Discussion Paper No. 686

COALITIONALLY STRATEGY-PROOF

RULES IN ALLOTMENT ECONOMIES WITH HOMOGENEOUS

INDIVISIBLE GOODS

Kentaro Hatsumi and

Shigehiro Serizawa

March 2007 Revised July 2008

Revised February 2009

The Institute of Social and Economic Research Osaka University

6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Coalitionally Strategy-Proof Rules in Allotment

Economies with Homogeneous Indivisible Goods∗

Kentaro Hatsumi† and Shigehiro Serizawa‡

First Version: January 12, 2006This Version: January 13, 2009

Abstract: We consider the allotment problem of homogeneous indivisi-ble goods among agents with single-peaked and risk-averse von Neumann-Morgenstern expected utility functions. We establish that a rule satisfiescoalitional strategy-proofness, same-sideness, and strong symmetry if andonly if it is the uniform probabilistic rule. By constructing an example,we show that if same-sideness is replaced by respect for unanimity, thisstatement does not hold even with the additional requirements of no-envy,anonymity, at most binary, peaks-onlyness and continuity.

Keywords: allotment problem, coalitional strategy-proofness, homogeneousindivisible goods, single-peaked preference, uniform probabilistic rule

JEL Classification Numbers: C72, D71, D81

∗We would like to thank the associate editor and two anonymous referees whose com-ments and suggestions significantly improve this paper. We are also grateful to MasakiAoyagi, Kazuhiko Hashimoto, Herve Moulin, Hiroo Sasaki, Koji Takamiya, WilliamThomson, Takuma Wakayama as well as other participants at the Eighth InternationalMeeting of the Society for Social Choice and Welfare in Istanbul, the 2006 Autumn AnnualMeeting of the Japanese Economic Association in Osaka, and Yokohama National Univer-sity Seminar of Economic Theory for their helpful comments. We acknowledge financialsupports from the Japan Society for the Promotion of Science via the Research Fellowshipfor Young Scientist (Hatsumi) and the Grant-in-Aid for Scientific Research (Serizawa).

†Graduate School of Economics, Osaka University. 1-7, Machikaneyama, Toyonaka,560-0043, Japan. E-Mail: egp020hk@mail2.econ.osaka-u.ac.jp

‡Institute of Social and Economic Research, Osaka University. 6-1, Mihogaoka, Ibaraki,567-0047, Japan. E-Mail: serizawa@iser.osaka-u.ac.jp

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1 Introduction

Situations exist that can be interpreted as allotment problems of goods with-out any disposal. For example, consider the situation where the managerof a firm assigns staff overtime with a fixed wage. Each staff member hasher own ideal overtime working hours: some staff may not want any over-time, others who need extra spending money may want to do some but nottoo much overtime. The manager has the power to assign all staff overtimework in a just proportion. We can interpret this situation as an allotmentproblem of overtime work for the manager.

Sprumont (1991) initiates an axiomatic analysis of this kind of allotmentproblem. He analyzes a model in which there is a perfectly divisible good andallocation rules are deterministic. In this model, he assumes that agents have“single-peaked” preferences over their consumption levels, and characterizes“the uniform rule”1. A preference is single-peaked if there is some ideal pointcalled a “peak”, and welfare is strictly decreasing on either direction awayfrom the peak. The uniform rule is the rule such that agents are allowed tochoose their preferred consumption subject to a common bound, which ischosen to satisfy feasibility. Sprumont (1991) shows that the uniform rule isa unique allocation rule satisfying strategy-proofness, Pareto-efficiency, andanonymity. Strategy-proofness is a frequently employed incentive compati-bility property. It requires that it is a weakly dominant strategy for eachagent to represent her true preference. Anonymity requires that the nameof each agent does not matter for the outcome allocation. Many subsequentstudies follow Sprumont (1991) in analyzing the uniform rule from differentperspectives by employing various axioms2.

In the real world, the goods to be allocated are not often perfectly divis-ible. In the above example of assigning overtime work, working time is ofteninstitutionally restricted to hour units, even though time is perfectly divisi-ble. If the number of units is sufficiently large, the assumption of a perfectlydivisible good may approximate the situation. However, if the number ofunits is small, it is doubtful.

Based on the works by Sasaki (1997) and Kureishi (2000), Ehlers andKlaus (2003) investigate the problem of probabilistically allocating finiteunits of homogeneous indivisible goods. They show that when agents havesingle-peaked preferences with a probabilistic extension3, “the uniform prob-

1The uniform rule is first considered by Benassy (1982) for the analysis of fixed priceeconomies.

2For example, Thomson (1994a, 1994b, 1995), Barbera, Jackson and Neme (1997),Ching and Serizawa (1998), Masso and Neme (2001, 2004, 2007), Chun (2006), Kesten(2006), Klaus (2006), and Mizobuchi and Serizawa (2006).

3Ehlers and Klaus (2003) define preferences with a probabilistic extension as follows.A marginal (probability) distribution is weakly preferred to another if the first marginaldistribution assigns the upper contour set of each consumption level at least the sameprobability as that assigned by the second. This preference drops completeness over the

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abilistic rule”, a probabilistic variant of the uniform rule, is a unique rulesatisfying strategy-proofness, Pareto-efficiency, and no-envy4. They alsoshow that when agents have single-peaked and “risk-averse” utility func-tions satisfying the von Neumann-Morgenstern expected utility property,the uniform probabilistic rule is a unique rule satisfying strategy-proofness,Pareto-efficiency, and “symmetry”5. The latter is the counterpart of Ching(1994)’s result showing that anonymity in the uniqueness result of Sprumont(1991) can be weaken to symmetry in the deterministic model. Ehlers andKlaus (2003) demonstrate that the latter result is obtained as a corollaryof Ching (1994) especially due to at most binary property, which is a prop-erty implied by Pareto-efficiency and requires that each agent has strictlypositive probabilities over at most two adjacent consumption levels.

This probabilistic model of allotment economies developed by Ehlers andKlaus (2003) is very important and interesting. However, research in thisarea is relatively underdeveloped6. The target of the present paper is toprovide richer knowledge on this probabilistic model, and to investigate itusing other interesting axioms other than strategy-proofness.

In this paper, we focus on the property of “coalitional strategy-proofness”.Coalitional strategy-proofness is a stronger concept than strategy-proofness,and requires that no coalition can increase the utilities of all members atthe same time. In situations where agents are likely to cooperate in mis-representing their preferences, the property of coalitional strategy-proofnessis beneficial for making agents reveal their true preferences without worry-ing about cooperative manipulation. Our main result is that when agentspossess single-peaked and risk-averse von Neumann-Morgenstern expectedutility functions, the uniform probabilistic rule is a unique probabilistic al-location rule satisfying coalitional strategy-proofness, “same-sideness”, and“strong symmetry”7. Same-sideness in this probabilistic model is a weakerefficiency property than Pareto-efficiency, and requires that if the indivisiblegoods are in excess demand, that is, if the sum of agents’ peaks is greaterthan the endowment, then no agent receives an amount more than her peakwith positive probability, and conversely, if the goods are in excess supply,then no agent receive an amount less than her peak with positive proba-bility. Thus, our result is an alternative characterization of the uniformprobabilistic rule in Ehlers and Klaus (2003).

set of marginal distributions.4No-envy requires that no agent strictly prefers the consumption of any other agent to

her own.5Symmetry requires that whenever two agents have the same preferences, they receive

the indifferent consumptions.6As far as we know, Kureishi and Mizukami (2007) is the only other research article

on this environment. They show that symmetry in the second result of Ehlers and Klaus(2003) can be replaced by “equal probability for the best” property.

7Strong symmetry requires that whenever two agents have the same preferences, thegoods are distributed to them by the same probability distribution.

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We also show by constructing an example that if same-sideness is re-placed by “respect for unanimity”8, which is a weaker axiom than same-sideness, the uniqueness of the uniform probabilistic rule is violated evenwith “strong coalitional strategy-proofness”9, no-envy, anonymity and atmost binary. Furthermore, we find that even with the additional require-ments of “peaks-onlyness”10 and “continuity”11, the uniqueness of the uni-form probabilistic rule does not hold. That is, the uniform probabilistic ruleis not a unique allocation rule satisfying peaks-onlyness and continuity inaddition to the above requirements.

In his recent work, Serizawa (2006) shows that in Sprumont’s (1991) de-terministic model of a perfectly divisible good, the uniform rule is a uniquerule satisfying “effectively pairwise strategy-proofness”, respect for unanim-ity and symmetry. Effectively pairwise strategy-proofness is a property thatrules are strategy-proof and that if a pair of agents can make both agentsbetter off by pairwise manipulation, one of them has an incentive to betrayher partner. It requires rules to be immune only to unilateral manipulationsand to pairwise manipulations in which agents have no incentives to be-tray, while coalitional strategy-proofness requires rules to be immune to allcoalitional manipulations. In this point, the former is much weaker than thelatter. Our results provide a remarkable comparison to Serizawa’s (2006) re-sult, and demonstrate that when dropping the property of same-sideness, theprobabilistic assignment problem is much different from Sprumont’s (1991)deterministic problem.

This paper is organized as follows. Section 2 describes the model andthe results. Section 3 concludes the paper. Technical discussions includingthe proofs of facts and the theorem are in the Appendix.

2 The model and the results

There are k ∈ Z++12 units of homogeneous indivisible goods. We con-

sider the problem of allotting k units of the goods to a set of agents N ={1, · · · , n}. A coalition is a subset N ′ of N . Given a coalition N ′ ⊆ N andan agent i ∈ N , we denote the coalition N\N ′ by −N ′, and the coalitionN\{i} by −i . Let K = {0, 1, · · · , k}, which is the set of consumption levels.We call a = (x1, · · · , xn) ∈ Kn a feasible allocation if

∑i∈N xi = k. Let A

8Respect for unanimity requires that if the sum of agents’ peaks equals the endowment,all agents receive their peak consumption.

9Strong coalitional strategy-proofness requires that by coalitional manipulation, nocoalition can increase the utility of any member in the coalition without decreasing theutility of some other member.

10Peaks-onlyness requires that the outcome allocation depends only on the peak profile.11Continuity requires that small changes in the utility profile cause only small changes

in the outcome allocation.12Z++ is the set of positive integers and Z+ is the set of nonnegative integers.

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denote the set of all feasible allocations.A (probability) distribution over A is interpreted as a lottery on A.

For A = {a1, · · · , a|A|}13, we denote such a distribution over A by [p1 ◦a1, · · · , p|A| ◦ a|A|] where for all l ∈ {1, · · · , |A|}, pl ∈ [0, 1] is the probabilityof al, and

∑|A|l=1 pl = 1. For convenience, to express a distribution, we only

write feasible allocations al that occur with a strictly positive probabilitypl > 0. For example, instead of [12 ◦ a1, 1

2 ◦ a2, 0 ◦ a3, · · · , 0 ◦ a|A|], we write[12 ◦ a1, 1

2 ◦ a2]. Let P denote the set of all distributions over A.Let Pi denote the set of all marginal (probability) distributions for i ∈ N

over her allotments in K, induced by all p ∈ P . Each agent i ∈ N only caresfor her marginal distribution pi ∈ Pi on K. Given pi ∈ Pi and K ′ ⊆ K,pi(K ′) denotes the probability that the marginal distribution pi places overK ′. If K ′ = {x}, we write simply pi(x) instead of pi(K ′) to refer to theprobability that agent i receives x units through the marginal distributionpi.

Each agent i ∈ N has a utility function ui : K → R that satisfies the(von Neumann-Morgenstern) expected utility property. Given a marginaldistribution pi ∈ Pi, we denote the expected utility by

E(pi; ui) =∑x∈K

pi(x) · ui(x).

It is well known that any affine transformation of an expected utilityfunction represents the same preference relation over Pi. Therefore, wenormalize any utility function ui as ui(0) = 0 and ||ui(1)−ui(0)|| = 114. Weassume that the utility functions possess the following two properties.

Definition. A utility function ui is single-peaked if there exists a uniquepeak b(ui) ∈ K such that for all x, y ∈ K with x > y ≥ b(ui) or b(ui) ≥ y >x, u(y) > u(x).

Definition. A utility function ui is risk-averse if for all x ∈ K\{0, k},ui(x) − ui(x − 1) > ui(x + 1) − ui(x).

Let U denote the class of all single-peaked and risk-averse von Neumann-Morgenstern utility functions15. Let Un denote the set of all von Neumann-Morgenstern utility profiles u = (ui)i∈N such that for all i ∈ N , ui ∈ U .Given a coalition N ′ ⊆ N , let uN ′ denote a partial utility profile of thecoalition (ui)i∈N ′ such that for all i ∈ N ′, ui ∈ U . (uN ′ , u−N ′) representsthe utility profile such that agent i ∈ N ′ has ui and agent j ∈ −N ′ has uj .

13|A| is the number of feasible allocations.14For a ∈ R, ||a|| represents the absolute value of a.15If a utility function exhibits risk-aversion, then it is weakly single-peaked, i.e., there

exist at most two adjacent peaks b(ui), b(ui) + 1 ∈ K, and for all x, y ∈ K, if x > y ≥b(ui) + 1 or b(ui) ≥ y > x, then u(y) > u(x). However, (strict) single-peakedness andrisk-averseness are independent.

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Note that the two distributions need not be equal, even though theirmarginal distributions are the same, as illustrated by Example 1 below.

Example 1 (Ehlers and Klaus, 2003). Let N = {1, 2, 3}, k = 9, p = [13 ◦(3, 6, 0), 1

3 ◦(0, 3, 6), 13 ◦(6, 0, 3)], and p′ = [13 ◦(3, 0, 6), 1

3 ◦(6, 3, 0), 13 ◦(0, 6, 3)].

Let pi and p′i be the marginal distributions for i ∈ N induced by p and p′.Then, for all i ∈ N , pi = p′i, but p 6= p′. ♦

If two distributions p, p′ ∈ P have the same marginal distribution profile,i.e., pi = p′i for all i ∈ N , then p and p′ are equivalent from the viewpointof agents. Thus, we focus on marginal distribution profiles instead of distri-butions on A. A marginal distribution profile p = (p1, · · · , pn) ∈

∏i∈N Pi is

feasible if there is a probability distribution p ∈ P such that for all i ∈ N , pi

is induced by p. We denote by P the set of all feasible marginal distributionprofiles.

We define two efficiency properties of the marginal distribution profiles.The first is “Pareto-efficiency”, one of the most common efficiency proper-ties in economics. This requires that there are no other feasible marginaldistributions where all agents are weakly better-off and some are strictlybetter-off. The other is “same-sideness.” This requires that if the indivisiblegoods are in excess demand, that is, if the sum of agents’ peaks is greaterthan the endowment, then no agent receives an amount more than her peakwith positive probability, and conversely, if the goods are in excess supply,then no agent receive an amount less than her peak with positive probability.Same-sideness has the following merits as an alternative efficiency property.First, it is a necessary condition for Pareto-efficiency. Second, it is muchsimpler and easier to check. Moreover, violating same-sideness results inthe imposition of amounts more than wished on agents when the goods arein excess demand. Such allocations can hardly be justified16.

Definition. A marginal distribution profile p ∈ P satisfies Pareto-efficiencywith respect to u ∈ Un if there is no p′ ∈ P such that for all i ∈ N ,E(p′i; ui) ≥ E(pi; ui) and for some j ∈ N , E(p′j ; uj) > E(pj ; uj).

Definition. A marginal distribution profile p ∈ P satisfies same-sidenesswith respect to u ∈ Un if

∑i∈N b(ui) ≥ k implies that for all i ∈ N ,

pi([0, b(ui)]) = 1, and∑

i∈N b(ui) ≤ k implies that for all i ∈ N , pi([b(ui), k]) =1.

In the deterministic model with a perfectly divisible good, Pareto-efficiencyis equivalent to same-sideness. In this probabilistic model, Pareto-efficiencyimplies same-sideness, however, Example 2 illustrates that the inverse im-plication is not true.

16Amoros (2002) investigates the allotment economies with two perfectly divisible goodswhere same-sideness is a weaker condition of Pareto-efficiency, and employs same-sidenessin his main characterization. Amoros (2002) refers to same-sideness as “condition E”.

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Example 2. Let N = {1, 2} and k = 2. Let u ∈ U2 be such that u1 = u2,u1(0) = 0, u1(1) = 1 and u1(2) = 3

2 . Let p ∈ P be such that pi(0) = pi(2) =12 for i = 1, 2 and p′ ∈ P be such that p′i(1) = 1 for i = 1, 2.

Then, E(pi; ui) = 34 and E(p′i; ui) = 1 for i = 1, 2. p is same-sided with

respect to u but is not Pareto-efficient with respect to u. ♦

The next property called “at most binary” states that each agent hasstrictly positive probabilities over at most two adjacent elements of K. Fact1 below implies that this property has an important role in the model.

Definition. A marginal distribution profile p ∈ P satisfies at most binaryif for all i ∈ N , there exists x ∈ K\{k} such that pi(x) + pi(x + 1) = 1.

Fact 1 (Sasaki 1997, Kureishi 2000). A marginal distribution profile p ∈ Psatisfies Pareto-efficiency with respect to u if and only if it satisfies same-sideness with respect to u and at most binary17.

A probabilistic (allocation) rule is a function f : Un → P . Given a prob-abilistic rule f , u ∈ Un and i ∈ N , fi(u) denotes the marginal distributionof agent i when the utility profile is u under the rule f . Given K ′ ⊆ K,fi(u)(K ′) denotes the probability that fi(u) places over K ′, and if K ′ = {x},fi(u)(x) denotes fi(u)(K ′).

We introduce several properties of f . The first properties relate to theefficiency of f .

Definition. A probabilistic rule f satisfies Pareto-efficiency if for all u ∈Un, f(u) is Pareto-efficient with respect to u.

Definition. A probabilistic rule f satisfies same-sideness if for all u ∈ Un,f(u) is same-sided with respect to u.

Definition. A probabilistic rule f satisfies at most binary if for all u ∈ Un,f(u) satisfies at most binary.

Definition. A probabilistic rule f satisfies respect for unanimity if for allu ∈ Un such that

∑i∈N b(ui) = k, and all i ∈ N , fi(u)(b(ui)) = 1.

Note that a probabilistic rule f satisfies Pareto-efficiency if and only if itsatisfies same-sideness and at most binary, due to Fact 1. In addition, notethat same-sideness implies respect for unanimity.

The following three properties relate to the incentive compatibility foragents to reveal their true utility functions. “Strategy-proofness” requiresthat no agent can increase her utility by manipulating her revealed utility.“Coalitional strategy-proofness” is a stronger condition; it requires that no

17The only if part of Fact 1 is given by Sasaki (1997) and the if part is by Kureishi(2000). Unfortunately, these two are rarely promulgated. Therefore, we reconstruct theproof in an online supplementary note (Hatsumi and Serizawa 2009).

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coalition can increase the utilities of all members at the same time. Fur-ther, “strong coalitional strategy-proofness” requires that no coalition canincrease the utility of any member in the coalition via coalitional manipula-tion without decreasing the utility of some other member in the coalition.

Definition. A probabilistic rule f satisfies strategy-proofness if for all u ∈Un, for all i ∈ N , and all ui ∈ U , E(fi(u);ui) ≥ E(fi(ui, u−i);ui).

Definition. A probabilistic rule f satisfies coalitional strategy-proofness iffor all u ∈ Un, all N ′ ⊆ N , and all uN ′ ∈ UN ′

, there exists i ∈ N ′ such thatE(fi(u);ui) ≥ E(fi(uN ′ , u−N ′);ui).

Definition. A probabilistic rule f satisfies strong coalitional strategy-proofnessif for all u ∈ Un, all N ′ ⊆ N , and all uN ′ ∈ UN ′

, whenever there is i ∈ N ′

such that E(fi(uN ′ , u−N ′);ui) > E(fi(u);ui), there exists j ∈ N ′ such thatE(fj(u);uj) > E(fj(uN ′ , u−N ′);uj).

In addition, we introduce several properties relating to fairness. “Sym-metry” requires that agents with the same utility functions obtain the sameexpected utilities. “Strong symmetry” requires that agents with the sameutility functions have the same marginal distributions. “No-envy” requiresthat no agent strictly prefers the marginal distribution of any other agentto her own. “Anonymity” requires that the name of each agent does notmatter. The relationships between these properties are as follows. No-envyand anonymity are independent. No-envy and strong symmetry are alsoindependent. Anonymity implies strong symmetry. Strong symmetry orno-envy implies symmetry.

Definition. A probabilistic rule f satisfies symmetry if for all u ∈ Un andall i, j ∈ N such that ui = uj , E(fi(u);ui) = E(fj(u);uj).

Definition. A probabilistic rule f satisfies strong symmetry if for all u ∈ Un

and all i, j ∈ N such that ui = uj , fi(u) = fj(u).

Definition. A probabilistic rule f satisfies no-envy if for all u ∈ Un and alli, j ∈ N , E(fi(u);ui) ≥ E(fj(u);ui).

Definition. Let Πn be the class of all permutations on N . For all u ∈ Un

and all π ∈ Πn, let uπ = (uπ(i))i∈N . A probabilistic rule f satisfies anonymityif for all u ∈ Un, all π ∈ Πn, and all i ∈ N , fπ(i)(u) = fi(uπ).

We note Fact 2 below related to the feasibility of the marginal distribu-tion profile. Given that all marginal distribution profiles other than thosein Examples 4, 5, and 6 satisfy the conditions of Fact 2, we do not explicitlydiscuss the feasibilities.

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Fact 2. Let Na ⊆ N and N b = −Na. Let na ∈ N ∪ {0} be the number ofagents in Na and nb ∈ N ∪ {0} be that in N b. Let {xi}i∈Na ∈ Kna

be suchthat

∑i∈Na xi ≤ k. Let µ ∈ R+ be such that µ = (k −

∑i∈Na xi)/nb, and

xµ ∈ K be such that µ ∈ [xµ, xµ+1). Let a marginal distribution profile p besuch that for all i ∈ Na, pi(xi) = 1, and for all j ∈ N b, pj(xµ) = 1− (µ−xµ)and pj(xµ + 1) = µ − xµ. Then, p is feasible, i.e., p ∈ P .

The proof of Fact 2 is in the appendix.We introduce the “uniform probabilistic rule”. The uniform probabilistic

rule assigns agents the marginal distribution profile subject to the commonbound depending on each utility profile to satisfy feasibility. Before theformal description of the uniform probabilistic rule, we introduce an exampleof an algorithm to find the outcome of the uniform probabilistic rule.

Example 3 (Sasaki, 1997). (i) Let N = {1, 2, 3, 4} and k = 15. Assumeu ∈ U4 is such that b(u1) = 1, b(u2) = b(u3) = 2, and b(u4) = 5. Note that∑

i∈N b(ui) < k (excess supply). First, consider that the common bound iskn = 15

4 . Then, in the case of excess supply, search for agents with peakslarger than the common bound. In this case, it is agent 4. Then, let agent 4take her peak consumption level 5 with probability 1, and shift the commonbound for the remaining three agents to (k − b(u4))/(n − 1) = 10

3 . Iteratethis operation until no remaining agent’s peak is larger than the commonbound. In this example, since the peaks of agents 1, 2, and 3 are less than103 , the iteration stops at 10

3 . Thus, the common bound finally fixed is 103 .

The outcome of the uniform probabilistic rule is that agent 4 is assigned5(= b(u4)) with probability 1, and the remaining agents are assigned 3 withprobability 2

3 and 4 with probability 13 to satisfy feasibility.

(ii) In the excess demand case (∑

i∈N b(ui) > k), the symmetric algorithm of(i) can be applied. Let N = {1, 2, 3, 4} and k = 12. Assume u ∈ U4 is suchthat b(u1) = 4, b(u2) = 2, b(u3) = 10, and b(u4) = 3. The final commonbound is calculated as 7

2 . Agents 2 and 4 are assigned their peak levels withprobability 1 since their peaks are less than the common bound, and agents1 and 3 are assigned 3 with probability 1

2 and 4 with probability 12 . ♦

The formal definition of the uniform probabilistic rule is as follows. Letλ : Un → R+ be the function such that if

∑i∈N b(ui) ≥ k,

∑i∈N min{b(ui), λ(u)} =

k, and if∑

i∈N b(ui) < k,∑

i∈N max{b(ui), λ(u)} = k. λ is the function todetermine the common bound. Let xλ : Un → K be the function suchthat if

∑i∈N b(ui) ≥ k, λ(u) ∈ [xλ(u), xλ(u) + 1) and if

∑i∈N b(ui) < k,

λ(u) ∈ (xλ(u), xλ(u) + 1].

Definition (Sasaki, 1997). The uniform probabilistic rule is the probabilis-tic rule f such that for all u ∈ Un, the following holds:(i) If

∑i∈N b(ui) > k (excess demand) , then for all i ∈ N ,

b(ui) ≤ xλ(u) =⇒ fi(u)(b(ui)) = 1, and

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b(ui) ≥ xλ(u) + 1 =⇒

{fi(u)(xλ(u) + 1) = λ(u) − xλ(u)fi(u)(xλ(u)) = 1 − (λ(u) − xλ(u)).

(ii) If∑

i∈N b(ui) = k (balanced demand), then for all i ∈ N , fi(u)(b(ui)) =1.(iii) If

∑i∈N b(ui) < k (excess supply), then for all i ∈ N ,

b(ui) ≥ xλ(u) + 1 =⇒ fi(u)(b(ui)) = 1, and

b(ui) ≤ xλ(u) =⇒

{fi(u)(xλ(u) + 1) = λ(u) − xλ(u)fi(u)(xλ(u)) = 1 − (λ(u) − xλ(u)).

Notice that in Example 3 (i), λ(u) = 103 and xλ(u) = 3, and in (ii),

λ(u) = 72 and xλ(u) = 3.

In a previous investigation of the probabilistic model, Sasaki (1997)shows that the uniform probabilistic rule is the only rule satisfying strategy-proofness, Pareto-efficiency, and anonymity. Kureishi (2000) and Ehlers andKlaus (2003) weaken anonymity to symmetry and show the uniqueness ofthe uniform probabilistic rule satisfying these properties. These results forthe probabilistic model are parallel to those of Sprumont (1991) and Ching(1994), respectively, who originally studied a deterministic model in whichthe goods are perfectly divisible.

In the deterministic model with a perfectly divisible good, Serizawa(2006) recently showed that the uniform rule is the only rule satisfying ef-fectively pairwise strategy-proofness, respect for unanimity, and symmetry.Thus, it is an interesting question as to whether a result parallel to Ser-izawa (2006) also holds in the probabilistic model. However, Example 4below illustrates that Serizawa’s (2006) uniqueness result does not hold inthe probabilistic model even though effectively pairwise strategy-proofnessand symmetry are respectively strengthened to strong coalitional strategy-proofness and no-envy or anonymity with an additional requirement of atmost binary.

Example 4. Let n = 3 and k = 2. We define the probabilistic rule f asfollows.If u ∈ U3 is such that for one agent, say i, b(ui) = 1 and for any other agentj ∈ N\{i}, b(uj) = 0, then, (i) in the case of ui(1) − ui(0) ≥ ui(1) − ui(2),{

fi(u)(1) = 1820 , fi(u)(2) = 2

20

fj(u)(0) = 1120 , fj(u)(1) = 9

20

and (ii) in the case of ui(1) − ui(0) < ui(1) − ui(2),{fi(u)(0) = 2

20 , fi(u)(1) = 1820

fj(u)(0) = 920 , fj(u)(1) = 11

20 .

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Otherwise, f induces the same marginal distribution profile as the uniformprobabilistic rule.

Then, although the probabilistic rule f satisfies strong coalitional strategy-proofness, respect for unanimity, no-envy, anonymity, and at most binary,it is not the uniform probabilistic rule18 ♦

We introduce two additional properties. “Peaks-onlyness” requires thatthe outcome marginal distribution profile depends only on the peak profile.If a rule satisfies peaks-onlyness, we can reduce the necessary informationfor a planner to the peak profile. “Continuity” requires that small changesin the utility profile cause only small changes in the outcome allocation.

Definition. A probabilistic rule f satisfies peaks-onlyness if for all u, u′ ∈Un such that for all i ∈ N , b(ui) = b(u′

i), f(u) = f(u′).

Definition. A probabilistic rule f satisfies continuity if for all u ∈ Un andany ε > 0, there exists δ > 0 such that for all u′ ∈ Un,

[∀i ∈ N,∀x ∈ K, ‖ ui(x) − u′i(x) ‖< δ]

=⇒[∀i ∈ N,∀x ∈ K, ‖ fi(u)(x) − fi(u′)(x) ‖< ε].

In the deterministic model with a perfectly divisible good, these twoproperties are standard and often obtained from strategy-proofness withauxiliary properties. However, note that the rule in Example 4 does not sat-isfy peaks-onlyness or continuity, even though it satisfies coalitional strategy-proofness, respect for unanimity, no-envy, anonymity, and at most binary.Thus, in the probabilistic model, these properties do not imply peaks-onlyness or continuity.

In this model, peaks-onlyness implies continuity.

Fact 3. If a probabilistic rule f satisfies peaks-onlyness, then it satisfiescontinuity.

The proof of Fact 3 is in the appendix.Example 5 below illustrates that even though we impose peaks-onlyness

as well as the previous properties, we cannot characterize the uniform prob-abilistic rule as a unique rule satisfying such properties. In addition, becauseof Fact 3, adding continuity to these properties has no effect.

Example 5. Let n = 4 and k = 2. We define a probabilistic rule f asfollows.If u ∈ U4 is such that for one agent, say i, b(ui) = 0, and for any other agentj ∈ N\{i}, b(uj) ≥ 1, then{

fi(u)(0) = 2730 , fi(u)(1) = 3

30

fj(u)(0) = 1130 , fj(u)(1) = 19

30 .18The feasibility is given in the appendix. A detailed explanation is given in an online

supplementary note (Hatsumi and Serizawa 2009).

11

Otherwise, f induces the same marginal distribution profile as the uniformprobabilistic rule.

Then, although the rule f satisfies the properties of strong coalitionalstrategy-proofness, respect for unanimity, no-envy, anonymity, at most bi-nary, and peaks-onlyness, it is not the uniform probabilistic rule19. ♦

In the deterministic model, coalitional strategy-proofness is a much strongerrequirement than strategy-proofness in that the uniform rule is character-ized as a unique rule satisfying coalitional strategy-proofness together withmild auxiliary conditions. On the other hand, in the probabilistic model, asExample 5 illustrates, some rules other than the uniform probabilistic rulesatisfy coalitional strategy-proofness and the many auxiliary conditions thatcharacterize the uniform rule in the deterministic model. The existence ofsuch rules makes a remarkable contrast between the probabilistic and deter-ministic models. Therefore, it is worthwhile to discuss why such rules existin the probabilistic model.

In the probabilistic model, the peaks of the utility functions and peakprofiles are finite. Thus, peaks-only rules such as the uniform probabilisticrule have finite ranges. Note that coalitional strategy-proofness is a systemof inequalities of allocations imposed on rules. Because of the finiteness ofthe range, there is slackness of the system of inequalities, i.e., there is roomto change the range of the uniform probabilistic rule without violating theseinequalities and the above conditions. That is, in the probabilistic model,we can construct rules from the uniform probabilistic rule by changing therange within such room.

It is also worthwhile to compare the fact discussed above to the resultsin Ehlers and Klaus (2003). They point out that if we impose at mostbinary, we can reduce the probabilistic model to the deterministic modelsince any marginal distribution satisfying at most binary is represented bya nonnegative real number. In addition, they mention that the domain ofutility profiles in the probabilistic model is rich enough to apply Ching’s(1994) proof to show that the uniform probabilistic rule is a unique rulesatisfying strategy-proofness, Pareto-efficiency, and symmetry. However, theabove discussion reveals that the domain in the probabilistic model is notsufficiently rich to obtain Serizawa’s (2006) parallel result.

Since coalitional strategy-proofness is not so strong in the probabilisticmodel, to characterize the uniform probabilistic rule by coalitional strategy-proofness, we require a stronger efficiency property than respect for una-nimity. Our main characterization employs same-sideness instead of respectfor unanimity.

Theorem. A probabilistic rule f satisfies coalitional strategy-proofness, same-sideness and strong symmetry if and only if it is the uniform probabilistic

19The feasibility is given in the appendix. A detailed explanation is in an online sup-plementary note (Hatsumi and Serizawa 2009).

12

rule.

The proof of the theorem is in the appendix.In this characterization, we do not employ strong coalitional strategy-

proofness but the weaker version of coalitional strategy-proofness. Notethat since same-sideness is weaker than Pareto-efficiency in the probabilis-tic model, this characterization is independent of Sasaki (1997), Kureishi(2000), and Ehlers and Klaus (2003).

Although coalitional strategy-proofness is stronger than strategy-proofness,we emphasize that even strong coalitional strategy-proofness and same-sideness do not imply at most binary. This fact is illustrated by Example 6below.

Example 6. Let n = 3 and k = 2. We define the probabilistic rule f asfollows.For all u ∈ U3, if b(u1) = 2 and b(u2) = b(u3) ≥ 1,{

f1(u)(0) = 115 , f1(u)(1) = 1

15 , f1(u)(2) = 1315

f2(u)(0) = f3(u)(0) = 2730 , f2(u)(1) = f3(u)(1) = 3

30 ,

and if b(u1) = 1 and b(u2) = b(u3) ≥ 1,{f1(u)(0) = 1

15 , f1(u)(1) = 1415

f2(u)(0) = f3(u)(0) = 715 , f2(u)(1) = f3(u)(1) = 8

15 .

Otherwise, f induces the same marginal distribution profile as the uniformprobabilistic rule.

Then, the rule f satisfies strong coalitional strategy-proofness and same-sideness, even though it violates at most binary20. ♦

In the deterministic model, an agent’s consumption set is one dimen-sional, and the feasible allocation set is n − 1 dimensional. On the otherhand, in the probabilistic model, an agent’s consumption set, i.e., the setof the marginal distributions is k dimensional, and so the set of the feasiblemarginal distribution profiles is considerably higher than n− 1 even thoughthe feasibility constraint makes its dimension less than k · (n − 1). Further-more, for any utility function ui ∈ U , at a marginal distribution with supportof more than or equal to three consumption levels, no “Maskin monotonictransformation”21 of ui can be taken from U . That is, the technique ofMaskin monotonic transformation, which plays an important role in the

20The feasibility is given in the appendix. A detailed explanation is given in an onlinesupplementary note (Hatsumi and Serizawa 2009).

21A utility function u′i ∈ U is Maskin monotonic transformation of ui ∈ U at a marginal

distribution pi if p′i 6= pi and E(p′

i; u′i) ≥ E(pi; u

′i) together imply E(p′

i; ui) > E(pi; ui).This property was developed by Maskin (1999).

13

strategy-proofness literature, cannot be applied to this marginal distribu-tion. Accordingly, analyzing the probabilistic model is much more difficultthan analyzing the deterministic model.

As Ehlers and Klaus (2003) explain, when at most binary is assumed, theprobabilistic model is reduced to the deterministic model, that is, at mostbinary makes the probabilistic model tractable. However, Example 6, whereagent 1 is assigned a marginal distribution with positive probabilities of allthe consumption levels, demonstrates that without at most binary, even ifcoalitional strategy-proofness and same-sideness are assumed, analyzing theprobabilistic model is still difficult.

3 Concluding Remarks

We have established that a rule satisfies coalitional strategy-proofness, same-sideness, and strong symmetry if and only if it is the uniform probabilisticrule. This result implies that the uniform probabilistic rule retains a veryimportant role in the probabilistic model of homogeneous indivisible goodswhen a planner wishes to coalitionally strategy-proof property, similarly tothe consequence in the deterministic model. We also show, by constructingexamples, that if same-sideness is replaced by respect for unanimity, thestatement does not hold even with the additional requirements of no-envy,anonymity, at most binary, peaks-onlyness and continuity. This fact em-phasizes the difference between the probabilistic and deterministic models,and suggests that matters of interest remain in the probabilistic model ofassigning homogeneous indivisible goods. We anticipate the current paperwill encourage further study of this model.

Appendix

Proof of Fact 2. Without loss of generality, assume Na = {1, · · · , na} andN b = {na+1, · · · , n}. Let y = k−

∑i∈Na xi−nb ·xµ. Note that y

nb = µ−xµ.We construct a distribution inducing the marginal distribution profile of thestatement.

Consider the distribution below.

[1nb

◦ (x1, · · · , xna , xµ + 1, · · · , xµ + 1︸ ︷︷ ︸y

, xµ, · · · , xµ),

1nb

◦ (x1, · · · , xna ,xµ, xµ + 1, · · · , xµ + 1︸ ︷︷ ︸y

, xµ, · · · , xµ),

· · · ,

1nb

◦ (x1, · · · , xna ,xµ, · · · , xµ, xµ + 1, · · · , xµ + 1︸ ︷︷ ︸y

),

14

1nb

◦ (x1, · · · , xna , xµ + 1︸ ︷︷ ︸1

, xµ, · · · , xµ, xµ + 1, · · · , xµ + 1︸ ︷︷ ︸y−1

),

1nb

◦ (x1, · · · , xna , xµ + 1, xµ + 1︸ ︷︷ ︸2

, xµ, · · · , xµ, xµ + 1, · · · , xµ + 1︸ ︷︷ ︸y−2

),

· · · ,

1nb

◦ (x1, · · · , xna , xµ + 1, · · · , xµ + 1︸ ︷︷ ︸y−1

, xµ, · · · , xµ, xµ + 1︸ ︷︷ ︸1

)]

This distribution is constructed by assigning allocations with probability1nb to each of nb allocations. Notice that each allocation is feasible. It inducesthat for all i ∈ Na, xi is assigned with probability 1, and for all j ∈ N b,xµ + 1 is assigned with probability y

nb (= µ − xµ) and xµ with probability1 − y

nb . Thus, we have the statement.

Proof of Fact 3. We introduce a lemma at first, and then prove the fact.

Lemma 1. For all u, u′ ∈ Un and all i ∈ N , if b(ui) 6= b(u′i), then there

exists x ∈ K such that ‖ ui(x) − u′i(x) ‖> [ui(b(ui)) − ui(b(u′

i))]/2.

Proof of Lemma 1. Let u, u′ ∈ Un, i ∈ N and b(ui) 6= b(u′i). First, we show

that

‖ ui(b(ui)) − u′i(b(ui)) ‖ + ‖ ui(b(u′

i)) − u′i(b(u

′i)) ‖

> ui(b(ui)) − ui(b(u′i)). (1)

Case 1. ui(b(u′i)) ≥ u′

i(b(u′i))

Note that ui(b(ui)) − u′i(b(ui)) > ui(b(ui)) − u′

i(b(u′i)) ≥ ui(b(ui)) −

ui(b(u′i)). Thus, (1) holds.

Case 2. ui(b(ui)) ≤ u′i(b(ui))

Note that u′i(b(u

′i)) − ui(b(u′

i)) > u′i(b(ui)) − ui(b(u′

i)) ≥ ui(b(ui)) −ui(b(u′

i)). Thus, (1) holds.

Case 3. ui(b(u′i)) < u′

i(b(u′i)) and ui(b(ui)) > u′

i(b(ui))

In this case, ‖ ui(b(ui)) − u′i(b(ui)) ‖ + ‖ ui(b(u′

i)) − u′i(b(u

′i)) ‖=

ui(b(ui))−u′i(b(ui))+u′

i(b(u′i))−ui(b(u′

i)) > ui(b(ui))−ui(b(u′i)). Thus, (1)

holds.

By the above three cases, we have (1). Thus max{‖ ui(b(ui))−u′i(b(ui)) ‖

, ‖ ui(b(u′i))− u′

i(b(u′i)) ‖} > [ui(b(ui))− ui(b(u′

i))]/2 and we have the state-ment.

15

Let a probabilistic rule f satisfy peaks-onlyness. Let u ∈ Un and ε > 0.Given i ∈ N , let yi = arg maxy∈K\b(ui) ui(y). Let δ > 0 be such that for alli ∈ N , δ ≤ [ui(b(ui)) − ui(yi)]/2, and let u′ ∈ Un be such that for all i ∈ Nand all x ∈ K, ‖ ui(x) − u′

i(x) ‖< δ. We show that for all i ∈ N and allx ∈ K, ‖ fi(u)(x) − fi(u′)(x) ‖< ε.

First, we show that for all i ∈ N , b(ui) = b(u′i). Suppose there exists

i ∈ N such that b(ui) 6= b(u′i), and we derive a contradiction. By the

assumption, for all x ∈ K, ‖ ui(x) − u′i(x) ‖< δ ≤ [ui(b(ui)) − ui(yi)]/2. By

the definition of yi, [ui(b(ui))−ui(y)]/2 ≤ [ui(b(ui))−ui(b(u′i))]/2. Thus we

have that for all x ∈ K, ‖ ui(x) − u′i(x) ‖< [ui(b(ui)) − ui(b(u′

i))]/2. It is acontradiction to Lemma 1. Thus, we have that for all i ∈ N , b(ui) = b(u′

i).Therefore, peaks-onlyness implies f(u) = f(u′), and we have the state-

ment of the fact.

Proof of the Theorem. It is easy to check the if part of the theorem. Here,we show the only if part. We first introduce three lemmas. Since themarginal distribution profiles in this section all satisfy the condition of Fact2, we do not explicitly check the feasibilities.

Lemma 2. For all u ∈ Un, if p, p′ ∈ P are both Pareto-efficient with respectto u, and for all i ∈ N , E(pi; ui) = E(p′i; ui), then p = p′.

Proof of Lemma 2. Let u ∈ Un, let p, p′ ∈ P be Pareto-efficient with respectto u, and let E(pi; ui) = E(p′i; ui) for all i ∈ N . We show p = p′.

Suppose, on the contrary, that there exists i ∈ N such that pi 6= p′i, andwe derive a contradiction.

Since both p and p′ are Pareto-efficient with respect to u, Fact 1 impliesthat p and p′ satisfy same-sideness with respect to u and at most binary.

From at most binary, there exist x ∈ K such that pi(x) > 0 and pi(x) +pi(x + 1) = 1, and y ∈ K such that p′i(y) > 0 and p′i(y) + p′i(y + 1) = 122.

Case 1: x 6= y.

Without loss of generality, assume x > y. If∑

i∈N b(ui) ≥ k, by same-sideness, y < y + 1 ≤ x < x + 1 ≤ b(ui)23. Then by single-peakedness andp′i(y) > 0, E(pi; ui) = pi(x) · u(x) + pi(x + 1) · u(x + 1) > p′i(y) · u(y) +p′i(y + 1) · u(y + 1) = E(pi; ui). It is a contradiction to the assumptionE(pi; ui) = E(p′i; ui).

If∑

i∈N b(ui) < k, by same-sideness, b(ui) ≤ y < y + 1 ≤ x. Then bysingle-peakedness and p′i(y) > 0, E(pi; ui) = pi(x)·u(x)+pi(x+1)·u(x+1) <p′i(y) · u(y) + p′i(y + 1) · u(y + 1) = E(p′i; ui). It is a contradiction to theassumption E(pi; ui) = E(p′i;ui).

22In the case of x = k, pi(x) = 1. Similarly, in the case of y = k, pi(y) = 1.23If x = b(ui), then same-sideness implies pi(x) = 1 and pi(x + 1) = 0 even though

b(ui) < x + 1. Thus, the proof still works.

16

Case 2: x = y.

Without loss of generality, assume pi(x) > p′i(x). If∑

i∈N b(ui) ≥ k, thenby same-sideness and single-peakedness, E(pi; ui) = pi(x) · u(x) + pi(x + 1) ·u(x+1) < p′i(x) ·u(x)+ p′i(x+1) ·u(x+1) = E(p′i; ui). It is a contradictionto the assumption E(pi; ui) = E(p′i; ui).

If∑

i∈N b(ui) < k, then by same-sideness and single-peakedness, E(pi; ui) =pi(x) ·u(x)+pi(x+1) ·u(x+1) > p′i(y) ·u(y)+p′i(y+1) ·u(y+1) = E(p′i;ui).It is a contradiction to the assumption E(pi; ui) = E(p′i; ui).

From Cases 1 and 2, we have p = p′.

Lemma 3. Let f be a rule satisfying coalitional strategy-proofness andsymmetry. For all u ∈ Un such that u1 = · · · = un and all u′ ∈ N such thatfor all i ∈ N , b(u′

i) = b(ui), if f(u) is Pareto-efficient with respect to u, thenf(u) = f(u′).

Proof of Lemma 3. Let u, u′ ∈ Un be such that u1 = · · · = un, for alli ∈ N , b(u′

i) = b(ui), and f(u) is Pareto-efficient with respect to u. Weshow f(u) = f(u′) by mathematical induction.

Step A: If u′1 = · · · = u′

n, then f(u) = f(u′).

By symmetry, E(f1(u);u1) = · · · = E(fn(u); un) and E(f1(u′);u′1) =

· · · = E(fn(u′);u′n). Since f(u) is Pareto-efficient with respect to u, Fact

1 implies that f(u) satisfies same-sideness with respect to u and at mostbinary. Since b(ui) = b(u′

i) for all i ∈ N , f(u) also satisfies same-sidenesswith respect to u′. Thus, f(u) is Pareto-efficient with respect to u′.

If for some j ∈ N , E(fj(u); u′j) < E(fj(u′);u′

j), then by symmetry, forall i ∈ N , E(fi(u);u′

i) < E(fi(u′);u′i). It contradicts Pareto-efficiency of

f(u) with respect to u′. Thus, for all i ∈ N , E(fi(u);u′i) ≥ E(fi(u′);u′

i).If for some j ∈ N , E(fj(u); u′

j) > E(fj(u′);u′j), then by symmetry, for

all i ∈ N , E(fi(u);u′i) > E(fi(u′);u′

i). Then, the coalition of all agents Nwith profile u′ manipulates the rule via u and increases the utilities of allmembers. It is a contradiction to coalitional strategy-proofness.

Therefore, for all i ∈ N , E(fi(u);u′i) = E(fi(u′);u′

i). By Lemma 2,f(u) = f(u′).

Step B: Let h ∈ N . Assume that if u′1 = · · · = u′

h, f(u) = f(u′). Then, ifu′

1 = · · · = u′h−1, f(u) = f(u′).

Let u′ ∈ Un be such that u′1 = · · · = u′

h−1. Then by symmetry,E(f1(u′);u′

1) = · · · = E(fh−1(u′);u′h−1). Thus, if for some i ∈ {1, · · · , h−1},

E(fi(u);u′i) > E(fi(u′);u′

i), then for all i ∈ {1, · · · , h − 1}, E(fi(u);u′i) >

E(fi(u′);u′i). Then, the coalition {1, · · · , h − 1} with u′

{1,··· ,h−1} manipu-lates the rule via u{1,··· ,h−1} such that for all i ∈ {1, · · · , h − 1}, ui = u′

h.Then, any i ∈ {1, · · · , h − 1} obtains fi(u) and increases her utility by the

17

induction hypothesis. It is a contradiction to coalitional strategy-proofness.Therefore, for all i ∈ {1, · · · , h − 1}, E(fi(u);u′

i) ≤ E(fi(u′);u′i).

If, for some j ∈ {h, · · · , n}, E(fj(u);u′j) > E(fj(u′);u′

j), then j withu′

j manipulates the rule via uj = u′1 and obtains fj(u) by the induction

hypothesis. It is a contradiction to strategy-proofness. Thus, for all j ∈{h, · · · , n}, E(fj(u);u′

j) ≤ E(fj(u′);u′j).

Therefore, for all i ∈ N , E(fi(u);u′i) ≤ E(fi(u′);u′

i). Similarly to StepA, We can show that f(u) is Pareto-efficient with respect to u′. Thus, for alli ∈ N , E(fi(u); u′

i) = E(fi(u′);u′i). Therefore, by Lemma 2, f(u) = f(u′).

From Step A and B, we have the statement of the lemma.

Lemma 4. If f satisfies same-sideness, then it respects unanimity.

Proof of Lemma 4. By same-sideness,∑

i∈N b(ui) = k implies that for alli ∈ N , fi(u)([0, b(ui)]) = 1 and fi(u)([b(ui), k]) = 1. Thus for all i ∈ N ,fi(u)(b(ui)) = 1.

We prove the theorem by five steps. Hereafter, let f be a rule satisfyingcoalitional strategy-proofness, same-sideness, and strong symmetry.

Step 1. For all u ∈ Un such that∑

i∈N b(ui) = k and all i ∈ N , fi(u)(b(ui)) =1.

Proof of Step 1. By Lemma 4, the statement is directly implied.

Step 2. Let x ∈ K be such that kn ∈ [x, x + 1). Let u ∈ Un be such that

for all i ∈ N , b(ui) = x. Then for all i ∈ N , fi(u)(x) = x + 1 − kn and

fi(u)(x + 1) = kn − x.

Proof of Step 2. For all z ∈ K such that x + 2 ≤ z ≤ k , let rz(ui) ∈ R besuch that rz(ui) · [ui(x)−ui(x+1)] = ui(x+1)−ui(z). Note that by single-peakedness of ui with b(ui) = x, for all z ∈ K such that x + 2 ≤ z ≤ k − 1,rz(ui) < rz+1(ui). By single-peakedness and risk-averseness, we also havethat for all z ∈ K such that x + 2 ≤ z ≤ k − 1,

0 < rz(ui) − [z − (x + 1)] < rz+1(ui) − [(z + 1) − (x + 1)]. (2)

[Figure 1 enters around here.]

Let p ∈ P be such that for all i ∈ N , pi(x) = x + 1 − kn and pi(x + 1) =

kn − x. We show f(u) = p.

Case A: n · x = k.

18

By Step 1, for all i ∈ N , fi(u)(x) = 1 and fi(u)(x + 1) = 0. Thus thestatement holds.

Case B: n · x < k

Step B-1. First, we consider the case where u1 = · · · = un. By same-sideness, for all i ∈ N , fi(u)([x, k]) = 1. Let u′ ∈ Un be such that b(u′

1) =· · · = b(u′

k−nx) = x + 1 and b(u′k−nx+1) = · · · = b(u′

n) = x. Then, we have∑i∈N b(u′

i) = k. Thus, by Step 1, for all i ∈ {1, · · · , k−nx}, fi(u′)(x+1) = 1and for all for all i ∈ {k − nx + 1, · · · , k}, fi(u′)(x) = 1.

Thus, coalitional strategy-proofness and symmetry imply that for alli ∈ {1, · · · , k − nx}, E(fi(u);ui) ≥ E(fi(u′);ui) = ui(x + 1). By symmetry,for all i ∈ N , E(fi(u);ui) ≥ ui(x + 1). Note that

E(fi(u); ui) ≥ ui(x + 1)

⇐⇒∑

z∈[x,k]

fi(u)(z) · ui(z) ≥ ui(x + 1) (by same-sideness)

⇐⇒∑

z∈[x,k]

fi(u)(z) · [ui(z) − ui(x + 1)] ≥ 0

⇐⇒ fi(u)(x) · [ui(x) − ui(x + 1)]

−∑

z∈[x+2,k]

fi(u)(z) · [ui(x + 1) − ui(z)] ≥ 0. (3)

By using the notation r, we rewrite (3) as: for all i ∈ N ,

fi(u)(x) −∑

z∈[x+2,k]

fi(u)(z) · rz(ui) ≥ 0. (4)

We show that for all i ∈ N , fi(u)([x+2, k]) = 0 by mathematical induction.

Step B-1-1: For all i ∈ N , fi(u)(k) = 0.

Suppose, on the contrary, for some j ∈ N , fi(u)(k) > 0. Then, bystrong symmetry and u1 = · · ·un, for all i ∈ N , fi(u)(k) > 0. We derive acontradiction.

Let u ∈ Un be such that u1 = · · · = un, for all i ∈ N , b(ui) = x, andrx+2(ui) > 1

fi(u)(k) . By strong symmetry, f1(u) = · · · = fn(u).Suppose for some j ∈ N , fj(u)([x + 2, k]) ≥ fj(u)(k). Then,

fj(u)(x) −∑

z∈[x+2,k]

fj(u)(z) · rz(uj)

≤ fj(u)(x) −∑

z∈[x+2,k]

fj(u)(z) · rx+2(uj)

(by rx+2(uj) ≤ rz(uj) for all z ∈ [x + 2, k])

19

= fj(u)(x) − fj(u)([x + 2, k]) · rx+2(uj)≤ fj(u)(x) − fj(u)(k) · rx+2(uj)

< fj(u)(x) − 1 (by rx+2(uj) >1

fj(u)(k))

≤ 0.

It is a contradiction since u ∈ Un also has to satisfy (4). Thus, for all i ∈ N ,∑z∈[x+2,k]

fi(u)(z) < fi(u)(k). (5)

By feasibility,∑

i∈N

∑z∈K fi(u)(z) · z =

∑i∈N

∑z∈K fi(u)(z) · z = k.

Thus, by strong symmetry, for all i ∈ N ,∑z∈K

fi(u)(z) · z =∑z∈K

fi(u)(z) · z =k

n.

Then, by same-sideness,∑

z∈[x,k] fi(u)(z) · z =∑

z∈[x,k] fi(u)(z) · z. Notethat ∑

z∈[x,k]

fi(u)(z) · z =∑

z∈[x,k]

fi(u)(z) · z

⇐⇒∑

z∈[x,k]\{x+1}

[fi(u)(z) − fi(u)(z)] · z = −[fi(u)(x + 1) − fi(u)(x + 1)] · (x + 1)

⇐⇒∑

z∈[x,k]\{x+1}

[fi(u)(z) − fi(u)(z)] · z

= −{[1 −∑

z∈[x,k]\{x+1}

fi(u)(z)] − [1 −∑

z∈[x,k]\{x+1}

fi(u)(z)]} · (x + 1)

⇐⇒∑

z∈[x,k]\{x+1}

[fi(u)(z) − fi(u)(z)] · z

= {∑

z∈[x,k]\{x+1}

fi(u)(z) −∑

z∈[x,k]\{x+1}

fi(u)(z)} · (x + 1)

⇐⇒∑

z∈[x,k]\{x+1}

[fi(u)(z) − fi(u)(z)] · [z − (x + 1)] = 0

⇐⇒∑

z∈[x+2,k]

[fi(u)(z) − fi(u)(z)] · [z − (x + 1)] = fi(u)(x) − fi(u)(x). (6)

Thus,

E(fi(u);ui) − E(fi(u);ui)

=∑

z∈[x,k]\{x+1}

[fi(u)(z) − fi(u)(z)] · ui(z)

20

+ {[1 −∑

z∈[x,k]\{x+1}

fi(u)(z)] − [(1 −∑

z∈[x,k]\{x+1}

fi(u)(z)]} · ui(x + 1)

=∑

z∈[x,k]\{x+1}

[fi(u)(z) − fi(u)(z)] · [ui(z) − ui(x + 1)]

= {∑

z∈[x+2,k]

[fi(u)(z) − fi(u)(z)] · [z − (x + 1)]} · [ui(x) − ui(x + 1)]

+∑

z∈[x+2,k]

[fi(u)(z) − fi(u)(z)] · [ui(z) − ui(x + 1)] (by (6) )

= {∑

z∈[x+2,k]

[fi(u)(z) − fi(u)(z)] · [z − (x + 1)]} · [ui(x) − ui(x + 1)]

+∑

z∈[x+2,k]

[fi(u)(z) − fi(u)(z)] · {−rz(ui) · [ui(x) − ui(x + 1)]}

(by the definition of r)

= [ui(x) − ui(x + 1)] · {∑

z∈[x+2,k]

[fi(u)(z) − fi(u)(z)] · [z − (x + 1) − rz(ui)]}

= [ui(x) − ui(x + 1)]

· {∑

z∈[x+2,k]

[fi(u)(z) − fi(u)(z)] · [rz(ui) − {z − (x + 1)}]} (7)

Note that ∑z∈[x+2,k]

[fi(u)(z) − fi(u)(z)] · [rz(ui) − {z − (x + 1)}]

= [fi(u)(k) − fi(u)(k)] · [rk(ui) − {k − (x + 1)}]

+∑

z∈[x+2,k−1]

[fi(u)(z) − fi(u)(z)] · [rz(ui) − {z − (x + 1)}]

≥ [fi(u)(k) − fi(u)(k)] · [rk(ui) − {k − (x + 1)}]

−∑

z∈[x+2,k−1]

fi(u)(z) · [rz(ui) − {z − (x + 1)}]

(by (2), for all z ∈ [x + 2, k − 1], rz(ui) − {z − (x + 1)} > 0)≥ [fi(u)(k) − fi(u)(k)] · [rk(ui) − {k − (x + 1)}]

−∑

z∈[x+2,k−1]

fi(u)(z) · [rk(ui) − {k − (x + 1)}] (by (2))

= [fi(u)(k) −∑

z∈[x+2,k]

fi(u)(z)] · [rk(ui) − {k − (x + 1)}]

> 0 (by (5) and (2).) (8)

Then (7) and (8) together imply that for all i ∈ N , E(fi(u);ui)−E(fi(u); ui) >0. It is a contradiction to coalitional strategy-proofness. Thus, for all i ∈ N ,

21

fi(u)(k) = 0.

Step B-1-2. Let y ∈ K be such that y ≥ x + 2. Assume that for all i ∈ N ,fi(u)([y + 1, k]) = 0. Then, for all i ∈ N , fi(u)([y, k]) = 0.

By same-sideness and the induction hypothesis, for all i ∈ N , fi([x, y]) =1. Then we apply a similar argument to Step B-1-1 by replacing k with y,and we have that for all i ∈ N , fi(u)([y, k]) = 0.

Now, we have for all u ∈ Un such that u1 = · · · = un and b(ui) = x,fi(u)([x, x + 1]) = 1. Symmetry and feasibility imply that for all i ∈ N ,fi(u)(x) = x + 1 − k

n and fi(u)(x + 1) = kn − x, i.e., f(u) = p.

Step B-2. Note that for all u ∈ Un such that for all i ∈ N , b(ui) = x, p isPareto-efficient with respect to u. Thus by Lemma 3 and Step B-1, for allu ∈ Un such that for all i ∈ N , b(ui) = x, f(u) = p. We finish Case B.

From Cases A and B, the statement is established.

Step 3. Let x ∈ K be such that kn ∈ [x, x + 1). Let u ∈ Un be such

that b(u1) = · · · = b(un). Then for all i ∈ N , fi(u)(x) = x + 1 − kn and

fi(u)(x + 1) = kn − x.

Proof of Step 3. Step A. First, we consider the case where u1 = · · · = un.Let p ∈ P be such that for all i ∈ N , pi(x) = x + 1 − k

n and pi(x + 1) =kn −x. Then, p satisfies same-sideness with respect to u and at most binary.Thus, it is Pareto-efficient with respect to u.

Let u ∈ Un be such that for all i ∈ N , b(ui) = x. Then, by Step2, f(u) = p. Since p is Pareto-efficient with respect to u, and symmetryimplies E(f1(u);u1) = · · · = E(fn(u); un), it follows that for all i ∈ N ,E(fi(u);ui) = E(pi; ui) ≥ E(fi(u);ui). If E(fi(u);ui) > E(fi(u);ui), itis a contradiction to coalitional strategy-proofness. Thus, for all i ∈ N ,E(fi(u);ui) = E(fi(u);ui) = E(pi, ui).

Therefore, by Lemma 2, f(u) = p.

Step B. From Lemma 3 and Step A, for all u ∈ Un such that b(u1) = · · · =b(un), we have f(u) = p.

Step 4. Let x ∈ K be such that kn ∈ [x, x + 1). Let u ∈ Un be such that

for all i ∈ N , b(ui) ≤ x, or for all i ∈ N , b(ui) ≥ x + 1. Then for all i ∈ N ,fi(u)(x) = x + 1 − k

n and fi(u)(x + 1) = kn − x.

Proof of Step 4. Assume that for all i ∈ N , b(ui) ≤ x, since the othercase can be treated symmetrically. Let p ∈ P be such that for all i ∈ N ,pi(x) = x+1− k

n and pi(x+1) = kn −x. We prove f(u) = p by mathematical

induction.

22

Step A: If u1 = · · · = un, then f(u) = p.

The statement is from Step 3.

Step B: Let h ∈ N . Assume that for all u′ ∈ Un such that for all u′1 =

· · · = u′h, we have f(u) = p. Then, if u is such that u1 = · · · = uh−1, we

have f(u) = p.

Let u1 = · · ·uh−1. By symmetry, E(f1(u);u1) = · · · = E(fh−1(u);uh−1).Thus, if, for some j ∈ {1, · · · , h − 1}, E(pj ; uj) > E(fj(u);uj), then for alli ∈ {1, · · · , h − 1}, E(pi; ui) > E(fi(u);ui). Then, coalition {1, · · · , h − 1}with u{1,··· ,h−1} manipulates the rule via u{1,··· ,h−1} such that for all i ∈{1, · · · , h − 1}, ui = uh, and any i ∈ {1, · · · , h − 1} obtains pi by theinduction hypothesis and increases her utility. It is a contradiction to coali-tional strategy-proofness. Therefore, for all i ∈ {1, · · · , h − 1}, E(pi; ui) ≤E(fi(u);ui).

On the other hand, if, for some j ∈ {h, · · · , n}, E(pj ; uj) > E(fj(u);uj),then j with uj manipulates the rule via uj = u1. Then, j obtains pj by in-duction hypothesis and increases her utility. It is a contradiction to strategy-proofness. Therefore, for all j ∈ {h, · · · , n}, E(pj ; uj) ≤ E(fj(u);uj).

Thus, for all i ∈ N , E(pi; ui) ≤ E(fi(u);ui). Since p satisfies same-sideness with respect to u and at most binary, Fact 1 implies that p is Pareto-efficient with respect to u. Therefore, for all i ∈ N , E(fi(u);ui) = E(pi;ui).By Lemma 2, f(u) = p.

By Step A and Step B, we have the statement of this step.

Step 5. (i) For all u ∈ Un, if∑

i∈N b(ui) < k, then for all i ∈ N such thatb(ui) ≥ xλ(u)+1, fi(u)(b(ui)) = 1 and for all i ∈ N such that b(ui) ≤ xλ(u),fi(u)(xλ(u) + 1) = λ(u) − xλ(u) and fi(u)(xλ(u)) = (xλ(u) + 1) − λ(u).(ii) For all u ∈ Un, if

∑i∈N b(ui) > k, then for all i ∈ N such that b(ui) ≤

xλ(u), fi(u)(b(ui)) = 1 and for all i ∈ N such that b(ui) ≥ xλ(u) + 1,fi(u)(xλ(u) + 1) = λ(u) − xλ(u) and fi(u)(xλ(u)) = (xλ(u) + 1) − λ(u).

Proof of Step 5. Given u ∈ Un, let N(u) = {i ∈ N : b(ui) ≥ xλ(u) + 1}and N(u) = {i ∈ N : b(ui) ≤ xλ(u)}. In addition, let n(u) be the numberof agents in N(u), and n(u) be the number of agents in N(u). Note thatN(u) ∪ N(u) = N .

Without loss of generality, let u ∈ Un be such that∑

i∈N b(ui) < k, sincethe other case is symmetrically proved. We prove this step by mathematicalinduction on n(u).

Step A: For all u ∈ Un, if n(u) = 0, then for all i ∈ N(u) = N , fi(u)(xλ(u)+1) = λ(u) − xλ(u) and fi(u)(xλ(u)) = (xλ(u) + 1) − λ(u).

In this case, λ(u) = kn . Thus by Step 4, the statement is directly implied.

23

Step B: Let l ∈ N\{0, n}24. Assume that for all u ∈ Un, if n(u) ≤ l − 1,then for all i ∈ N(u), fi(u)(b(ui)) = 1 and for all i ∈ N(u), fi(u)(xλ(u)) =xλ(u) + 1 − λ(u) and fi(u)(xλ(u) + 1) = λ(u) − xλ(u). Then for all u ∈Un, if n(u) = l, for all i ∈ N(u), fi(u)(b(ui)) = 1 and for all i ∈ N(u),fi(u)(xλ(u)) = xλ(u) + 1 − λ(u) and fi(u)(xλ(u) + 1) = λ(u) − xλ(u).

Let n(u) = l. First, we show that for all i ∈ N(u), fi(u)(b(ui)) = 1.Here, we start a new mathematical induction within Step B. Note that bysame-sideness, we have that for all i ∈ N(u), fi(u)([b(ui), k]) = 1.

Step B-i: For all i ∈ N(u), fi(u)(k) = 0.

Suppose, on the contrary, that for some i ∈ N(u), fi(u)(k) > 0. Wederive a contradiction.

Let u′i ∈ U be such that b(u′

i) = 0. Then, fi(u′i, u−i)(xλ(u′

i, u−i)) =xλ(u′

i, u−i) + 1 − λ(u′i, u−i) and fi(u′

i, u−i)(xλ(u′i, u−i) + 1) = λ(u′

i, u−i) −xλ(u′

i, u−i) by the induction hypothesis of Step B. Note that by the definitionof xλ, b(ui) ≥ xλ(u′

i, u−i) + 125.

[Figure 2 enters around here. ]

Since fi(u)(k) > 0, there is ui ∈ U such that b(ui) = b(ui) and ui(0) >{1−fi(u)(k)}· ui(b(ui))+fi(u)(k) · ui(b(ui)+1). Suppose fi(ui, u−i)([b(ui)+1, k]) ≥ fi(u)(k). Then, by same-sideness, we have that

fi(ui, u−i)(b(ui)) ≤ 1 − fi(u)(k). (9)

Then,

E(fi(u′i, u−i); ui)

= fi(u′i, u−i)(xλ(u′

i, u−i)) · ui(xλ(u′i, u−i))

+ fi(u′i, u−i)(xλ(u′

i, u−i) + 1) · ui(xλ(u′i, u−i) + 1)

> ui(0) (by b(ui) ≥ xλ(u′i, u−i) + 1 and single-peakedness)

> {1 − fi(u)(k)} · ui(b(ui)) + fi(u)(k) · ui(b(ui) + 1)(by the definition of ui)

≥ fi(ui, u−i)(b(ui)) · ui(b(ui)) + fi(ui, u−i)([b(ui) + 1, k]) · ui(b(ui) + 1)(by (9) and ui(b(ui)) > ui(b(ui) + 1))

≥ fi(ui, u−i)(b(ui)) · ui(b(ui)) +∑

z∈[b(ui)+1,k]

fi(ui, u−i)(z) · ui(z)

24We need not consider the case of l = n since ifP

i∈N b(ui) < k, n(u) cannot be equalto n.

25Suppose, on the contrary, that b(ui) ≤ xλ(u′i, u−i). Then, xλ(u′

i, u−i) = xλ(u) bythe definition of xλ in the case

P

i∈N b(ui) < k. This contradicts the assumption thatb(ui) ≥ xλ(u) + 1. Thus, b(ui) ≥ xλ(u′

i, u−i) + 1.

24

(by single-peakedness)= E(fi(ui, u−i); ui)

It is a contradiction to strategy-proofness. Thus

fi(ui, u−i)([b(ui) + 1, k]) < fi(u)(k). (10)

Then,

E(fi(ui, u−i);ui)

= fi(ui, u−i)(b(ui)) · ui(b(ui)) +∑

z∈[b(ui)+1,k]

fi(ui, u−i)(z) · ui(z)

≥ fi(ui, u−i)(b(ui)) · ui(b(ui)) + fi(ui, u−i)([b(ui) + 1, k]) · ui(k)(by single-peakedness)

> {1 − fi(u)(k)} · ui(b(ui)) + fi(u)(k) · ui(k) (by (10))

≥∑

z∈[b(ui),k−1]

fi(u)(z) · ui(z) + fi(u)(k) · ui(k) (by single-peakedness)

= E(fi(u); ui).

It is a contradiction to strategy-proofness. Thus, we have fi(u)(k) = 0 forall i ∈ N(u).

Step B-ii: Let x ∈ K be such that b(ui) + 1 ≤ x ≤ k − 1. Assume that forall i ∈ N(u), fi(u)([x + 1, k]) = 0. Then for all i ∈ N(u), fi(u)([x, k]) = 0.

Suppose, on the contrary, that for some i ∈ N(u), fi(u)([x, k]) > 0, andwe derive a contradiction. By fi(u)([x + 1, k]) = 0 (induction hypothesis),fi(u)([b(ui), x]) > 0. Then we apply a similar argument to Step B-i byreplacing k with x, and we have that for all i ∈ N , fi(u)([x, k]) = 0.

Now, we have that for all i ∈ N(u), fi(u)(b(ui)) = 1. Next, we showthat for all i ∈ N(u), fi(u)(xλ(u)) = xλ(u)+1−λ(u) and fi(u)(xλ(u)+1) =λ(u) − xλ(u).

Let k′ = k −∑

i∈N(u) b(ui). Since for all i ∈ N(u), fi(u)(b(ui)) = 1,∑i∈N(u)

∑xi∈K fi(u)(xi)xi = k′. Note that λ(u) = k′

n(u) and for all i ∈ N(u),b(ui) < λ(u).

Then we can use a similar argument to Step 4 by replacing k with k′

and kn by k′

n(u) . We omit the detailed proof.

By Steps A and B, we have for all i ∈ N such that b(ui) ≥ xλ(u) + 1,fi(u)(b(ui)) = 1 and for all i ∈ N such that b(ui) ≤ xλ(u), fi(u)(xλ(u)+1) =λ(u) − xλ(u) and fi(u)(xλ(u)) = (xλ(u) + 1) − λ(u).

Finally, by Step 1 and Step 5, a probabilistic rule satisfies coalitionalstrategy-proofness, same-sideness, and strong symmetry if and only if it isthe uniform probabilistic rule.

25

Feasibilities in Example 4, 5 and 6. Due to Fact 2, it is sufficient to show thatin each example, there is a distribution over feasible allocations inducing themarginal distribution profile f(u) when f(u) is different from the outcomeof the uniform probabilistic rule.Example 4. Without loss of generality, let u ∈ U3 be such that b(u1) = 1and b(u2) = b(u3) = 0. In the case of u1(1) − u1(0) ≥ u1(1) − u1(2),[ 220 ◦ (2, 0, 0), 9

20 ◦ (1, 1, 0), 920 ◦ (1, 0, 1)] induces f(u). In the case of u1(1) −

u1(0) < u1(1) − u1(2), [ 220 ◦ (0, 1, 1), 9

20 ◦ (1, 1, 0), 920 ◦ (1, 0, 1)] induces f(u).

♦Example 5. Without loss of generality, let u ∈ U4 be such that b(u1) = 0and for any other agent j ∈ N\{1}, b(uj) ≥ 1. Then, [ 1

30 ◦ (1, 1, 0, 0), 130 ◦

(1, 0, 1, 0), 130 ◦(1, 0, 0, 1), 9

30 ◦(0, 1, 1, 0), 930 ◦(0, 1, 0, 1), 9

30 ◦(0, 0, 1, 1)] inducesf(u). ♦Example 6. Let u ∈ Un be such that b(u2) ≥ 1, b(u3) ≥ 1. Whenb(u1) = 2, [ 1

15 ◦ (0, 1, 1), 130 ◦ (1, 1, 0), 1

30 ◦ (1, 0, 1), 1315 ◦ (2, 0, 0)] induces f(u).

When b(u1) = 1, [ 115 ◦ (0, 1, 1), 7

15 ◦ (1, 1, 0), 715 ◦ (1, 0, 1)] induces f(u). ♦

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28

b(ui) b(ui) + 1 z

1

rz(ui)

ui

·

0

·

Figure 1. Illustration of rz(ui) in the proof of Step 2.

b(ui) b(ui) + 1

ui

0ui

u′i

Figure 2. Illustration of ui, u′i and ui in the proof of Step B-i of Step 5

29