Strategyproof Social Decision Schemes on Super Condorcet

Draft – October 27, 2021

Strategyproof Social Decision Schemes on Super Condorcet Domains

Felix Brandt, Patrick Lederer, Sascha TauschTechnical University of Munich

[email protected], [email protected], [email protected]

Abstract

One of the central economic paradigms in multi-agent systemsis that agents should not be better off by acting dishonestly. Inthe context of collective decision-making, this axiom is knownas strategyproofness and turned out to be rather prohibitive,even when allowing for randomization. In particular, Gibbard’srandom dictatorship theorem shows that only rather unattrac-tive social decision schemes (SDSs) satisfy strategyproofnesson the full domain of preferences. In this paper, we obtainmore positive results by investigating strategyproof SDSs onthe Condorcet domain, which consists of all preference pro-files that admit a Condorcet winner. In more detail, we showthat, if the number of voters n is odd, every strategyproof andnon-imposing SDS on the Condorcet domain can be repre-sented as a mixture of dictatorial SDSs and the Condorcetrule (which chooses the Condorcet winner with probability1). Moreover, we prove that, if n is odd, only mixtures ofdictatorial SDSs satisfy strategyproofness and non-impositionon every sufficiently connected superset of the Condorcet do-main. This strengthens the random dictatorship theorem andestablishes that the Condorcet domain is essentially a maxi-mal domain that allows for attractive strategyproof SDSs. Wefinally extend our results to an even number of voters by char-acterizing the set of group-strategyproof and non-imposingSDSs on the Condorcet domain and its supersets.

IntroductionStrategyproofness—no agent should be better of by actingdishonestly—is one of the central economic paradigms inmulti-agent systems (see, e.g., Nisan et al. 2007; Shoham andLeyton-Brown 2009; Brandt et al. 2016). A significant prob-lem for such systems is collective decision-making, whichaims at identifying socially desirable outcomes by lettingvoters express their preferences over the possible alternatives.A multitude of theorems has shown that even rather basicproperties of voting rules cannot be satisfied simultaneously.In this context, strategyproofness is known to be a particu-larly restrictive axiom. This is exemplified by the Gibbard-Satterthwaite theorem which states that dictatorships are theonly deterministic voting rules that satisfy strategyproofnessand non-imposition (i.e., every alternative is elected for somepreference profile). Since dictatorships are not acceptablefor most applications, this result is usually considered animpossibility theorem.

One of the most successful escape routes from the Gibbard-Satterthwaite impossibility is to restrict the domain of feasi-ble preference profiles. For instance, Moulin (1980) promi-nently showed that there are attractive strategyproof votingrules on the domain of single-peaked preference profiles, andvarious other restricted domains of preferences have beenconsidered since then (see, e.g., Barbera, Gul, and Stacchetti1993; Peremans and Storcken 1999; Saporiti 2009; Elkind,Lackner, and Peters 2017). The idea behind domain restric-tions is that the voters’ preferences often obey structuralconstraints and thus, not all preference profiles are likely orplausible. A particularly significant constraint is the existenceof a Condorcet winner, which is an alternative that is favoredto every other alternative by a majority of the voters. Apartfrom its natural appeal, this concept is important becausethere is strong empirical evidence that real-world electionsusually admit Condorcet winners (see, e.g., Regenwetter et al.2006; Laslier 2010; Gehrlein and Lepelley 2011). This mo-tivates the study of the Condorcet domain, which consistsprecisely of the preference profiles that have a Condorcetwinner. Note that the Condorcet domain is a superset of sev-eral important domains such as those of single-peaked andsingle-dipped preferences when the number of voters is odd.There are several results showing the existence of attractivestrategyproof voting rules on the Condorcet domain. In partic-ular, Campbell and Kelly (2003) characterize the Condorcetrule, which always picks the Condorcet winner, as the onlystrategyproof, non-imposing, and non-dictatorial voting ruleon the Condorcet domain if the number of voters is odd.

In this paper, we focus on randomized voting rules, so-called social decision schemes (SDSs). Gibbard (1977) hasshown that randomization unfortunately does not allow formuch more leeway beyond the negative consequences of theGibbard-Satterthwaite theorem: random dictatorships, whichselect each voter with a fixed probability and elect the fa-vorite alternative of the chosen voter, are the only SDSson the full domain that satisfy strategyproofness and non-imposition (which in the randomized setting requires thatevery alternative is chosen with probability 1 for some pref-erence profile). Thus, these SDSs are merely “mixtures ofdictatorships”. In order to circumvent this negative result, weare interested in large domains that allow for strategyproofand non-imposing SDSs apart from random dictatorships.A natural candidate for this is the Condorcet domain and,


indeed, we show that the Condorcet domain is essentially amaximal domain that allows for strategyproof, non-imposing,and “non-randomly dictatorial” social choice. In more detail,we prove that, if the number of voters n is odd, every strat-egyproof and non-imposing SDS on the Condorcet domaincan be represented as a mixture of dictatorial SDSs and theCondorcet rule (which chooses the Condorcet winner withprobability 1). This result entails that the Condorcet rule isthe only strategyproof, non-imposing, and completely “non-randomly dictatorial” SDS on the Condorcet domain for oddn. Moreover, we show that—under a mild connectednesscondition on the domain—random dictatorships are the onlystrategyproof and non-imposing SDS on every superset ofthe Condorcet domain. Unfortunately, our theorems do notextend to an even number of voters because a single voter can-not change the Condorcet winner in this case. We address thisproblem by characterizing the set of group-strategyproof andnon-imposing SDSs on the Condorcet domain and most ofits supersets. Our characterizations on the Condorcet domaincan be seen as attractive complements to classic negativeresults for the full domain, whereas our results for supersetsof the Condorcet domain significantly strengthen these nega-tive results. In particular, our theorems imply statements byBarbera (1979) and Campbell and Kelly (2003) as well asthe Gibbard-Satterthwaite theorem (Gibbard 1973; Satterth-waite 1975) and the random dictatorship theorem (Gibbard1977) when the number of voters is odd. The relationships toresults for deterministic voting rules hold because every suchrule can be seen as an SDS that always returns degeneratelotteries.

Related WorkRestricting the domain of preference profiles in order tocircumvent classic impossibility theorems has a long tra-dition and remains an active research area to date. In par-ticular, the existence of attractive deterministic voting rulesthat satisfy strategyproofness has been shown for a num-ber of domains. Classic examples include the domains ofsingle-peaked (Moulin 1980), single-dipped (Barbera, Berga,and Moreno 2012), and single-crossing (Saporiti 2009) pref-erence profiles. More recent possibility results focus onbroader but more technical domains such as the domainsof multi-dimensionally single-peaked or semi single-peakedpreference profiles (e.g., Barbera, Gul, and Stacchetti 1993;Nehring and Puppe 2007; Chatterji, Sanver, and Sen 2013;Reffgen 2015). On the other hand, domain restrictions arealso used to strengthen impossibility results by proving themfor smaller domains (e.g, Aswal, Chatterji, and Sen 2003;Sato 2010; Gopakumar and Roy 2018). In more recent re-search, the possibility and impossibility results converge bygiving precise conditions under which a domain allows forstrategyproof and non-dictatorial deterministic voting rules(e.g., Chatterji and Sen 2011; Chatterji, Sanver, and Sen 2013;Roy and Storcken 2019; Chatterji and Zeng 2021).

While similar results have also been put forward for SDSs,this setting is not as well understood. For instance, Ehlers,Peters, and Storcken (2002) have shown the existence ofattractive strategyproof SDSs on the domain of single-peakedpreference profiles (see also Peters et al. 2014; Pycia and

Unver 2015). The existence of strategyproof and non-impo-sing SDSs other than random dictatorships has also beeninvestigated for a variety of other domains (e.g., Peters et al.2017; Roy and Sadhukhan 2020; Peters, Roy, and Sadhukhan2021). Chatterji, Sen, and Zeng (2014) and Chatterji andZeng (2018) pursue a more general approach and identifycriteria for deciding whether a domain admits such SDSs.

The strong interest in restricted domains also led to thestudy of many computational problems for restricted do-mains (e.g., Conitzer 2009; Faliszewski et al. 2011; Bred-ereck, Chen, and Woeginger 2013; Brandt et al. 2015; Elkind,Lackner, and Peters 2016; Peters 2017; Peters and Lackner2020). For instance, Bredereck, Chen, and Woeginger (2013)give an algorithm for recognizing whether a preference pro-file is single-crossing, which can be used to decide whetherthe positive results apply for a given domain.

Finally, observe that all aforementioned results are re-stricted to Cartesian domains, i.e., domains of the formD = Xn, where X is a set of preference relations. How-ever, the Condorcet domain is not Cartesian. In this sense,the only results directly related to ours are the ones by Camp-bell and Kelly (2003) and their follow-up work (Merrill 2011;Campbell and Kelly 2015, 2016). These papers can be seen aspredecessors of our work since they investigate strategyproofdeterministic voting rules on the Condorcet domain.

PreliminariesLet N = {1, . . . , n} denote a finite set of voters andA = {a, b, . . . } a finite set of m alternatives. Every voteri ∈ N is equipped with a preference relation Ri, which is acomplete, transitive, and anti-symmetric binary relation onA. We defineR as the set of all preference relations on A. Apreference profileR ∈ Rn consists of one preference relationfor each voter i ∈ N . A domain of preference profiles D is asubset of the full domainRn. When writing preference pro-files, we represent preference relations as comma-separatedlists and indicate the set of voters who share a preferencerelation directly before the preference relation. For instance,{1, 2} : a, b, c indicates that voters 1 and 2 prefer a to b to c.We omit the brackets for singleton sets.

The main object of study in this paper are social decisionschemes (SDSs) which are voting rules that may use random-ization to determine the winner of an election. More formally,an SDS maps every preference profile R of a domain D to alottery over the alternatives, which determines the winningchance of every alternative. A lottery p is a probability dis-tribution over the alternatives, i.e, p(x) ≥ 0 for all x ∈ Aand

∑x∈A p(x) = 1, and the set of all lotteries over A is

denoted by ∆(A). An SDS on a domain D is then a functionof the form f : D → ∆(A). The term f(R, x) denotes theprobability assigned to x by the lottery f(R) and for everyset X ⊆ A, we define f(R,X) =

∑x∈X f(R, x). Note that

SDSs are a generalization of deterministic voting rules whichchoose an alternative with probability 1 in every preferenceprofile. A natural requirement for an SDS f : D → ∆(A) isnon-imposition which states that for every alternative x ∈ A,there is a preference profile R ∈ D such that f(R, x) = 1.Less formally, this axiom merely requires that every alterna-


tive is chosen with probability 1 in some preference profile.For example, an alternative could be chosen with probability1 if it is top-ranked by all voters.

Stochastic Dominance and Strategyproofness

Strategic manipulation is one of the central issues in socialchoice theory: voters might be better off by voting dishon-estly. Since satisfactory collective decisions require the vot-ers’ true preferences, SDSs should incentivize honest vot-ing. In order to formalize this, we need to specify how vot-ers compare lotteries over alternatives. The most prominentapproach for this is based on stochastic dominance (see,e.g, Gibbard 1977; Ehlers, Peters, and Storcken 2002; Pe-ters, Roy, and Sadhukhan 2021). Let the upper contour setU(Ri, x) = {y ∈ A : yRix} be the set of alternatives thatvoter i weakly prefers to x in R. Then, stochastic dominancestates that a voter i prefers a lottery p to another lotteryq, denoted by p %i q, if p(U(Ri, x)) ≥ q(U(Ri, x)) forall x ∈ A. Note that the stochastic dominance relation istransitive but not complete. Using stochastic dominance tocompare lotteries is appealing because p %i q holds iff pguarantees voter i at least as much expected utility than qfor every utility function that is ordinally consistent with hispreference relation Ri (see, e.g., Brandl et al. 2018).

Based on stochastic dominance, we define multiple vari-ants of strategyproofness. The standard notion requires that asingle voter cannot benefit by lying about his true preferences.Formally, we say an SDS f : D → ∆(A) is strategyproofif f(R) %i f(R′) for all preference profiles R,R′ ∈ D andvoters i ∈ N such that Rj = R′j for all j ∈ N \ {i}. Con-versely, an SDS is called manipulable if it is not strategyproof.A convenient property of strategyproofness is that mixturesof strategyproof SDSs are again strategyproof, i.e., the set ofstrategyproof SDSs is convex for every domain.

A natural weakening of strategyproofness is local strate-gyproofness, which only disincentivizes manipulations byswapping two alternatives. In order to formalize this con-cept, we define Ri:yx as the preference profile derived fromanother profile R by reinforcing y against x in the prefer-ence relation of voter i. Note that this definition requiresthat x 6= y and that voter i ranks x directly above y in R.Then, an SDS f : D → ∆(A) is locally strategyproof iff(R) %i f(R′) for all voters i ∈ N , alternatives x, y ∈ A,and preference profiles R,R′ ∈ D such that R′ = Ri:yx.Local strategyproofness is an attractive weakening of strate-gyproofness since empirical results suggest that voters oftenmanipulate by reporting such inconspicuous lies (see, e.g.,Fischbacher and Follmi-Heusi 2013; Mennle et al. 2015).

In order to disincentivize groups of voters from manipu-lating, we need a stronger strategyproofness notion: an SDSf : D → ∆(A) is group-strategyproof if for all prefer-ence profiles R,R′ ∈ D and all non-empty sets of votersI ⊆ N with Rj = R′j for all j ∈ N \ I , there is a voteri ∈ I such that f(R) %i f(R′). Conversely, an SDS isgroup-manipulable if it is not group-strategyproof. Note thatgroup-strategyproofness implies strategyproofness and strat-egyproofness implies local strategyproofness.

γ-Randomly Dictatorial SDSs

The random dictatorship theorem shows that random dictator-ships are the only non-imposing and strategyproof SDSs onthe full domain (see, e.g., Gibbard 1977). Informally, a ran-dom dictatorship f is a mixture of dictatorial SDSs. A dicta-torial SDSs di always chooses the most preferred alternativeof a single voter i ∈ N with probability 1. Formally, an SDSf is a random dictatorship if there are weights αi ≥ 0 for allvoters i ∈ N such that

∑i∈N αi = 1 and f =

∑i∈N αidi.

While they are more attractive than dictatorships, randomdictatorships are often undesirable because they do not allowfor any compromise. For instance, if the voters agree on asecond best alternative but strongly disagree on the best one,it seems reasonable to compromise on the second best alterna-tive. However, this is not possible with random dictatorshipsand thus, we here interpret the random dictatorship theoremas a negative result.

An escape route from the random dictatorship theorem is toconsider restricted domains of preference profiles. However,random dictatorships are still strategyproof and non-imposingon any reasonable subdomain and thus, the strategyproof andnon-imposing SDSs that are no random dictatorships areof particular interest. Unfortunately, it is not straightforwardhow to define these “non-randomly dictatorial” SDSs becausemixtures of strategyproof SDSs are again strategyproof. Forinstance, the SDS that returns with probability 0.99 the out-come of a random dictatorship and otherwise the outcome ofanother strategyproof SDS is strategyproof and no randomdictatorship but it clearly feels “randomly dictatorial”. Toaddress this problem, Brandt, Lederer, and Romen (2021)introduced the notion of γ-randomly dictatorial SDSs. A strat-egyproof SDS f : D → ∆(A) is γ-randomly dictatorial ifγ ∈ [0, 1] is the maximal value such that f can be representedas f = γd+(1−γ)g, where d is a random dictatorship and gis a strategyproof SDS onD. Note that if γ < 1, the maximal-ity of γ implies that g is 0-randomly dictatorial. On the otherhand, if γ = 1, f is a random dictatorship and the choice of gdoes not matter. Intuitively, the notion of γ-random dictator-ships quantifies how close an SDS is to a random dictatorship.Hence, 0-randomly dictatorial SDSs can be seen as a rigorousrandomized equivalent to deterministic non-dictatorial votingrules.

Note that γ-random dictatorships are defined with respectto a specific strategyproofness notion, but the general ideaof this axiom is independent of such details. Thus, we canuse other strategyproofness notions to derive new variants ofγ-dictatorships. In particular, we call a locally strategyproofSDS f : D → ∆(A) locally γ-randomly dictatorial if γ ∈[0, 1] is the maximal value such that f can be represented asf = γd + (1 − γ)g, where d is a random dictatorship andg is a locally strategyproof SDS on D. The intuition behindlocally γ-randomly dictatorial SDSs is the same as for γ-random dictatorships, but local γ-random dictatorships areeasier to handle because local strategyproofness is a muchmore restricted concept than strategyproofness. Also, sincestrategyproofness implies local strategyproofness, every γ-randomly dictatorial SDS is locally γ′-randomly dictatorialfor γ′ ≥ γ.


Super Condorcet DomainsSince there are no attractive strategyproof rules on the fulldomain, we investigate the Condorcet domain and its su-per domains with respect to the existence of such SDSs.For defining these domains, we first have to introduce addi-tional terminology. The majority margin gR(x, y) = |{i ∈N : xRiy}| − |{i ∈ N : yRix}| indicates how many morevoters prefer x to y in the profile R than vice versa. Based onthe majority margins, we define the Condorcet winner of aprofile R as the alternative x such that gR(x, y) > 0 for ally ∈ A \ {x}. Since the existence of a Condorcet winner isnot guaranteed, we are interested in the Condorcet domainDC = {R ∈ Rn : there is a Condorcet winner in R} whichcontains all profiles with a Condorcet winner. A particularlynatural SDS on the Condorcet domain is the Condorcet rule(COND) which assigns always probability 1 to the Con-dorcet winner.

In this paper, we also investigate super Condorcet domains,which are domains D such that DC ⊆ D. For our analysis,we need to define additional properties of super Condorcetdomains. First, a super Condorcet domain D is strict if thereis a profileR ∈ D such that for every alternative x ∈ A, thereis another alternative y ∈ A\{x}with gR(y, x) > 0. In otherwords, every strict super Condorcet domain contains at leastone profile that does not even admit a weak Condorcet winner,i.e., an alternative that beats or ties every other alternative. Ifn is odd, every super Condorcet domain D with D 6= DC isstrict, but this is not true when there are majority ties.

Super Condorcet domains D can be disconnected withrespect to strategyproofness: there can be a profile R ∈ Dthat differs from every other profile R′ ∈ D \ {R} in at leasttwo preference relations. This is problematic for our analy-sis because strategyproofness has no implications for suchan isolated profile R. We address this issue by introducingconnected super Condorcet domains. To this end, we firstintroduce the idea of ad-paths: an ad-path from a profile Rto another profile R′ in a domain D is a sequence of pref-erence profiles (R1, . . . , Rl) such that R1 = R, Rl = R′,Rk ∈ D for all k ∈ {1, . . . , l}, and the profile Rk+1 evolvesout of Rk by swapping two alternatives x, y ∈ A in the pref-erence relation of a voter i ∈ N , i.e, Rk+1 = (Rk)i:yx for allk ∈ {1, . . . , l−1}. Then, a super Condorcet domainD is con-nected if for all alternatives x ∈ A and profiles R ∈ D \ DC ,there is an ad-path from R to a profile R′ ∈ DC on whichx is never swapped with another alternative. Less formally,this means that for every profile R outside of the Condorcetdomain and every alternative x ∈ A, we can apply a sequenceof swaps, none of which involves x, to transform R into aprofile in the Condorcet domain. Furthermore, observe thatthe notion of connected super Condorcet domains is onlyreasonable for an odd number of voters n; for even n, theCondorcet domain is the only connected super Condorcetdomain because of majority ties.

Finally, we introduce the concept of Condorcet-consistencybecause the Condorcet rule is not well-defined on super Con-dorcet domains: an SDS f on a super Condorcet domain D isCondorcet-consistent if f(R) = COND(R) for all profilesR ∈ DC ⊆ D. Hence, Condorcet-consistent SDSs extendthe Condorcet rule to larger domains.

CharacterizationsIn this section, we present our characterizations of strate-gyproof and group-strategyproof SDSs on super Condorcetdomains. First, we discuss characterizations of strategyproofand non-imposing SDSs on connected super Condorcet do-mains for odd n and show that the Condorcet domain isessentially the only super Condorcet domain that allows for astrategyproof, 0-randomly dictatorial, and non-imposing SDS.We then extend these result to an even number of voters bystrengthening strategyproofness to group-strategyproofness.We defer the proofs of all results to the supplementary ma-terial because of space restrictions and only discuss proofsketches here.

Results based on StrategyproofnessOur first result shows that every strategyproof and non-imposing SDS on the Condorcet is a mixture of a randomdictatorship and the Condorcet rule if n is odd. As a byprod-uct, we obtain a characterization of the Condorcet rule as theonly strategyproof, non-imposing, and 0-randomly dictatorialSDS on the Condorcet domain for odd n. By contrast, weshow for every strict and connected super Condorcet domainD that random dictatorships are the only strategyproof andnon-imposing SDSs on D if n is odd.

In order to derive these characterizations, we rely on thenotion of locally γ-randomly dictatorial SDSs. In particu-lar, we are interested in the locally 0-randomly dictatorial,locally strategyproof, and non-imposing SDSs on a domainD because these are the new SDSs gained by consideringa restricted domain. Moreover, these SDSs characterize theset of all locally strategyproof and non-imposing SDSs forall sufficiently connected domains. We prove this claim forconnected super Condorcet domains, but it holds for all com-monly studied domains.

Lemma 1. Assume n ≥ 3 is odd and m ≥ 3. An SDS f on aconnected super Condorcet domain is locally strategyproofand non-imposing iff it is a random dictatorship or there areγ ∈ [0, 1), a random dictatorship d, and a locally 0-randomlydictatorial, locally strategyproof, and non-imposing SDS gon D such that f = γd+ (1− γ)g.

The direction from left to right follows from the defini-tion of local γ-random dictatorships: for every locally strat-egyproof SDS f , there is a maximal γ such that f can berepresented as f = γd + (1 − γ)g, where d is a randomdictatorship and g is a locally strategyproof SDS. If γ = 1,f is a random dictatorship. On the other hand, if γ < 1,the maximality of γ entails that g is locally 0-randomly dic-tatorial. Since g also inherits non-imposition from f , thereis a suitable representation for every non-imposing and lo-cally strategyproof SDS on a connected super Condorcetdomain. For the other direction, note that every random dic-tatorship is locally strategyproof and non-imposing on everyconnected super Condorcet domain. On the other hand, iff = γd + (1 − γ)g for γ ∈ [0, 1), a random dictatorshipd, and a locally strategyproof and non-imposing SDS g, itfollows that f is also locally strategyproof because mixturesof locally strategyproof SDSs are locally strategyproof on


every domain. Moreover, it is easy to see that f is also non-imposing since both g and d satisfy this axiom.

As a consequence of Lemma 1, the set of locally 0-randomly dictatorial, locally strategyproof, and non-imposingSDSs G of a connected super Condorcet domain D charac-terizes the set of all locally strategyproof and non-imposingSDSs on D because each such SDS is a random dictatorshipor a mixture of a random dictatorship and an SDS g ∈ G.Hence, a characterization of the set G for a connected superCondorcet domain D immediately implies a characteriza-tion of the set of all locally strategyproof and non-imposingSDSs on D. In particular, if there is no locally strategyproof,non-imposing, and locally 0-randomly dictatorial SDS on D,random dictatorships are the only locally strategyproof andnon-imposing SDSs on this domain.

Motivated by Lemma 1, we investigate the set of locally 0-randomly dictatorial, locally strategyproof, and non-imposingSDSs for connected super Condorcet domains. As a firststep, we present a characterization of locally 0-randomlydictatorial SDSs on these domains.

Lemma 2. Assume n ≥ 3 is odd and m ≥ 3. A non-imposing and locally strategyproof SDS f on a connectedsuper Condorcet domain is locally 0-randomly dictatorial iffit is Condorcet-consistent.

The direction from right to left is trivial: if f is a Condorcet-consistent SDS, the best alternative of every voter can haveprobability 0 because it might not be the Condorcet win-ner. Consequently, every Condorcet-consistent and locallystrategyproof SDS on a connected super Condorcet domainis locally 0-randomly dictatorial. The inverse direction ismuch more difficult to prove. The key insight is that everylocally strategyproof, locally 0-randomly dictatorial, and non-imposing SDS on a connected super Condorcet domain Dhas to choose an alternative with probability 1 whenever itis top-ranked by n − 1 voters. Departing from this insight,we derive inductively that an alternative is also chosen withprobability 1 whenever it is top-ranked by more than halfof the voters. Finally, we infer from this observation thatevery locally strategyproof, locally 0-randomly dictatorial,and non-imposing SDS on D is Condorcet-consistent.

Lemma 2 has multiple important consequences. First ofall, it identifies Condorcet-consistent SDSs as counterpartsto random dictatorships on the large class of connected su-per Condorcet domains because every locally strategyproof,non-imposing, and locally 0-randomly dictatorial SDS isCondorcet-consistent, whereas only random dictatorshipsare locally 1-randomly dictatorial. Even more, Theorem 1and Theorem 2 show that this insight is also true if we usestrategyproofness and 0-random dictatorships instead of thelocal variants. This observation has an intuitive explanation:while random dictatorships never compromise, Condorcet-consistent SDSs can be seen as maximally compromising.Furthermore, we can use Lemma 2 to characterize the setof strategyproof and non-imposing SDSs on the Condorcetdomain for odd n.

Theorem 1. Assume n is odd and m ≥ 3. An SDS on theCondorcet domain is strategyproof and non-imposing iff it isa mixture of a random dictatorship and the Condorcet rule.

The proof of this result follows almost directly fromLemma 1 and Lemma 2. In particular, the definition ofCondorcet-consistency implies that the Condorcet rule is theonly Condorcet-consistent SDS on the Condorcet domain andthus, Lemma 2 shows that it is also the only SDS that can si-multaneously satisfy local strategyproofness, non-imposition,and local 0-random dictatorship. Since the Condorcet ruleindeed satisfies all these axioms, it is the only locally strat-egyproof, locally 0-randomly dictatorial, and non-imposingSDS on the Condorcet domain for odd n. Hence, we canuse Lemma 1 to derive that every locally strategyproof andnon-imposing SDS is a mixture of dictatorial SDSs and theCondorcet rule. Finally, since all these SDSs are also strat-egyproof and strategyproofness implies local strategyproof-ness, the theorem follows.

Note that Theorem 1 immediately implies that the Con-dorcet rule is the only 0-randomly dictatorial SDS on theCondorcet domain that satisfies strategyproofness and non-imposition if n ≥ 3 is odd and m ≥ 3. This corollary gener-alizes Theorem 1 of Campbell and Kelly (2003) who havecharacterized the Condorcet rule with equivalent axioms inthe deterministic setting. Furthermore, this insight highlightsthe appeal of the Condorcet rule on the Condorcet domainbecause every other strategyproof and non-imposing SDS ismerely a mixture of the Condorcet rule and a random dicta-torship. Hence, there is a unique, attractive, and strategyproofmethod for choosing the winner of an election if we restrictour attention to the Condorcet domain.

A natural follow-up question to Theorem 1 is whether wecan find larger domains for which there are strategyproofand non-imposing SDSs other than random dictatorships. Wefind a negative answer to this problem if n is odd: on everystrict and connected super Condorcet domain D, only ran-dom dictatorships are strategyproof and non-imposing. Notethat here, strictness implies only that D 6= DC because nis odd. Hence, the subsequent theorem shows that the Con-dorcet domain is essentially a maximal domain that allowsfor attractive strategyproof SDSs.

Theorem 2. Assume n is odd and m ≥ 3. An SDS on a strictand connected super Condorcet domain is strategyproof andnon-imposing iff it is a random dictatorship.

The proof of this theorem is an easy consequence ofLemma 1 and Lemma 2. In particular, we only need toshow that there is no locally strategyproof and Condorcet-consistent SDS on any strict and connected super Condorcetdomain D because Lemma 2 then shows that there is no lo-cally strategyproof, non-imposing, and locally 0-randomlydictatorial SDS on this domain. Thus, we can use Lemma 1to derive that only random dictatorships satisfy local strate-gyproofness and non-imposition, and consequently, the theo-rem follows because all these SDSs are also strategyproof.

First, note that Theorem 2 generalizes the random dic-tatorship theorem from the full domain to every strict andconnected super Condorcet domain if n is odd. Moreover,since deterministic voting rules can be seen as SDSs whichchoose in every preference profile an alternative with proba-bility 1, our result also generalizes the Gibbard-Satterhwaitetheorem to these smaller domains if n is odd. In particu-


lar, Theorem 2 shows that adding even a single profile tothe Condorcet domain can turn the positive result of Theo-rem 1 into a negative one because the resulting domain isa strict and connected super Condorcet domain. This fol-lows, for instance, by considering the subsequent domainD1 = DC ∪ {R∗}. The preference profile R∗ is shown be-low, where I = {4, 6, . . . , n − 1}, J = {5, 7, . . . , n}, andthe ∗ indicates that all missing alternatives can be orderedarbitrarily below a, b, and c.R∗: 1: a, b, c, ∗ 2: b, c, a, ∗ 3: c, a, b, ∗

I: a, b, c, ∗ J : c, b, a, ∗It is easy to verify that D1 is a strict and connected super

Condorcet domain because we can go from R∗ to a profilein the Condorcet domain if voter 1 swaps a and b, if voter 2swaps b and c, and if voter 3 swaps a and c. Hence, Theorem 2entails that random dictatorships are the only strategyproofand non-imposing SDSs on D1.

Remark 1. An important consequence of Theorems 1 and 2is that, if n is odd, every strategyproof and non-imposing SDSon a connected super Condorcet domain can be representedas a mixture of deterministic voting rules, each of which isstrategyproof and non-imposing. This is sometimes calleddeterministic extreme point property and it is remarkable thatmany important domains satisfy this condition (see Roy andSadhukhan 2020). On one side, this shows that randomizationdoes not lead to completely new strategyproof SDSs. On theother hand, the deterministic extreme point property allowsfor a natural interpretation of strategyproof and non-imposingSDSs: we decide by chance which deterministic voting ruleis executed. Also, note that there are domains that violate thiscondition (see Chatterji, Sen, and Zeng 2014).

Remark 2. There are domains D with DC ( D that allowfor non-imposing and strategyproof SDSs that are no randomdictatorships. For example, consider the domain D2 whichis derived by adding a single preference profile R1 to theCondorcet domain. If R1 differs from every profile in DC inthe preference relations of at least two voters, an arbitraryoutcome can be returned for R1 without violating strate-gyproofness. This shows that the connectedness condition isrequired for Theorem 2.

Remark 3. The proofs of Theorem 1 and Theorem 2 showthat these characterizations also hold if we use local strate-gyproofness instead of strategyproofness. Consequently, lo-cal strategyproofness is equivalent to strategyproofness forevery connected super Condorcet domain if n is odd. Equiva-lent results have been shown for a number of other domains,e.g., the domain of single-peaked profiles, the domain ofsingle-dipped profiles, or the full domain (see, e.g., Gibbard1977; Carroll 2012; Sato 2013; Kumar et al. 2021). Further-more, since local strategyproofness is equivalent to strate-gyproofness on every connected super Condorcet domain,the notions of local γ-random dictatorships and γ-randomdictatorships are equivalent, too. Hence, all local and globalnotions can be used interchangeably for our results.

Remark 4. If n is even, there are larger domains thanthe Condorcet domain that allow for strategyproof, non-

imposing, and 0-randomly dictatorial SDSs. An exampleof such a domain is the shifted Condorcet domainDRi

C whichcontains a profileR iff there is a Condorcet winner in (R,Ri),i.e., in the profile derived by adding a fixed preference re-lation Ri to R. An SDS on DRi

C that satisfies all our re-quirements is f(R) = COND(R,Ri) because the domain{(R,Ri) : R ∈ DRi

C } is a subset of the Condorcet domainfor n+ 1 voters.

Results based on Group-strategyproofnessMaybe the strongest restrictions of Theorem 1 and Theo-

rem 2 is that both results only apply if the number of votersn is odd. Indeed, Theorem 1 does not hold for even n be-cause the counterexamples by Merrill (2011) show that thereare strategyproof, non-imposing, and 0-randomly dictatorialSDSs f : DC → ∆(A) other than the Condorcet rule ifn is even. The reason for this is that a single voter cannotchange the Condorcet winner if n is even and consequently,the Condorcet domain breaks down into several disconnectedsubdomains. A natural approach to restore the connectednessis to use group-strategyproofness instead of strategyproofnessbecause this axiom allows to change the Condorcet winnerwith a group-manipulation. We therefore characterize nowthe set of group-strategyproof and non-imposing SDSs onsuper Condorcet domains independently of the parity of n.In more detail, we show that only dictatorial SDSs and theCondorcet rule satisfy these axioms on the Condorcet do-main, whereas only dictatorships are group-strategyproof andnon-imposing on strict super Condorcet domains.

The proofs of these results are very similar to the proofs ofTheorem 1 and Theorem 2. In particular, we first generalizeLemma 2 by characterizing when a group-strategyproof andnon-imposing SDS is non-dictatorial.

Lemma 3. Assume n ≥ 3 andm ≥ 3. A group-strategyproofand non-imposing SDS on a super Condorcet domain is non-dictatorial iff it is Condorcet-consistent.

It is obvious that a Condorcet-consistent SDS is non-dictatorial as the best alternative of a voter gets probability0 if it is not the Condorcet winner. For the other direction,we prove that every group-strategyproof, non-imposing, andnon-dictatorial SDS on a super Condorcet domain choosesan alternative x with probability 1 whenever n − 1 votersreport x as their favorite option. Based on this insight, wederive—analogously to the proof of Lemma 2—that suchSDSs are Condorcet-consistent.

Similar to Lemma 2, Lemma 3 identifies Condorcet-consistent SDSs as counterparts to dictatorships since allgroup-strategyproof, non-imposing, and non-dictatorial SDSson super Condorcet domains are Condorcet-consistent. How-ever, Lemma 3 is more general as we characterize non-dictatorial SDSs instead of local 0-random dictatorships andno connectedness condition on the domain is required. Thisis possible since we use group-strategyproofness instead oflocal strategyproofness. Next, we employ this lemma to char-acterize the set of group-strategyproof and non-imposingSDSs on the Condorcet domain and every strict super Con-dorcet domain.


Full domain Strict super Condorcet domains Condorcet domain

Deterministic,strategyproof, andnon-imposing voting rules

Dictatorships (Gibbard1973; Satterthwaite 1975)

Dictatorships� (Consequence ofTheorem 2)

Dictatorships and theCondorcet rule∗ (Campbelland Kelly 2003)

Strategyproof andnon-imposing SDSs

Random dictatorships(Gibbard 1977)

Random dictatorships�(Theorem 2)

Mixtures of randomdictatorships and theCondorcet rule∗ (Theorem 1)

Group-strategyproof andnon-imposing SDSs

Dictatorial SDSs (Barbera1979) Dictatorial SDSs (Theorem 3) Dictatorial SDSs and the

Condorcet rule (Theorem 3)

Table 1: Comparison of results for the full domain, strict super Condorcet domains, and the Condorcet domain. Each rowcharacterizes a set of SDSs for the full domain, every strict super Condorcet domain, and the Condorcet domain, respectively. Allresults require that there are m ≥ 3 alternatives and results marked with an asterisk (∗) only hold if there is an odd number ofvoters. The results marked with a diamond (�) require that n is odd and that the strict super Condorcet domain is additionallyconnected. New results are italicized.

Theorem 3. Assume m ≥ 3. An SDS on the Condorcetdomain is group-strategyproof and non-imposing iff it is adictatorial SDS or the Condorcet rule. An SDS on a strictsuper Condorcet domain is group-strategyproof and non-imposing iff it is a dictatorial SDS.

The proof of the claim for the Condorcet domain followsimmediately from Lemma 3 since the Condorcet rule is theonly Condorcet-consistent SDS on the Condorcet domain.Hence, only dictatorial SDSs and the Condorcet rule can begroup-strategyproof and non-imposing and it is easy to verifythat all these SDSs indeed satisfy both axioms. Furthermore,we prove the claim for strict super Condorcet domains byshowing that there is no group-strategyproof and Condorcet-consistent SDS on these domains because Lemma 3 entailsthen that every group-strategyproof and non-imposing SDSis dictatorial.

Theorem 3 generalizes many consequences of Theorem 1and Theorem 2 to super Condorcet domains for an evennumber of voters by using group-strategyproofness. For in-stance, it entails that the Condorcet rule is the only group-strategyproof, non-imposing, and non-dictatorial SDS on theCondorcet domain and thus generalizes an analogous im-plication of Theorem 1. Moreover, the second part of thetheorem shows that the Condorcet rule is essentially a max-imal domain that allows for a group-strategyproof and non-imposing SDS apart of dictatorships. In particular, if n isodd, every super Condorcet domain D with D 6= DC is strictbecause no majority ties are possible. Hence, Theorem 3shows for odd n that no superset of the Condorcet domainadmits group-strategyproof and non-imposing SDSs otherthan dictatorships.

Remark 5. The results of Barbera (1979) imply that ev-ery group-strategyproof and non-imposing SDS on the fulldomain is a dictatorship. Hence, Theorem 3 and Barbera’sresults share a common idea: group-strategyproof and non-imposing SDSs cannot use randomization to determine thewinner. However, whereas only undesirable SDSs are group-strategyproof and non-imposing on the full domain, the at-tractive Condorcet rule satisfies these axioms on DC .

ConclusionWe studied strategyproof and non-imposing social decisionschemes (SDSs) on the Condorcet domain (which consistsof all preference profiles with a Condorcet winner) and itssupersets. In contrast to the full domain, there are attractivestrategyproof SDSs on the Condorcet domain. In particular,we show that, if the number of voters n is odd, every strat-egyproof and non-imposing SDS on the Condorcet domaincan be represented as a mixture of a random dictatorshipand the Condorcet rule. An immediate consequence of thisinsight is that the Condorcet rule is the only strategyproof,non-imposing, and 0-randomly dictatorial SDS on the Con-dorcet domain if n is odd. Furthermore, we show that, ifn is odd, only random dictatorships are strategyproof andnon-imposing on every sufficiently connected superset of theCondorcet domain. This shows that the Condorcet domain isessentially a maximal domain which allows for a 0-randomlydictatorial, non-imposing, and strategyproof SDS. Finally,we extend our results to an even number of voters by usinggroup-strategyproofness: we show that only the Condorcetrule and dictatorial SDSs are group-strategyproof and non-imposing on the Condorcet domain, whereas only dictatorialSDSs satisfy these axioms on larger domains.

Our results for the Condorcet domain show an astonishingsimilarity to classic results for the full domain, but have amore positive flavor. For instance, while the random dictator-ship theorem shows that only mixtures of dictatorial SDSsare strategyproof and non-imposing on the full domain, weprove in Theorem 1 that mixtures of dictatorial SDSs and theCondorcet rule are the only strategyproof and non-imposingSDSs on the Condorcet domain (if the number of voters isodd). A more exhaustive comparison between results for thefull domain and for the Condorcet domain is given in Table 1.In particular, our results highlights the important role of theCondorcet rule in the Condorcet domain and thus, they makea strong case for choosing the Condorcet winner wheneverit exists. Furthermore, our results for strict super Condorcetdomains strengthen the classic results by showing that theyeven hold in small domains.


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Appendix A: Proofs of Theorems 1 and 2In this appendix, our goal is to prove Theorems 1 and 2. Asthe proofs of these results are rather involved, we introduceseveral auxiliary lemmas which are organized in subsectionsto indicate related ideas. In particular, we first investigate theimplications of local strategyproofness and non-impositionfor connected super Condorcet domains. Based on these in-sights, we next analyze the behavior of local 0-randomlydictatorial SDSs and prove Lemma 2. As a last step, wederive Lemma 1, Theorem 1, and Theorem 2.

In many of the subsequent proofs, we use additional nota-tion. In particular, we use the ∗ symbol in preference relationsto indicate that all missing alternatives can be ordered arbi-trarily. For instance, 1 : a, b, ∗ means that voter 1 prefersa to b to every other alternative, and that the alternatives inA \ {a, b} can be ordered arbitrarily. Furthermore, the rankr(Ri, x) = |{y ∈ A : xRiy}| of an alternative x with respectto a preference relation Ri denotes the number of alternativesthat voter i weakly prefers to x. Less formally, if an alterna-tive is the k-th best alternative of voter i, its rank with respectto Ri is k. In particular, r(Ri, x) = r(Ri, y) + 1 means thatvoter i places alternative y directly over alternative x in hispreference relation.

Local Strategyproofness on Connected SuperCondorcet DomainsAs first step, we investigate the consequences of local strate-gyproofness for connected super Condorcet domains. Hence,we present a characterization of local strategyproofness basedon non-perversity and localizedness. An SDS f : D → ∆(A)is non-perverse if f(Ri:yx, y) ≥ f(R, y) for all preferenceprofiles R ∈ D, alternatives x, y ∈ A, and voters i ∈ N suchthat Ri:yx ∈ D. Furthermore, an SDS f : D → ∆(A) islocalized if f(Ri:yx, z) = f(R, z) for all preference profilesR ∈ D, distinct alternatives x, y, z ∈ A, and voters i ∈ Nsuch that Ri:yx ∈ D. Intuitively, non-perversity requires thatthe probability of an alternative increases if it is reinforced,and localizedness demands that the probability of an alterna-tive does not change if it is not involved in a swap. Gibbard(1977), who first used these axioms, has proven that the con-junction of non-perversity and localizedness is equivalent tostrategyproofness on the full domain. We show a variant ofthis claim for restricted domains: non-perversity and local-izedness are equivalent to local strategyproofness for everydomain.

Lemma 4. Let D ⊆ Rn denote an arbitrary domain. AnSDS f : D → ∆(A) is locally strategyproof iff it is localizedand non-perverse.

Proof. Consider an arbitrary domain D and let f denotean SDS on this domain. We need to show that f is locallystrategyproof if it is localized and non-perverse, and that f isnon-perverse and localized if it is locally strategyproof.

Claim 1: If f is locally strategyproof, it is localized andnon-perverse.

First, assume that f : D → ∆(A) is a locally strategyproofSDS. Moreover, consider two arbitrary preference profilesR,R′ ∈ D, a voter i ∈ N , and two alternatives x, y ∈ A such

that R′ = Ri:yx. Note that local strategyproofness impliesthat f(R) %i f(R′) and f(R′) %′i f(R).

We first show that f satisfies localizedness, i.e., thatf(R, z) = f(R′, z) for all alternatives z ∈ A \ {x, y}.Observe for this that the assumption R′ = Ri:yx en-tails that U(Ri, z) = U(R′i, z) for all z ∈ A \ {x, y}and that U(Ri, y) = U(R′i, x). Thus, local strategyproof-ness from R to R′ and from R′ to R implies thatf(R,U(Ri, z)) = f(R′, U(R′i, z)) for all z ∈ A \ {x, y}and that f(R,U(Ri, y)) = f(R′, U(R′i, x)). For instance,the latter is true because local strategyproofness implies thatf(R,U(Ri, y)) ≥ f(R′, U(Ri, y)) and f(R′, U(R′i, x)) ≥f(R,U(R′i, x)). Furthermore, note that for every alternativez ∈ A and every preference relation Ri, the setU(Ri, z)\{z}is either empty (if z is the best alternative of voter i) orequal to the upper contour set U(Ri, z

′) of the alternativez′ with r(Ri, z

′) = r(Ri, z) − 1. Hence, it holds for allalternatives z ∈ A \ {x, y} with r(Ri, z) 6= r(Ri, y) + 1that f(R,U(Ri, z) \ {z}) = f(R′, U(R′i, z) \ {z}) becauseeither U(Ri, z) \ {z} = U(R′i, z) \ {z} = ∅ or there is an-other alternative z′ ∈ A \ {x, y} such that U(Ri, z) \ {z} =U(Ri, z

′) = U(R′i, z′) = U(R′i, z) \ {z}. On the other

hand, it also holds for the alternative z ∈ A \ {x, y}with r(Ri, z) = r(Ri, y) + 1 that f(R,U(Ri, z) \ {z}) =f(R′, U(R′i, z)\{z}) because U(Ri, z)\{z} = U(Ri, y) =U(R′i, x) = U(R′i, z) \ {z}. Hence, f is localized becausewe can calculate for all alternatives z ∈ A \ {x, y} that

f(R, z) = f(R,U(Ri, z))− f(R,U(Ri, z) \ {z})= f(R′, U(R′i, z))− f(R′, U(R′i, z) \ {z}) = f(R′, z).

Finally, we show that f is non-perverse, i.e., thatf(R′, y) ≥ f(R, y). Using local strategyproofness fromR′ to R, we derive that f(R′, U(R′i, y)) ≥ f(R,U(R′i, y)),and localizedness implies that f(R′, z) = f(R, z) for allz ∈ A \ ({x, y}). Since U(R′i, y) \ {y} ⊆ A \ {x, y}, itfollows that f is non-perverse because

f(R′, y) = f(R′, U(R′i, y))− f(R′, U(R′i, y) \ {y})≥ f(R,U(R′i, y))− f(R,U(R′i, y) \ {y}) = f(R, y).

Claim 2: If f is localized and non-perverse, it is locallystrategyproof.

Suppose that f : D → ∆(A) is a localized and non-perverse SDS. We need to show that this SDS is locally strate-gyproof and consider thus two preference profilesR,R′ ∈ D,a voter i ∈ N , and two alternatives x, y ∈ A such that R′ =Ri:yx. Our goal is to prove that f(R) %i f(R′) which re-quires that f(R,U(Ri, z)) ≥ f(R′, U(Ri, z)) for all z ∈ A.First, note that f(R, z) = f(R′, z) for all z ∈ A \ {x, y} be-cause of localizedness. This implies that f(R,U(Ri, z)) =f(R′, U(Ri, z)) for all z ∈ A \ {x} with zRix becausex, y 6∈ U(Ri, z) and thus U(Ri, z) ⊆ A \ {x, y}. Further-more, for all z ∈ A \ {x} with xRiz, we can compute thatf(R,U(Ri, z)) = 1− f(R,A \U(Ri, z)) = 1− f(R′, A \U(Ri, z)) = f(R′, U(Ri, z)) because {x, y} ⊆ U(Ri, z)and thus A \ U(Ri, z) ⊆ A \ {x, y}. Finally, non-perversityshows for x that f(R, x) ≥ f(R′, x). Since localizednessimplies that f(R,w) = f(R′, w) for all w ∈ U(Ri, x)\{x},it follows that f(R,U(Ri, x)) ≥ f(R′, U(Ri, x)). Hence, it


indeed holds that f(R,U(Ri, z)) ≥ f(R′, U(Ri, z)) for allz ∈ A, which shows that f is locally strategyproof.

Lemma 4 clarifies the consequences of local strategyproof-ness on restricted domains and shows that we can use equiva-lently non-perversity and localizedness, which are easier tohandle. Next, we investigate locally strategyproof and non-imposing SDSs on connected super Condorcet domains. Forphrasing and proving the next lemma, we need to introducesome additional terminology. We say an alternative x ∈ APareto-dominates any other alternative y ∈ A \ {x} in apreference profileR if xRiy for all voters i ∈ N . Conversely,an alternative is Pareto-optimal if it is not Pareto-dominatedby another alternative. Finally, an SDS f : D → ∆(A) isPareto-optimal if f(R, x) = 0 for all alternatives x ∈ A andpreference profiles R ∈ D such that x is Pareto-dominatedin R. Pareto-optimality is maybe the most prominent effi-ciency notion in economic theory. Furthermore, we call anSDS f : D → ∆(A) k-unanimous if f(R, x) = 1 for allalternatives x ∈ A and preference profiles R ∈ D suchthat at least n − k voters top-rank x in R. k-unanimity isa natural strengthening of the common idea of unanimity,which requires that an alternative is chosen with probability1 if it is unanimously top-ranked. We show next that everynon-imposing and locally strategyproof SDS on a connectedsuper Condorcet domain is Pareto-optimal and therefore also0-unanimous if n is odd.Lemma 5. Assume n is odd andm ≥ 3. Every non-imposingand locally strategyproof SDS on a connected super Con-dorcet domain satisfies Pareto-optimality.

Proof. Assume n is odd and let f denote a locally strate-gyproof and non-imposing SDS on an arbitrary connectedsuper Condorcet domain D. We assume for contradictionthat f fails Pareto-optimality and derive a contradiction intwo steps: first, we show f needs to be 0-unanimous, andsecondly, we show that this cannot be the case if a Pareto-dominated alternative is assigned positive probability in aprofile. Since these claims contradict each other, it followsthat every locally strategyproof and non-imposing SDS onD satisfies Pareto-optimality. In the subsequent proof, wealways ensure that we stay in the Condorcet domain, whichalso means that we stay in D since DC ⊆ D.

Claim 1: Every locally strategyproof and non-imposingSDS f : D → ∆(A) satisfies 0-unanimity.

Choose an arbitrary alternative x ∈ A and let R ∈ Ddenote a profile in which all voters report x as their favoritechoice. We need to show that f(R, x) = 1 and consider forthis a profile R1 ∈ D with f(R1, x) = 1. Such a profileexists since f is non-imposing. If R1 ∈ D \ DC , we first usethe connectedness of D to go into the Condorcet domain. Inmore detail, this axiom states that there is an ad-path fromR1 to a profile R2 ∈ DC such that x is not swapped withany alternative along this ad-path. Consequently, a repeatedapplication of localizedness along this ad-path shows thatf(R2, x) = f(R1, x) = 1. On the other hand, if R1 is al-ready in the Condorcet domain, we just define R2 = R1.

Let c denote the Condorcet winner in R2. We proceedwith a case distinction with respect to whether c = x or not.

First, assume that x is the Condorcet winner in R2. In thiscase, we let all voters repeatedly swap up x until it is the bestalternative of every voter. Since we only reinforce x, it staysthe Condorcet winner during these steps. Moreover, non-perversity entails that the probability of x is non-decreasing.Thus, this process results in a profile R3 in which x is unan-imously top-ranked and f(R3, x) = 1. As last step, we letall voters reorder all alternatives in A \ {x} one after an-other to derive the profile R. It is easy to see that we can usepairwise swaps for this, and thus, localizedness implies thatf(R, x) = 1. Also, these steps stay in the Condorcet domainbecause x is always unanimously top-ranked. Hence, the casex = c is proven.

For the second case, we assume that c 6= x and defineI = {i ∈ N : cR2

i x} as the set of voters who prefer c to x inR2. Next, we let all voters in I repeatedly swap up c until itis their best alternative. Because c is only reinforced duringthese steps, it stays the Condorcet winner. Furthermore, wenever swap c and x because of the definition of I . Hence,this process terminates in a profile R3 with f(R3, x) = 1because of localizedness. As second step, we let the votersi ∈ I one after another swap up x until it is their second bestalternative and the voters in N \ I swap up x until it is theirbest alternative. During these steps, c stays the Condorcetwinner as all voters in I report it as their best alternative andthus, we do not leave the Condorcet domain. Moreover, non-perversity entails that the probability of x cannot decreaseduring these steps, and therefore, f(R4, x) = 1 for the re-sulting profile R4. As next step, we let the voters in I swapx and c one after another, which results in a profile R5 inwhich x is the Condorcet winner because all voters reportit as their favorite choice. This process does not leave theCondorcet domain as every voter top-ranks either c or x andthus, one of these alternatives is top-ranked by more than halfof the voters during each step because n is odd. Finally, wededuce that f(R5, x) = 1 because non-perversity requiresthat f(R5, x) ≥ f(R4, x). As last step, we transformR5 intoR by reordering the alternatives in A \ {x} in the preferencerelations of all voters. Since x is not involved in any of theseswaps, it follows that f(R, x) = 1 because of localizedness,and we do not leave the Condorcet domain as x is alwaysunanimously top-ranked. Because x and R are arbitrarilychosen, it follows that f is 0-unanimous.

Claim 2: Every locally strategyproof SDS f : D →∆(A) that violates Pareto-optimality fails 0-unanimity.

Next, assume that f violates Pareto-optimality, whichmeans that there is a profile R ∈ D and alternatives x, y ∈ Asuch that xRiy for all voters i ∈ N but f(R, y) > 0. Weshow that x is not chosen with probability 1 if it is top-rankedby every voter. If R ∈ D \ DC , we use again the connected-ness of D to go to a profile R1 ∈ DC . In particular, there isan ad-path from R to a profile R1 ∈ DC such that y is notswapped with any other alternative along this path. Hence, lo-calizedness implies that f(R1, y) = f(R, y) > 0, and x stillPareto-dominates y inR1 as we never swap these alternatives.If R ∈ DC , we define R1 = R.

Next, we use a case distinction with respect to the Con-dorcet winner c in R1. First, assume that x is the Condorcet


winner in R1. In this case, we can sequentially swap up xin the preference relation of every voter until it is unani-mously top-ranked. Because y is not affected by these steps,localizedness implies that f(R2, y) = f(R1, y) > 0 for theresulting profile R2. However, this means that f violates 0-unanimity as f(R2, x) 6= 1 even though x is unanimouslytop-ranked in R2.

As second case, assume that c 6= x and let I = {i ∈N : cR1

i x} denote the voters who prefer c to x in R1. Notealso that c 6= y because all voters prefer x to y. As firststep, we let the voters i ∈ I swap up c until it is their bestalternative. We do not leave the Condorcet domain duringthese steps because we only reinforce the Condorcet winner.Furthermore, y is not involved in any swap since cR1

i x andxR1

i y for all voters i ∈ I , and thus, this step leads to a profileR2 with f(R2, y) = f(R1, y) > 0 because of localizedness.As next step, we let all voters i ∈ I reinforce x until it is theirsecond best alternative and all voters i ∈ N \ I reinforcex until it is their best alternative. Alternative c stays theCondorcet winner during these steps since it is always top-ranked by all voters in I , and localizedness implies for theresulting profile R3 that f(R3, y) = f(R2, y) > 0 becauseall voters prefer x to y in R2. Finally, we derive the profileR4 by sequentially swapping c and x in the preferences ofthe voters i ∈ I . This process does not leave the Condorcetdomain because in every intermediate profile, only c and xare top-ranked. This means that one of these alternatives isthe Condorcet winner because n is odd and thus, either c orx is top-ranked by more than half of the voters. Moreover,y is not involved in these swaps and localizedness impliestherefore that f(R4, y) = f(R3, y) > 0. However, x is nowtop-ranked by all voters but f(R4, x) 6= 1, which means thatf fails 0-unanimity.

Lemma 5 is one of our basic tools that will be used inalmost all subsequent proofs. For instance, we use this re-sult now to show that every locally strategyproof and non-imposing SDS on a connected super Condorcet domain satis-fies strong symmetry conditions between profiles in whichonly two alternatives are top-ranked. For formally stating thislemma, we denote with NR

x = {i ∈ N : r(Ri, x) = 1} theset of voters who report x as their favorite alternative in R.Moreover, we define SxIy = {R ∈ Rn : NR

x = I ∧NRy =

N \ I} for all subsets of voters I ⊆ N and alternativesx, y ∈ A as the domain that contains precisely the prefer-ence profiles in which all voters in I report x as their bestalternative and all voters in N \ I report y as their best al-ternative. Note that SxIy ⊆ DC for all subsets of voters Iand alternatives x, y ∈ A if n is odd because either x or y istop-ranked by more than half of the voters and thus, one ofthese alternatives is the Condorcet winner for every profileR ∈ SxIy.

Lemma 6. Assume n ≥ 3 is odd and m ≥ 3. Furthermore,consider a locally strategyproof and non-imposing SDS f ona connected super Condorcet domain. For all sets of votersI ⊆ N with ∅ ( I ( N , alternatives w, x, y, z ∈ A withw 6= x and y 6= z, and preference profiles R ∈ SwIx, R′ ∈SyIz , it holds that f(R,w) = f(R′, y), f(R, x) = f(R′, z),and f(R,w) + f(R, x) = 1.

Proof. Consider a connected super Condorcet domain Dand let f : D → ∆(A) denote a locally strategyproof andnon-imposing SDS. Furthermore, consider an arbitrary set ofvoters I with ∅ ( I ( N . We prove the lemma in three steps:first, we show that f(R) = f(R′) and f(R, x)+f(R, y) = 1for all alternatives x, y ∈ A with x 6= y and all preferenceprofilesR,R′ ∈ SxIy . This is helpful as we now only need toshow that there are preference profilesR ∈ SwIx,R′ ∈ SyIzsuch that f(R,w) = f(R′, y), f(R, x) = f(R′, z), andf(R, x) + f(R, y) = 1. As second step, we consider threedistinct alternatives x, y, z ∈ A and show that f(R, x) =f(R′, z) for all R ∈ SxIy, R′ ∈ SzIy. Using these twoinsights, we prove the lemma as last step.

Step 1: f(R) = f(R′) and f(R, x) + (R, y) = 1 forall distinct alternatives x, y ∈ A and preference profilesR,R′ ∈ SxIy

Let x and y denote two distinct alternatives and con-sider a profile R∗ ∈ DC ⊆ D such that r(R∗i , x) = 1and r(R∗i , y) = 2 for all voters i ∈ I , and r(R∗i , y) = 1and r(R∗i , x) = 2 for all voters i ∈ N \ I . Note thatf(R∗, x) + f(R∗, y) = 1 because f is Pareto-optimal (seeLemma 5) and all alternatives z ∈ A \ {x, y} are Pareto-dominated in R∗. Moreover, consider an arbitrary preferenceprofile R ∈ SxIy. We will show that f(R∗) = f(R) whichproves this step since R is chosen arbitrarily. Note that wedo not have to worry about leaving D during the subsequenttransformations because we do not change the favorite al-ternatives of the voters. Since either I or N \ I containsmore than half of the voters, it follows therefore that thereis always a Condorcet winner. For transforming R∗ into R,we consider two auxiliary profiles R1 and R2: in R1, thevoters i ∈ I report R∗i and the voters i ∈ N \ I reportRi. The profile R2 is constructed inversely: the voters ini ∈ I report Ri and the voters i ∈ N \ I report R∗i . Next,we show that f(R, x) = f(R1, x) = f(R∗, x). Note forthis that we can use pairwise swaps to transform R∗ to R1

and that no swap involves y because we only need to re-order the alternatives A \ {y} in the preference relations ofthe voters i ∈ N \ I . Thus, it follows from localizednessthat f(R1, y) = f(R∗, y). Moreover, y Pareto-dominatesall other alternatives but x in R1 as it is second-ranked byall voters in I and top-ranked by the voters in N \ I . Hence,Lemma 5 implies that f(R1, x)+f(R1, y) = 1 which meansthat f(R1, x) = 1 − f(R1, y) = 1 − f(R∗, y) = f(R∗, x).As next step, we transfrom R1 into R by reordering the alter-natives in A \ {x} in the preference relations of voters i. Wecan use for this again pairwise swaps that do not involve xand thus, localizedness implies that f(R, x) = f(R1, x) =f(R∗, x). Moreover, we can use a symmetric argument toderive that f(R, y) = f(R2, y) = f(R∗, y). Therefore, weconclude that f(R) = f(R∗) and f(R, x) +f(R, y) = 1 forall preference profiles R ∈ SxIy.

Step 2: f(R, x) = f(R′, z) for all distinct alternativesx, y, z ∈ A and preference profiles R ∈ SxIy, R′ ∈ SzIy

For proving this claim, let x, y, z denote three distinctalternatives and consider the profiles R3 and R4.R3: I: x, z, y, ∗ N\I: y, x, z, ∗R4: I: z, x, y, ∗ N\I: y, x, z, ∗


We start the analysis at the profileR3 ∈ SxIy . In particular,note that Step 1 implies for R3 that f(R3, x) + f(R3, y) =1. Next, we derive the profile R4 by letting every voteri ∈ I swap x and z one after another. If y is the Con-dorcet winner in R3, it is clear that this process does notleave the Condorcet domain as y is not involved in thesesteps. If x is the Condorcet winner, then x or z are theCondorcet winner during each step. The reason for thisis that gR3(z, w) = gR3(x,w) > 0 for every alternativew ∈ A\{x, z}. This means that x is the Condorcet winner aslong as a majority of the voters prefers x to z and otherwise, zis the Condorcet winner. Moreover, localizedness implies thatf(R4, y) = f(R3, y) since y was not involved in any swapduring these steps. Finally, Step 1 implies that f(R4, w) = 0for all w ∈ A \ {y, z} because R4 ∈ SzIy. Hence, we inferthat f(R4, z) = 1 − f(R4, y) = 1 − f(R3, y) = f(R3, x).Since R3 ∈ SxIy and R4 ∈ SzIy, we can now use the in-sights of the first step to deduce that f(R, x) = f(R′, z) forall distinct alternatives x, y, z ∈ A and preference profilesR ∈ SxIy and R′ ∈ SzIy.

Step 3: f(R,w) = f(R′, y), f(R, x) = f(R′, z) andf(R,w) + f(R, x) = 1 for all alternatives w, x, y, z ∈ Awith w 6= x and y 6= z and all preference profiles R ∈SwIx, R′ ∈ SyIz

Let w, x, y, z denote arbitrary alternatives with w 6= xand y 6= z. First note that Step 1 proves the claim ifw = y and x = z. Furthermore, the case that x = z andw 6= y follows from Step 1 and Step 2: Step 2 entails thatf(R,w) = f(R′, y) for all R ∈ SwIx and R′ ∈ SyIx andthen, we infer from Step 1 that f(R, x) = f(R′, x) becauseall other alternatives must have probability 0 in both R andR′. Note that we can use the same argument also for thecase that w = y and x 6= z. The reason for this is thatSwIx = SxN\Iw and SyIz = SzN\Iy, i.e., we can just re-vert the roles of I and N \ I to apply the same argumentas in the previous case. If {w, x} ∩ {y, z} = ∅, the lemmafollows by applying the previous ideas twice: we know thatf(R,w) = f(R, y) = f(R′, y) and f(R, x) = f(R, x) =f(R′, z) for all R ∈ SwIx, R ∈ SyIx, and R′ ∈ SyIz . Fi-nally, the last case is that w = z or x = y or both. We focuson the case that w = z, i.e., we assume that R ∈ SwIx

and R′ ∈ SyIw; the case x = y follows by a symmet-ric argument. In this case, we use an auxiliary alternativev and two auxiliary profiles R1 ∈ SvIx, R2 ∈ SvIw: weknow that f(R,w) = f(R1, v) = f(R2, v) = f(R′, y) andf(R, x) = f(R1, x) = f(R2, w) = f(R′, w) for all prefer-ence profilesR ∈ SwIx,R′ ∈ SyIw by applying the previousargument three times. Note that this argument also coversthe case that w = z and x = y because y can be chosenarbitrarily in A \ {w}. This concludes the proof as all casesare covered.

Lemma 6 is helpful for our analysis because determiningthe probabilities for a single preference profile R ∈ SxIy

specifies the outcome for a whole class of profiles. As lastresult of this section, we show a technical auxiliary lemmathat discusses how the probabilities change if a voter swapsto alternatives w, x ∈ A in two related preference profiles Rand R.

Lemma 7. Assume n ≥ 2 and m ≥ 3, and let f denote alocally strategyproof SDS on an arbitrary domain D ⊆ Rn.Moreover, consider preference profilesR, R,R′, R′ ∈ D, twodistinct voters i, j ∈ N , and alternatives w, x, y, z ∈ A suchthat w 6= x, y 6= z, |{w, x} ∩ {y, z}| ≤ 1, R = Ri:xw, R′ =Rj:zy, and R′ = Rj:zy. It holds that f(R, w)− f(R,w) =f(R′, w) − f(R′, w) and f(R, x) − f(R, x) = f(R′, x) −f(R′, x).

Proof. Assume that n ≥ 2 and m ≥ 3, and let f denote alocally strategyproof SDS on an arbitrary domain D. It isfor this lemma crucial that local strategyproofness implieslocalizedness for every domain (see Lemma 4). Next, supposethat there are preference profiles R, R,R′, R′ ∈ D, votersi, j ∈ N , and alternatives w, x, y, z ∈ A such that w 6= x,y 6= z, |{w, x} ∩ {y, z}| ≤ 1, R = Ri:xw, R′ = Rj:zy,and R′ = Rj:zy. First, note that the assumption |{w, x} ∩{y, z}| ≤ 1 means that w 6∈ {y, z} or x 6∈ {y, z} or both.We assume without loss of generality that w 6∈ {y, z} asboth cases are symmetric. Hence, localizedness between Rand R′ implies that f(R′, w) = f(R,w) and localizednessbetween R and R′ that f(R′, w) = f(R, w) because R′ =Rj:zy and R′ = Rj:zy . Therefore, we infer immediately thatf(R, w)−f(R,w) = f(R′, w)−f(R′, w), proving the firstclaim of this lemma.

Next, observe that our assumptions entail that R′ =(R′)i:xw and we suppose that R = Ri:xw. Thus, local-izedness implies that f(R′, v) = f(R′, v) and f(R, v) =f(R, v) for all v ∈ A \ {w, x}. Hence, we can compute that

f(R, x)− f(R, x)

=(

1− f(R, A \ {x}))−(

1− f(R,A \ {x}))

=(f(R,w)− f(R, w)

)=(f(R′, w)− f(R′, w)

)=(

1− f(R′, A \ {x}))−(

1− f(R′, A \ {x}))

= f(R′, x)− f(R′, x).

This proves the second equality of the lemma.

Local 0-randomly dictatorships on ConnectedSuper Condorcet DomainsAs next step, we discuss what it means to be locally 0-randomly dictatorial on connected super Condorcet domains.In particular, we show Lemma 2 in this section: if n ≥ 3 isodd and m ≥ 3, a locally strategyproof and non-imposingSDS on a connected super Condorcet domain is locally 0-randomly dictatorial iff it is Condorcet-consistent. Hence, weinvestigate first the behavior of local 0-random dictatorshipson connected super Condorcet domains.

Lemma 8. Assume n ≥ 3 is odd andm ≥ 3 and let f denotea locally strategyproof and locally 0-randomly dictatorialSDS on a connected super Condorcet domain D. For everyvoter i ∈ N , there are preference profiles R,R′ ∈ D andalternatives x, y ∈ A such that R′ = Ri:yx, r(Ri, x) = 1,r(Ri, y) = 2, and f(R′, y) = f(R, y).


Proof. Consider a connected super Condorcet domain D andlet f denote a locally strategyproof and locally 0-randomlydictatorial SDS onD. Furthermore, suppose for contradictionthat there is a voter i ∈ N such that f(R′, y)− f(R, y) 6= 0for all preference profiles R,R′ ∈ D and alternatives x, y ∈A such that R′ = Ri:yx, r(Ri, x) = 1, and r(Ri, y) = 2.First note that non-perversity entails for all these profiles thatf(R′, y) − f(R, y) ≥ 0. Consequently, there is an ε > 0such that f(R′, y)− f(R, y) ≥ ε for all preference profilesR,R′ ∈ D and alternatives x, y ∈ A such that R′ = Ri:yx,r(Ri, x) = 1, and r(Ri, y) = 2. Our goal is to show that f islocally γ-randomly dictatorial for γ ≥ ε. This contradicts theassumption that f is a local 0-random dictatorship and hence,there must be profiles for voter i that satisfy all conditions ofthe lemma.

For deriving the contradiction, we proceed in multiplesteps. In Step 1, we show that f(R, x) ≥ ε for all preferenceprofiles R ∈ D and alternatives x ∈ A such that voter iprefers x the most in R. Based on this observation, it followsthat ε < 1 because otherwise, f chooses always the bestalternative of voter i with probability 1. However, this meansthat f is the dictatorial SDS of voter i and therefore, it is1-randomly dictatorial if ε = 1. Since this contradicts theassumption that f is locally 0-randomly dictatorial, we canassume that 0 < ε < 1. For this case, we show in Step 2 thatg = 1

1−εf −ε

1−εdi (where di is the dictatorial SDS of voteri) is a well-defined and locally strategyproof SDS. Hence, fcan be represented as f = εdi + (1− ε)g, which shows thatf is locally γ-randomly dictatorial for some γ ≥ ε. This is inconflict with our assumptions and thus, there must be profilesR,R′ ∈ D and alternatives x, y ∈ A such that f(R, y) =f(R′, y), R′ = Ri:yx, r(Ri, x) = 1, and r(Ri, y) = 2.

Step 1: f(R, x) ≥ ε for every preference profile R ∈ Dand every alternative x ∈ A such that r(Ri, x) = 1.

As first step, we show that f(R, x) ≥ ε for all preferenceprofiles R ∈ D and alternatives x ∈ A such that voter itop-ranks x in R. Thus, consider an arbitrary profile R ∈ Dand let x denote voter i’s best alternative. As first step, weconstruct a profile R1 ∈ DC such that f(R, x) = f(R1, x)and r(Ri, x) = 1. If R is already in the Condorcet domain,we just define R1 = R. Otherwise, we can find an ad-pathfrom R ∈ D\DC to a profile R1 ∈ DC along which x is notswapped with any other alternative because D is connected.Consequently, a repeated application of localizedness alongthe path from R to R1 implies that f(R1, x) = f(R, x) andx is still voter i’s best alternative in R1.

In the sequel, we show that f(R1, x) ≥ ε, which provesthis step since f(R, x) = f(R1, x). For this, we use a casedistinction with respect to the Condorcet winner c in R1 andassume first that c 6= x. In this case, consider the profile R2

derived fromR1 by swapping voter i’s best alternative x withhis second best alternative y. Since x is not the Condorcetwinner in R1, it follows that R2 ∈ DC ⊆ D: either theCondorcet winner is not affected by swapping x and y orit is reinforced if c = y. Our assumptions on f imply thatf(R1, x) − f(R2, x) ≥ ε and it holds that f(R2, x) ≥ 0because of the definition of SDSs. Hence, f(R1, x) ≥ εwhich proves this case.

As second case, suppose that c = x. In this case, we choosean arbitrary alternative y ∈ A \ {x} and define J = {j ∈N : xR1

jy} as the set of voters who prefer x to y in R1. Asfirst step, consider the profile R2 derived from R1 by lettingall voters j ∈ J reinforce y until it is directly below x. Sinceno swap in the construction ofR2 involves x, we do not leavethe Condorcet domain. Also, this observation implies thatf(R2, x) = f(R1, x) because of localizedness. Moreover,gR2(y, z) = gR2(x, z) > 0 for all alternatives z ∈ A\{x, y}because xR2

jz implies yR2z and x is the Condorcet winner inR2. We use this fact to construct the next profileR3 by lettingthe voters j ∈ J \ {i} swap x and y one after another. Thisconstruction does not leave the Condorcet domain: since nis odd and gR2(x, z) = gR2(y, z) > 0 for all z ∈ A \ {x, y},x or y is the Condorcet winner in every intermediate profile.Moreover, in the final profile R3, all voters in N \ {i} prefery to x and thus, y is now the Condorcet winner becausen ≥ 3. Hence, it follows from the last case that f(R3, x) ≥ εbecause voter i top-ranks x in R3 and x is not the Condorcetwinner in this profile. Furthermore, we only weaken x inthe construction of R3, and thus, non-perversity implies thatf(R3, x) ≤ f(R2, x) = f(R1, x). Combining the last twoobservations shows that f(R1, x) ≥ ε and thus, f alwayschooses the best alternative of voter i with a probability of atleast ε.

Step 2: If ε < 1, then g = 11−εf−

ε1−εdi is a well-defined

and locally strategyproof SDS.Assume that ε < 1 and consider the function g =

11−εf −

ε1−εdi, where di is the dictatorial SDS of voter i.

We need to show that g is a well-defined and locally strate-gyproof SDS because f is then not locally 0-randomly dic-tatorial. First, we show that g is a well-defined SDS, i.e.,that g(R, x) ≥ 0 for all alternatives x ∈ A and preferenceprofiles R ∈ D, and that

∑x∈A g(R, x) = 1 for all prefer-

ence profiles R ∈ D. For proving the first claim, considera preference profile R ∈ D and an alternative x ∈ A. Weuse a case distinction with respect to whether x is voteri’s best alternative in R. If this is the case, step 1 impliesthat f(R, x) ≥ ε and di(R, x) = 1 because di choosesthe best alternative of voter i. Hence, we can compute thatg(R, x) = 1

1−εf(R, x) − ε1−εdi(R, x) ≥ ε

1−ε −ε

1−ε = 0.On the other hand, if x is not voter i’s best alternative,it follows that di(R, x) = 0. This means that g(R, x) =

11−εf(R, x)− ε

1−εdi(R, x) = 11−εf(R, x) ≥ 0, which shows

that g only assigns non-negative values to the alternatives.Next, we show that the probabilities assigned by g sum upto 1 for every preference profile R ∈ D. This follows imme-diately by the definition of this SDS since

∑x∈A g(R, x) =

11−ε

∑x∈A f(R, x)− ε

1−ε∑x∈A di(R, x) = 1

1−ε−ε

1−ε = 1for every profile R ∈ D. Hence, g is a well-defined SDS.

Next, we show that g is locally strategyproof. BecauseLemma 4 shows that local strategyproofness is equivalentto non-perversity and localizedness, it suffices to show thatg satisfies the latter two axioms. Hence, consider two pref-erence profiles R,R′ ∈ D, two alternatives x, y ∈ A, anda voter j ∈ N such that R′ = Rj:yx. First, note that g islocalized because


g(R, z) =1

1− εf(R, z)− ε

1− εdi(R, z)

=1

1− εf(R′, z)− ε

1− εdi(R

′, z) = g(R′, z)

for all z ∈ A \ {x, y}. This is true since f and di are locallystrategyproof and therefore localized. As second point, weneed to show that g is non-perverse, i.e., that g(R′, y) ≥g(R, y) for all preference profiles R,R′ ∈ D, alternativesx, y ∈ A, and voters j ∈ N such that R′ = Rj:yx. Note forthis that f(R′, y) ≥ f(R, y) for all these profiles because fis locally strategyproof and thus also non-perverse. Moreover,if r(Ri, x) > 1 or j 6= i, it follows that di(R, y) = di(R

′, y).Combining these observations shows that

g(R′, y) =1

1− εf(R′, y)− ε

1− εdi(R

′, y)

≥ 1

1− εf(R, y)− ε

1− εdi(R, y) = g(R, y).

Hence, g is non-perverse in this case. On the other hand, ifr(Ri, x) = 1 and j = i, it holds that di(R, y) = 0 6= 1 =di(R

′, y). Moreover, our assumptions on f imply for thiscase that f(R′, y)− f(R, y) ≥ ε. Thus, we can compute that

g(R′, y) =1

1− εf(R′, y)− ε

1− εdi(R

′, y)

≥ 1

1− ε(ε+ f(R, y))− ε

1− ε

=1

1− εf(R, y)

= g(R, y).

This inequality shows also for the second case that g isnon-perverse and consequently, it is locally strategyproof.Hence, f is not locally 0-randomly dictatorial because f =εdi+(1−ε)g. In particular, this representation entails that f islocally γ-randomly dictatorial for γ ≥ ε. This means that theinitial assumption is false and there are for every voter i ∈ Ntwo preference profiles R,R′ ∈ D and alternatives x, y ∈ Asuch that f(R, y) = f(R′, y), R′ = Ri:yx, r(Ri, x) = 1,and r(Ri, y) = 2.

As a consequence of Lemma 8, there are for every locally0-randomly dictatorial and locally strategyproof SDS f onthe Condorcet domain and every voter i ∈ N two preferenceprofiles R,R′ ∈ DC and two alternatives x, y ∈ A such thatR′ = Ri:yx, f(R′, y) − f(R, y) = 0, and voter i prefers xthe most and y the second most in R. We show next that thisis also true for every connected super Condorcet domain D,i.e., even if DC ( D, we can find for every voter i suitableprofiles R and R′ in the Condorcet domain.

Lemma 9. Assume n ≥ 3 is odd andm ≥ 3 and let f denotea locally strategyproof and locally 0-randomly dictatorialSDS on a connected super Condorcet domain D. For everyvoter i ∈ N , there are preference profiles R,R′ ∈ DC ⊆ Dand alternatives x, y ∈ A such that R′ = Ri:yx, r(Ri, x) =1, r(Ri, y) = 2, and f(R′, y) = f(R, y).

Proof. Assume that n ≥ 3 is odd and m ≥ 3 and let Ddenote a connected super Condorcet domain. Furthermore,consider a locally strategyproof and locally 0-randomly dic-tatorial SDS f : D → ∆(A). We need to show that for everyvoter j ∈ N there are preference profiles R,R′ ∈ DC and al-ternatives x, y ∈ A such that R′ = Rj:yx, r(Rj , x) = 1,r(Rj , y) = 2, and f(R, y) = f(R′, y). First, note thatLemma 8 proves that there are such profiles R,R′ ∈ Dfor every voter. If these profiles are for all j ∈ N also inthe Condorcet domain, the lemma is already proven. Hence,suppose that there is a voter i ∈ N such that {R,R′} 6⊆ DCfor all profiles R,R′ ∈ D and alternatives x, y ∈ A withR′ = Ri:yx, r(Ri, x) = 1, r(Ri, y) = 2, and f(R, y) =f(R′, y). Next, we consider two such profiles R and R′ andwe suppose that R 6∈ DC because the case that R′ 6∈ DCis symmetric. Our goal is to transform the profiles R,R′into two new profiles R and R′, respectively, that satisfy allconditions of the lemma, i.e., {R, R′} ⊆ DC , R′ = Ri:yx,r(Ri, x) = 1, r(Ri, y) = 2, and f(R, y) = f(R′, y). Notefor this that the assumption f(R, y) = f(R′, y) implies alsothat f(R, x) = f(R′, x) because of localizedness.

As first step in the construction of R and R′, we use theconnectedness of D: there is an ad-path from R to a profileR1 ∈ DC that does not swap x with another alternative. Sim-ilarly, if R′ 6∈ DC , there is also a profile R2 ∈ DC derivedfrom R′ without swapping x; if R′ ∈ DC , we set R2 = R′.Since R1 is derived from R without swapping x with an-other alternative, it follows that U(R1

j , x) = U(Rj , x) forall voters j ∈ N . In particular, this means that x is not theCondorcet winner in R1 because it is not the Condorcet win-ner in R. Analogously, it follows that U(R2

j , x) = U(R′j , x)

for all voters j ∈ N . This entails that U(R2j , x) = U(R1

j , x)for all j ∈ N \ {i} because Rj = R′j for these voters, andU(R1

i , x) ⊆ U(R2i , x) because R′ = Ri:yx. In particular,

these observations imply that x is also not the Condorcetwinner in R2, and that r(R1

i , x) = 1, r(R2i , y) = 1, and

r(R2i , x) = 2. Finally, localizedness shows that f(R1, x) =

f(R, x) = f(R′, x) = f(R2, x) because x is not swappedwith another alternative in the construction of R1 or R2.

Next, let c denote the Condorcet winner in R1 and definethe set I = {j ∈ N : cR1

jx} of voters who prefer c to x inR1. Analogously, we define c′ as the Condorcet winner inR2 and J = {j ∈ N : c′R2

jx} as the set of voters who preferc′ to x in R2. Note that i 6∈ I because voter i top-ranks xin R1. We derive the profile R from R1 as follows: we letall voters j ∈ I swap up c until it is their best alternative.This step leads to an intermediate profile R3, and we do notleaveDC ⊆ D during the construction of this profile becausewe only reinforce the Condorcet winner c. Furthermore, itfollows from the definition of I that x is not involved in anyswap, which means that and f(R3, x) = f(R1, x) becauseof localizedness. Finally, we construct the profile R basedon R3 by reordering the alternatives z ∈ A \ {x} in voteri’s preference relation as in R2, i.e., voter i’s preferencerelation only differs from R2

i in the fact that he prefers x to y.Since all voters in I prefer c the most in R3, we do not leaveDC during this step. Furthermore, localizedness implies thatf(R, x) = f(R3, x) = f(R, x).


As second point, we construct the profile R′ based on R2.In particular, we need to ensure that R′j = Rj for all vot-ers j ∈ N \ {i} and introduce therefore multiple auxiliaryprofiles. Hence, consider the profile R4 derived from R2 byletting all voters j ∈ J push up c′ until it is their best al-ternative, all voters j ∈ I ∩ J push up c until it is directlybelow c′, and all voters j ∈ I \ J push up c until it is theirbest alternative. Note that voter i does not change his prefer-ence relation during these steps: it holds that i 6∈ I becauser(R1

i , x) = 1, and if i ∈ J , then c′ = y because alternative yis the only alternative in A \ {x} with yR2

i x. However, voteri already top-ranks y inR2 and thus, it follows thatR4

i = R2i .

Furthermore, we do not leave the Condorcet domain duringthese steps because we never weaken the Condorcet winner c′.Finally, observe that x is not involved in any swap: we havealready shown that R4

i = R2i , and the definition of the sets

I and J combined with the fact that U(R1j , x) = U(R2

j , x)for all j ∈ N \ {i} implies that these voters do not swapx with another alternative. Hence, localizedness entails thatf(R4, x) = f(R2, x).

As next step, we construct the profile R5 based on R4 byletting all voters in I∩J swap c and c′ one after another. Notethat this intersection is non-empty because |I| > n

2 and |J | >n2 . Furthermore, this process does not leave the Condorcetdomain because n is odd and gR4(c, z) > 0, gR4(c′, z) > 0for all z ∈ A \ {c, c′}, which means that either c or c′ is theCondorcet winner in every intermediate profile. In particular,c is the Condorcet winner in the final profile as it is top-ranked by all voters in I . As final points on R5, observe thatlocalizedness implies that f(R5, x) = f(R4, x), and thatR5i = R4

i = R2i because i 6∈ I .

As last step, we use the observation that all voters in Ireport c as their best alternative in both R5 and R to con-struct the final profile R′. In more detail, we derive thisprofile from R5 by letting all voters j ∈ N \ {i} reordertheir preference relations as in R. We can transform R5 intoR′ without leaving DC by only using pairwise swaps be-cause all voters in I always report c as their best alternative,which ensures that c is the Condorcet winner. Furthermore,observe that U(R5

j , x) = U(Rj , x) for all voters j ∈ N \{i}because x was never swapped with another alternative dur-ing the construction of these profiles. Hence, we can con-struct the profile R′ without swapping x with another alter-native in the preference relations of these voters and local-izedness implies consequently that f(R′, x) = f(R5, x).Finally, observe that, by construction, R′j = Rj for allj ∈ N \ {i}, and R′i = R2

i = Ri:yxi . Hence, it indeedholds that R′ = Ri:yx and R, R′ ∈ DC . Furthermore,f(R, x) = f(R, x) = f(R′, x) = f(R′, x) because wenever swapped x with another alternative in our constructionand it follows therefore from localizedness that f(R′, y) =1−∑z∈A\{y} f(R′, z) = 1−

∑z∈A\{y} f(R, z) = f(R, y).

This proves the lemma.

Lemma 9 is important because it shows that the behaviorof 0-randomly dictatorial SDSs does not depend on the exactconnected super Condorcet domain. In particular, regardlessof the considered super Condorcet domain, there are for every

voter preference profiles in the Condorcet domain that satisfyall conditions of Lemma 8 and thus, our subsequent analysiscan focus on profiles in DC .

In the next lemma, we show that every 0-randomly dic-tatorial, locally strategyproof, and non-imposing SDS on aconnected super Condorcet domain is 1-unanimous if n ≥ 3is odd. Recall that an SDS f : D → ∆(A) is k-unanimousif f(R, x) = 1 for all alternatives x ∈ A and preferenceprofiles R ∈ D such that |NR

x | ≥ n− k.Lemma 10. Assume n ≥ 3 is odd and m ≥ 3. Every lo-cally 0-randomly dictatorial, locally strategyproof, and non-imposing SDS on a connected super Condorcet domain is1-unanimous.

Proof. Assume n ≥ 3 is odd and m ≥ 3 and let D denote aconnected super Condorcet domain. Furthermore, let f de-note a locally 0-randomly dictatorial SDS on D that satisfieslocal strategyproofness and non-imposition. We need to showthat f(R, x) = 1 for every preference profile R ∈ D andevery alternative x ∈ A such that |NR

x | ≥ n− 1. First, notethat f meets all requirements of Lemma 5, which implies thatan alternative is chosen with probability 1 if it is top-rankedby all voters because every other alternative is then Pareto-dominated. Therefore, it suffices to focus on profiles R ∈ Dand alternatives x ∈ A such that |NR

x | = n− 1.Consider for this an arbitrary voter i ∈ N . We show in the

sequel that f(R, x) = 1 for all preference profiles R ∈ Dand alternatives x ∈ A such that NR

x = N \ {i}. Since i ischosen arbitrarily, this claim proves the lemma. As next step,we use the insights of Lemma 9: for the considered voter i,there are preference profiles R, R ∈ DC and two alternativesx, y ∈ A such that R = Ri:yx, r(Ri, x) = 1, r(Ri, y) = 2,and f(R, y) = f(R, y). We proceed with a case distinctionwith respect to the profile R and its Condorcet winner c.

Case 1: There is z ∈ A such that N \ {i} ⊆ NzR.

As first case, suppose that there is an alternative z ∈ Asuch that N \ {i} ⊆ Nz

R, i.e., all voters in N \ {i} reportz as their best alternative in R. If z = x, then x Pareto-dominates every other alternative because every voter top-ranks x. Hence, f(R, x) = 1 because of Lemma 5. Then,our assumptions imply that f(R, y) = f(R, y) = 0, eventhough R ∈ SxN\{i}y. Thus, Lemma 6 applies and showsthat f(R′, w) = 1 for all profiles R′ ∈ D and alternativesw ∈ A such that NR′

w = N \ {i}. The reason for this is thatfor all these profiles R′, there is an alternative v ∈ A \ {w}such that R′ ∈ SwN\{i}v. Moreover, the case that z = yfollows by exchanging the roles of R and R in the aboveargument.

Hence, suppose that z ∈ A \ {x, y}. In this case, we useLemma 6 to deduce that f(R, y) = 0 since R ∈ SzN\{i}x.Our assumptions entail therefore that f(R, y) = f(R, y) =0. Since R ∈ SzN\{i}y , we can again use Lemma 6 to derivethat f(R′, w) = 1 for all preference profiles R′ ∈ D andalternatives w ∈ A such that N \ {i} is precisely the set ofvoters who report w as their favorite choice in R. Hence, theclaim is in this case proven.

Case 2: There is no z ∈ A such that N \ {i} ⊆ NRx and

c ∈ A \ {x, y}.


As second case, suppose that the voters in N \ {i} do notagree on a best alternative and suppose that the Condorcetwinner c in the profile R satisfies that c ∈ A \ {x, y}. In thiscase, we derive two new the profiles R′ and R′ from R andR, respectively, such that R′ ∈ ScN\{i}x, R′ = (R′)i:yx,r(R′i, x) = 1, r(R′i, y) = 2, and f(R′, y) = f(R′, y). Then,the insights of Case 1 show that f is 1-unanimous. For con-structing these new preference profiles, we repeatedly ap-ply Lemma 7 to push up c in the preference of every voterj ∈ N \{i} until it is their best alternative. In more detail, weconstruct two sequences of preference profiles (R1, . . . , Rk)and (R1, . . . , Rk) such that R1 = R, Rk = R′, R1 = R,Rk = R′, and for every index l ∈ {1, . . . , k − 1} there isa voter j ∈ N \ {i} and an alternative z ∈ A \ {c} suchthatRl+1 = (Rl)j:cz and Rl+1 = (Rl)j:cz . Less formally, ineach step, the new profile is derived by swapping up c in thepreference of some voter j in both Rl and Rl. It is straight-forward to construct such a sequence of preference profiles:we repeatedly identify a voter in j ∈ N \ {i} who does nottop-rank c and reinforce this alternative in his preferencerelation.

Since R = Ri:yx, our assumptions imply that Rl =(Rl)i:yx because we use the same sequence of swaps to de-rive Rl from R and Rl from R. Hence, we can use for eachstep Lemma 7 to derive that f(Rl+1, y) − f(Rl+1, y) =f(Rl, y)− f(Rl, y). In particular, it is important for this thatc 6∈ {x, y} to ensure that |{c, z} ∩ {x, y}| ≤ 1 (where zdenote the alternatives with which c is swapped in the currentstep). By repeatedly using the last insight, it follows thatf(R′, y)− f(R′, y) = f(R, y)− f(R, y) = 0. Furthermore,since we only reinforce the Condorcet winner, we do notleave the Condorcet domain. Hence, it is easy to verify thatthe profiles R′, and R′ indeed satisfy all requirements ofCase 1, which means that f is 1-unanimous.

Case 3: There is no z ∈ A such that N \ {i} ⊆ NRx and

c ∈ {x, y}.As last case, we assume again that the voters j ∈ N \ {i}

do not agree on a best alternative. However, this time wesuppose additionally that c ∈ {x, y}, i.e., the Condorcetwinner is affected in the derivation of R. For this case, wedefine the sets I = {j ∈ N \ {i} : xRiy} and J = {j ∈ N \{i} : yRix}, which partition the voters in N \ {i} accordingto their preference on x and y in R. We use these sets toconstruct auxiliary profiles just as in the last case. In moredetail, we derive two new profiles R1 and R1 from R and R,respectively, by pushing up x and y in the preference relationsof the voters j ∈ N \ {i} without changing their relativeorder until they are the best two alternatives. These profilesare shown below and, just as in Case 2, a repeated applicationof Lemma 7 proves that f(R1, y) − f(R1, y) = f(R, y) −f(R, y) = 0. Furthermore, Lemma 5 implies that f(R1, x) +f(R1, y) = 1 and f(R1, x) + f(R1, y) = 1 because allother alternatives are Pareto-dominated. Hence, it followsthat f(R1) = f(R1). As last point, it should be mentionedthat this process does not leave the Condorcet domain as theCondorcet winner c ∈ {x, y} is never weakened.R1: I: x, y, ∗ J : y, x, ∗ i: x, y, ∗R1: I: x, y, ∗ J : y, x, ∗ i: y, x, ∗

As next step, we choose an arbitrary alternative z ∈ A \{x, y}, and let the voter i reinforce this alternative until it ishis second best one. Applying this step to R1 and R1 resultsin the profilesR2 and R2, respectively. SinceR1, R2 ∈ SyJxand R1, R2 ∈ SxIy , it follows from Lemma 6 that f(R2) =f(R1) = (R1) = f(R2). Finally, note that we do not leavethe Condorcet domain in the construction ofR2 andR3 sincex or y is always top-ranked by more than half of the voters.R2: I: x, y, ∗ J : y, x, ∗ i: x, z, y, ∗R2: I: x, y, ∗ J : y, x, ∗ i: y, z, x, ∗

Thereafter, we consider the profiles R3 and R3 derivedfrom R2 and R2, respectively, by letting voter i swap hisbest two alternatives. First note that R3, R3 ∈ DC becausegR3(x, z) = gR3(x, z) = gR3(y, z) = gR3(y, z) = n− 2 >0 and thus, this swap does not affect whether x or y is theCondorcet winner. Furthermore, localizedness between R2

and R3 implies that f(R3, y) = f(R2, y) and localized-ness between R2 = R3 implies that f(R3, x) = f(R2, x).Moreover, we already know that f(R2) = f(R2) andnon-perversity between R3 and R3 shows that f(R3, x) ≥f(R3, x). Hence, we conclude that f(R3, x) ≥ f(R3, x) =f(R2, x) = f(R2, x). Finally, this means that f(R3) =f(R2) because f(R2, y) + f(R2, x) = 1, f(R3, y) =f(R2, y), and f(R3, x) ≥ f(R2, x). In particular, this showsthat f(R3, z) = 0.R3: I: x, y, ∗ J : y, x, ∗ i: z, x, y, ∗R3: I: x, y, ∗ J : y, x, ∗ i: z, y, x, ∗As last point, we let all voters j ∈ J swap x and y to derive

the profile R4 from R3. If x is the Condorcet winner in R3,it is obvious that this process stays in the Condorcet domain.On the other hand, if y is the Condorcet winner in R3, wealso do not leave the Condorcet domain because gR3(x,w) =gR3(y, w) > 0 for all alternatives w ∈ A \ {x, y}. Hence,y is the Condorcet winner during these steps as long as amajority of voters prefers y to x, and x is the Condorcetwinner otherwise. Next, note that f(R4, z) = f(R3, z) = 0because of localizedness.R4 : I: x, y, ∗ J : x, y, ∗ i: z, x, y, ∗

Since R4 ∈ SxN\{i}z , f(R4, z) = 0 implies thatf(R′, w) = 1 for all preference profiles R ∈ D and alterna-tives w ∈ A such that NR′

w = N \ {i} because of Lemma 6.Since voter i is chosen arbitrarily, it follows from the lastthree cases that f is indeed 1-unanimous.

As next step, we strengthen the consequences of Lemma 10by showing that every 1-unanimous and locally strategyproofSDS on a connected super Condorcet domain also satisfiesn−1

2 -unanimity.

Lemma 11. Assume n ≥ 3 is odd and m ≥ 3. Every 1-unanimous and locally strategyproof SDS on a connectedsuper Condorcet domain is n−1

2 -unanimous.

Proof. Let f denote a 1-unanimous and locally strategyproofSDS on a connected super Condorcet domain D. We provethe lemma by an induction on k ∈ {1, . . . , n−3

2 } and showthat if f is k-unanimous, it is also k + 1-unanimous. As first


step, observe that the induction basis k = 1 is true becausewe assume that f is 1-unanimous.

Next, suppose that f is k-unanimous for a fixed k ∈{1, . . . , n−3

2 }. We need to show that f is k + 1-unanimous,i.e., that f(R, x) = 1 for every profile R ∈ D and everyalternative x ∈ A such that |NR

x | = n − k − 1. Note thatk + 1-unanimity also applies for profiles R ∈ D and alter-natives x ∈ A with |NR

x | > n − k − 1, but k-unanimityimmediately guarantees for these profiles that f(R, x) = 1.Hence, it suffices to focus on profiles R ∈ D and alternativesx ∈ A such that |NR

x | = n− k − 1.As first step, we show that f(R, x) = 1 for all alternatives

x, y ∈ A and profiles R ∈ SxIy with |I| = n − k − 1. Forproving this, consider an arbitrary set of voters I ⊆ N with|I| = n−k−1, let h denote an arbitrary voter inN\I , and de-fine the set J = N\(I∪{h}). Note that J 6= ∅ because k ≥ 1implies that |I| ≤ n − 2. Finally, consider the profiles R1

to R4 shown below and note that {R1, R2, R3, R4} ⊆ DCbecause |I| > n

2 and the top-ranked alternatives of thesevoters is thus the Condorcet winner. Our goal is to show thatf(R4, x) = 1 because Lemma 6 implies then our intermedi-ate claim.R1: I: x, z, y, ∗ J : z, y, x, ∗ h: y, x, z, ∗R2: I: x, z, y, ∗ J : z, y, x, ∗ h: x, y, z, ∗R3: I: z, x, y, ∗ J : z, y, x, ∗ h: y, x, z, ∗R4: I: x, z, y, ∗ J : y, z, x, ∗ h: y, x, z, ∗First, observe that f satisfies Pareto-optimality because

of Lemma 5 and thus, f(R1, w) = 0 for all alternativesw ∈ A\{x, y, z}. Next, f(R1, z) = 0 because voter h couldmanipulate by deviating to R2 otherwise. In more detail,f(R2, x) = 1 because of k-unanimity and thus, localizednessfromR2 toR1 requires that f(R1, z) = 0. We can use a simi-lar argument betweenR1 andR3 to derive that f(R1, y) = 0.Note for this that f(R3, z) = 1 because of k-unanimity andthat we can derive R1 from R3 by letting the voters i ∈ Isequentially swap x and z. This process does not leave theCondorcet domain as either x or z are the Condorcet winnerin every intermediate profile. This is true because n is oddand |I| > n

2 implies that gR3(x,w) > 0 and gR3(z, w) > 0for all alternatives w ∈ A \ {x, z}. Moreover, no swap in-volves y and thus, we derive that f(R1, y) = f(R3, y) = 0because of localizedness. Hence, it follows that f(R1, x) = 1because f(R1, w) = 0 for all other alternatives w ∈ A \ {x}.Finally, we derive the profile R4 from R1 by swapping yand z in the preference relations of the voters i ∈ I . Becausethese swaps do not affect x, it stays the Condorcet winner andlocalizedness implies that f(R4, x) = f(R1, x) = 1. Finally,observe that R4 ∈ SxIy , which means that Lemma 6 appliesand shows that f(R, x) = 1 for all alternatives x, y ∈ A andpreference profiles R ∈ SxIy.

As next step, consider an arbitrary alternative x ∈ A and aprofileR ∈ D such that |NR

x | = n−k−1. It remains to showthat f(R, x) = 1 and we consider for this an auxiliary profileR′ which satisfies thatR′i = Ri for all voters i ∈ NR

x , and allvoters i ∈ N \NR

x report an alternative y ∈ A \ {x} as theirfavorite option and x as their least preferred option. It followsfrom the previous observations that f(R′, x) = 1 becauseR′ ∈ SxNR

x y. Starting at R′, we let each voter i ∈ N \NRx

use swaps to transform his preference relation into Ri. Inparticular, we never need to weaken x during these swapsas all voters in N \ NR

x report it as their least preferred al-ternative in R′. Hence, it follows from non-perversity andlocalizedness that f(R, x) = f(R′, x) = 1. Also, it shouldbe mentioned that all intermediate profiles are in the Con-dorcet domain because x is always top-ranked by all votersin NR

x . Finally, since the alternative x and the profile R arechosen arbitrarily, it follows that f satisfies k + 1-unanimity.Hence, the induction step is proven and thus, we derive thatf is n−1

2 -unanimous.

Note that the inductive argument in this proof breaks downfor k = n+1

2 because the profile R1 is not in the Condorcetdomain anymore. However, for larger domains, such as thefull domain, one can repeat this inductive argument until|I| = 1 to derive that there is no 1-unanimous and locallystrategyproof SDS on these domains.

Finally, we use the insights of this section to proveLemma 2.

Lemma 2. Assume n ≥ 3 is odd and m ≥ 3. A non-imposing and locally strategyproof SDS f on a connectedsuper Condorcet domain is locally 0-randomly dictatorial iffit is Condorcet-consistent.

Proof. Assume n ≥ 3 is odd and consider an arbitrary con-nected super Condorcet domain D. Moreover, let f denotea locally strategyproof and non-imposing SDS on D. If fis Condorcet-consistent, it is easy to see that it locally 0-randomly dictatorial because the best alternative of everyvoter can have probability 0. In more detail, it holds for allvoters i ∈ N and alternatives x, y ∈ A that f(R, y) = 0for all preference profiles R ∈ SxN\{i}y. Consequently, fcannot be represented as a mixture of a random dictatorshipand another locally strategyproof SDS because this repre-sentation implies that there is a voter whose best alternativealways gets positive probability. This shows that every lo-cally strategyproof and Condorcet-consistent SDS is locally0-randomly dictatorial.

For the inverse direction, suppose that f is locallystrategyproof, locally 0-randomly dictatorial, and non-imposing. Moreover, suppose for contradiction that f violatesCondorcet-consistency, i.e., that there is a profile R ∈ DCwith Condorcet winner c such that f(R, c) 6= 1. This meansthat there is another alternative d ∈ A\{c}with f(R, d) > 0.As next step, we let all voters i ∈ N with cRid swap up cuntil it is their best alternative to derive the profile R′. Sincewe only reinforce the Condorcet winner, we do not leave theCondorcet domain and f(R′, d) > 0 follows from localized-ness because no swap involves d. Consequently, f(R′, c) < 1even though c is top-ranked by more than half of the votersin R′. This is true as more than half of the voters prefer cto d in R because it is the Condorcet winner. However, thiscontradicts our previous insights: f satisfies all requirementsof Lemma 10 and Lemma 11, and thus it has to choose analternative with probability 1 if it is top-ranked by more thanhalf of the voters. Hence, we have derived a contradiction,which implies that the assumption that f fails Condorcet-consistency is wrong.


Proofs of Theorems 1 and 2We are almost ready to show Theorems 1 and 2. The last aux-iliary lemma required for proving these results is Lemma 1.Lemma 1. Assume n ≥ 3 is odd and m ≥ 3. An SDS f on aconnected super Condorcet domain is locally strategyproofand non-imposing iff it is a random dictatorship or there areγ ∈ [0, 1), a random dictatorship d, and a locally 0-randomlydictatorial, locally strategyproof, and non-imposing SDS gon D such that f = γd+ (1− γ)g.

Proof. Assume n ≥ 3 is odd and consider a connected superCondorcet domain D. We first show the direction from leftto right and let f therefore denote a locally strategyproofand non-imposing SDS on D. Using the definition of localγ-random dictatorships, there is a maximal γ ∈ [0, 1] suchthat f can be represented as f = γd + (1 − γ)g, whered is a random dictatorship and g is another locally strate-gyproof SDS on D. If γ = 1, this means that f is a randomdictatorship and thus, the lemma holds in this case. On theother hand, if γ < 1, g is locally 0-randomly dictatorialbecause of the maximality of γ; otherwise, we could rep-resent g as g = γ′d′ + (1 − γ′)h, where γ′ > 0, d′ is arandom dictatorship, and h is another locally strategyproofSDS onD. Consequently, f = γd+(1−γ)(γ′d′+(1−γ′)h)which means that f is γ + (1 − γ)γ′-randomly dictatorial.This contradicts the maximality of γ and thus, g must belocally 0-randomly dictatorial. Finally, observe that g inher-its non-imposition form f . In more detail, there is for everyalternative x ∈ A a profile R ∈ D such that f(R, x) = 1and it holds for these profiles also that g(R, x) = 1; other-wise, f(R, x) = γd(R, x) + (1 − γ)g(R, x) < 1 becauseg(R, x) < 1. Hence, if γ < 1, f can be indeed represent as amixture of a random dictatorship and a locally strategyproof,locally 0-randomly dictatorial, and non-imposing SDS on D.

For the other direction, note first that every random dic-tatorship is locally strategyproof on D because they satisfythis axiom even on the full domain, and these SDSs are non-imposing on D because they choose every alternative withprobability 1 if it is unanimously top-ranked. Hence, if anSDS f is random dictatorship on D, it satisfies all require-ments of this lemma. Therefore, suppose that the SDS f is amixture of a random dictatorship d and another locally strat-egyproof, locally 0-randomly dictatorial, and non-imposingSDS g, i.e., there is γ ∈ [0, 1) such that f = γd+ (1− γ)g.First note that f is locally strategyproof on D because mix-tures of locally strategyproof SDSs are also locally strate-gyproof. In more detail, consider two arbitrary preferenceprofiles R,R′ ∈ D for which there are two alternativesx, y ∈ A and a voter i ∈ N such that R′ = Ri:yx. Sinced and g are locally strategyproof, it follows that g(R) %ig(R′) and d(R) %i d(R′). This means equivalently thatg(R,U(Ri, x)) ≥ g(R′, U(Ri, x) and d(R,U(Ri, x)) ≥d(R′, U(Ri, x)) for all x ∈ A. Hence, it is straightfor-ward to verify that f(R,U(Ri, x)) = γd(R,U(Ri, x)) +(1 − γ)g(R,U(Ri, x)) ≥ γd(R′, U(Ri, x)) + (1 −γ)g(R′, U(Ri, x)) = f(R′, U(Ri, x)) for all x ∈ A, whichshows that f(R) %i f(R′). Moreover, f is non-imposingbecause both d and g are non-imposing and locally strate-gyproof. Hence, Lemma 5 implies that d and g choose an al-

ternative x with probability 1 if it is unanimously top-rankedand thus, f also chooses x with probability 1 in such profiles.This shows that every mixture of a random dictatorship anda locally strategyproof, locally 0-randomly dictatorial, andnon-imposing SDS on a connected super Condorcet domainis again locally strategyproof and non-imposing.

Next, we discuss the proof of Theorem 1, which followsalmost immediately from Lemma 1 and Lemma 2.

Theorem 1. Assume n is odd and m ≥ 3. An SDS on theCondorcet domain is strategyproof and non-imposing iff it isa mixture of a random dictatorship and the Condorcet rule.

Proof. Assume that the number of voters is odd and m ≥3. First, note that the theorem is trivial if n = 1 becauseevery single voter profile has a Condorcet winner. Therefore,Gibbard’s random dictatorship theorem shows that an SDSis strategyproof and non-imposing iff it always chooses thebest alternative of the single voter with probability 1. Sincethis SDS is equivalent to both the Condorcet rule and thedictatorial SDS of this voter, the theorem holds if n = 1.

Next, suppose that n ≥ 3. We first show that every strat-egyproof and non-imposing SDS on the Condorcet domainis a mixture of a random dictatorship and the Condorcetrule. For this, we characterize the Condorcet rule as the onlylocally strategyproof, locally 0-randomly dictatorial, and non-imposing SDS on the Condorcet domain. Hence, note thatthe Condorcet rule is non-imposing because an alternativeis chosen with probability 1 if it is the Condorcet winner,and it satisfies even full strategyproofness because a votercan only change the Condorcet winner to a less preferredalternative. In more detail, a voter i ∈ N can only changethe Condorcet winner c of a profile R ∈ DC to an alter-native c′ with cRic′ because gR(c, c′) > 0 and the sign ofthis majority margin flips if c′ is the Condorcet winner af-ter the manipulation. Hence, Lemma 2 shows that an SDSon the Condorcet domain is locally strategyproof, locally0-randomly dictatorial, and non-imposing iff it is the Con-dorcet rule because no other SDS on the Condorcet domainis Condorcet-consistent. Next, we can use Lemma 1 to derivethat mixtures of random dictatorships and the Condorcet ruleare the only locally strategyproof and non-imposing SDSs onthe Condorcet domain. Finally, this means that every strate-gyproof and non-imposing SDS on DC can be representedas a mixture of a random dictatorship and the Condorcet rulebecause strategyproofness implies local strategyproofness.

For the inverse direction, observe that mixtures of strate-gyproof SDSs are strategyproof. Since the Condorcet ruleand all random dictatorships are strategyproof on the Con-dorcet domain, it follows that mixtures of these SDSs are alsostrategyproof. Moreover, the Condorcet rule and all randomdictatorships choose an alternative with probability 1 if itis unanimously top-ranked. Hence, mixtures of these SDSsalso choose an alternative with probability 1 if every votertop-ranks it and therefore, these SDSs satisfy non-imposition.Thus, an SDS on the Condorcet domain is strategyproof andnon-imposing iff it is a mixture of a random dictatorship andthe Condorcet rule.


Similar to Theorem 1, we can also derive Theorem 2 fromLemma 1 and Lemma 2. In particular, these lemmas implyTheorem 2 if we show that there is no locally strategyproofand Condorcet-consistent SDS on a strict and connected superCondorcet domain because then Lemma 2 shows that there isalso no locally strategyproof, locally 0-randomly dictatorial,and non-imposing SDS on this domain.

Theorem 2. Assume n is odd and m ≥ 3. An SDS on a strictand connected super Condorcet domain is strategyproof andnon-imposing iff it is a random dictatorship.

Proof. Assume n is odd and m ≥ 3 and consider an arbi-trary connected and strict super Condorcet domain D. Notethat strictness only requires that D 6= DC because n isodd. Hence, there is no strict super Condorcet domain ifn = 1 because every single voter profile has a Condorcetwinner. Therefore, we suppose in the sequel that n ≥ 3.Next, it is straightforward that random dictatorships satisfynon-imposition and strategyproofness on D since they satisfythese axioms even on the full domain. Hence, we focus on theinverse direction: every strategyproof and non-imposing SDSon D is a random dictatorship. We prove this by showing thatthere is no locally strategyproof and Condorcet-consistentSDS. Then, Lemma 2 implies that there is no locally strate-gyproof, locally 0-randomly dictatorial, and non-imposingSDS, and Lemma 1 shows therefore that random dictatorshipsare the only locally strategyproof and non-imposing SDSs onD. Since strategyproofness implies local strategyproofness,this proves the theorem.

Hence, assume for contradiction that there is a locallystrategyproof and Condorcet-consistent SDS f : D → ∆(A).Since D is a strict super Condorcet domain, there is a profileR ∈ D \ DC . We show in the sequel that there is no feasibleoutcome forR. Note for this thatD is also connected and thus,there is for every alternative x ∈ A an ad-path π from R to aprofile R′ ∈ DC such that x is not swapped with any otheralternative on π. Consider this profileR′ for an arbitrary alter-native z ∈ A. Since R′ ∈ DC and f is Condorcet-consistent,it follows for the Condorcet winner c in R′ that f(R′, c) = 1.Moreover, c 6= z because z is not the Condorcet winner in Rand it is not swapped with any alternative in the constructionof R′. Hence, a repeated application of localizedness impliesthat f(R, z) = f(R′, z) = 0. Finally, since z was chosenarbitrarily, we derive that f(R, x) = 0 for all alternativesx ∈ A. This contradicts the definition of an SDS and thus,there is no locally strategyproof and Condorcet-consistentSDS on D.

Appendix B: Proof of Theorem 3The last result that we need to prove is Theorem 3. For show-ing this result, we proceed similar to to the proofs of Theo-rem 1 and Theorem 2: first, we investigate the consequencesof group-strategyproofness on super Condorcet domains andnext what it means to be non-dictatorial on these domains.Finally, we use these insights to characterize the set of group-strategyproof and non-imposing SDSs for the Condorcet do-main and every strict super Condorcet domain.

Group-strategyproofness on Super CondorcetDomainsAnalogous to the proofs of Theorem 1 and Theorem 2, wediscuss first the consequences of group-strategyproofness onsuper Condorcet domains. As first point, note that group-strategyproofness implies local strategyproofness and there-fore also non-perversity and localizedness. Furthermore, weshow next that every group-strategyproof and non-imposingSDS on a super Condorcet domain is Pareto-optimal.

Lemma 12. Assume m ≥ 3. Every non-imposing and group-strategyproof SDS on a super Condorcet domain is Pareto-optimal.

Proof. Consider an arbitrary super Condorcet domain D andlet f denote a group-strategyproof and non-imposing SDS onD. Moreover, suppose that f violates Pareto-optimality, i.e.,that there are a profile R ∈ D and two alternatives x, y ∈ Asuch that xRiy for all voters i ∈ N but f(R, y) > 0. First,note that there is a profile R′ ∈ D such that f(R, x) =1 because f is non-imposing. As next step, consider theprofile R∗ ∈ DC ⊆ D derived from R′ by letting all votersmake x to their best alternative. Group-strategyproofnessfrom R∗ to R′ implies that f(R∗, x) = 1 because otherwise,the voters could manipulate by reverting this step. However,this means that the set of all voters N can group-manipulateby deviating from R to R∗: it holds for all voters i ∈ N thatf(R,U(Ri, x)) < 1 = f(R∗, U(Ri, x)) because f(R, y) >0 and y 6∈ U(Ri, x). Hence, no voter prefers f(R) to f(R∗),i.e., f(R) 6%i f(R∗) for all i ∈ N , which means that this stepis indeed a successful group-manipulation. This contradictsthe assumption that f is group-strategyproof, which impliesthat f satisfies Pareto-optimality.

Note that Lemma 12 uses group-strategyproofness to gen-eralize Lemma 5: firstly, the new result holds also for evenn and secondly, it is not restricted to connected super Con-dorcet domains. As next step, we discuss a generalization ofLemma 6 to our new setting. Note that the proof of the newlemma can be easily derived from the proof of Lemma 6 andtherefore, we only explain the differences. In particular, notefor the subsequent lemma that SxIy ⊆ DC if |I| 6= n

2 .

Lemma 13. Assume n ≥ 3 and m ≥ 3 and let f denotea group-strategyproof and non-imposing SDS on a superCondorcet domain D. For all sets of voters I ⊆ N with|I| 6∈ {0, n2 , n}, alternatives w, x, y, z ∈ A with w 6= x andy 6= z, and preference profiles R ∈ SwIx, R′ ∈ SyIz , itholds that f(R,w) = f(R′, y), f(R, x) = f(R′, z), andf(R,w) + f(R, x) = 1.

Proof. We prove this result by pointing out how to changethe proof of Lemma 6. Hence, let D denote a super Con-dorcet domain and let f denote a group-strategyproof andnon-imposing SDS on D. Furthermore, consider a set of vot-ers I ⊆ N with |I| 6∈ {0, n2 , n}. Just as for Lemma 6, weprove this result in three steps. First, we show that for alldistinct alternatives x, y ∈ A that f(R, x) + f(R, y) = 1and f(R) = f(R′) for all profiles R,R′ ∈ SxIy. We canuse exactly the same argument as in Lemma 6 to prove this


claim because the applied construction does not change theCondorcet winner.

As second step, we show that f(R, x) = f(R′, z) forall distinct alternatives x, y, z ∈ A and preference profilesR ∈ SxIy and R′ ∈ SzIy. In particular, we prove this stepby considering the same profilesR3 andR4 as in the proof ofLemma 6. First, note thatR3 ∈ SxIy and thus, the insights ofthe first step imply that f(R3, w) = 0 for all w ∈ A \ {x, y}.Next, we let all voters in I simultaneously swap x and z toderive R4. The insights of the first step apply again and showthat f(R4, w) = 0 for all w ∈ A \ {y, z}. Also, since we letall voters in I change their preference relation simultaneously,we do not leave the Condorcet domain since I 6= n

2 . Finally,note that group-strategyproofness implies that f(R3, y) =f(R4, y): if f(R3, y) > f(R4, y), then the voters i ∈ Ican group-manipulate by deviating from R3 to R4 and iff(R3, y) < f(R4, y), these voters can group-manipulate bydeviating from R4 to R3. Combining all observations showsthat f(R4, z) = 1 − f(R4, y) = 1 − f(R3, y) = f(R3, x).Finally, using the insights of the first step, it follows thatf(R, x) = f(R′, z) for all R ∈ SxIy, R′ ∈ SzIy.R3: I: x, z, y, ∗ N\I: y, x, z, ∗R4: I: z, x, y, ∗ N\I: y, x, z, ∗As last step, we repeatedly use the insights of the last two

paragraphs to prove the lemma. Here, we can use again thesame argumentation as in Step 3 of the proof of Lemma 6because all considered profiles are in the Condorcet domain.

Non-dictatorial SDSs on Super Condorcet DomainsOur next goal is to investigate the consequences of what itmeans to be non-dictatorial and group-strategyproof on asuper Condorcet domain. In particular, we show Lemma 3in this section. As first step, we prove that every group-strategyproof, non-dictatorial, and non-imposing SDS ona super Condorcet domain is 1-unanimous.Lemma 14. Assume n ≥ 3 and m ≥ 3. Every group-strategyproof, non-dictatorial, and non-imposing SDS ona super Condorcet domain is 1-unanimous.

Proof. Assume that f is a group-strategyproof, non-dictatorial, and non-imposing SDS on a super Condorcetdomain D. Furthermore, suppose for contradiction that f isnot 1-unanimous. Since Lemma 12 shows that f satisfiesPareto-optimality, this means that there is a profile R∗ ∈ D,a voter i ∈ N , and two alternatives a, b ∈ A such that voteri reports a as his favorite alternative in R∗, all other votersreport b as their favorite alternative, but f(R∗, a) > 0. Inthe sequel, we show that this assumption means that f is thedictatorial SDS for voter i. This contradicts our assumptionsand therefore f has to satisfy 1-unanimity.

As first point, we show that f(R, x) = 1 for all alternativesx, y ∈ A and preference profiles R ∈ SyN\{i}x. Considerfor this the profiles R1 and R2 depicted in the sequel andnote that R1, R2 ∈ DC ⊆ D since all voters in N \{i} agreeon a best alternative in these profiles.

R1: i: x, y, z, ∗ N\{i}: z, y, x, ∗R2: i: y, x, z, ∗ N\{i}: y, z, x, ∗

We show in the sequel that f(R1, x) = 1 by contra-diction, i.e., we suppose that f(R1, x) < 1. First, ob-serve that f(R1, x) > 0 because Lemma 13 shows thatf(R1, x) = f(R∗, a) > 0 since R1 ∈ SzN\{i}x andR∗ ∈ SbN\{i}a. Moreover, this lemma also entails thatf(R1, w) = 0 for all alternatives w ∈ A \ {x, z}. Thus,f(R1, x) < 1 implies that f(R1, z) > 0. In particular, thismeans that f(R,U(Rj , y)) < 1 for all voters j ∈ N . Hence,the set of all voters N can group-manipulate by deviating toR2 because Lemma 12 implies that f(R2, y) = 1. Conse-quently, it indeed holds that f(R1, x) = 1 because otherwise,f is group-manipulable. Finally, Lemma 13 implies now thatf(R, x) = 1 for all alternatives x, y ∈ A and preferenceprofiles R ∈ SyN\{i}x.

As second step, we show that f always chooses voteri’s best alternative with probability 1. Hence, consider anarbitrary preference profile R ∈ D and let x denote voter i’sbest alternative in R. Furthermore, we choose an arbitraryalternative y ∈ A \ {x} and consider a profile R′ ∈ DC ⊆ Dsuch that R′i = Ri, and r(R′j , x) = m and r(R′j , y) = 1 forall voters j ∈ N \ {i}. Since R′ ∈ SyN\{i}x, it follows fromour previous observations that f(R′, x) = 1 even though allvoters in N \ {i} report x as their worst alternative. Thisimplies that f(R, x) = 1 because otherwise, the voters inN \ {i} can group-manipulate by deviating from R′ to R.This is true since x is the worst alternative of these voters inR′. Because the profile R ∈ D is chosen arbitrarily, f alwayschooses the best alternative of voter i with probability 1 andthus, it is the dictatorial SDS of this voter. This contradictsthe assumption that f is non-dictatorial, which implies that fis 1-unanimous.

Next, we use Lemma 14 to prove Lemma 3. Note thatwe use Lemma 11 in the proof because it is straightforwardto adapt this lemma to the case for even n if we use group-strategyproofness instead of strategyproofness.Lemma 3. Assume n ≥ 3 andm ≥ 3. A group-strategyproofand non-imposing SDS on a super Condorcet domain is non-dictatorial iff it is Condorcet-consistent.

Proof. Consider an arbitrary super Condorcet domain D forn ≥ 3 and suppose that f is a group-strategyproof and non-imposing SDS on D. First, note that if f is also Condorcet-consistent, it is obviously non-dictatorial. The reason forthis is that f(R, x) = 1 holds for every preference profileR ∈ DC in which only n − 1 voters prefer x the most.Consequently, no voter i ∈ N is a dictator for f as there is aprofile in which voter i’s best alternative gets probability 0.

For the inverse direction, suppose that f is group-strategyproof, non-imposing, and non-dictatorial. First, notethat Lemma 14 shows that f is 1-unanimous. Next, it isstraightforward to adapt the proof of Lemma 11 to derive thatf(R, x) = 1 for all preference profiles R ∈ DC and alterna-tives x ∈ A such that more than half of the voters report xas their favorite alternative in R. Based on this insight, it iseasy to show by contradiction that f is Condorcet-consistent:suppose that there is a profile R ∈ DC with Condorcetwinner c but f(R, c) 6= 1. Hence, there is an alternatived ∈ A \ {c} with f(R, d) > 0. This means in particular that


f(R,U(Ri, c)) < 1 for all voters i ∈ N with cRid. As aconsequence, these voters can group-manipulate by makingc into their best alternative. This step results in a profile R′in which more than half of the voters top-rank c and hence,our previous observation implies that f(R′, c) = 1. Thus,f(R,U(Ri, c)) < f(R′, U(Ri, c)) for all i ∈ N with cRid,which shows that f is group-manipulable. This means that theinitial assumption is wrong and every group-strategyproof,non-imposing, and non-dictatorial SDS on a super Condorcetdomain is Condorcet-consistent.

Proof of Theorem 3Finally, we use our previous insights to prove Theorem 3.

Theorem 3. Assume m ≥ 3. An SDS on the Condorcetdomain is group-strategyproof and non-imposing iff it is adictatorial SDS or the Condorcet rule. An SDS on a strictsuper Condorcet domain is group-strategyproof and non-imposing iff it is a dictatorial SDS.

Proof. The theorem consists of two independent characteri-zations of group-strategyproof and non-imposing SDSs forthe Condorcet domain and for strict super Condorcet domains,respectively. We prove these results separately.

Claim 1: An SDS on the Condorcet domain is group-strategyproof and non-imposing iff it is a dictatorial SDSor the Condorcet rule.

We first show the direction from left to right and thus,consider a group-strategyproof and non-imposing SDS f onthe Condorcet domain. We need to show that f is the Con-dorcet rule or a dictatorial SDS. First, note that if n ≥ 3, thisfollows immediately from Lemma 3 because the Condorcetrule is the only Condorcet-consistent SDS on the Condorcetdomain. Hence, this lemma shows that f is either dictato-rial if it is not Condorcet-consistent, or the Condorcet rule.Moreover, if n ≤ 2, our claim follows from the fact that f isPareto-optimal (see Lemma 12). In more detail, if n = 1, thebest alternative x of the single voter always Pareto-dominatesevery other alternative and thus, f(R, x) = 1. If n = 2, aprofile R is only in the Condorcet domain if the two votersagree on the best alternative x. Hence, alternative x Pareto-dominates every other alternative, which means again thatf(R, x) = 1. In both cases, the single Pareto-optimal alter-natives is also the Condorcet winner, which means that f isthe Condorcet rule (which is equivalent to every dictatorshipin this case).

For the other direction, we need to show that all dictatorialSDSs and the Condorcet rule are group-strategyproof andnon-imposing. First, note that all these SDSs are by definitionnon-imposing on the Condorcet domain because they areeven deterministic. Next, every dictatorial SDS di is group-strategyproof because only a single voter i can change theoutcome. However, this voter has no incentive to manipulateas his best alternative is already chosen with probability 1,which means that any other outcome makes this voter worse.Thus, every dictatorial SDS is group-strategyproof.

Finally, we show that the Condorcet rule is group-strategyproof. Assume for contradiction that this is not thecase, i.e., there are preference profiles R,R′ ∈ DC and a

set of voters I ⊆ N such that Rj = R′j for all j ∈ N \ Iand COND(R) 6%i COND(R′) for all voters i ∈ I . If theCondorcet winner in R and R′ is the same, this is clearlyno manipulation since COND(R) = COND(R′). Hence,assume that the Condorcet winner c of R is not the samealternative as the Condorcet winner c′ of R′. This meansthat gR(c, c′) > 0 and gR′(c′, c) > 0, which is only possibleif there is a manipulator i ∈ I with cRic′. However, thisvoter prefers COND(R) to COND(R′), contradicting thatCOND(R) 6%j COND(R′) for all j ∈ I . Thus, the Con-dorcet rule is group-strategyproof on the Condorcet domain.

Claim 2: An SDS on a strict super Condorcet domain isgroup-strategyproof and non-imposing iff it is dictatorial.

First, note that there is not strict super Condorcet domain ifn ≤ 2. In more detail, if n = 1, every profile has a Condorcetwinner, which means that there is no super Condorcet domainwithD 6= DC . Furthermore, if n = 2, the best alternative x ofthe first voter satisfies in every preference profileR ∈ R2 thatgR(x, y) ≥ 0 for all y ∈ A \ {x} and thus, there cannot be aprofile in which every alternative strictly loses the majoritycomparison to another alternative. Hence, we suppose in thesequel that n ≥ 3 and consider an arbitrary strict super Con-dorcet domainD. Observe first that the same arguments as forClaim 1 show that every dictatorship is group-strategyproofand non-imposing onD. Thus, we focus on proving that everygroup-strategyproof and non-imposing SDS on D is dictato-rial. For this, we show that there is no group-strategyproof,non-imposing, and Condorcet-consistent SDS on D becauseLemma 3 shows then that only dictatorial SDSs can be group-strategyproof and non-imposing on D.

Therefore, suppose for contradiction that f is a group-strategyproof and Condorcet-consistent SDS on D. Subse-quently, we use the fact that D is a strict super Condorcetdomain: there is profile R ∈ D such that every alternativex ∈ A strictly loses the pairwise comparison against an-other alternative y ∈ A \ {x}, i.e., such that gR(y, x) > 0.Next, consider an alternative x ∈ A such that f(R, x) > 0;such an alternative must exist because

∑x∈A f(R, x) = 1.

We show that the voters I = {i ∈ N : yRix} can group-manipulate by reporting y as their favorite alternative. Inparticular, this manipulation leads to a profile R′ ∈ DCsince gR(y, x) > 0 implies that |I| > n

2 . In more de-tail, y is the Condorcet winner in R′ and thus, Condorcet-consistency requires that f(R, y) = 1. However, it holdsthat f(R,U(Ri, y)) < 1 = f(R′, U(Ri, y)) for all i ∈ Ibecause x 6∈ U(Ri, y) and f(R, x) > 0. Hence, this stepis indeed a group-manipulation for the voters i ∈ I , contra-dicting that f is group-strategyproof. Therefore, there is nogroup-strategyproof and Condorcet-consistent SDS onD andconsequently, Lemma 3 entails that only dictatorial SDSs cansatisfy group-strategyproofness and non-imposition on strictsuper Condorcet domains.

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