universally beneficial manipulation: a characterization

Soc Choice WelfDOI 10.1007/s00355-013-0790-7

ORIGINAL PAPER

Universally beneficial manipulation: a characterization

Donald E. Campbell · Jerry S. Kelly

Received: 26 August 2010 / Accepted: 21 December 2013© Springer-Verlag Berlin Heidelberg 2014

Abstract There exist social choice rules for which every manipulation benefits every-one. This paper constructs a large variety of rules with this property and provides twocharacterizations of such rules.

This, and two companion papers (Campbell and Kelly 2013a,b), study two variationsin the Gibbard–Satterthwaite Theorem, which tells us that strategy-proof rules on afull domain are undesirable: They either have a very restricted range (one alternativeor two alternatives) or are dictatorial.

Our first variation allows some manipulations but only if they benefit everyone. Anymanipulation of a social choice rule obviously benefits at least one person, the manip-ulator. Usually, at least one person is made worse off. But Campbell and Kelly (2010b)presented an example of a rule which is manipulable and for which all manipulationsbenefit everyone. We show here that the class of such rules is large and complex andwe provide two characterizations of such rules. This will enable us in the final sec-tion to discuss the desirability of such rules. Along the way, we will present a secondvariation that uses strategy-proofness but on a smaller domain than usually used.

There have been many studies of the consequences of strategy-proofness. In manyways, our approach is most similar to the literature on counterthreats (e.g., Pattanaik1976) where one allows manipulations but only where there would be a countermovethat makes the original manipulator worse off. For that literature, manipulations are

D. E. CampbellDepartment of Economics and the Program in Public Policy,The College of William and Mary, Williamsburg, VA 23187-8795, USAe-mail: [email protected]

J. S. Kelly (B)Department of Economics, Syracuse University, Syracuse, NY 13244-1020, USAe-mail: [email protected]

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D. E. Campbell, J. S. Kelly

allowed, but only when they would not be carried out. Here, manipulation are allowedthat would be carried out, but only when those manipulations benefit everyone.

To be clear in advance, we are not advocating the rules satisfying our relaxationof strategy-proofness. Our results will have a strong Gibbard–Satterthwaite flavor.We are determining that none of the members of the class of full domain rules whichhave the property that all manipulations benefit everyone are desirable even thoughthis class is much richer than the class of strategy-proof rules.

Consider the following scenario: An academic department of 7 individuals has tochoose from three options: (A) Hire candidate A; (B) Hire candidate B; (F) Stoppingthe current search, redesign the department to include a new field quite different fromthe existing fields and starting a new search next year in the new field. A social choicerule is proposed: Choose the new field only if everyone prefers that to both A and B;otherwise, choose the simple majority winner between A and B. This rule satisfiesuniversally beneficial manipulation (UBM). No one has an incentive to manipulatefrom one of A or B to the other since simple majority voting is strategy-proof. No onehas an incentive to manipulate from F to either A or B since each must prefer F to Aand B for F to have been chosen. If someone is able to manipulate from either A or Bto F it must benefit everyone else as well.

Notice that this rule is not neutral. F is a different kind of alternative and it getstreated in a special way. Also notice that this procedure violates a standard Paretocondition. Consider the following preferences of the n voters.

1 2 3 4 5 6 7F F F F F F AB B B B B B FA A A A A A B

At that profile of preferences, B is chosen even though everyone prefers F to B.But this violation of Pareto does not seem very serious since this outcome is unstable:at this profile we would expect individual 7 to manipulate, raising F to the top to getPareto optimal F, which he prefers to B. Clearly, all UBM rules except strategy-proofones will violate Pareto in this less important way. But this rule fails Pareto in a moreserious fashion. Consider the next profile of preferences:

1 2 3 4 5 6 7F F F F F A AB B B B B F FA A A A A B B

At this profile, B is again chosen even though everyone prefers F to B. But thistime no single individual can manipulate to a preferred alternative and the violationof Pareto would persist.

Of course UBM is at least as desirable as strategy-proofness, since it is a strictlyweaker condition, but we will have to address how it interacts with other social choiceconditions. We will pay some attention to neutrality, but the immediately obviousinteraction is with the Pareto condition. By the Gibbard–Satterthwaite theorem (Gib-

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Universally beneficial manipulation

bard 1973; Satterthwaite 1975), within the class of rules with a full domain, and a rangeof at least three alternatives, the only non-manipulable rule is dictatorship. So everynondictatorial rule in the UBM class is somewhere manipulable and each manipula-tion reveals a violation of the Pareto condition. But as we discuss in the final section,if all violations of Pareto are at profiles where someone would manipulate away, thefailure of Pareto will seem fairly benign: The Pareto-dominated outcomes would neverbe realized. Whether that is possible will be discussed at the end. Even if in a smallnumber of situations Pareto-dominated alternatives are chosen, we might be willingto tolerate the few failures of Pareto if that permits a rule satisfying UBM and is oth-erwise desirable. (Following the seminal paper of Wilson (1972), there has arisen asubstantial literature on social choice without the Pareto principle, e.g., Bandyopad-hyay (1982), Fountain and Suzumura (1982), Border (1983), Kelsey (1984), Lensbergand Thomson (1988), Gekker (1988), Campbell (1989, 1990), Campbell and Kelly(1993, 2003), Malawski and Zhou (1994), Tanaka (2003, 2007), and Powers and White(2008).

1 Framework and first examples

We take as given a finite set X of alternatives with |X | = m ≥ 2 and a finite setN = {1, 2, . . . , n} of individuals with n ≥ 2. A (strong) ordering on X is a complete,asymmetric, transitive relation on X and the set of all such orderings is L(X). Givenan ordering R in L(X), its inverse, R−1, is the ordering in L(X) given by a R−1b iffbRa. A profile p = (p(1), p(2), . . . , p(n)) is a map from N to L(X), and we writex �p

h y if individual h strongly prefers x to y at profile p. The set of all profiles isL(X)N . A social choice rule is a function f : L(X)N → X . Thus we are making afull domain assumption throughout this paper.

Given a subset S of X and an ordering R on X , the restriction, R|S, of R to S, isR ∩ (S × S). Given a profile p = (p(1), p(2), . . . , p(n)) in L(X)N , the restrictionof p to S, is

p|S = (p(1)|S, p(2)|S, . . . , p(n)|S)

Two profiles p and q are h-variants, where h ∈ N , if q(i) = p(i) for all i �= h.Individual h can manipulate the social choice rule f : L(X)N → X at p via p′ if (1)p and p′ belong to L(X)N ; (2) p and p′ are h -variants; and (3) f (p′) �p

h f (p). Andf is strategy-proof or non-manipulable if no one can manipulate f at any profile.Rule f satisfies universally beneficial manipulation (UBM) if for every profile p,every individual h, and every h-variant profile p∗ with f (p∗) �p

h f (p), then we havef (p∗) �p

j f (p) for all j ∈ N .UBM is a solidarity condition, and as a referee has pointed out, that makes it

bear a family resemblance to the condition of welfare domination under preferencereplacement (wdupr) [see, for example, Thomson (1999)]. For a comparison, supposethe current preference profile is (R1, R2, . . . , Rn) and that yields outcome x . Then #1’spreferences change to R∗

1 so the new preference profile is (R∗1 , R2, . . . , Rn) and that

yields outcome y. Then wdupr requires that either x Ri y for all i > 1 or y Ri x for all

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i > 1. Nothing is said about comparing x and y according to R1 or according to R∗1 ;

#1’s preferences play no role in wdupr except that they change. By contrast, our UBMcondition comes into play only when y R1x (so #1 has an incentive to manipulate) andwe then don’t allow x Ri y for everyone else.

We next introduce some examples and several methods for generating new UBMrules from old. These methods will lead to a large and complex class of rules satisfyingUBM. The first example is trivial, but fundamental to the construction of all UBMrules.

Example 1 Any non-manipulable rule satisfies UBM.

There are manipulable rules satisfying UBM, but all such rules have, at their heart,a rule that is non-manipulable on a large subdomain. To see this, we start with severaldefinitions. At profile p, alternative x Pareto dominates (PDs) y if x �p

i y for all i .We say that x Pareto dominates (PDs) the set S ⊆ X if x Pareto dominates everyy ∈ S. Let NP be the subdomain of L(X)N consisting of all those profiles p on whichthere is no Pareto dominance, i.e., there do not exist alternatives x , y such that onePareto dominates the other. For f to be UBM, f restricted to NP must be strategy-proof. This is because starting at any profile in NP , any change in outcome hurts atleast one person; so if voter i manipulates, then—since i must gain—someone elsemust lose.

Since all UBM rules are extensions to L(X)N of strategy-proof rules on NP , weneed to know the structure of non-manipulable rules g on NP . To describe this struc-ture, we say a non-empty collection C of subsets of N is comprehensive if

(1) At least one singleton set {x} /∈ C;(2) C contains at least one proper subset of N ;(3) If C ∈ C and C ⊆ C∗ ⊆ N , then C∗ ∈ C.

If f is a strategy-proof rule on NP with range {a, b}, then the collection of coalitionswinning for a, i.e., the collection of coalitions C such for all u ∈ NP , whenever a �u

i bfor all i in C , then f (u) = a, is a comprehensive collection. Similarly, the collection ofall coalitions winning for b is comprehensive. Going in the other direction, given anycomprehensive collection C, if we define f by setting f (u) = a if {i/a �u

i b} ∈ C andf (u) = b otherwise is a strategy-proof rule on NP with range {a, b}. [These resultsare obtained by a simple adaptation to NP of an argument for L(X)N in Campbelland Kelly (2010b).] Note that while our definition of comprehensive implies N ∈ C,there is actually no profile u in NP with {i ∈ N/a �u

i b} = N .Most of what follows is dependent on the following theorem, which is our second

variation on Gibbard-Satterthwaite announced at the beginning of this paper. We provethat the Gibbard–Satterthwaite conclusions hold for rules that are strategy-proof onNP , rather than on all of L(X)N . A proof of this is carried out in the companionpapers Campbell and Kelly (2013a,b).

Non-paretian domain theorem Suppose g : NP → X is strategy-proof. Then oneof the following holds:

1. g is constant;2. g is dictatorial;

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3. |Range(g)| = 2, say Range(g) = {a, b}, and there exists a comprehensive col-lection C of subsets of N satisfying the condition

g(u) = a if and only if {i ∈ N/a �ui b} ∈ C

Our standard example of a rule in this third class—for an odd number ofindividuals—will be simple majority voting (SMV) where C is the collection of allcoalitions with more than n

2 members. Because of this domain theorem, our exampleswill often come in groups of three; one where f |NP , the restriction of f to subdomainNP , has range of just one alternative; a second, where f |NP is non-dictatorial, usinga range with two alternatives; and a third where f |NP is dictatorial on a subset of twoor more alternatives.

To introduce our first technique for constructing new UBM rules from old ones, westart with simple examples of manipulable UBM rules.

Example 2a Fix a ∈ X . For u ∈ L(X)N ,

f (u) ={

x, if x is the unique alternative that Pareto dominates a;a, otherwise

Example 2b For odd n, fix a, b ⊆ X . For u ∈ L(X)N

f (u) ={

x, if x is the unique alternative that Pareto dominates {a, b};The SMV winner between a and b, otherwise.

Example 2c Fix a, b, c ⊆ X . For u ∈ L(X)N ,

f (u) ={

x, if x is the unique alternative that Pareto dominates {a, b, c};The highest ranked alternative in u(1)|{a, b, c}, otherwise.

These examples have the element x Pareto dominating all the alternatives in {a, b}or in {a, b, c}, not just the alternative chosen by the second criterion in each ruledefinition. To help clarify this issue, compare the next two examples.1

Example 3a For odd n, let {a, b} ⊆ X ; then set

f (u) =⎧⎨⎩

x, if x Pareto dominates {a, b} and a majority have xranked first;

The SMV winner between a and b, otherwise.

Example 3b For odd n, let {a, b} ⊆ X ; then set

f (u) =⎧⎨⎩

x, if x Pareto dominates the SMV winner between a and band a majority have x ranked first;


1 For these examples, we thank Malte Lierl and Asha Sundaram.

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Example 3a is easily seen to be a UBM rule. But 3b is not. Consider profile u:

1 2 3a c cc a db d bd b a

Here a is the majority winner between a and b and no alternative is Pareto superiorto a, so f (u) = a. If #2 manipulates to profile u∗:

1 2 3a c cc b db a bd d a

then b is the majority winner between a and b. Alternative c is top-ranked by amajority and everyone prefers c to b; hence f (u∗) = c. Individual #2 gains by thismanipulation, but because #1 is made worse off, Example 3b fails UBM.

However we do not always require Pareto-dominance of all of Range( f |NP).Consider the next two examples.

Example 4a X = {x, y, z}. We define f (u) by two cases:

1. f (u) = z if either everyone has z at the top, or u = u∗ for the following specialprofile u∗:

1 2 3z z yx x zy y x

2. Otherwise f (u) = x if either #1 or #2 prefers x to y and f (u) = y if both #1 and#2 prefer y to x .

Careful analysis will show that Example 4a, where Range( f |NP) = {x, y}, is aUBM rule even though z is chosen at u∗ when it Pareto dominates x but not y.

Example 4b X = {x, y, z}. We define f (u) by two cases:

1. f (u) = z if either everyone has z at the top, or u = u∗ for the following specialprofile u∗:

1 2 3z z xy y zx x y

2. Otherwise f (u) = x if either #1 or #2 prefers x to y and f (u) = y if both #1 and#2 prefer y to x .

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Example 4b differs from 4a only in that x and y switch places in u∗ for everyone.But, while 4a is a UBM rule, the same is not true for 4b. Consider profile u∗∗, a2-variant of u* at which x is selected:

1 2 3z z xy x zx y y

Individual #2 would precipitate the selection of z by manipulating from u∗∗ to u∗,making #3 worse off.

To help distinguish among these examples, to determine when the chosen alternativehas to be Pareto superior to all of Range( f |NP), we use the following, proven in thecompanion paper (Campbell and Kelly 2013a):

Equivalence Theorem Assume m ≥ 3 and n ≥ 3. Let h be a rule on NP that isstrategy-proof on that domain, with SdenotingRange(h). Then h(u) = h(u∗) forany two profiles u and u∗ such that u|S = u∗|S.

This result will enable us to identify new functions, g and g∗, that will play acentral role in relating f (u) to Range( f |NP). First, if h is a strategy-proof rule onNP , we define g on NP|S = {v ∈ L(S)n/v = u|S f or some u in NP}, by settingg(v) = h(u) for any u such that u|S = v. Making use of the Domain Theorem, wethen extend g to g∗ on L(S)N by the following conditions:

(a) If the range of g is a singleton {x}, let g∗ be constant on L(S)N with range {x};(b) If |Range(g)| > 1 and g is dictatorial with dictator i , let g∗ be dictatorial onL(S)N with dictator i ;(c) If g is non-dictatorial and has |Range(g)| = 2, say Range(g) = {a, b}, weneed only extend g to two more profiles; let g∗(u) = a if everyone prefers a to bat u; let g∗(u) = b if everyone prefers b to a.

We will call g∗ the natural extension of g from NP|S to L(S)N .So, in general, we start with a UBM rule f defined on all of L(X)N , and then

examine its restriction h = f |NP . Rule h is strategy-proof on NP and has range S.This induces (by the Equivalence Theorem), a strategy-proof rule g on NP|S. Finally,g is extended to g∗ on all of L(S)N in a natural way. Much of what we say in the restof the paper will require comparison of f (u) and g∗(u|S).

f : L(X)N −→ X

↓h : NP −→ X

↓g : NP|S −→ S

↓g∗ : L(S)N −→ S

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In particular, the issue here, illustrated by the comparison of Examples 4a and 4b,about whether, for the UBM property, f (u) has to Pareto dominate only g∗(u|S) orinstead all of S has to do with properties of manipulations of g∗ at u|S.

For every strategy-proof rule g on NP|S, there is a UBM rule f such that g canbe obtained in the manner of the previous diagram. But the same is not true for everystrategy-proof rule g∗ on L(S)N . Consider the case S = {a, b} and the rule g∗ onL(S)N that selects a unless everyone prefers b to a. This is not the extension of astrategy-proof g on NP|S. The restriction of g∗ to NP|S is constant and would beextended to a constant rule on L(S)N according to the defining conditions for a naturalextension.

Now call a profile u grounded if g∗ has the property that at u|S, no one can changethe outcome by a unilateral change in his preference ordering; everyone’s option setis just {g∗(u|S)}. Not only can no one change to a more preferred outcome; no onecan change to a less preferred outcome. In Example 4a, at profile u∗, the restriction toS = {x, y} is

1 2 3x x yy y x

and g∗ is determined by: g∗(u) = x if either #1 or #2 prefer x to y and g∗(u) = y ifboth #1 and #2 prefer y to x . Thus u∗ is grounded with g∗(u∗) = x and this is why theUBM property is not violated by having z Pareto dominate only x without requiringz to Pareto dominate all of S. By way of contrast, at the u∗ of Example 4b, u∗|S is

1 2 3y y xx x y

This profile is not grounded and #2 can alter the outcome with:

1 2 3y x xx y y

and it is precisely this alteration that makes f manipulable at u∗∗ in Example 4b in away that violates UBM.

Similarly go back to Example 3b. At profile u∗,

1 2 3a c cc b db a bd d a

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the restriction to S = {a, b} is

1 2 3a b bb a a

and g∗ is SMV. The profile u∗ is not grounded as #2 can alter the outcome of g∗, andit is precisely the possibility of this alteration that makes f manipulable to u∗ fromthe following profile:

1 2 3a c cc a db d bd b a

in a way that violates UBM.With dictatorship, no profile is grounded (the dictator can always change the out-

come), therefore we also need to insist in the case of a UBM rule for which g = f |NPis dictatorial that f (u) /∈ S requires f (u) Pareto-dominate all the alternatives in S,not just g∗(u|S).

Example 5a Select alternatives a, b, and c from X ; then set

f (u) ={

x, if x is the unique alternative that Pareto dominates {a, b, c};The highest ranked alternative in u(1)|{a, b, c}, otherwise.

Example 5b Select alternatives a, b, and c from X ; then set

f (u) =

⎧⎪⎨⎪⎩

α, if α is the unique alternative that Pareto dominates the

highest ranked alternative in u(1)|{a, b, c};The highest ranked alternative in u(1)|{a, b, c}, otherwise.

We will see later that Example 5a is a UBM rule, but 5b is not. Consider profile u

1 2 3y x yx b ca a ab y xc c b

Neither x nor y Pareto-dominates a, the element of {a, b, c} ranked highest in u(1),so f (u) = a. But if #1 manipulates to u∗ by raising b just above a, x uniquely Pareto-dominates b and f (u∗) = x . This is a gain for #1, but makes #3 worse off, showingf is not a UBM rule.

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2 New UBM rules from old by stacking

We now present the first of three general methods for generating new UBM rules fromold. We start by examining a new example:

Example 6 For odd n, select four alternatives {a, b, c, d} and set

f (u) =⎧⎨⎩

α, if {a, b} is the set of alternatives that Pareto dominates {c, d} andα is the SMV winner on {a, b};

The highest ranked alternative in u(1)|{c, d}, otherwise.

This is a UBM rule that consists of

(1) A set S = {c, d} and a default strategy-proof rule g∗ on L(S)n ; here, g∗ is #1’sdictatorship on {c, d};

(2) A set S1 = {a, b} that we check to see if it consists of all alternatives that Paretodominate S;

(3) A strategy-proof rule h1 on S1; here h1 is SMV on {a, b}.Intuitively, stacking is a way of constructing rules that generalizes this example in

three ways:

(1) If the rule on S is not dictatorship, we might have to consider whether or not aprofile is grounded to determine if the alternatives in an S1 must Pareto-dominateall of S or only the element of S chosen by the default rule;

(2) There may be several sets S1, S2, . . . that we check to see if they equal the setof all alternatives that Pareto dominate S (or the element chosen from S); andassociated rules h1, h2, . . . on the respective S1, S2, . . . sets.

(3) The hi rules may themselves be UBM rules and not just strategy-proof rules.

Formalizing now,

Stacking Let h be a strategy-proof rule on NP with range S and let g∗ be the naturalextension of the restriction of g to S. For finite index set A, let {Sα : α ∈ A} be a(possibly empty) collection of pairwise disjoint non-empty subsets of X\S. We allowS or some of the Sα to be singletons. For each α ∈ A, let hα be a UBM rule on Sα .Let S∗(u) be the set of alternatives that

(i) Pareto dominate all the alternatives in S at u, if u is not grounded;(ii) Pareto dominate g∗(u), if u is grounded.

Then define f on L(X)N as follows,

f (u) ={

hα(u|Sα}, if there exists an α such that S∗(u) = Sα;g∗(u), otherwise.

The disjointness of the sets in the collection {Sα} is important. Consider

Example 7 For odd n, select an alternative a in X ; then

f (u) =⎧⎨⎩

x, if there are exactly two alternatives that Pareto dominatea and x is the SMV winner between those two;

a, otherwise.

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This may look like stacking where the Sα would be all pairs of distinct alternativesfrom X\{a}, but these pairs are not disjoint and so the defining condition for stackingis violated (the need for this occurs in Subcase 1-B in the proof below of the StackingTheorem) and in fact f is not a UBM rule. Consider profile u:

1 2 3y y zz z yx x aa a x

Here y and z Pareto-dominate a, and y defeats z by SMV: f (u) = y. If #3 manip-ulates to u∗:

1 2 3y y zz z xx x aa a y

then x and z Pareto-dominate a, and z defeats x by SMV: f (u) = z. The gain by #3causes a loss for the other two individuals and that means f is not a UBM rule.

Our next example shows how position can play a role in constructing UBM rulesby stacking.

Example 8 For odd n, select a, b, c from X and consider the rule

f (u) ={

The SMV winner on {b, c} if a is in everyone’s bottom rank;a, otherwise.

It is easy to see that this is also a UBM rule. To see that it can be defined by stacking,take S = {a}, let g∗ be the constant rule that selects a; while {Sα} consists of the singleset X\{a} and hα is simple majority voting between b and c.

Our final example of this section shows a sequential application of stacking, i.e.,some of the hα in the definition of stacking are themselves obtained from stacking.

Example 9 For odd n, select a, b, c, d, e from X and consider the rule,

f (u) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

The SMV winner on {a, b} unless {c, d, e} is the set ofalternatives that Pareto dominate {a, b};

The SMV winner on {c, d} if {c, d, e} is the set of alternativesthat Pareto dominate {a, b} but e does not PD {c, d};

e, otherwise.

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We first observe that the following is a UBM rule on {c, d, e}

h(u) =

⎧⎪⎨⎪⎩

The SMV winner between c and d unless e is in

everyone’s top rank;e if e is in everyone’s top rank.

(This is just a variation on the stacking and reduction of Example 10). Then applya second use of stacking by setting S = {a, b}, g∗ is simple majority voting on {a, b},{Sα} contains of the one set {c, d, e} with hα = h, as defined just above.

We now want to prove that a rule constructed by stacking is a UBM rule. But wewill need some new terminology and a few preliminary results.

Selection Theorem Let f be a UBM rule, with S = Range(g) for g = f |NP and g∗is the natural extension to L(S)N of strategy-proof g. For any u ∈ L(X)N , we havef (u) ∈ S implies f (u) = g∗(u|S).

Proof Case 1 |S| = 1. Then the conclusion is trivial as f (u) and g∗(u|S) must bothyield the element in S if f (u) is in S.

Case 2 g∗ is dictatorial. Let the dictator be individual #1. Assume f (u) ∈ S butf (u) �= g∗(u|S). Without loss of generality, u|S is

1 2 · · · n...

...

a f (u) a...

... · · · ...

f (u) a f (u)...

......

where a is the top element in u(1)|S. Suppose some i , other than #1, also has apreferred to f (u). Then if i changes u(i) to u(1)−1, the resulting u∗ is in NP andf (u∗) = a. If even one person prefers f (u) to a, they would have been made worseoff. So suppose everyone has alternative a preferred to f (u) at u.

1 2 · · · n...

...

a a a...

... · · · ...

f (u) f (u) f (u)...

......

then at the n-variant profile u∗ where u∗(n) = u(1)−1,

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1 2 · · · n...

......

a a f (u)

...... · · · ...

f (u) f (u) a...

......

a is chosen by #1’s dictatorship. So n would manipulate from u∗ to u, making everyoneelse worse off.

Now suppose the only person who prefers a to f (u) at u is #1:

1 2 · · · n...

...

a f (u) f (u)

...... · · · ...

f (u) a a...

......

Then at the n-variant profile u∗ where u∗(n) = u(1)−1, alternative a is chosen. Son would manipulate from u∗ to u, making #1 worse off.

Case 3 |S| = 2 and there exists a comprehensive collection of coalitions winning fora. Assume b = f (u) ∈ S but a = g∗(u|S) �= b. For a = g∗(u|S), some winningcoalition C has a �u

i b for all i ∈ C . Without loss of generality, C = {1, . . . , k}.Suppose some j /∈ C has b �u

j a.

1 · · · k · · · j · · · n...

......

...

a a b...

......

...

b b a...

......

...

Consider j-variant profile u∗ where u∗( j) = (u(1))−1. Then u∗ ∈ NP and f (u∗) =g∗(u|S) = a. Then j has an incentive to manipulate from u∗ to u, and everyone in Closes.

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So we may now suppose that at u everyone prefers a to b but that f (u) = b.

1 · · · k · · · n − 1 n...

......

...

a a a a...

......

...

b b b b...

......

...

There is a coalition C ′ winning for a where C ′ is a proper subset of N . Without loss ofgenerality, assume n /∈ C ′. Then look at the n-variant profile u′ where u′(n) = u(1)−1.

1 · · · k · · · n − 1 n...

......

...

a a a b...

......

...

b b b a...

......

...

Here, u′ ∈ NP and f (u′) = a. So n has an incentive to manipulate from u′ to uand everyone else loses. �

Now for the main result of this section:

Stacking Theorem A rule f obtained by stacking is a UBM rule.

Proof Let f be defined by stacking via (1) S, (2) a strategy-proof rule g on NP|S, (3)sets {Sα}, and (4) UBM rules {hα}. Suppose that, at profile u, individual i manipulatesto an i-variant u∗, so that f (u∗) �= f (u) and f (u∗) �u

i f (u). We split our analysis intotwo cases. Reminder: A profile u is grounded if it does not allow anyone to unilaterallychange the outcome from g∗(u).Case 1 g∗(u∗|S) �= g∗(u|S). Then neither u nor u∗ is grounded. Because u is notgrounded, f (u) is above everything in S for each i ∈ N . The case f (u) = x /∈ S andf (u∗) ∈ S can be ruled out as i would not manipulate to u∗.Subcase (1-A) Suppose f (u) ∈ S and f (u∗) = x /∈ S. Since u∗ is not grounded, xmust Pareto dominate all of S at u∗. So in the manipulation to u∗ everyone must havebeen made better off; i must be better off because she manipulated and for all j �= i ,x is preferred by u∗( j) and so also by u( j) to everything in S, including f (u).Subcase (1-B) Suppose f (u) = x /∈ S and f (u∗) = y /∈ S. If x and y are inthe same Sα , then everyone must have been made better off since hα is a UBM rule.So suppose f (u) = x ∈ Sα = S∗(u), the alternatives that Pareto dominate S at uand f (u∗) = y ∈ Sβ = S∗(u∗), the alternatives that Pareto dominate S at u∗ (recallthose two sets are disjoint). Then for x ∈ S∗(u) and y /∈ S∗(u), it must be that x is

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above all of S in u(i) but y is not above all of S in u(i). Thus x �ui y so individual i

would not manipulate at u to u∗.Subcase (1-C) f (u) ∈ S and also f (u∗) ∈ S, is not possible, since g∗ is a strategy-proof rule.

Case 2 g∗(u|S) = g∗(u∗|S) and f (u) = g∗(u|S) ∈ S. Also f (u∗) = g∗(u∗|S) ∈ S,is not possible, since f (u∗) �= f (u) but g∗(u|S) = g∗(u∗|S).Subcase (2-A) Suppose f (u) = g∗(u|S) ∈ S and f (u∗) = x /∈ S. So in the manipu-lation to u∗ everyone must have been made better off; i must be better off because shemanipulated and for all j �= i , since x = f (u∗) /∈ S, x is preferred by u∗( j) becauseit must be in an Sα and thus Pareto-dominate g∗(u|S) and thus x is also preferred byu( j) to g∗(u∗|S) = g∗(u|S).Subcase (2-B) If f (u) = x /∈ S and f (u∗) = g∗(u∗|S) ∈ S, then i would notmanipulate to u∗ as it would make her worse off. For at u, f (u) Pareto dominatesg∗(u|S) so f (u) �u

i g∗(u|S) = g∗(u∗|S) = f (u∗).Subcase (2-C) Suppose f (u) = x /∈ S and f (u∗) = y /∈ S. If x and y are in thesame Sα , then everyone must have been made better off since hα is a UBM rule. Sofinally, suppose f (u) = x ∈ Sα = S∗(u), the alternatives that Pareto dominate S at uand f (u∗) = y ∈ Sβ = S∗(u∗), the alternatives that Pareto dominate S at u∗ (recallthose two sets are disjoint). Then for x ∈ S∗(u) and S∗(u), it must be that x is aboveall of S in u(i) but y is not above all of S in u(i). Thus x �u

i y so individual i wouldnot manipulate at u. �

3 New UBM rules from old by reducing

Consider

Example 10 For odd n, fix {a, b, c} ⊆ X with |X | > 3. For u ∈ L(X)N

f (u) =⎧⎨⎩

c, if c is the unique alternative that Pareto dominates {a, b} andeveryone ranks c first;


This example, though easily seen to be a UBM rule, can not be shown to be UBMby any sequence of applications of stacking and strategy-proof rules. We need a newtechnique for deriving new UBM rules from old. Intuitively, Example 10 is closelyrelated to

f (u) ={

c, if c is the unique alternative that Pareto dominates {a, b} ;The SMV winner between a and b, otherwise.

which is constructible by stacking. Example 10 is obtained from this last rule byreverting to the default (SMV on {a, b}) at more profiles than this one does. Thisreversion to the default more often is the idea behind reducing, our second way ofobtaining new UBM rules from old ones.

But first we introduce an important class of rules. A rule f is a (g, S)-rule onL(X)N if there exists an S, ∅ �= S ⊆ X , and a strategy-proof rule g on L(S)N (with

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natural extension g∗) and a partition of L(X)N into �∗ and L(X)N \�∗ such that (1)for u ∈ �∗, we have f (u) = g∗(u|S); and (2) for u ∈ L(X)N \�∗, it’s the case thatf (u) has the property that

(i) If u is not grounded, f (u) Pareto dominates all the alternatives in S at u;(ii) If u is grounded, f (u) Pareto dominates g∗(u|S).

Notice that a rule f obtained by stacking via S, {Sα}, strategy-proof rule g, andUBM rules {hα}, is a (g, S)-rule.

Reducing Our second method of constructing new UBM rules is reduction. Let f bea (g, S)-rule. Then f ∗ is a reduction of f if

1. For u ∈ �∗, we have f ∗(u) = f (u);2. For u ∈ L(X)N \�∗, it’s the case that f ∗(u) is either f (u) or g∗(u|S).

That is, f ∗ agrees with f except that it may sometimes use the default g∗(u|S)

when f doesn’t.Returning to Example 10, we can view this as the reduction of Example 2b to the

cases where the unique Pareto superior alternative is at everyone’s top. Similarly wehave

Example 11 For odd n, let {a, b} ⊆ X . For u ∈ L(X)N

f (u) =⎧⎨⎩

x, if x is the unique alternative that Pareto dominates {a, b}and a majority rank x first;


This is the reduction of Example 2b to the cases where x is in the top rank for amajority. Also Example 10 is a reduction of Example 11.

Reduction Theorem If f is a (g, S)-rule that satisfies UBM and f ∗ is a reduction off , then f ∗ is also a UBM rule.

Proof Suppose that at profile u, individual i manipulates to an i-variant u∗. Thenf ∗(u) and f ∗(u∗) can’t both be in S, since g∗ is a strategy-proof rule. If f ∗(u) is in Sand f ∗(u∗) is not, then everyone benefits since i gains and everyone else moves fromS to an alternative he prefers at both u and u∗, to all alternatives in S. If f ∗(u∗) is in Sbut f ∗(u) is not, i wouldn’t manipulate. If both f ∗(u) and f ∗(u∗) are not in S, theneveryone benefits since f is a UBM rule. �

4 New UBM rules from old by pasting

We will want to paste two or more social choice rules together. The analogy is withpasting two real-valued functions, f on domain [a, b] and g on [c, d] together withto get a new function h. We require that f and g coincide on [a, b] ∩ [c, d] and seth(x) = f (x) if x ∈ [a, b] and h(x) = g(x) if x ∈ [c, d]. Then h is non-negative if fand g are; h is continuous if f and g are. (See, for example, the Pasting Lemma inMunkres (1975), p. 108):

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To illustrate pasting, which is our third and last way of getting new UBM rulesfrom old, contrast the following two apparently similar examples. The first we treatedearlier by stacking plus reduction; recall Example 10 from Sect. 1:

Example 10 For odd n, select alternatives, a, b and c, from X .

f (u) =⎧⎨⎩

c, if c is the unique alternative that Pareto dominates {a, b} andeveryone ranks c first;


In contrast, consider:

Example 12 For odd n, select alternatives a and b from X.

f (u) ={

x, if x is everyone’s top ranked alternative;The SMV winner between a and b, otherwise.

While Example 12 is easily seen to be a UBM rule, it can not be constructed byany sequence of stackings and reductions starting from strategy-proof rules. Instead,we will now show how the UBM rule of Example 12 can be constructed by pastingtogether other UBM rules.

The domain of f can be partitioned into m − 1 components: First, for each α ∈X\{a, b}, let �α be the subdomain of all profiles where α is everyone’s top—and sowhere f chooses α, i.e., �α = f −1(α). Let �∗ be L(X)N \∪α �α on which f selectsthe SMV winner between a and b. Next, let fα be the reduction of f that is

fα(u) ={

α if α is everyone’s top ranked alternative;The SMV winner between a and b, otherwise.

Each such fα is a UBM rule (compare with Example 10). We now express rule fas pasting together the fα:

f (u) ={

fα(u), if there exists an α such that u ∈ �α;The SMV winner between a and b if u ∈ �∗.

No one would effect a change from a profile in any �α to a profile in �∗. (ashe would lose) or from one profile in �α to another in �α (the outcome would notchange). No one would manipulate from one profile in �∗ to another in �∗ sinceSMV is strategy-proof. Any manipulation from a profile in �∗ to one in a �α makeseveryone better off. Finally, no one is able to effect a change from a profile in a �α toa profile in a �β for β �= α.

A slightly different aspect of pasting UBM rules is developed via the next twoexamples.

Example 13 For odd n, select alternatives a and b from X .

f (u) ={

α, if α is #1′s top ranked alternative and αPDs {a, b};The SMV winner between a and b, otherwise.

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The domain of f can be partitioned into m − 1 components: First, for each α ∈X\{a, b}, let �α be the subdomain of all profiles where α both Pareto dominates{a, b} and is #1’s top—and so where f chooses α, again, �α = f −1(α). Then let �∗beL(X)N \ ∪ �α on which f selects the SMV winner between a and b. Now, let fαbe the reduction of f that is defined by.

fα(u) ={

α, if u ∈ �α;The SMV winner between a and b, otherwise.

Each such fα is a UBM rule. We now express f as pasting together the fα:.

f (u) ={

fα(u), if there exists an α such that u ∈ �α;The SMV winner between a and b if u ∈ �∗.

No one would effect a change from a profile in a �α to a profile in �∗ (as he wouldlose) or from one profile in �α to another in �α (the outcome would not change). Noone would manipulate from one profile in �∗ to another in �∗ since SMV is strategy-proof. Any manipulation from a profile in �∗ to one in a �α makes everyone betteroff. Finally, only #1 can effect a change from a profile in a �α to a profile in a �β forβ �= α, and #1 has no incentive to do so.

For a issue concerning manipulation from a profile in one �α to another �β ,consider the following UBM rule.

Example 14 Select alternative a from X . Rule f will take the value a at all but twogiven profiles, v and v∗, with the following properties: At profile v, a is at everyone’sbottom; also for everyone except the first individual, x is at the top and y is in thesecond rank; finally, #1’s ordering starts zxy . . .. Profile v∗ is exactly like v exceptthat for individual #1, her ordering starts xzy . . .. Set f (v) = y and f (v∗) = x .

Now define two new rules: Rule fv will take the value y at profile v and otherwisetake the value a. Rule fv∗ will take the value x at profile v∗ and otherwise take thevalue a. We want to show that rule f can be obtained as the pasting of fv and fv∗.We partition the profiles into �1, �2, and �∗, where �1 = {v}, �2 = {v∗}, and �∗is L(X)N \{v, v∗}. Any manipulation from a profile in �∗ to one in �1 or �2 makeseveryone better off. No one has an incentive to manipulate from �1 or �2 to �∗.Unlike Example 12, but similar to Example 13, individual #1 can effect a change from�2 to �1, but has no incentive to do so. Unlike both Examples 12 and 13, individual#1 can effect a change from �1 to �2, and does have an incentive to do so, but thatmanipulation will make everyone better off. The rule that results from pasting togetherUBM rules fv and fv∗ is f , which is then UBM, as we shall see from the next theorem.(This pasting together of rules that deviate from a g∗(u|S) default at only single profileswill play a very important role in the proof of the (g,S)-to-Constructability Theoremin Sect. 5).

In the examples above, the subdomains �α have been f −1(α) for some α. But wewill find it useful to allow a finer partition of L(X)N in which several subdomainsmay map to the same α. These will be distinguished by a second subscript, so that inthe following definition of pasting we will have domains �α1, �α2,…, �β1, �β2, . . . .

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Let α, β, . . . be elements in X\S and let fα1, fα2, . . . , fβ1, fβ2, . . . be UBM(g, S)-rules (the same g, S for all) having the property that for each θ , fθ j (u) is eitherθ or an element of S. Construct a partitioning of L(X)N by the following specificationof components: for each θ , j , let �θ j = f −1

θ j (θ) and then let �∗ be L(X)N \ ∪ �θ . Arule f is said to be obtained from rules fα1, fα2, . . . , fβ1, fβ2, . . . by pasting if

f (u) ={

θ, if there exist θ and j such that u ∈ �θ j ;g∗(u|S), if u ∈ �∗.

satisfying the condition that if there is an individual t , and t -variants u and u∗ withu ∈ �α j , u∗ ∈ �βk , for β �= α, then if t has an incentive to manipulate from u to u∗,everyone gains.

Pasting Theorem Suppose rule f is obtained from rules fα1, fα2, . . . , fβ1, fβ2, . . .

by pasting; then f is also a UBM (g, S)-rule.

Proof (1) No one would effect a change from a profile in a �α j to a profile in �∗since α = fα j (u) would Pareto dominate all of S. (2) No one would effect a changefrom a profile in �α j to one in �αk (the outcome wouldn’t change). (3) If anyonemanipulates from one profile in �∗ to another in �∗ everyone benefits since g is aUBM rule. (4) Any manipulation from a profile in �∗ to one in a �α j makes everyonebetter off since α would Pareto dominate everything in S. (5) Finally, if anyone wouldmanipulate from a profile in a �α j to a profile in a �βk for β �= α, then everyonegains, by the definition of pasting. �

5 Every constructible rule is a UBM rule and every (g, S)-rule is constructible

We now summarize results from the last three sections. Say a rule f is constructible ifthere exists a sequence, f0, f1, f2, . . . , ft such that each fi is either a strategy-proofrule, or can be obtained from f0, f1, f2, . . . , fi−1 by stacking, reducing, or pasting.

Constructability-to-UBM Theorem Any constructible rule is a UBM rule.

Proof This is immediate from the Stacking, Reduction, and Pasting Theorems. � We now turn our attention to a new set of examples to illustrate demonstrating

constructibility:

Example 15a Select alternative a ∈ X ; then,

f (u) =⎧⎨⎩

x, if x Pareto dominates a and a majority ofindividuals rank x first;

a, otherwise.

Example 15b For odd n, select alternatives a, b ∈ X ; then

f (u) =⎧⎨⎩

x, if x Pareto dominates {a, b} and a majority ofindividuals rank x first;

The SMV winner between a and b, otherwise

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Example 15c Select alternative a, b, and c ∈ X ; then

f (u) =⎧⎨⎩

x, if x Pareto dominates {a, b, c} and a majority ofindividuals rank x first;

The alternative in {a, b, c} ranked highest by#1, otherwise.

Each of these (g, S)-rules is easily seen to be a UBM rule. We will illustrate withExample 15b that each of these can be constructed by a sequence of applications ofstacking, reduction, and pasting.

First, fix an alternative c ∈ X\{a, b}. Let T be any set satisfying c ∈ T ⊆ X\{a, b}.Then define fc,T and T as follows:

fc,T (u) ={

c, if T is the set of alternatives that Pareto dominate {a, b};The SMV winner between a and b, otherwise.

Rule fc,T is a UBM rule by stacking, where S = {a, b}, where g∗ is SMV, thecollection {Sα} contains only T and hα is the constant rule that selects c.

Second, let fc,T ∗ be the reduction of fc,T so that fc,T ∗(u) = c only at those profilesu where a majority of individuals have c in their top rank. Third, paste the fc,T ∗’s overall T such that c ∈ T ⊆ X\{a, b}. This yields as a UBM rule

fc(u) =⎧⎨⎩

c, if c Pareto dominates {a, b} and a majority ofindividuals rank c first;

The SMV winner between a and b, otherwise

Finally, paste the fc’s over all c ∈ X\{a, b} to get Example 15b.

(g,S)-to-Constructability Theorem If f is a (g, S)-rule for some non-empty S ⊆ Xand strategy-proof g on S, then there exists a t ≥ 1 and a sequence, f0, f1, f2, . . . , ft

such that f0 is given by f0(u) = g∗(u|S) and f = ft and each fi is either a strategy-proof rule, or can be obtained from f0, f1, f2, . . . , fi−1 by stacking, reducing, orpasting.

Proof Let f be a (g, S)-rule. Define f0 by f0(u) = g∗(u|S). Select a profile u1 forwhich x = f (u1) �= g∗(u1|S). If u1 is not grounded, let Su1 be the set of alternativesthat Pareto dominate all the alternatives in S at u1. If u1 is grounded, let Su1 be the setof alternatives that Pareto dominate g∗(u1|S). Then let hu1 be the constant functionthat selects x = f (u1) at every profile in Su1. Construct f1 from f0 by stacking andthen define fu1∗ by reduction:

fu1∗(u) ={

x, if u = u1;g∗(u|S), otherwise.

Continue now selecting another profile u2 for which f (u2) �= g∗(u2|S) and con-struct f2 from f0 by stacking and then define fu2∗ by reduction. As long as there areprofiles v for which f (v) doesn’t give g∗(v|S), we keep constructing in the same way.

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Since the numbers of individuals and alternatives are finite, this process ends and wehave a sequence

f0, fu1, fu1∗ , fu2, fu2∗ , . . . , fus, fus∗ .

Finally we will construct f from fu1∗ , fu2∗ , . . . , fus∗ by pasting. Set �1 = {u1},�2 = {u2}, . . . , �s = {us}, and �∗ = L(X)N \ ∪ � j .

Any manipulation from one subdomain to another benefits all since that manipula-tion is also available under f . �

6 Every UBM rule is a (g, S)-rule

The result in the section title requires two preliminary theorems.

Theorem 1 (First Pareto dominance theorem) Let f be a UBM rule, S = f (NP) andg∗ is the extension to L(S)N of g : NP|S → S, the restriction to S of the restrictionof f to NP. Then if at u, we have f (u) /∈ S, then f (u) Pareto dominates g∗(u|S).

Proof Case 1 |S| = 1, say S = {s}. So g∗(u|S) = s. Suppose f (u) �= s and alsof (u) doesn’t Pareto dominate s. There exists an i such that s �u

i f (u). Supposethere exists a j such that f (u) �u

j s. Then at i-variant profile u∗ obtained by setting

u∗(i) = u( j)−1, we have u∗ ∈ NP and so f (u∗) = s. Thus i would manipulate tou∗, causing a loss for j , and violating UBM.

So we may assume s �ui f (u) for all i . Then look at 1 -variant u∗ with u∗(1) =

u(2)−1. Profile u∗ ∈ NP and so f (u∗) = s, but f (u) �u∗1 s, so #1 would manipulate

from u∗ to u, hurting everyone else, violating UBM.

Case 2 g∗ is dictatorial. Assume #1 is the dictator, and a is at the top of u(1) restrictedto S, but f (u) = x /∈ S and x does not Pareto-dominate a. Then a �u

i x for some i .(Subcase 2-1) There is an i > 1 with a �u

i x .Then at i-variant profilev obtained by setting v(i) = u( j)−1, we have v ∈ NP and

so f (u∗) = a. Thus i would manipulate to v. Since f is · · · a UBM rule, everyone atu must prefer a to x .

But if every i has a �ui x , look at 2-variant v with v(2) = u(1)−1. Profile v ∈ NP

and so f (v) = a, but x �v2 a, so #2 would manipulate from v to u, causing a loss for

everyone else in violation of UBM.(Subcase 2-2) The only instance of a preferred to x is a �u

1 x for individual #1. Thenlook at 2-variant v with v(2) = u(1)−1. Profile v ∈ NP and so f (v) = a, but x �v

2 a,so 2 would manipulate from v to u, causing a loss for #1 in violation of UBM.

Case 3 |S| = 2, say S = {a, b}, and g∗ is not dictatorial. Assume g∗(u) = a because{1, 2, . . . , h} is a winning coalition for a and every one in that coalition prefers a tob, and that f (u) = x /∈ S does not Pareto-dominate a. So for some i , a �u

i x . Firstnotice that would have to be true for one of the individuals who have a �u

i b. Forsuppose it only held for an individual with b �u

j a. Without loss of generality, u is

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1 · · · h · · · j · · ·...

......

x x b...

......

a a a...

......

b b x...

......

Then consider j-variant profile v where v( j) = u(1)−1. Then v is in NP and{1, . . . , h, j} is winning for a, so f (v) = a. Individual j would manipulate from u tov, causing a loss for #1. So at least one individual has a �u

i b and a �ui x :

1 · · · h · · · j · · ·...

a...

...... a a

x...

...... b b

b...

......

(We have indicated that x is above b in u(1), but we don’t require that; we couldalso have x below b.) Then for some j > h, consider j -variant profile u∗, whereu∗( j) = u(1)−1. Note that u∗( j) ranks x above a, but u∗ ∈ NP , and f (u∗) = a.Individual j would manipulate from u∗ to u, causing a loss for #1. �

Theorem 2 (Second Pareto dominance theorem) Let f be a UBM rule, S = f (NP)

and g∗ is the extension to L(S)N of g = ( f |NP)|S. Then if at u, f (u) /∈ S, and u isnot grounded, then f (u) Pareto dominates S.

Proof Case 1 |S| = 1. Then the result follows from Theorem 1, since S = {g∗(u)}.Case 2 g∗ is dictatorial. To see the claim of the theorem, fix z /∈ S, suppose g isdictatorial (with dictator #1) on S = {a, b, . . . }, that f sometimes selects z, and atprofile u we have f (u) = z while (by Theorem 1) z Pareto dominates g∗(u|S), butit does not Pareto dominate S. Suppose a is highest in u(1)|S, a = g∗(u|S). Since zPareto dominates a, z is above S for #1. Since z does not Pareto dominate S, for somei > 1, without loss of generality #2, z is ranked below some element b of S.

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1 2 i > 2...

...

z b...

...... z

a z...

...... a

b a...

......

Consider 1-variant u*, where #1’s ordering has initial segment zb. . . :

1 2 i > 2

z...

b b...

...... z

a z...

...... a

a...

...

If f (u∗) /∈ S, it must, by the first Pareto dominance theorem, Pareto dominateb = g∗(u∗|S). But by u∗(1), that could only be z, but u∗(2) shows z does not Paretodominate b. Therefore, f (u∗) ∈ S; by the Selection theorem, f (u∗) = b. So #1 wouldmanipulate from u∗ to u, causing a loss for #2, showing f can not be UBM.

Case 3 |S| = 2, say S = {a, b}, and g∗ is not dictatorial with g∗(u|S) = a by thecoalition C = {i/a �u

i b}. Suppose f (u) = c at profile u, and u is not grounded,but c does not Pareto dominate all of S. Since c does Pareto dominate g∗(u|S) byTheorem 1, then without loss of generality, b �u

j c for some j . This can’t be forany j ∈ C , for a �u

j b �uj c can’t hold at u where c Pareto dominates a. So C is

a proper subset of N and without loss of generality we will take C = {1, 2, . . . , h}for h < n.

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1 · · · h h + 1 · · · j · · · n...

......

c c... b

......

... b... b

a a... c

......

... a... a

b b... a

......

......

Since u is not grounded, there must be some, i , necessarily in C , who, if he changeshis ordering to one with b �u∗

i a, changes the value of g∗ to b. Without loss ofgenerality, we take this to be individual h.

Consider h-variant profile u∗ (where we set c = the alternative in h’s top rank):

1 · · · h h + 1 · · · j · · · n... c

...

c...

... b...

... b b... b

a...

... c...

... a a... a

b...

... a...

......

As we observed regarding the choice of individual k, g∗(u|S) = b. By the selectiontheorem, f (u∗) �= a. Since c still does not Pareto dominate b = g∗(u∗|S), Theorem1 tells us f (u∗) �= c. If f (u∗) = b, individual h would manipulate from u∗ to u,causing a loss for j . If f (u) = d ∈ X\{a, b, c}, then d must Pareto dominate b. Butthen j prefers d to c at u∗; since again h would manipulate from u∗ to u, that wouldcause a loss for j .

UBM-to-(g, S) Theorem Every UBM rule is a (g, S)-rule.

Proof . By the Selection Theorem, if f (u) �= g∗(u), then f (u) /∈ S. Then our resultis immediate from Theorems 1 and 2. �

7 Final remarks

Combining the above results,UBM implies (g, S) implies Constructible implies UBM.

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So we have two characterizations of UBM rules; they are the (g, S)-rules and theyare also the constructible rules.

It is important to understand that we are not generally advocating the use of UBMrules. They all have properties that—for some interpretations—may be undesirable.In particular, on the restriction to the very large subdomain NP of profiles where noalternative Pareto-dominates any other, a UBM rule must have either very restrictedrange or be dictatorial.

We discuss here the interaction of UBM with, first, neutrality and, second, withPareto.

Regarding neutrality, for m = 2, there are many (non-dictatorial) neutral UBMrules. Majority rule, with person #1 to break a tie, if any, is strategy-proof and henceUBM. It satisfies both neutrality and the Pareto condition.

Now suppose m ≥ 3. If S = Range( f |NP) is a proper subset of X , then fviolates neutrality. If S = Range( f |NP) = X where f is a UBM rule, then fis—by the Domain Theorem—dictatorial on NP and so also on L(X)n . So all non-dictatorial UBM rules fail neutrality. But this observation must be placed in context.All resolute social choice rules, whether or not they are UBM rules, must fail at leastone of neutrality and anonymity (Kelly 1990). Requiring neutrality forces failureof anonymity and in some interpretations, anonymity may be more important thanneutrality.

As observed in the introduction, neutrality may be inappropriate anyway. In ourexample of a department making hiring decisions, one of the alternatives is distinctlydifferent from the others and we may want it treated differently in the social choiceprocess. We often use social choice procedures which are deliberately non-neutral;they treat the status quo differently or may use super-majority or even unanimity rules.

Turning to the Pareto condition, as we have just seen, for m = 2, there do exist(non-dictatorial) UBM rules satisfying Pareto, for example majority rule, with person#1 to break a tie, if any.

Now suppose m ≥ 3. For any rule, whether or not it is a UBM rule, if Range( f ) isa proper subset of X , then f fails Pareto (for example, at a profile where an alternativein X\Range( f ) is everyone’s topmost alternative). So we pay attention only to ruleswhere Range( f ) = X . As observed in the introduction, every nondictatorial rule inthis class is somewhere manipulable and each manipulation reveals a violation of thePareto condition. But this seems fairly benign. We would not expect to see suchfailure observed as individuals would manipulate to Pareto superior outcomes.

But, most UBM rule with full range also fail Pareto at some profile where therule can not be manipulated; see for an instance Example 10. But there do existfull domain, full range non-dictatorial UBM rules such that Pareto is only violated atprofiles where the rule is manipulable and so have the property that we would neverexpect a Pareto-dominated alternative to be chosen.

Example 16 Select an a ∈ X and let f be the rule that, at profile u, chooses a unlessexactly one alternative, x , in X\{a} Pareto-dominates a, in which case f (u) = x .

Example 16 has the property that, at every profile where f (u) is Pareto-dominated,there is a manipulation possible that leads to a Pareto-optimal alternative beingselected. Suppose f (u) is Pareto-dominated. Then f (u) can not be an x �= a for then

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two alternatives would Pareto-dominate a and f (u) would have to equal a. So we mayassume f (u) = a and a is Pareto-dominated. It must be that more than one alternativePareto-dominates a; but then every individual could manipulate by lowering all butone of those alternatives below a.

For another example of a rule that selects a Pareto-dominated alternative only whensomeone would manipulate away from that situation, examine the department hiringrule in the opening section of this paper.

Even without these examples, we might be willing to tolerate a few failures ofPareto if that permits a rule satisfying UBM and is otherwise desirable. Such trade-offs between amount of violation of Pareto and satisfaction of other social choiceproperties have been extensively studied; see, (Campbell and Kelly 2002; Powers2001; Gibson and Powers 2009).

Optimal manipulations For UBM rules, no one ever suffers a loss caused by amanipulation. In Campbell and Kelly (2010b), we distinguished between losses causedby any possible manipulation and those caused by optimal manipulations. There existrules where losses caused by optimal manipulations are smaller than those caused bynon-optimal manipulations. This leads to the possibility that there might be rules thatfail UBM according to the definition given in this paper but which have the propertythat everyone gains whenever optimal manipulations are carried out. We have beenunable to discover any.

Conjecture If a rule has the property that everyone gains as a result of any optimalmanipulation, then it is a UBM rule.

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universally beneficial manipulation: a characterization

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