a i ; a j ], contradicting envy-freeness on awards. Let S be the set of agents with the largest award from the peak: Namely, S = fi 0 2 N j a i 0 a j 0 for all j 0 2 N g: Since N is …nite, S 6 = ;: We consider two subcases. (R2.1) The rule f de…ned by assigning to each agent i 2 N her most preferred integer, satis…es individual rationality and justi…ed envy-freeness on losses but it is not e¢ cient.
(R2.2) Any rule f 2 F EL satis…es e¢ ciency and justi…ed envy-freeness on losses but is not individually rational.
(R2.3) Any rule f 2 F CEA satis…es individual rationality and e¢ ciency but it does not satisfy justi…ed envy-freeness on losses.
(R2.4) The rule f de…ned in (R2.1) satis…es individual rationality and envy-freeness on awards but it is not e¢ cient.
(R2.5) Any rule f 2 F EA satis…es e¢ ciency and envy-freeness on awards but it is not individually rational.
(R2.6) Any rule f 2 F CEL satis…es individual rationality and e¢ ciency but it is not envy-freeness on awards.

Individual rationality and strategy-proofness
We now study the set of rules satisfying individual rationality and strategy-proofness on the set of symmetric single-peaked preferences. There are many rules satisfying both properties. For instance, the rule that selects f ( ) = ([p i ]) i2N for all 2 P S is individually rational and strategy-proof. But there are many more, yet some of them are very di¢ cult to justify as reasonable solutions to the problem. Consider the following family of rules. For each vector x 2 R N + satisfying P i2N x i 2 N 0 ; de…ne f x as the rule that when x is at least as good as ([p i ]) i2N for each i 2 N; f x selects x: Otherwise f x selects ([p i ]) i2N : It is easy to see that each rule in the family ff x j x 2 R N + and P i2N x i 2 N 0 g is individually rational and strategy-proof. However, this family contains many arbitrary and non-interesting rules. 11 Thus, we ask whether it is possible to identify a subset of individually rational and strategy-proof rules satisfying additionally a basic, weak and desirable property. We interpret Proposition 4 below as giving a negative answer to this question: individual rationality and strategy-proofness are not compatible even with unanimity, a very weak form of e¢ ciency.
Proposition 4 There is no rule on P S satisfying individual rationality, strategy-proofness and unanimity.
Since we have obtained a contradiction in each of the possible cases, there does not exist a rule satisfying simultaneously the properties of individual rationality, strategyproofness and unanimity.

Final remarks
Before …nishing the paper we deal with two natural questions. First, are our results generalizable to rules de…ned on P, the set of problems where agents have single-peaked preferences? Second, how do well-known rules, used to solve the division problem with a …xed amount of the good, behave when the number of units to allot is endogenous? We partially answer the two questions separately in each of the next two subsections.

Results for general single-peaked preferences
Obviously, all the impossibility results we have obtained for rules operating on the domain of symmetric single-peaked preferences also hold when they operate on the larger domain.
Proposition 5 contains some results on rules operating on the full domain of singlepeaked preferences. But before stating it, we need some additional notation to refer to the extremes of the individually rational intervals for those preferences. Let i be a single-peaked preference with peak p i : De…ne By continuity and single-peakedness, there are two numbers b l i ; b u i 2 R + satisfying the following conditions: The following statements hold.
(P5.1) A rule f on P is individually rational if and only if, for all 2 P and i 2 N , If a rule f on P is e¢ cient, then 3) There exist rules on P satisfying individual rationality and e¢ ciency.
(P5.2) It is similar to the proof of (P2.2) in Proposition 2, and hence we omit it.
(P5.3) It is enough to prove that for each 2 P there is an allotment y in F A satisfying individual rationality. If y belongs to the Pareto frontier of F A, the statement follows. Otherwise, each allotment in F A that Pareto dominates y satis…es both properties. Consider now b i de…ned as in (20). Then (b i ) i2N 2 F A and satis…es individual rationality.
(P5.4) Consider the rule f that, for each 2 P and each i 2 N; f i ( ) = b i , where b i is de…ned as in (20): It is immediate to see that f is individually rational and strategy-proof.
Example 2 below shows that the rules in F CEL and F CEA are not e¢ cient on the larger domain of single-peaked preferences.

Other rules
In the classical division problem, where a …xed amount of the good has to be allotted, the uniform rule emerges as the one that satis…es many desirable properties. For instance, Sprumont (1991) shows that it is the unique rule satisfying strategy-proofness, e¢ ciency and anonymity. Sprumont (1991) also shows that in this characterization anonymity can be replaced by non-envyness and Ching (1994) shows that in fact anonymity can be replaced by the weaker requirement of equal treatment of equals. Sönmez (1994) shows that the uniform rule is the unique one satisfying consistency, monotonicity and individual rationality from equal division. Thomson (1994aThomson ( , 1994bThomson ( , 1995Thomson ( and 1997 contains alternative characterizations of the uniform rule using the properties of one sided resourcemonotonicity, converse consistency, weak population-monotonicity and replication invariance, respectively. On the other hand, if one is concerned mostly with incentives and e¢ ciency issues (and leaves aside any equity principle), sequential dictator rules emerge as natural ways of solving the classical division problem, since they are strategy-proof and e¢ cient. However, we brie ‡y argue below that the natural adaptations of all these rules to our setting with endogenous integer units of the good are far from being desirable since they are neither individually rational nor strategy-proof even on P S .

Uniform rule
We adapt the uniform rule to our setting. As before, there will be many extensions of the uniform rule. At pro…les 2 P S where either P j2N p j < p +0:5 or P j2N p j > p +0:5 all extensions coincide and allot the e¢ cient units of the good. However, at pro…les 2 P S where P j2N p j = p + 0:5, there are two e¢ cient integers that could be allotted. The family of extended uniform rules contains all these extensions.
Extended uniform. We say that f is an extended uniform rule if, for all 2 P S , if P j2N p j < p + 0:5 (max fp i ; g) i2N if P j2N p j > p + 0:5 (min fp i ; g) i2N or (max fp i ; g) i2N if P j2N p j = p + 0:5; where is the unique real number for which it holds that P j2N min fp j ; g = p or P j2N max fp j ; g = p + 1.
Denote by F EU the set of all extended uniform rules.
Proposition 6 Let f be an extended uniform rule. Then, f is e¢ cient on P S but it is neither individually rational nor strategy-proof on P S .
Constrained extended uniform. We say that f is a constrained extended uniform rule if, for all 2 P S , f ( ) = 8 > < > : (min fp i ; maxfl i ; gg) i2N if P j2N p j < p + 0:5 (max fp i ; minfu i ; gg) i2N if P j2N p j > p + 0:5 (min fp i ; maxfl i ; gg) i2N or (max fp i ; minfu i ; gg) i2N if P j2N p j = p + 0:5; where is the unique real number for which it holds that P j2N min fp j ; maxfl j ; gg = p or P j2N max fp j ; minfu j ; gg = p + 1. In contrast to Bergantiños, Massó and Neme (2015), constrained extended uniform rules are not strategy-proof in this new setting. Nevertheless, they are still appealing because they belong to the class of individually rational and e¢ cient rules, and moreover, they satisfy equal treatment of equals.

Sequential dictator
We adapt the sequential dictator rule to the setting where the integer number of units to be allotted is endogenous. Fix an ordering on the set of agents and let them select sequentially, following the ordering, the amount they want (their peak) among the set of all e¢ cient allocations. Formally, let : N ! f1; : : : ; ng be a one-to-one mapping de…ning an ordering on the set of agents N ; namely, for i; j 2 N , (i) < (j) means that i goes before j in the ordering : Sequential dictator. We say that f SD is the sequential dictator rule relative to the ordering if, for all 2 P S and i 2 N , p k ] P fj 0 2Sj (j 0 )< (i)g p j 0 ; 0g otherwise.
Proposition 7 Let be an ordering. Then, f SD is e¢ cient on P S but it is neither individually rational nor strategy-proof on P S .