Strategy-proof Provision of Two Public Goods:
the Lexmax Extension
Lars Ehlers
†and Sidartha Gordon
‡May 2011 (revised November 2013)
Abstract
This paper studies the problem of providing two public goods for agents with
single-peaked preferences. A decision rule selects two points on the segment
[0,1]for the public goods for every profile of reported preferences. Agents
com-pare public good pairs by the lexmax ordering over pairs induced by their
single-peaked preference over single locations. We derive implications of strategy-proofness in this setting and compare them with those in the model with one public good and in the model with two public goods under the max
exten-sion. We characterize the class of decision rules satisfying strategy-proofness,
anonymity and continuity with respect to preferences. We also characterize subclasses of rules that satisfy additional properties.
JEL classification: D63, D71, D81
Keywords: Single-peaked preferences, public goods, strategy-proofness.
This research is financially supported by FQRSC (Québec). We thank Salvador Barberà and
John Duggan for their insights and audiences at Universitat Autonoma de Barcelona, the Meetings of the Society for Social Choice and Welfare (2008), the Workshop on Prior-free Mechanism Design (2009) and the Workshop on Social Decisions (2010) for comments and questions.
†Département de Sciences Économiques and CIREQ, Université de Montréal, Canada; e-mail:
1
Introduction
We consider the problem of selecting two alternatives for the provision of two identical public goods. Agents have single-peaked preferences on a closed interval. Preferences are single-peaked if up to a certain point, the peak, preferences are strictly increasing, and strictly decreasing beyond that point. The feasible set of alternatives consists
of pairs providing for each public good a level. Agents compare alternatives by the lexmax ordering over these pairs induced by their single-peaked preference. More explicitly, when comparing two pairs, an agent first compares his preferred level in both pairs. Only if he is indi§erent between his preferred level in both pairs, he then compares the other two levels. A public good economy is completely described by a list of parameters such as the set of the agents and their preferences over the closed
interval. A decision rule is a systematic way to assign the levels of the two public goods for each economy, that is for every profile of reported preferences.
An important application of the model is the problem of locating two pure public good facilities, such as electricity generating plants, or telecommunication repeaters. The lexmax ordering over pairs captures the fact that each agent will only use his most preferred alternative in the provided pair. Thus, his first order preference is
determined by comparing the best alternatives in each pair. He will use the second option only as a plan B, in case of a breakdown of the first best option.1
We are interested in rules that satisfy three main properties. Strategy-proofness
requires that no agent can individually benefit by misreporting his preferences. Anonymity
says that the selection does not depend on agents’ labels. Finally we imposecontinuity
in the agent’s preferences. Our main contribution is the introduction of a new class of
rules for two public goods, the class ofdelta rules, and its characterization. Our main result is that the three above properties characterize the class ofdelta rules. We also characterize subclasses of rules that (separately) satisfy additional requirements, such asgroup strategy-proofness, unanimity, Pareto-optimality,peaks-selection, diversity, population-monotonicity and replacement-domination.
The organization of the paper is as follows. In Section 2, we discuss related
litera-1Most of the electricity consumed in Barcelona comes from a plant located in Girona. If this
ture. In Section 3 we present the model. In Section 4, we introduce the class ofdelta rules and characterize them as the rules that satisfy strategy-proofness, anonymity
and continuity. In Section 5, we characterize interesting subclasses of rules that sat-isfy additional requirements. In Section 6, we investigate the implications of
peaks-selection. In Section 7 we investigate the existence of Condorcet winner(s).
2
Related literature
2.1
One good
Black (1948) introduces the problem of selecting the level of a single public good, when agents have single-peaked preferences. Moulin (1980) introduces generalized median rules and shows that these rules are the only ones that satisfy strategy-proofness,
anonymity and peaks-onliness.2 in the one good model. Ching (1997) shows that in
this characterization peaks-onliness can be replaced by continuity. With n agents, such a rule is described byn+ 1fixed cardinal points. The rule selects the median of then agents peaks and then+ 1 cardinal points.
Our delta rules can be viewed as an adaptation of the generalized median rules for the problem of selecting multiple public good levels, under the lexmax extension. A delta rule essentially locates each public good according to a generalized median
rule, whose cardinal points are not fixed, but depend in a particular way (which we describe) on the entire preference profile. In particular, a double generalized median rule, that selects each good according to a generalized median rule with fixed cardinal points is a special instance of a delta rule.
2.2
Multiple public goods: the max extension
The axiomatic study of the provision of multiple public goods is initiated by Miyagawa (1998, 2001). Miyagawa studies a model where alternatives are compared according to the max extension: a pair (or tuple) is preferred to another pair if and only if the best element of the first pair is preferred to the best element of the second pair.3
2See also Barberà and Jackson (1994).
3The implication of several axioms have been studied in this model. Miyagawa (2001) studies
Among other questions, Miyagawa (1998) examines the implications of strategy-proofnessin this model. He shows in particular that the only rule that satisfies Pareto-optimality,strategy-proofness andcontinuity is the extreme peaks rule, which selects the lowest peak and the highest peak for any preference profile. Ehlers (2001)
pro-vides an alternative characterization of the extreme peaks rule using “independence” properties. Bochet and Gordon (2012) characterize a class of rules that allocate pub-lic goods on agents’ peaks according to a priority ordering over unanimous profiles and satisfy strategy-proofness, the hierarchical rules.
The closest paper to ours in the max extension literature is by Heo (2013). Heo defines, for a given pair of public good levels and a given preference profile, the users
of a good as the agents who prefer it to the other and, possibly, the agents who are indi§erent between this good and the other. She then introduces a new property,
users-only, which says that the level of each good should only depend on the prefer-ences of its own users. Heo (2013) shows than the double generalized medians rules, that allocate each good according to a generalized median rule with fixed cardinal
points are the only rules that satisfy strategy-proofness, continuity, anonymity and
users-only.
The class characterized by Heo (2013) is a subclass of ours. However, her result is not logically implied by ours, because strategy-proofness is a stronger requirement under the lexmax than under the max extension. In particular the set of rules that satisfy our three main properties (strategy-proofness, continuity, and anonymity)
in the max extension model contains the delta rules, but also other complicated ones, which do not depend only on the agents’ peaks, but also on agents’ non-peak preference information. Heo (2013) also considers group strategy-proofness, which also has di§erent implications in the two models. While only some of the rules characterized by Heo (2013) are group strategy-proof with the max extension, all
delta rules satisfy this requirement under the lexmax extension.
2.3
Multiple public goods: the lexmax extension
Ehlers (2002, 2003) introduces the lexmax extension in the study of the provision of multiple public goods. He focuses on solidarity axioms and identifies and characterizes two interesting classes of rules, the single-plateaued rules and the single-peaked rules. Both of these classes are subsets of the class of delta rules.4 We discuss the exact
relation between Ehlers (2002, 2003) and our results in Section 5.
2.4
Other public good models with single-peaked preferences
Following Moulin (1980), a large literature studies the implications ofstrategy-proofness
in more general, sometimes abstract, common domains of single-peaked preferences.
It shows that Moulin’s (1980) generalized median voter rules can be adapted for these settings as well and characterized bystrategy-proofness and other properties. For ex-ample, Border and Jordan (1983) study the location of a public good on an Euclidean space. Barberà, Sonnenschein and Zhou (1990) study the selection of a subset from a set of candidates. Ehlers, Peters and Storcken (2002) study probabilistic decision rules. Schummer and Vohra (2002) studies the location of one public good on a
network. For a survey of this large literature, see Barberà (2011).
Our model departs from the models in this literature in two ways. The first one is that these models usually consider continuous preferences (sometimes by assuming that the set of alternatives is finite). This is not the case here, as lexmax preferences are never continuous. The second di§erence is crucial. These models generally assume that the common domain satisfies a richness condition. In particular, they assume
that each feasible alternative is the preferred alternative of some preference in the common domain. This is not the case here, since only alternatives on the diagonal (x, x) 2 [0,1]2 are the preferred alternative of some preference in the domain, even though all alternatives in [0,1]2 are feasible. The absence of the richness condition allows for a large set of strategy-proof rules.
4Gordon (2007a,b) has shown that in any pure public good model, of which the provision of
3
The Model
Let N ={1, . . . , n}, n 2N, be the set of agents. Each agent i2N is equipped with a single-peaked continuous preference relationRi defined over[0,1].5 We denote the
associated strict relation by Pi and the indi§erence relation by Ii. Single-peakedness
means that there exists a point p(Ri)2[0,1], called the peak of Ri, such that for all
x, y 2[0,1], ifx < y p(Ri)orx > y p(Ri), then y Pi x. LetRdenote the class of
all single-peaked preferences over[0,1]. LetRN denote the set of(preference) profiles
R= (Ri)i2N such that for alli2N, Ri 2R. Let p(R) = (p(Ri))i2N denote the peak
profile of R. For S N, the restriction (Ri)i2S of R to S is denoted by RS. Given
a profile R 2 RN, the smallest peak is denoted by p(R), and the largest peak by p(R). For S N, two profiles R, R0 2 RN are called S-deviations if R
N\S = R0N\S.
For i 2 N, two profiles are i-deviations if they are {i}-deviations. Note that for
i-deviations R and R0, we have R0 = (R0i, RN\{i}) and we sometimes write (R0i, Ri)
instead of R0.
Two public goods have to be selected. An alternative is a pair (x, y) such that 0xy1. The set of alternatives is denoted by [0,1]2.
A decision rule is a mapping' associating with every profileR 2RN an
alterna-tive. We write '(R) = ('1(R),'2(R))(where 0'1(R)'2(R)1).
Preferences are extended from [0,1] to the set of alternatives [0,1]2. With the
interpretation that each alternative is an option set, we assume that each agent com-pares two alternatives via thelexmax ordering induced by his single-peaked preference over[0,1]. Abusing notation, we use the same symbols to denote preferences over al-ternatives. Thus, for two pairs (x, y), and (x0, y0), and two permutations of {x, y}
and0 of{x0, y0}such that(x)Ri (y)and 0(x0)Ri 0(y0)we have(x, y)Pi (x0, y0)
if either (x) Pi 0(x0) or both (x) Ii 0(x0) and (y) Pi 0(y0). If (x) Ii 0(x0)
and (y)Ii 0(y0), then (x, y)Ii (x0, y0).
4
Strategy-proofness, anonymity and continuity:
a characterization
In this section, we first introduce our main properties,strategy-proofness,anonymity
and continuity. Second, we characterize the class of one-agent decision rules that satisfy strategy-proofness and continuity. Third, we turn to the n agent problem. We introduce delta rules and show that this class is characterized by our three main properties.
4.1
Main properties
First we introduce our main properties. Strategy-proofness means that an agent does not gain by misreporting his true preference.
Strategy-proofness: For alli2N and all i-deviationsR, R0 2RN,'(R)Ri '(R0).
Remark 1 Note that strategy-proofness with respect to the lexmax extension implies
strategy-proofnesswith respect to the max extension, studied by Miyagawa (1998) and
Heo (2013): for two alternatives (x, y) and (x0, y0), and two permutations of {x, y}
and0 of{x0, y0}such that(x)Ri(y)and0(x0)Ri 0(y0), under the max-extension Rmax
i we have (x, y) Rmaxi (x0, y0) i§ (x) Ri 0(x0).
Anonymity means that the decision rule is symmetric in its arguments.
Anonymity: For all permutations of N, '(R) ='((R)).6
To introduce continuity, we need to first define a metric on R. A preference Ri
is uniquely represented by a function ri : [0,1] ! [0,1] which denotes the closest
substitute on the other side of the peak of Ri (Sprumont, 1991): for all x 2 [0,1],
(i) if x p(Ri), then ri(x) 2 [p(Ri),1] is such that there exists no y 2 [p(Ri),1]
with ri(x)PiyRix or xRiyPiri(x) and (ii) if x p(Ri), then ri(x)2 [0, p(Ri)] is such
that there exists no y 2 [0, p(Ri)] with ri(x)PiyRix or xRiyPiri(x). Note that ri is
uniquely defined becauseRi is continuous and single-peaked. Let the metric onR be
such that
d(Ri, R0i) := max x2[0,1]|
ri(x)ri0(x)|.
Continuity: A rule ' is continuous if it is continuous for the metric d(·,·) onR.
4.2
One agent
Moulin (1980) first determines the one-agent decision rules and uses this result for his characterization of decision rules with n agents. We follow his route and first characterize the decision rules satisfying strategy-proofness and continuity for one agent. This result will be used later.
Recall from Ching (1997) and Moulin (1980) that a one-agent decision rule ' :
R![0,1]for one public good satisfiesstrategy-proofness andcontinuity if and only if there exist µ, µ2[0,1]such that
'(R1) = med(µ, p(R1), µ) for all R1 2R.
Back to the two good problem, let [x, x],[y, y][0,1]be such that
xy and xy. (a)
The range of the smallest public good will be [x, x] and[y, y]will be the range of the greatest public good. A pair of continuous functions
b: [x, x]7![y, y]
a: [y, y]7![x, x]
is called feasible if
xy=)a(y) =x and b(x) = y; (b1)
for all z 2[x, x], z a(b(z)); (b2)
for all z 2[y, y], b(a(z))z; (b3)
for all z 2[x, x]\[y, y], a(z) = z =b(z). (b4)
(b2), (b3) and (b4)). Given (a, b) 2 F, we define the one-agent decision rule '(a,b)
based on(a, b)as follows:
'(1a,b)(R1) =med(x, p(R1), a(med(y, p(R1), y)))
and
'(2a,b)(R1) =med(b(med(x, p(R1), x)), p(R1), y).
Observe that for the smallest public good, we have '(1a,b)(R1) = med(x, p(R1), a(y))
whenever p(R1)< y. This is just as in Moulin (1980) wherex and a(y) play the role
of cardinal points. Only whenp(R1)is large enough, or more precisely when p(R1)is
entering the range [y, y] of the greatest public good, then p(R1) is translated by the
function a and we have '(1a,b)(R1) =a(p(R1)) whenever p(R1)2[y, y]. If p(R1)> y,
then '(1a,b)(R1) = a(y).
An analogous description applies to the largest public good.
By Theorem 1, these decision rules are characterized by strategy-proofness and
continuity.
Theorem 1 Let ' be a one-agent decision rule. Then, ' satisfies strategy-proofness
and continuity if and only if for some (a, b)2F, '='(a,b).
We first show that if ' satisfies strategy-proofness and continuity, then the se-lected levels only depend on the peaks of the agents in the profile. This result is true for all decision rules (with an arbitrary number of agents).
Peaks-onliness: For all R, R0 2 RN, if for all i 2 N, p(Ri) = p(R0i), then '(R) =
'(R0).
Lemma 1 Let ' be a decision rule that satisfies strategy-proofness and continuity.
Then ' satisfies peaks-onliness.
Proof. Let R 2 RN, i
2 N and R0
i 2 R be such that p(Ri0) = p(Ri). We will
show that '(R) ='(R0
i, Ri). If for some R00i 2R, '(R00i, Ri) = (p(Ri), p(Ri)), then
by strategy-proofness, '(R) = (p(Ri), p(Ri)) = '(R0i, Ri). Otherwise, suppose that
for all R00
Let R
i : ! R be a continuous path (with respect to d) such that Ri (!) = Ri, R
i (!) = R0i, and for all ! 2 , p(Ri(!)) = p(Ri). Let x(!) and y(!) be two
functions such that '(R
i (!), Ri) = (x(!), y(!)) for all ! 2. By continuity, the
functions x(·) and y(·) are also continuous.
Consider four sets of parameters on the path:
1 :={!2[!,!] :p(Ri)x(!)y(!)},
2 :={!2[!,!] :x(!) Ri (!) y(!) and x(!)< p(Ri)< y(!)},
3 :={!2[!,!] :y(!) Pi(!) x(!) and x(!)< p(Ri)< y(!)},
4 :={!2[!,!] :x(!)y(!)p(Ri)}.
Because for each ! 2, (x(!), y(!))6= (p(Ri), p(Ri)), {1,2,3,4} is a partition
of. Then for all! 21,the preferenceRi (!)induces the same linear ordering over
the pairs in the set{(x(!0), y(!0)) :!0 2
1}.A pair is preferred to another if either
its first coordinate is lower, or the first coordinates are equal but its second coordinate is lower. Let (x1, y1) be the top element of the set {(x(!0), y(!0)) :!0 21} for
any preference Ri (!) with ! 2 1. By strategy-proofness, we have (x(!), y(!)) =
(x1, y1) for all !
2 1, i.e. the function (x(·), y(·)) is constant on 1. Similarly,
this function is constant on 2, 3 and 4. Therefore, each of these sets is closed.
If (x(!), y(!)) 6= (x(!), y(!)), then there are distinct k, l 2 {1,2,3,4} such that
k 6= ; 6= l (whereby k 2 {1,4} can be chosen if for all !0,! 2 2 [3 we have
x(!) = x(!0) and y(!) = y(!0)). This contradicts the fact that {1,2,3,4} is
a partition of [!,!] because a closed interval cannot be written as a disjoint union of several closed sets. Thus, (x(!), y(!)) = (x(!), y(!)) and '(R) = '(R0
i, Ri). Peaks-onliness follows then directly from this property.
This implication holds in a large class of domains, as shown by Weymark (2008). Note however that our model does not satisfy the richness condition of the domain and the continuity of preferences under which he obtains this result. Interestingly,
with respect to the max extension, this implication does not hold but Heo (2013) shows that strategy-proofness, continuity and users-only imply peaks-only in that model. We are now ready to prove Theorem 1.
Define for all R1 2 R, 'ˆ(p(R1)) := '(R1). Note that by peaks-onliness of ', 'ˆ :
[0,1]![0,1]2 is well-defined. Obviously, 'ˆ inherits continuity from'.
Let
x:= min
z2[0,1] ˆ
'1(z), x:= max
z2[0,1] ˆ
'1(z), y := min
z2[0,1] ˆ
'2(z), and y:= max
z2[0,1] ˆ
'2(z).
By continuity of ', these numbers are well-defined. By definition, x y and x y, and condition (a) is satisfied.
First, let [x, x]\ [y, y] 6= ;, i.e. y x. We show that for all z 2 [y, x] we have 'ˆ1(z) = ˆ'2(z) = z. By definition and strategy-proofness, z 2 {'ˆ1(z),'ˆ2(z)}.
Suppose z = ˆ'1(z) and z < 'ˆ2(z). Since z 2 [y, y], there exists some z0 2 [0,1]
such that 'ˆ2(z0) = z and 'ˆ1(z0) z. But then we may choose R1 2 R such that
ˆ
'1(z0)P1'ˆ2(z) and p(R1) = z. Now this contradicts strategy-proofness of ' because
'(R1) = ˆ'(z)and by reporting R01 such that p(R01) = z0 we have '(R01) = ˆ'(z0) and
ˆ
'(z0)P
1'ˆ(z). Hence, for all z 2[x, x]\[y, y],
ˆ
'1(z) = ˆ'2(z) = z. (1)
Second, we define a and b. Let
a(z) := ˆ'1(z)for all z 2[y, y].
and
b(z) := ˆ'2(z) for all z 2[x, x],
Note that by definition of 'ˆ, (1), and strategy-proofness of ', for all z 2 [x, x], ˆ
'1(z) = z, and for all z 2 [y, y], 'ˆ2(z) = z. Thus, a : [y, y]! [x, x] and b : [x, x] !
[y, y] are well-defined.
Third, we show that the pair(a, b)is feasible. Bycontinuity of'ˆ, bothaandb are continuous. By (1) and the definitions, for allz 2[x, x]\[y, y],b(z) = z=a(z), which
is condition (b4). Furthermore, x = ˆ'1(x) and by definition b(x) max{x, y}. If
the inequality is strict, then by (1), x < y. Furthermore, by definition and strategy-proofness, 'ˆ2(y) = y and 'ˆ1(y) < x. But now our definitions, x < y, the above
facts and strategy-proofness imply 'ˆ(12(x+y)) = (x, y). This is a contradiction to
strategy-proofness since for R1, R01 2 R such that p(R1) = x and p(R01) = 1
2(x+y)
manipulates atR1 viaR01, a contradiction to strategy-proofness. Thus, ifx < y, then
b(x) = y and similarly,a(y) = x which together with (1) yields condition (b1).
By strategy-proofness, peaks-onliness, and the definitions, it is straightforward that for allz 2[x, x],z a(b(z)), which is condition (b2). Similarly, for allz 2[y, y],
b(a(z))z, which is condition (b3).
Fourth, we show that ' = '(a,b). Let R
1 2 R. If p(R1) < x, then by
strategy-proofness, continuity, and peaks-onliness, '1(R1) = ˆ'1(p(R1)) = x = ' (a,b) 1 (R1).
If p(R1) 2 [x, x], then '1(R1) = p(R1) = ' (a,b)
1 (R1). If p(R1) 2 ]x, y[, then by
definition and strategy-proofness, '1(R1) = x. Since by conditions (b1) and (b4),
a(y) = min{x, y}, we also have '(1a,b)(R1) = x. If p(R1) 2 [y, y], then '1(R1) =
a(p(R1)) = ' (a,b)
1 (R1). If p(R1) > y, then by strategy-proofness, continuity, and
peaks-onliness,'1(R1) = ˆ'1(p(R1)) =a(y) =' (a,b)
1 (R1). Hence, '1(R1) =' (a,b) 1 (R1).
Similarly, it follows '2(R1) = ' (a,b)
2 (R1), the desired conclusion.
(If) If for some(a, b)2F we have' ='(a,b), then it is straightforward to verify that
' satisfies strategy-proofness and continuity.
The following example gives some insight to the extent of “arbitrariness” of feasible pairs of functions.
Example 1 Let a(z) = 1
2 for all z 2 [ 1
2,1] and b : [0, 1 2] ! [
1
2,1] be an arbitrary
continuous function such that b(12) = 12. Then [x, x] = [0,12] and [y, y] = [12,1]. Obviously, b(x) = 12 = max{x, y} and a(y) = 12 = min{x, y}. Furthermore, for all
z 2 [0,12], z 21 a(b(z)); and for all z 2 [12,1], both b(a(z)) = b(12) = 12 z and
b(]a(z),12]) =b(;) =;. Hence, (a, b) is feasible and '(a,b) satisfies strategy-proofness
and continuity.
4.3
Many agents
We first introduce the class ofdelta rules (fornagents). We then show that our three main properties characterize it.
It will helpful to recall from Ching (1997) and Moulin (1980) that a decision
rule ' : RN ![0,1] for one public good satisfiesanonymity, strategy-proofness and
µ0 µ1 · · ·µn and
'(R) = med(p(R1), . . . , p(Rn), µ0, µ1, . . . , µn) for all R2RN.
Here for anystrategy-proof rule the numberµk is calibrated via profiles wherenk
agents have their peak at zero and k agents have their peak at one. From
strategy-proofness it follows that when more agents move to the right, these numbers must move to the right, i.e. we have µ0 µ1 · · ·µn.
4.3.1 Delta rules
LetN0 =N[{0} and let throughout(s, t)be a pair such that s, t 2N0. Similarly to
the location of one public good, for two public goods and the lexmax extension any rule satisfying our properties will be calibrated by considering certain profiles: more precisely, for any pair(s, t)it will be important to consider the induced rule whereby
s agents have their peak at zero, t agents have their peak at one, and the remaining
agents move together from zero to one. As we show, any such rule is a one-agent decision rule satisfying strategy-proofness and continuity and thus, by Theorem 1, determined by a pair (as,t, bs,t)2F.
Conditions (A1), (A2), and (A3) below (analogous to condition (a) for the one agent case) require that the ranges of these functions move to the right as more agents move to the right. Second we show that the functionas,tcan be equivalently obtained
whereby s agents have their peak at zero and all other agents move together from zero to one (and this function we denote by as instead of as,0). Similarlybs,t can be
obtained whereby t agents have their peak at one and all other agents move together from zero to one (and this function we denote bybt instead of b0,t). Conditions
(B1)-(B4) are the same conditions as in the one agent case on the functions as,t and bs,t
expressed in terms ofasandbt. Condition (C) is inclusion of ranges of the functionsas
andbtfor certain parameters. Conditions (D) and (E) are monotonicity requirements on the functionsas and bt when more agents are located at one of the extremes zero
or one. The formal description is below.
are elements of [0,1]{(s,t)2N20:0s+tn} such that, for each(s, t)with s+t
n1,
xs+1,t xs,t and ys,t ys+1,t (A1)
xs,t+1 xs,t and ys,t ys,t+1 (A2)
and for each(s, t)such that s1, tn1, and s+tn,
(xs,t, ys,t)(xs1,t+1, ys1,t+1) ; (A3)
and a= (a0, ..., an1) and b = (b0, ..., bn1) are continuous functions
as : [ys,0, ys,ns]![xn,0, xs,0]
bt : [xnt,t, x0,t]![y0,t, y0,n].
that satisfy the following conditions:
For all (s, t) such that s+tn1 :
xs,tys,t =) as(ys,t) = xs,t and bt(xs,t) =ys,t; (B1)
for all z 2[xnt,t, xs,t], zas(bt(z)) ; (B2)
for all z 2[ys,t, ys,ns], z bt(as(z)) ; (B3)
for all z 2[ys,t, ys,ns]\[xnt,t, xs,t], as(z) =z =bt(z). (B4)
Conditions (B2) and (B3) are in fact equivalent. They say on the rectangle to the north-west of the pair (xs,t, ys,t), i.e.
{(z, z0) : (xs,t, ys,t)(z, z0)(0,1)},
the graph of as is always to the north-east of the graph of bt. The two graphs have no strict crossings in this rectangle, although they may touch without crossing (for example, they may be tangent). Moreover:
as([ys,t, ys,t+1])[min{xs+1,t, ys,t},min{xs,t, ys,t+1}] ; (C)
bt([xs+1,t, xs,t])[max{xs+1,t, ys,t},max{xs,t, ys,t+1}].
For all z 2ys+1,0, ys+1,n(s+1)
, as+1(z)as(z); (D)
For all z 2xn(t+1),t+1, x0,t+1
For all z 2[xs,ns, xs,0], if bns(z)2[ys,0, ys,ns], then z as(bns(z)); (E)
For all z 2[ys,0, ys,ns], if as(z)2[xs,ns, x0,ns], then z bns(as(z)).
The two inequalities (E) are in fact equivalent. They complement the conditions (B2) and (B3). They say on the rectangle to the south-east of the pair(xs,ns, ys,ns),
i.e.
{(z, z0) : (1,0)(z, z0)(xs,t, ys,t)},
the graph ofasis always to the south-west relative to the graph ofbt.The two graphs
have no strict crossings in this rectangle, although they may touch without crossing (for example, they may be tangent). Moreover, the graphs of the functions as and
bns do not (strictly) cross anywhere else in[0,1]2 except at(xs,t, ys,t).
These restrictions are illustrated in Figure 3. For (a, b, x, y) 2 Fn we define
the delta rule '(a,b,x,y) associated with (a, b, x, y) as follows. For all R
2 RN, let p(R) =:p= (p1, ..., pn) be the vectors the peaks in p(R) ranked in increasing order,
so that p1 ... pn. The analogue of the cardinal points in the single public good
case are here cardinal functions, defined as follows:
0(p) := a0(med(p1, ..., pn, y0,0, ..., y0,n))
...
s(p) := as(med(ps+1, ..., pn, ys,0, ..., ys,ns))
...
n1(p) := an1(med(pn, yn1,0, yn1,1)).
and n(p) := xn,0. Note that for instance for 0, we have by conditions (A), y0,0
y0,1 · · · y0,n and y1,n1 y0,n (and observe condition (D) in case the ranges of
the median of 2(ns) + 1 numbers. Similarly, let
0(p) :=b0(med(p1, ..., pn, x0,0, ..., xn,0))
...
t(p) :=bt(med(p1, ..., pnt, x0,t, xnt,t))
...
n1(p) :=bn1(med(p1, x0,n1, x1,n1))
and n(p) :=y0,n. Finally, let
'(1a,b,x,y)(R) = med(p1, ..., pn,0(p), ...,n(p))
'(2a,b,x,y)(R) = med(p1, ..., pn,0(p), ...,n(p)).
These formulas will be the “right” generalizations of the one-agent decision rules in Theorem 1 to n agents. Interestingly, the (s+ 1)-th highest cardinal function (with index s) for the first good does not depend on the s lowest peaks and the (t+ 1)-th lowest cardinal function (with index t) for the second good does not depend on the t
highest peaks.
4.3.2 Double generalized median rules
A important subset of the class of delta rules is the one where each good is selected according to some fixed generalized median rule (Moulin, 1980). A rule is a double generalized median rule if its parameters (a, b, x, y) are such that there are two fixed
vectors (µ0, ..., µn) and (0, ...,n)such that
1µ0 · · ·µn 0
00 · · ·n 1
and for alls, t such thats+t n, we have
xs,t =µs
ys,t =t
for all z 2[ys,0, ys,ns], as(z) = min{µs, z}
Among these rules, the quantile rules are those such that (µ
0, ..., µn)2 {0,1} n+1
and (
0, ...,n) 2 {0,1} n+1
. They select each good according to a fixed quantile.
An important quantile rule is the extreme peaks rule, which selects the lowest and highest peaks in the profile and corresponds to the parameters(µ
0, ..., µn) = (0, ...,0)
and (0, ...,n) = (1, ...,1).
We will discuss properties of these subclasses in Section 5.
4.3.3 Main characterization
We are now ready to present our main result, which says that the delta rules are characterized byanonymity, strategy-proofness and continuity.
Theorem 2 Let'be a decision rule. Then,'satisfiesanonymity, strategy-proofness
and continuity if and only if for some (a, b, x, y)2Fn,
'='(a,b,x,y).
It is clear from their definition that the delta rules satisfy anonymity and conti-nuity. In the next lemma, we verify that these rules also satisfy strategy-proofness.
We then show conversely that any rule satisfying these three properties must be a
delta rule.
Lemma 2 For all (a, b, x, y)2Fn, the rule'(a,b,x,y) satisfies strategy-proofness.
Proof. Let i 2 N. Without loss of generality, we will assume that i = n, so that
N \ {n}={1, ..., n1}. Let RN\{n} be a fixed profile and letp1 ...pn1 be the
peaks inpRN\{n}
ranked in nondecreasing order. We will show that
:Rn!
'(1a,b,x,y)(Rn, RN\{n}),'(2a,b,x,y)(Rn, RN\{n})
.
is a strategy-proof one-agent decision rule, i.e. is of the form '(a,b) for some
(a, b)2 F. Using the conventions p0 := 0 and pn+1 := 1, let j be the unique index
in{0, ..., n}such that for allRn,1(Rn) =p(Rn)only ifp(Rn)2[pj, pj+1]. Similarly,
letk be the unique index in{1, ..., n1} such that for all Rn, 2(Rn) =p(Rn)only
if p(Rn)2[pk, pk+1]. Then the formulas defining simplify as follows
1(Rn) =med
max{pj,j+1(max{pj+1, p(Rn)}, ..., pn1)}, p(Rn),
min{pj+1,j(max{p(Rn), pj}, pj+1, ..., pn1)}
and
2(Rn) =med
maxpk,n(k+1)(p1, ..., pk,min{pk+1, p(Rn)})
,
p(Rn),min{pk+1,nk(p1, ...,min{pk, p(Rn)})} !
which further simplify as follows
1(Rn) =med
max{pj, xj+1,k}, p(Rn),
min (pj+1, aj(med(max{pk, yj,k}, p(Rn),min{pk+1, yj,k+1})))
!
2(Rn) =med
max{pk, bk+1(med(max{pj, xj+1,k}, p(Rn),min{pj+1, xj,k}))},
p(Rn),min{pk+1, yj,k+1}
! .
The rule is then the rule '(a,b) with functions b
: [x, x] 7! [y, y] and a :
[y
, y]7![x, x]such that
x = max{pj, xj+1,k}
x = min{pj+1, xj,k}
y
= max{pk, yj,k} y = min{pk+1, yj,k+1}
for all z 2[y
, y], a
(z) = min
{pj+1, aj(z)}
for all z 2[x, x], b(z) = max{pk, bk+1(z)}.
Note that pj pj+1 and xj+1,k xj,k (by (A1)) imply x x; and pk pk+1 and
yj,k yj,k+1 (by (A2)) imply y y. Conditions (A), (B), (C) and (D) ensure
that (a, b) 2 F. Therefore and '(a,b,x,y) satisfy strategy-proofness, the desired
conclusion.
4.3.4 Proof of the “only if” implication in Theorem 2.
Throughout the proof, let ' : RN ! [0,1]2 be a decision rule satisfying strategy-proofness,continuity, andanonymity. We will show that there is a list(a, b, x, y)2Fn
such that '='(a,b,x,y).
Step 1: Preliminaries We first establish two useful lemmas. The next lemma says
Lemma 3 Let ' : RN
! [0,1]2 be strategy-proof. Then for all
; 6= S N, for all
S-deviations R, R0 2RN,such that R
i =Rj and R0i =R0j for all i, j 2S, we have for
all i2S, '(R) Ri '
R0
S, RN\S
.
Proof. Without loss of generality, suppose that S = {1, ..., s}. By
strategy-proofness, '(R)R1'(R01, R1) and for eachi2{1, ..., s1},
'R0{1,...,i}, R{i+1,...,n}
Ri '
R0{1,...,i+1}, R{i+2,...,n}
.
SinceR1 =R2 =· · ·=RsandR1 is transitive, it follows that'(R)R1 '
R0
S, RN\S
,
the desired conclusion.
Next, we observe that any strategy-proof and continuous rule must satisfy the following properties. The first property says that if some agents’ peaks are initially to the left of the left good in the alternative selected by the rule at the initial profile, and their preferences change in such a way that their peak remains weakly to the left of the left good in the initially selected alternative, the rule selects the same alterna-tive after and before the change.7
Left-uncompromisingness: For all R, R0 2 RN, and M N such that for all
i2M, p(Ri)<'1(R) and p(R0i)'1(R), we have'
R0
M, RN\M
='(R).
We also define the symmetric property and other similar properties.
Right-uncompromisingness: For all R, R0 2 RN, and M N such that for all
i2M, '2(R)< p(Ri) and '2(R)p(Ri0), we have'
RM0 , RN\M
='(R).
Center-uncompromisingness: For all R, R0 2 RN, and M
N such that for all i 2 M, '1(R) < p(Ri) < '2(R) and '1(R) p(Ri0) '2(R), we have
'R0
M, RN\M
='(R).
7Ching (1997) introduces a similar property in the one good problem. Heo (2013) defines a
Left-trans-uncompromisingness: For allR, R0 2 RN, and M
N such that for all i 2 M, p(Ri) '1(R) and p(R0i) = '2(R), we have '1(R) '1
R0
M, RN\M
and '2
R0
M, RN\M
='2(R).
Right-trans-uncompromisingness: For all R, R0 2 RN, and M N such that
for alli2M,'2(R)p(Ri)and p(R0i) ='1(R),we have '1
R0
M, RN\M
='1(R)
and '2
R0
M, RN\M
'2(R).
Lemma 4 Let ' be a rule that satisfies strategy-proofness and continuity. Then '
satisfies peaks-onliness and the five forms of uncompromisingness listed above.
Proof. We know from the results of Section 4.2 that all one-agent decision rules,
which arestrategy-proof andcontinuous, satisfy these properties. Decision rules with
n agents, which arestrategy-proof andcontinuous, inherit these properties, since the
profile transformations that are considered can be achieved in|M|steps by replacing the preference of one agent at a time.
Step 1: We calibrate the rule ' and obtain a list (a, b, x, y) from'. We first
calibrate the rule. For each M N and z 2 [0,1], let Rz
M 2 RM be a preference
profile such thatp(Rzi) =z for alli2M.From' we derive 2n+ 2one-agent decision rules as follows: for all(s, t)such that s+t n1, let 's,t :
R![0,1]2 be defined
by
's,t(Rzn) = '(R0{1,...,s}, R1{s+1,...,s+t}, Rz{s+t+1,...,n}) for all z 2[0,1].
This function describes the locus of the allocation, as n(s+t) agents peaks move together from0 to 1, while s agents peaks stay at 0 and t agents, who have already travelled, wait for the newcomers at 1.
Now by Lemma 3, for any pair (s, t) such that s+t n1, the rule 's,t is a
strategy-proof one-agent decision rule. Thus by Theorem 1, for any such pair, there exists (as,t, bs,t) 2 F such that 's,t = '(as,t,bs,t). Let xs,t be the upper end of the
domain ofbs,t and let ys,t be the lower end of the domain of as,t.
For all (s, t) such that s+t=n, let
(xs,t, ys,t) :='(R0{1,...,s}, R
1
Instead of xs,t we often write xnt,t and instead of ys,t we often write ys,ns. These
definitions imply that for any such pair, xs,t is the lower end of the domain ofbs0,t for
all s0 2 {0,1, . . . , s1} and that y
s,t is the upper end of the domain of as,t0 for all
t0 2{0,1, . . . , t1}. Thus, for alls+t n1,we obtain
as,t : [ys,t, ys,ns]![xnt,t, xs,t]
bs,t : [xnt,t, xs,t]![ys,t, ys,ns].
Last, for alls 2{0, ..., n1}, let
as :=as,0 (whereas: [ys,0, ys,ns]![xn,0, xs,0]),
and for allt 2{0, ..., n}, let
bt:=b0,t (wherebt[xnt,t, x0,t]![y0,t, y0,n]).
Step 2: We verify that (a, b, x, y) is indeed an element of Fn.
Lemma 5 For all (s, t) such thats+t n1,the inequalities (A1), (A2) and (A3)
hold.
Proof. Let (s, t) be such that s+t n1. Let i1, ..., is+1 and j1, ..., jt be distinct
agents inN,and letM0 :={i1, ..., is+1}, M1 :={j1, ..., jt}andM2 :=N\(M0[M1).
First, we show that xs+1,t xs,t. Let z := xs+1,t. If xs+1,t = 0, the inequality is
true. Suppose that xs+1,t > 0. Let R be a profile such that RM0 := R
0
M0, RM1 :=
R1
M1 and if M2 6= ;, RM2 := R
z
M2. By definition of xs+1,t, we have '1(R) = xs+1,t.
Since xs+1,t > 0, by left-uncompromisingness, we have '1(Rzis+1, RN\{is+1}) = xs+1,t.
Therefore 's,t1 (Rz
n) = xs+1,t. By definition of xs,t, this implies that xs+1,t xs,t, the
desired inequality.Via a symmetric argument we obtainys,tys,t+1, and (A1) holds.
Second, we show that ys,t ys+1,t. Letz :=ys+1,t. LetR be the profile such that RM0 :=R
0
M0, RM1 :=R
1
M1 and ifM2 6=;, RM2 :=R
z
M2.By definition ofys+1,t,we have '2(R) = ys+1,t. By left-trans-uncompromisingness, we have '2(Rzis+1, RN\{is+1}) =
ys+1,t.Therefore 's,t2 (Rzn) =ys+1,t.By definition of ys,t, this implies thatys,t ys+1,t,
the desired inequality. Via a symmetric argument we obtain xs,t+1 xs,t, and (A2)
Last, we show that xs+1,t xs,t+1. Let z := xs+1,t. Let R be a profile such that RM0 := R
0
M0, RM1 := R
1
M1 and if M2 6= ;, RM2 := R
z
M2. By definition of xs+1,t,
we have'1(R) =xs+1,t. Bystrategy-proofness, '(R) R0is+1 '
R1
is+1, RN\{is+1}
, and
therefore, '1(R) '1
R1
is+1, RN\{is+1}
. If M2 = ;, then by definition of xs,t+1, we
have '1
Ri1s+1, RN\{is+1}
= xs,t+1. Otherwise '1
Ri1s+1, RN\{is+1}
= 's,t1 +1(Rzn) and by definition ofxs,t+1, we have's,t1 +1(Rzn)xs,t+1. In both cases
xs+1,t ='s1+1,t(R
z
n) ='1(R)'1
R1i
s+1, RN\{is+1}
xs,t+1,
the desired inequality. Via a symmetric argument we obtainys+1,t ys,t+1, and (A3)
holds.
Lemma 6 For all z 2 [ys,t, ys,ns], as,t(z) = as(z) and for all z 2 [xnt,t, xs,t],
bs,t(z) =bt(z).
Proof. The first equality follows immediately from left-uncompromisingness, the
second fromright-uncompromisingness.
Lemma 7 The list (a, b, x, y) satisfies conditions(B1), (B2), (B3) and (B4).
Proof. By definition, the parameters of 's,t are
as,t : [ys,t, ys,ns]![xnt,t, xs,t]
bs,t : [xnt,t, xs,t]![ys,t, ys,ns],
and (as,t, bs,t)2F.From Lemma 6, we know that the restriction of as to [ys,t, ys,ns]
coincides with as,t and the restriction of bt to [xnt,t, xs,t] coincides with bs,t. Thus,
as|[ys,t,ys,ns], bt|[xnt,t,xs,t]
2 F. Conditions (B1), (B2) and (B3) follow easily from this observation. Furthermore, for all z 2 [ys,t, ys,ns]\[xnt,t, xs,t], we must have as(z) =z =bt(z), which is (B4). Note that xnt,t ys,t and xs,t ys,ns.
Lemma 8 For all (s, t) such thats+t n1, the inclusions
as([ys,t, ys,t+1])[min{xs+1,t, ys,t},min{xs,t, ys,t+1}] ; (C)
bt([xs+1,t, xs,t])[max{xs+1,t, ys,t},max{xs,t, ys,t+1}] ;
Proof. We prove the second inclusion. Note that (as,t, bs,t)2F. Now by Lemma 6,
the restriction of as to [ys,t, ys,ns]and the restriction of bt to[xnt,t, xs,t] form a pair
inF. Therefore,bt maps[xnt,t, xs,t]to[ys,t, ys,ns]. Moreover, for allz 2[xs+1,t, xs,t]
we havebt(z)z xs+1,t. It follows that
bt([xs+1,t, xs,t])[max{xs+1,t, ys,t}, ys,ns].
We still need to show that for all z 2 [xs+1,t, xs,t], bt(z) max{xs,t, ys,t+1}. Let
:R![0,1]2 be the one-agent decision rule defined by
(Rzn) = '(R0{1,...,s}, R1{s+1,...,s+t}, Rxs,t
{s+t+1,...,n1}, R
z n).
By Theorem 1 there is a pair (a , b ) 2 F such that = '(a ,b ). Let [x , x ] and [y , y ][0,1] be such that
b : [x , x ] 7![y , y ]
a : [y , y ]7![x , x ].
One shows that[x , x ] = [xs+1,t, xs,t]and[y , y ] = [max{xs+1,t, ys,t},max{xs,t, ys,t+1}]
(otherwise we obtain a contradiction to the definition of these numbers using
un-compromisingness), and for all z 2 [xs+1,t, xs,t], b (z) = bt(z) and thus, bt(z)
max{xs,t, ys,t+1},which proves the second inclusion. The first inclusion follows from
a symmetric argument.
Lemma 9 The list (a, b, x, y) satisfies condition (D).
Proof. We prove the first inequality of condition (D). Note that the domain of
as+1 is [ys+1,0, ys+1,n(s+1)]. By (A1), ys,0 ys+1,0, and by (A3), ys+1,n(s+1)
ys,ns]. Thus, [ys+1,0, ys+1,n(s+1)] [ys,0, ys,ns] which is the domain of as. Let
z 2ys+1,0, ys+1,n(s+1)
. Suppose by contradiction that as(z)< as+1(z). At the
pro-file
R01, ..., R0s+1, Rzs+2, ..., Rzn,
agent s + 1 can change the outcome from (as+1(z), z) to (as(z), z) (by reporting Rz
s+1) and strictly benefits from it, which contradicts strategy-proofness. The second
Lemma 10 The list (a, b, x, y) satisfies condition (E).
Proof. We prove the first inequality of condition (E). Note that
as : [ys,0, ys,ns]![xn,0, xs,0]
bns : [xs,ns, x0,ns]![y0,ns, y0,n].
Letz 2[xs,ns, x0,ns] be such thatbns(z)2[ys,0, ys,ns]. Letz0 =bns(z). Suppose
by contradiction thatz < as(bns(z)).Then
'Rz1, ..., Rzs, R1s+1, ..., R1n= (z, bns(z)).
By strategy-proofness (and using Lemma 3), this implies that
'Rz1, ..., Rzs, Rzs+10 , ..., Rnz0= (z, bns(z)). (2)
We also have
'R01, ..., R0s, Rzs+10 , ..., Rnz0= (as(z0), z0).
By strategy-proofness and z < as(bns(z)), this implies that
'Rz1, ..., Rzs, Rzs+10 , ..., Rnz0= (as(z0), z0)
which contradicts (2) and proves the first inequality of condition (E). The second inequality of condition (E) follows from a symmetric argument.
Step 3: We now show that ' = '(a,b,x,y), where (a, b, x, y) 2 F is obtained
from' in Step 1. The proof is by induction on the numbernRof distinct interior
peaks in the profile R, i.e.
nR :=|{p(Ri) :i2N and 0< p(Ri)<1}|.
If nR = 0 or nR = 1, then by peaks-onliness, R is one of the calibrating profiles
for obtaining (a, b, x, y), and by definition,'(R) = '(a,b,x,y)(R). Thus, the induction
basis holds for k = 1.
By induction suppose that'and'(a,b,x,y)coincide on all profilesRhaving at most
Let R 2 RN be such that n
R = k + 1. By peaks-onliness of ' and '(a,b,x,y), we
may suppose that for all i, j 2 N, if p(Ri) = p(Rj), then Ri = Rj. First, suppose
that for some i 2 N, 0 < p(Ri) < '
(a,b,x,y)
1 (R). Let M N be such that for all
j 2M, p(Rj) =p(Ri), and for all j 2N\M, p(Rj) 6=p(Ri). Let R0 = (R0M, RN\M).
By definition, nR0 =k. Thus, by the induction hypothesis, '(R0(a,b,x,y)(R0). Because both' and '(a,b,x,y) satisfyleft-uncompromisingness, we have
'(R) ='(R0(a,b,x,y)(R0(a,b,x,y)(R),
which is the desired conclusion. Using right-uncompromisingness and an analogous argument, we obtain '(R) ='(a,b,x,y)(R)if for some i
2N, '(2a,b,x,y)(R)< p(Ri)<1.
Similarly, using center-uncompromisingness and an analogous argument, we obtain
'(R) ='(a,b,x,y)(R) if for some i, j 2N, '1(a,b,x,y)(R)< p(Ri)< p(Rj)<'(2a,b,x,y)(R).
We are left with the case where there exists z 2 ]'(1a,b,x,y)(R),'(2a,b,x,y)(R)[ such that
{p(Ri) :i2N and 0< p(Ri)<1}{'
(a,b,x,y)
1 (R), z,' (a,b,x,y)
2 (R)}. (3)
Since nR 2, there exists j 2 N such that p(Rj) 2 {'
(a,b,x,y)
1 (R),'
(a,b,x,y)
2 (R)}, say
p(R1) = ' (a,b,x,y)
1 (R).
Similarly, by interchanging the roles of ' and '(a,b,x,y) we obtain that there exists
z0 2 ]'
1(R),'2(R)[such that
{p(Ri) :i2N and 0< p(Ri)<1}{'1(R), z0,'2(R)}. (4)
Since p(R1) =' (a,b,x,y)
1 (R), we now have '1(R)p(R1).
Suppose that '(R)6='(a,b,x,y)(R).
If '1(R) 6= ' (a,b,x,y)
1 (R), then '1(R) < p(R1) = ' (a,b,x,y)
1 (R) and by nR 2, for
some j 2N, p(R1)< p(Rj)'2(R). By (4), we have z0 =p(R1) and
{p(Ri) :i2N and 0< p(Ri)<1}={p(R1),'2(R)}.
Letl 2N be such thatp(Rl) ='2(R) (where '2(R)<1).
Let M N be such that M = {i 2 N : 0 < p(Ri) < 1} and R0 2 RN be such that R0N\M = RN\M and for all i 2 M, R0i = Rl. By center-uncompromisingness of
','(R0) = '(R). Because n
(observing Lemma 3), '(2a,b,x,y)(R) 6= '2(R). If ' (a,b,x,y)
2 (R) < p(Rl), then this
con-tradicts (3). Thus, p(R1) < '2(R) = p(Rl) < '
(a,b,x,y)
2 (R). Let R00 2 RN be such
that R00N\M =RN\M and for all i 2 M, R00i =R1. By center-uncompromisingness of
'(a,b,x,y), we have '(a,b,x,y)(R00(a,b,x,y)(R). Because nR00 = k, the induction hypothesis
implies '(R00(a,b,x,y)(R00).
Summarizing, we have
'1(R)< p(R1) =' (a,b,x,y)
1 (R)< p(Rl) ='2(R)<' (a,b,x,y)
2 (R)
and '(R00(a,b,x,y)(R). Let M0 ={i2N :p(R
i) =p(Rl)} and for allv 2[p(R1), p(Rl)],
let Rv = (Rv
M0, R00N\M0). By peaks-onliness of ', we have '(Rp(Rl)) = '(R) and
'(Rp(R1)) ='(a,b,x,y)(R). Using '(Rp(R1)) ='(a,b,x,y)(R), strategy-proofness of ' and
peaks-onlinessof ', for allv 2[p(R1), p(Rl)]it is not possible that'1(Rv)< p(R1)<
v < '2(Rv). Thus, by strategy-proofness, if '1(Rv) < p(R1), then v = '2(Rv).
Similarly, if p(Rl)<'2(Rv), then v ='1(Rv).
By continuity of ', for some v 2 ]p(R1), p(Rl)[, '1(R) < '1(Rv) < p(R1) or
p(Rl) < '2(Rv) < ' (a,b,x,y)
2 (R), say p(Rl) < '2(Rv) < ' (a,b,x,y)
2 (R). By the above,
for any such v we have '1(Rv) = v. Hence, by continuity of ', there exists ˆv 2
]p(R1), p(Rl)[such that'(Rvˆ) = (ˆv, p(Rl)). Because'1(R)< p(R1)<ˆv and'2(R) =
p(Rl), now this is a contradiction to strategy-proofness of ' (because '(Rp(Rl)) =
'(R)).
Thus, if '(R)6='(a,b,x,y)(R), then '
1(R) =' (a,b,x,y)
1 (R) and '2(R)6=' (a,b,x,y)
2 (R),
say'(2a,b,x,y)(R)<'2(R). By (3) and (4), there exists l2N such that
{p(Ri) :i2N and 0< p(Ri)<1}={p(R1), p(Rl)}
andp(R1)< p(Rl)'
(a,b,x,y)
2 (R). LetM ={i2N :p(Ri) = p(Rl)}. LetR0 2RN be
such thatR0
N\M =RN\M and for all i2M, R0i =R1. By center-uncompromisingness
of','(R0) ='(R). Becausen
R0 =k, the induction hypothesis implies'(a,b,x,y)(R0) =
'(R0). But now'(a,b,x,y)(R)P
1'(a,b,x,y)(R0)which contradictsstrategy-proofness using
Lemma 3 and the factR0
i =R1 for all i2M.
Hence, '(R) = '(a,b,x,y)(R) for all R
2 RN such that n
R k+ 1. By induction
this is now true for all profileR 2RN, the desired conclusion.
formula” for defining the rule'(a,b,x,y). Here it su¢ces to show this for the calibrating
profiles.
We will use the following convention. For anyz 2[0,1]and any(s, t), let(0s, zns)
denote the peak profile where s agents have their peak at 0 and ns agents have
their peak atz; and similarly, let (znt,1t)denote the peak profile wherent agents have their peak at z and t agents have their peak at 1.
Lemma 11 Let (a, b, x, y)2Fn.
(i) For all s2{0, . . . , n1} and all z 2[ys,0, ys,ns] we have
as(z) = med(0s, zns,0(0s, zns), . . . ,n(0s, zns))
z =med(0s, zns,0(0s, zns), . . . ,n(0s, zns)).
(ii) For all t2{0, . . . , n1} and all z 2[xnt,t, x0,t] we have
z =med(znt,1t,0(znt,1t), . . . ,n(znt,1t)),
bt(z) = med(znt,1t,0(znt,1t), . . . ,n(znt,1t)).
Proof. We only show (i) (because (ii) follows from a symmetric argument). Take
l 2{0, . . . , n1}. By definition (using the convention p= (0s, zns))
l(0s, zns) =al(med(pl+1, . . . , pn, yl,0, . . . , yl,nl)).
Note that the argument of al is the median of 2(nl) + 1 numbers, and by (A2), yl,0 · · ·yl,nl. Thus,
(*) ifz < yl,0, thenmed(pl+1, . . . , pn, yl,0, . . . , yl,nl) =yl,0;
(**a) if l s and z 2[yl,0, yl,nl], thenmed(pl+1, . . . , pn, yl,0, . . . , yl,nl) =z;
(**b) if l < sand z 2[yl,0, yl,ns], then med(pl+1, . . . , pn, yl,0, . . . , yl,nl) = z;
(***a) l s and yl,nl < z, thenmed(pl+1, . . . , pn, yl,0, . . . , yl,nl) =yl,nl.
(***b) l < sand yl,ns < z, then med(pl+1, . . . , pn, yl,0, . . . , yl,nl) = yl,ns.
• Ifl =s,then by asssumptionz 2[yl,0, yl,nl], so thatmed(pl+1, . . . , pn, yl,0, . . . , yl,nl) = z. From now on, we only consider the cases wherel 6=s.
• (*) andz ys,0 and (A1) imply that l > s.
We now show that for all l > s, we have
l(med(pl+1, . . . , pn, yl,0, . . . , yl,nl))as(z)
and for alll < s, we have
l(med(pl+1, . . . , pn, yl,0, . . . , yl,nl))as(z).
Case 1. Let l > s. In this case
l(med(pl+1, . . . , pn, yl,0, . . . , yl,nl)) = al(med(yl,0, z, yl,nl)).
If (**a) holds, then by condition (D) (check the domain condition for this inequality),
we obtain
l(med(pl+1, . . . , pn, yl,0, . . . , yl,nl)) =al(z)as(z),
the desired conclusion.
Case 2. Let l < s. In this case
l(med(pl+1, . . . , pn, yl,0, . . . , yl,nl)) = al(med(yl,0, z, yl,ns))
=al(max (z, yl,ns)).
l(med(pl+1, . . . , pn, yl,0, . . . , yl,nl)) = al(med(yl,0, z, yl,nl)).
• yl,nl< ys,ns and by (A3),s < l, and med(pl+1, . . . , pn, yl,0, . . . , yl,nl) = yl,nl.
By (A1), we have y0,0 y1,0 · · · yn,0. Because ys,0 z, it follows that
for all l 2 {0, . . . , s} (noting that by (A3), ys,ns yl,nl and (***) is not
possi-ble), med(pl+1, . . . , pn, yl,0, . . . , yl,nl) =z. Thus, for alll 2 {0, . . . , s}, l(0s, zns) = al(z). By condition (D), we have as(z) as1(z) · · · a0(z), and by
defini-tion as(z) z. Because ns agents have their peak at z in the profile (0s, zns)
(0s, zns,
0(0s, zns), . . . ,n(0s, zns)). Similarly, for all l 2 {s + 1, . . . , n}, from ys,0 yl,0,
med(pl+1, . . . , pn, yl,0, . . . , yl,nl) = med(yl,0, z, yl,nl).
Ifmed(pl+1, . . . , pn, yl,0, . . . , yl,nl) =z, then by condition (D), we haveal(z)as(z).
Let z < yn,0 and t 2{s, s+ 1, . . . , n} be such that z 2[yt,0, yt+1,0[. Note that by
(A2) and (A3), for all l 2{s, s+ 1, . . . , t},
z < yt+1,0 yt+1,n(t+1) yl,nl
and med(pl+1, . . . , pn, yl,0, . . . , yl,nl) = z. Thus, using condition (D), for all l 2
{s, s+ 1, . . . , t}, l(0s, zns) = al(z)as(z).
Note that for all l 2 {t + 1, . . . , n}, we have z < yt+1,0 yl,0 yl,nl and med(pl+1, . . . , pn, yl,0, . . . , yl,nl) = yl,0, i.e. (***) never occurs. Ifxt+1,0 yt+1,0, then
by (B1) we have at+1,0(yt+1,0) = xt+1,0. Suppose to the contrary thatas(z)< xt+1,0.
Byat(z)as(z)(notingyt,0 z < yt+1,0 yt,1) now at(z)< xt+1,0. By z 2[yt,0, yt,1]
and condition (C), now yt,0 as(z)< xt+1,0.
If med(pl+1, . . . , pn, yl,0, . . . , yl,nl) = yl,0 and z < yl,0, then al(yl,0) = xl,0 and
by l > s and (A1), xl,0 xs,0 as(z). If med(pl+1, . . . , pn, yl,0, . . . , yl,nl) = yl,nl,
then yl,nl < z ys,ns and by (A1) and (A2), ys,0 < yl,0 yl,nl, now (D) implies
al(yl,nl) as(yl,nl). Thus, for all l 2{s+ 1, . . . , n}, l(0s, zns) as(z). Because
s agents have their peak at 0, now at least n+ 1 numbers are smaller than or equal
toas(z) in(0s, zns,0(0s, zns), . . . ,n(0s, zns)). Hence,
as(z) = med(0s, zns,0(0s, zns), . . . ,n(0s, zns)),
the desired conclusion. The second part of (i) can be shown in a similar way.
Note that the “median formula” used for defining'(a,b,x,y)satisfiescontinuity, and
by Lemma 2, strategy-proofness. Thus, if any strategy-proof and continuous rule chooses the same alternatives for the calibrating profiles as the “median formula”
'(a,b,x,y), then by Step 3, this rule and the “median formula”'(a,b,x,y) must coincide.
5
Subclasses
5.1
Group strategy-proofness
Group strategy-proofnessmeans that no coalition of agents can jointly misreport their preferences in such a way that each member of the coalition weakly gains with at least one strict gain.
Group strategy-proofness: For all S N, all S-deviations R, R0 2 RN, if for
some i2S,'(R0)Pi '(R), then for some j 2S,'(R) Pj '(R0).
One can easily verify that all the rules characterized in the previous section also satisfy group strategy-proofness.
Theorem 3 For all (a, b, x, y) 2 Fn, the rule ' = '(a,b,x,y) satisfies group
strategy-proofness.
Remark 2 There is no logical relation between group strategy-proofness in the
lex-max and the lex-max extension. However, within the rules that are strategy-proof in the lexmax extension, one can show that the set of rules that are group strategy-proof
in the lexmax extension contains the set of rules that are group-strategy-proof in the max extension. This is because, except in the case of a coalition of agents with the same preferences, there are more group improvements (which are strict gains for at least one agent) under the max extension than under the lexmax extension. Theorem
3 contrasts with the result in Heo (2013) which shows that only a subset of the double generalized median rules (with fixed cardinal points) are group strategy-proof.
5.2
Unanimity
Byunanimity, for every preference profile where all agents have the same preference, both goods must be located at the agents’ common peak.
Unanimity: For allR 2RN such thatR
1 =· · ·=Rn,we have'(R) = (p(R1), p(R1)).
Theorem 4 For all (a, b, x, y)2Fn, the rule '='(a,b,x,y) satisfiesunanimity if and
5.3
Pareto-optimality
By Pareto-optimality, for every preference profile the assigned alternative cannot be changed such that no agent is worse o§ and some agent is better o§.
Pareto-optimality: For allR 2RN and allx
2[0,1]2, if for somei
2N,x Pi'(R),
then for some j 2N, '(R) Pj x.
Notice that for Pareto-optimality it is not su¢cient that the assigned alternative is an element of[p(R), p(R)]2. It is necessary for every assigned alternative that each
closed interval with endpoints of two assigned locations contains at least one reported peak of the agents.
Lemma 12 Let ' be a decision rule. Then, ' satisfies Pareto-optimality if and
only if for all R 2 RN, '(R) 2 [p(R), p(R)]2 and there exists i 2 N such that
p(Ri)2['1(R),'2(R)].
We now characterize the subclass of rules that satisfy Pareto-optimality.
Theorem 5 For all (a, b, x, y)2Fn, the rule' ='(a,b,x,y) satisfiesPareto-optimality
if and only if
1. For all r2{0, . . . , n},yr,0 = 0 and x0,r = 1;
2. For all r2{1, . . . , n1}, either xr,nr = 0 or yr,nr= 1.
The first condition says that (i) when the set of the preferred locations of the agents is {0, x}, for some x 2 [0,1], the rule locates both goods in [0, x] ; (ii) when
the set of the preferred locations of the agents is{x,1}, for some x2[0,1],the rule locates both goods in [x,1] ; (iii) when all agents have the same preferred location, the rule locates both goods at this location. The second condition says that if all agents are extremists, i.e. their preferred location is either 0 or 1, then at least one of the two locations is at an extreme, either 0 or 1. Therefore, Conditions 1. and 2. are necessary for Pareto-optimality.
in the profile. The second condition ensures that the rule never locates both goods in the open interval defined by two consecutive peaks. These are precisely the conditions that characterize Pareto-optimality in this model. Therefore, Conditions 1. and 2. are su¢cient for Pareto-optimality.
Remark 3 Except at unanimous profiles, Pareto-optimalitywith respect to the max
extension implies Pareto-optimalitywith respect to the lexmax extension.
Remark 4 For the location of one public good,unanimityandstrategy-proofness
im-ply Pareto-optimality. For the location of two public goods and the lexmax extension, by Theorem 4 and Theorem 5 this implication does not hold.
5.4
Generalized medians
In the max extension domain, Heo (2013) shows that the double generalized median rules (with fixed cardinal points) are the only ones that satisfy strategy-proofness,
anonymity,continuity andusers-only, a property that requires that the location of a good only depends on the preferences of the agents who prefer it in the menu.
In the max extension domain, Miyagawa (1998) provides various characterizations of the extreme peaks rule. In particular, he shows that it is the only one that satisfies on the one hand, strategy-proofness, continuity and Pareto-e¢ciency, and on the other hand group strategy-proofness, peaks-selection and citizen sovereignty.8 We
provide a new axiomatization of this rule in the next subsection.
5.5
Diversity
Bydiversity,the two goods cannot coincide, unless all agents have the same preferred location.
Diversity: For all R 2RN,
|{p(Ri) :i2N}|>1)'1(R)6='2(R).
We have the following results.
8Bycitizen sovereignty, all alternatives are attainable via some reported profile. Peaks-selection
Theorem 6 For all (a, b, x, y) 2 Fn, the rule ' = '(a,b,x,y) satisfies diversity if and
only if x0,1 < y0,1 and x1,0 < y1,0.
Moreover, the only rule within this class that satisfies Pareto-optimality is the extreme peaks rule 'e (where for all R
2RN, 'e(R) = (p(R), p(R)).
Theorem 7 For all (a, b, x, y)2Fn, the rule' ='(a,b,x,y) satisfiesPareto-optimality
and diversity and if and only if it is the extreme peaks rule.
Remark 5 Instead of requiring diversity one may restrict the set of alternatives by
not allowing both public goods to be located at the same location, i.e. for all R2RN,
'1(R)6='2(R) (*). With this requirement, it is easy to see that for (a, b, x, y)2Fn,
'(a,b,x,y) satisfies (*) if and only if for all (s, t) such that s+t
n1 we have (i) there exists no z 2 [y0,0, y0,n] such that as(z) = z (note that from conditions (A1)
and (A3), y0,0 · · · ys,0 · · · y0,n) and (ii) there exists no z 2 [xn,0, x0,n]
such that bt(z) = z (note that from conditions (A1) and (A3), xn,0 · · · xt,nt
· · · · x0,t). By Theorem 2, those rules are characterized by anonymity, strategy-proofness, continuity and requirement (*). Furthermore, above (i) and (ii) just say that the range of '(a,b,x,y) never touches the main diagonal
{(z, z) :z 2[0,1]}.
5.6
Population-monotonicity
In a model with a variable population, Ehlers (2003) characterizes a class of rules, which he calls the single-plateaued rules. In this context, a rule satisfies population-monotonicity (Thomson, 1983a,b) if when new agents are added to the population and given preferences, and the preferences of the agents initially present are kept fixed, these agents either all weakly gain or they all weakly lose. Ehlers (2003) shows that in his model, a rule satisfies Pareto-optimality and population-monotonicity if
and only if it is a single-plateaued rule.9
Next, we describe the class of single-plateaued rules, and show that it is a subclass of the Pareto-optimal delta rules, characterized in Section 5.3. A rule '(a,b,x,y) is
9Miyagawa (1998) characterizes the set of rules that satisfy Pareto-optimality and
single-plateaued, if
xn,0 =yn,0 = 0
x0,n =y0,n = 1
ys,0 = 0 for all s= 1, ..., n1
and there are two pairs (x, y) and (x, y) in[0,1]2 such that x y, x x and
y y and either x = 0 ory = 1 and
xs,t =x for all s, t such that 0< s+t < n
ys,t =y for all s, t such that s+t < n and t >0
xs,ns =x for all s= 1, ..., n1
ys,ns =y for all s= 1, ..., n1
bt(z) =f(z) for all t= 2, ..., n2,
wheref : [x, x]![y, y]is a continuous and strictly decreasing function such that
a0(z) =z
a1(z) =an1(z) = min
z, y, f1(z)
as(z) =f1(z) for all s = 2, ..., n2
an(z) = 0
and
b0(z) =z
b1(z) =bn1(z) = max{f(x), y, z}
bt(z) =f(z) for all t= 2, ..., n2
bn(z) = 1.
Gordon (2007a,b) has shown that in any pure public good model (fixed set of alterna-tives that does not depend on the set of agents; symmetric preference domain) with a variable population, under Pareto-optimality, population-monotonicity is equivalent to strategy-proofness and represented-types-only, a property that requires that the
