Relaxed Notions of Condorcet-Consistency and Efficiency for Strategyproof Social Decision Schemes

Social decision schemes (SDSs) map the preferences of a group of voters over some set of $m$ alternatives to a probability distribution over the alternatives. A seminal characterization of strategyproof SDSs by Gibbard implies that there are no strategyproof Condorcet extensions and that only random dictatorships satisfy ex post efficiency and strategyproofness. The latter is known as the random dictatorship theorem. We relax Condorcet-consistency and ex post efficiency by introducing a lower bound on the probability of Condorcet winners and an upper bound on the probability of Pareto-dominated alternatives, respectively. We then show that the SDS that assigns probabilities proportional to Copeland scores is the only anonymous, neutral, and strategyproof SDS that can guarantee the Condorcet winner a probability of at least 2/m. Moreover, no strategyproof SDS can exceed this bound, even when dropping anonymity and neutrality. Secondly, we prove a continuous strengthening of Gibbard's random dictatorship theorem: the less probability we put on Pareto-dominated alternatives, the closer to a random dictatorship is the resulting SDS. Finally, we show that the only anonymous, neutral, and strategyproof SDSs that maximize the probability of Condorcet winners while minimizing the probability of Pareto-dominated alternatives are mixtures of the uniform random dictatorship and the randomized Copeland rule.


INTRODUCTION
Multi-agent systems are often faced with problems of collective decision making: how to find a group decision given the preferences of multiple individual agents.These problems, which have been traditionally studied by economists and mathematicians, are of increasing interest to computer scientists who employ the formalisms of social choice theory to analyze computational multiagent systems [see, e.g., 8,9,25,29].
A pervasive phenomenon in collective decision making is strategic manipulation: voters may be better off by lying about their preferences than reporting them truthfully.This is problematic since all desirable properties of a voting rule are in doubt when voters act dishonestly.Thus, it is important that voting rules incentivize voters to report their true preferences.Unfortunately, Gibbard [18] and Satterthwaite [27] have shown independently that dictatorships are the only non-imposing voting rules that are immune to strategic manipulations.However, these voting rules are unacceptable for most applications because they invariably return the most preferred alternative of a fixed voter.A natural question is whether more positive results can be obtained when allowing for randomization.Gibbard [19] hence introduced social decision schemes (SDSs), which map the preferences of the voters to a lottery over the alternatives and defined SDSs to be strategyproof if no voter can obtain more expected utility for any utility representation that is consistent with his ordinal preference relation.He then gave a complete characterization of strategyproof SDSs in terms of convex combinations of two types of restricted SDSs, so-called unilaterals and duples.An important consequence of this result is the random dictatorship theorem: random dictatorships are the only ex post efficient and strategyproof SDSs.Random dictatorships are convex combinations of dictatorships, i.e., each voter is selected with some fixed probability and the top choice of the chosen voter is returned.In contrast to deterministic dictatorships, the uniform random dictatorship, in which every agent is picked with the same probability, enjoys a high degree of fairness and is in fact used in many subdomains of social choice [see, e.g., 1,11].As a consequence of these observations, Gibbard's theorem has been the point of departure for a lot of follow-up work.In addition to several alternative proofs of the theorem [e.g., 13,23,30], there have been extensions with respect to manipulations by groups [4], cardinal preferences [e.g., 15,22,24], weaker notions of strategyproofness [e.g., 2, 5, 7, 28], and restricted domains of preference [e.g., 10,14].
Random dictatorships suffer from the disadvantage that they do not allow for compromise.For instance, suppose that voters strongly disagree on the best alternative, but have a common second best alternative.In such a scenario, it seems reasonable to choose the second best alternative but random dictatorships do not allow for this compromise.On a formal level, this observation is related to the fact that random dictatorships violate Condorcetconsistency, which demands that an alternatives that beats all other alternatives in pairwise majority comparisons should be selected.Motivated by this observation, we analyze the limitations of strategyproof SDSs by relaxing two classic conditions: Condorcetconsistency and ex post efficiency.To this end, we say that an SDS is -Condorcet-consistent if a Condorcet winner always receives a probability of at least and -ex post efficient if a Pareto-dominated alternative always receives a probability of at most .Moreover, we say a strategyproof SDS is -randomly dictatorial if it can be represented as a convex combination of two strategyproof SDSs, one of which is a random dictatorship that will be selected with probability .All of these axioms are discussed in more detail in Section 2.2.
Building on an alternative characterization of strategyproof SDSs by Barberà [3], we then show the following results ( is the number of alternatives and the number of voters): • Let , ≥ 3.There is no strategyproof SDS that satisfies -Condorcet-consistency for > 2 / .Moreover, the randomized Copeland rule, which assigns probabilities proportional to Copeland scores, is the only strategyproof SDS that satisfies anonymity, neutrality, and 2 / -Condorcet-consistency.
• Let 0 ≤ ≤ 1 and ≥ 3. Every strategyproof SDS that is1− -ex post efficient is -randomly dictatorial for ≥ .If we require additionally anonymity, neutrality, and ≥ 4, then only mixtures of the uniform random dictatorship and the uniform lottery satisfy this bound tightly.
• Let ≥ 4 and ≥ 5.No strategyproof SDS that is -Condorcet-consistent is -ex post efficient for < −2 −1 .If we additionally require anonymity and neutrality, then only mixtures of the uniform random dictatorship and the randomized Copeland rule satisfy = −2 −1 .
The first statement characterizes the randomized Copeland rule as the "most Condorcet-consistent" SDS that satisfies strategyproofness, anonymity, and neutrality.In fact, no strategyproof SDS can guarantee more than 2 / probability to the Condorcet winner, even when dropping anonymity and neutrality.The second point can be interpreted as a continuous strengthening of Gibbard's random dictatorship theorem: the less probability we put on Pareto-dominated alternatives, the more randomly dictatorial is the resulting SDS.In particular, this theorem indicates that we cannot find appealing strategyproof SDSs by allowing that Paretodominated alternatives gain a small probability since the resulting SDS will be very similar to random dictatorships.The last statement identifies a tradeoff between -Condorcet-consistency and -ex post efficiency: the more probability a strategyproof SDS guarantees to the Condorcet winner, the less efficient it is.Thus, we can either only maximize the -Condorcet-consistency or the -ex post efficiency of a strategyproof SDS, which again highlights the central roles of the randomized Copeland rule and random dictatorships.

THE MODEL
Let = {1, 2, . . ., } be a finite set of voters and let = { , , . . .} be a finite set of alternatives.Every voter has a preference relation ≻ , which is an anti-symmetric, complete, and transitive binary relation on .We write ≻ if voter prefers strictly to and if ≻ or = .The set of all preference relations is denoted by R. A preference profile ∈ R contains the preference relation of each voter ∈ .We define the supporting size for against in the preference profile by ( ) = |{ ∈ : ≻ }|.
Given a preference profile, we are interested in the winning chance of each alternative.We therefore analyze social decision schemes (SDSs), which map each preference profile to a lottery over the alternatives.A lottery is a probability distribution over the set of alternatives , i.e., it assigns each alternative a probability ( ) ≥ 0 such that ∈ ( ) = 1.The set of all lotteries over is denoted by Δ( ).Formally, a social decision scheme (SDS) is a function : R → Δ( ).We denote with ( , ) the probability assigned to alternative by for the preference profile .
Since there is a huge number of SDSs, we now discuss axioms formalizing desirable properties of these functions.Two basic fairness conditions are anonymity and neutrality.Anonymity requires that voters are treated equally.Formally, an SDS is anonymous if ( ) = ( ( )) for all preference profiles and permutations : → .Here, ′ = ( ) denotes the profile with ≻ ′ ( ) = ≻ for all voters ∈ .Neutrality guarantees that alternatives are treated equally and formally requires for an SDS that ( , ) = ( ( ), ( )) for all preference profiles and permutations : → .This time, ′ = ( ) is the profile derived by permuting the alternatives in according to , i.e, ( ) ≻ ′ ( ) if and only if ≻ for all alternatives , ∈ and voters ∈ .

Stochastic Dominance and Strategyproofness
This paper is concerned with strategyproof SDSs, i.e., social decision schemes in which voters cannot benefit by lying about their preferences.In order to make this formally precise, we need to specify how voters compare lotteries.To this end, we leverage the well-known notion of stochastic dominance: a voter (weakly) prefers a lottery to another lottery , written as , if ( ) for every alternative ∈ .Less formally, a voter prefers a lottery weakly to a lottery if, for every alternative ∈ , returns a better alternative than with as least as much probability as .Stochastic dominance does not induce a complete order on the set of lotteries, i.e., there are lotteries and such that a voter neither prefers to nor to .
Based on stochastic dominance, we can now formalize strategyproofness.An SDS is strategyproof if ( ) ( ′ ) for all preference profiles and ′ and voters ∈ such that ≻ = ≻ ′ for all ∈ \ { }.Less formally, strategyproofness requires that every voter prefers the lottery obtained by voting truthfully to any lottery that he could obtain by voting dishonestly.Conversely, we call an SDS manipulable if it is not strategyproof.While there are other ways to compare lotteries with each other, stochastic dominance is the most common one [see, e.g, 2,3,6,16,19].This is mainly due to the fact that implies that the expected utility of is at least as high as the expected utility of for every vNM utility function that is ordinally consistent with voter 's preferences.Hence, if an SDS is strategyproof, no voter can manipulate regardless of his exact utility function [see, e.g., 7,28].This observation immediately implies that the convex combination ℎ = + (1 − ) (for some ∈ [0, 1]) of two strategyproof SDSs and is again strategyproof: a manipulator who obtains more expected utility from ℎ( ′ ) than ℎ( ) prefers ( ′ ) to ( ) or ( ′ ) to ( ).
Gibbard [19] shows that every strategyproof SDS can be represented as convex combinations of unilaterals and duples. 1 The terms "unilaterals" and "duples" refer here to special classes of SDSs: a unilateral is a strategyproof SDS that only depends on the preferences of a single voter , i.e., ( ) = ( ′ ) for all preference profiles and ′ such that ≻ = ≻ ′ .A duple, on other hand, is a strategyproof SDS that only chooses between two alternatives and , i.e., ( , ) = 0 for all preference profiles and alternatives ∈ \ { , }.
Theorem 1 (Gibbard [19]).An SDS is strategyproof if and only if it can be represented as a convex combination of unilaterals and duples.
Since duples and unilaterals are by definition strategyproof, Theorem 1 only states that strategyproof SDSs can be decomposed into a mixture of strategyproof SDSs, each of which must be of a special type.In order to circumvent this restriction, Gibbard proves another characterization of strategyproof SDSs.
Theorem 2 (Gibbard [19]).An SDS is strategyproof if and only if it is non-perverse and localized.
Non-perversity and localizedness are two axioms describing the behavior of an SDS.For defining these axioms, we denote with : the profile derived from by only reinforcing against in voter 's preference relation.Note that this requires that ≻ and that there is no alternative ∈ such that ≻ ≻ .Then, an SDS is non-perverse if ( : , ) ≥ ( , ) for all preference profiles , voters ∈ , and alternatives , ∈ .Moreover, an SDS is localized if ( : , ) = ( , ) for all preference profiles , voters ∈ , and distinct alternatives , , ∈ .Intuitively, non-perversity-which is now often referred to as monotonicityrequires that the probability of an alternative only increases if it is reinforced, and localizedness that the probability of an alternative does not depend on the order of the other alternatives.Together, Theorem 1 and Theorem 2 show that each strategyproof SDS can be represented as a mixture of unilaterals and duples, each of which is non-perverse and localized.Since Gibbard's results can be quite difficult to work with, we now state another characterization of strategyproof SDSs due to Barberà [3].This author has shown that every strategyproof SDS that satisfies anonymity and neutrality can be represented as a convex combination of a supporting size SDS and a point voting SDS.A point voting SDS is defined by a scoring vector ∈ \{ } ( ) .Note that point voting SDSs can be seen as a generalization of (deterministic) positional scoring rules and supporting size SDSs can be seen as a variant of Fishburn's C2 functions [17].
Theorem 3 (Barberà [3]).An SDS is anonymous, neutral, and strategyproof if and only if it can be represented as a convex combination of a point voting SDS and a supporting size SDS.
Many well-known SDSs can be represented as point voting SDSs or supporting size SDSs.For example, the uniform random dictatorship RD , which chooses one voter uniformly at random and returns his best alternative, is the point voting SDS defined by the scoring vector 1 , 0, . . ., 0 .An instance of a supporting size SDS is the randomized Copeland rule , which assigns probabilities proportional to the Copeland scores ( , This SDS is the supporting size SDS defined by the vector = ( , −1 , . . ., 0 ), where = , and = 0 otherwise.Furthermore, there are SDSs that can be represented both as point voting SDSs and supporting size SDSs.An example is the randomized Borda rule , which randomizes proportional to the Borda scores of the alternatives.This SDS is the point voting SDS defined by the vector 2( −1) ( −1) , 2( −2) ( −1) , • • • ,

Relaxing Classic Axioms
The goal of this paper is to identify attractive strategyproof SDSs other than random dictatorships by relaxing classic axioms from social choice theory.In more detail, we investigate how much probability can be guaranteed to Condorcet winners and how little probability must be assigned to Pareto-dominated alternatives by strategyproof SDSs.In the following we formalize these ideas using -Condorcet-consistency and -ex post efficiency.
Let us first consider -ex post efficiency, which is based on Pareto-dominance.An alternative Pareto-dominates another alternative in a preference profile if ≻ for all ∈ .The standard notion of ex post efficiency then formalizes that Pareto-dominated alternatives should have no winning chance, i.e., ( , ) = 0 for all preference profiles and alternatives that are Pareto-dominated in .As first shown by Gibbard, random dictatorships are the only strategyproof SDSs that satisfy ex post efficiency.These SDSs choose each voter with a fixed probability and return his best alternative as winner.However, this result breaks down once we allow that Pareto-dominated alternatives can have a non-zero chance of winning > 0. For illustrating this point, consider a random dictatorship and another strategyproof SDS .Then, the SDS * = (1− ) + is strategyproof for every ∈ (0, 1] and no random dictatorship, but assigns a probability of at most to Pareto-dominated alternatives.We call the last property -ex post efficiency: an SDS is -ex post efficient if ( , ) ≤ for all preference profiles and alternatives that are Pareto-dominated in .
A natural generalization of the random dictatorship theorem is to ask which strategyproof SDSs satisfy -ex post efficiency for small values of .If is sufficiently small, -ex post efficiency may be quite acceptable.As we show, the random dictatorship theorem is quite robust in the sense that all SDSs that satisfy -ex post efficiency for < 1 are similar to random dictatorships.In order to formalize this observation, we introduce -randomly dictatorial SDSs: a strategyproof SDS is -randomly dictatorial if ∈ [0, 1] is the maximal value such that can be represented as = + (1− ) , where is a random dictatorship and is another strategyproof SDS.In particular, we require that is strategyproof as otherwise, SDSs that seem "non-randomly dictatorial" are not 0randomly dictatorial.For instance, the uniform lottery , which always assigns probability 1 to all alternatives, is not 0-randomly dictatorial if is not required to be strategyproof because it can be represented as = 1 + −1 , where is the dictatorial SDS of voter and is the SDS that randomizes uniformly over all alternatives but voter 's favorite one.Moreover, it should be mentioned that the maximality of implies that is 0-randomly dictatorial if < 1.Otherwise, we could also represent as a mixture of a random dictatorship and some other strategyproof SDS ℎ, which means that is ′ -randomly dictatorial for ′ > .Due to the symmetry of ′ , we may assume without loss of generality that ( ′ , ) > 0. Since is Condorcet-consistent, it holds that ( , ) = 1.Thus, voter 1 can manipulate by swapping and in .
For a better understanding of -randomly dictatorial SDSs, we provide next a characterization of these SDSs.Recall for the following lemma that : denotes the profile derived from by only reinforcing against in voter 's preference relation.Lemma 1.A strategyproof SDS is -randomly dictatorial if and only if there are non-negative values 1 , . . ., such that: i) ii) ( : , ) − ( , ) ≥ for all alternatives , ∈ , voters ∈ , and preference profiles in which voter prefers the most and the second most.
iii) for every voter ∈ , there are alternatives , ∈ and a profile such that voter prefers the most and the second most in , and ( : , ) − ( , ) = .
The proof of this lemma can be found in the appendix.Lemma 1 gives an intuitive interpretation of -randomly dictatorial SDSs: this axiom only requires that there are voters who always increase the winning probability of an alternative by at least if they reinforce it to the first place.Hence, for small values of , this axiom is desirable as it only formulates a variant of strict monotonicity.However, for larger values of , -randomly dictatorial SDSs become more similar to random dictatorships.Furthermore, the proof of Lemma 1 shows that the decomposition of -randomly dictatorial SDSs is completely determined by the values 1 , . . ., : given these values for an strategyproof SDS , it can be represented as , where is a strategyproof SDS and the dictatorial SDS of voter .
Finally, we introduce -Condorcet-consistency.To this end, we first define the notion of a Condorcet winner.A Condorcet winner is an alternative that wins every majority comparison according to preference profile , i.e., ( ) > ( ) for all ∈ \ { }.Condorcet-consistency demands that ( , ) = 1 for all preference profiles and alternatives such that is the Condorcet winner in .Unfortunately, Condorcet-consistency is in conflict with strategyproofness, which can easily be derived from Gibbard's random dictatorship theorem.A simple two-profile proof for this fact when = = 3 is given in Figure 1.To circumvent this impossibility, we relax Condorcet-consistency: instead of requiring that the Condorcet winner always obtains probability 1, we only require that it receives a probability of at least .This idea leads to -Condorcetconsistency: an SDS satisfies this axiom if ( , ) ≥ for all profiles and alternatives ∈ such that is the Condorcet Table 1: Values of , , and for which specific SDSs are -Condorcet-consistent, -ex post efficient, and -randomly dictatorial.Each row shows the values of , , and for which a specific SDS satisfies the corresponding axioms.RD abbreviates the uniform random dictatorship, the uniform lottery, the randomized Borda rule, and the randomized Copeland rule.

SDS
-Condorcet -consistency -ex post efficiency -random dictatorship 0 winner in .For small values of , this axiom is clearly compatible with strategyproofness and therefore, we are interested in the maximum value of such that there are -Condorcet-consistent and strategyproof SDSs.
For a better understanding of -Condorcet-consistency, -ex post efficiency, and -random dictatorships, we discuss some of the values in Table 1 as examples.The uniform random dictatorship is 1-randomly dictatorial and 0-ex post efficient by definition.Moreover, it is 0-Condorcet-consistent because a Condorcet winner may not be top-ranked by any voter.The randomized Borda rule is 2( −2) ( −1) -ex post efficient because it assigns this probability to an alternative that is second-ranked by every voter.Moreover, it is 2 ( −1) -randomly dictatorial as we can represent it as 2 , where RD is the uniform random dictatorship and is the point voting SDS defined by the scoring vector 2( −2) ( ( −1)−2) , 2( −2) ( ( −1)−2) , 2( −3) ( ( −1)−2) , . . ., 0 .Finally, the randomized Copeland rule is 0-randomly dictatorial because there is for every voter a profile in which he can swap his two best alternatives without affecting the outcome.Moreover, it is 2 -Condorcet-consistent because a Condorcet winner satisfies that ( ) > 2 for all ∈ \ { } and hence, ( , Note that Table 1 also contains a row corresponding to the uniform lottery.We consider this SDS as a threshold with respect to -Condorcet-consistency and -ex post efficiency because we can compute the uniform lottery without knowledge about the voters' preferences.Hence, if an SDS performs worse than the uniform lottery with respect to -Condorcet-consistency or -ex post efficiency, we could also dismiss the voters' preferences.

RESULTS
In this section, we present our results about the -Condorcetconsistency and the -ex post efficiency of strategyproof SDSs.First, we prove that no strategyproof SDS satisfies -Condorcetconsistency for > 2 and that the randomized Copeland rule is the only anonymous, neutral, and strategyproof SDS that satisfies -Condorcet-consistency for = 2 .Moreover, we show that every 1− -ex post efficient and strategyproof SDS is -randomly dictatorial for ≥ .This statement can be seen as a continuous generalization of the random dictatorship theorem and implies, for instance, that every 0-randomly dictatorial and strategyproof SDS can only satisfy -ex post efficiency for ≥ 1 , i.e., such SDSs are at least as inefficient as the uniform lottery.Even more, when additionally imposing anonymity and neutrality, we prove that only mixtures of the uniform random dictatorship and the uniform lottery satisfy this bound tightly, which shows that relaxing ex post efficiency does not allow for appealing SDSs.In the last theorem, we identify a tradeoff between Condorcet-consistency and ex post efficiency: no strategyproof SDS that satisfies -Condorcet consistency is -ex post efficient for < −2 −1 .We derive these results through a series of lemmas.The proofs of all lemmas and Theorem 5 are deferred to the appendix and we only present short proof sketches instead.

-Condorcet-consistency
As discussed in Section 2.2, Condorcet-consistent SDSs violate strategyproofness.Therefore, we analyze the maximal such that -Condorcet-consistency and strategyproofness are compatible.Our results show that strategyproofness only allows for a small degree of Condorcet-consistency: we prove that no strategyproof SDS satisfies -Condorcet-consistency for > 2 .This bound is tight as the randomized Copeland rule is 2 -Condorcetconsistent, which means that it is one of the "most Condorcetconsistent" strategyproof SDSs.Even more, we can turn this observation in a characterization of by additionally requiring anonymity and neutrality: the randomized Copeland rule is the only strategyproof SDS that satisfies 2 -Condorcet-consistency, anonymity, and neutrality.
For proving these results, we derive next a number of lemmas.As first step, we show in Lemma 2 that we can use a strategyproof and -Condorcet-consistent SDS to construct another strategyproof SDS that satisfies anonymity, neutrality, and -Condorcet-consistency for the same .
The central idea in the proof of Lemma 2 is the following: if there is a strategyproof and -Condorcet-consistent SDS , then the SDS ( , ) = ( ( ( )), ( )) is also strategyproof and -Condorcet-consistent for all permutations : → and : → .Since mixtures of strategyproof and -Condorcet-consistent SDSs are also strategyproof and -Condorcet-consistent, we can therefore construct an SDS that satisfies all requirements of the lemma by averaging over all permutations on and .More formally, the SDS * = 1 !!∈Π ∈T (where Π denotes the set of all permutations on and T the set of all permutations on ) meets all criteria of the lemma.Due to Lemma 2, we investigate next the -Condorcetconsistency of strategyproof SDSs that satisfy anonymity and neutrality.The reason for this is that this lemma turns an upper bound on for these SDSs into an upper bound for all strategyproof SDSs.Since Theorem 3 shows that every strategyproof, anonymous, and neutral SDS can be decomposed in a point voting SDS and a supporting size SDS, we investigate these two classes separately in the following two lemmas.First, we bound the -Condorcetconsistency of point voting SDSs.
The proof of this lemma relies on the observation that there can be ⌈ 2 ⌉ Condorcet winner candidates, i.e., alternatives that can be made into the Condorcet winner by keeping at the same position in the preferences of every voter and only reordering the other alternatives.Since reordering the other alternatives does not affect the probability of in a point voting SDS, it follows that every Condorcet winner candidate has a probability of at least .Hence, we derive that ≤ 1 ⌈ 2 ⌉ ≤ 2 and a slightly more involved argument shows that the inequality is strict.
The last ingredient for the proof of Theorem 4 is that no supporting size SDS can assign a probability of more than 2 to any alternative.This immediately implies that no supporting size SDS satisfies -Condorcet-consistency for > 2 .
Lemma 4. No supporting size SDS can assign more than 2 probability to an alternative.
The proof of this lemma follows straightforwardly from the definition of supporting size SDSs.Each such SDS is defined by a scoring vector ( , . . ., 0 ) such that + − = 2 ( −1) for all ∈ {0, . . ., } and The probability of an alternative in a supporting size SDS is therefore bounded by Finally, we have all necessary lemmas for the proof of our first theorem.

P
. The theorem consists of two claims: the characterization of the randomized Condorcet rule and the fact that no other strategyproof SDS can attain -Condorcet-consistency for a larger than .We prove these claims separately.
The randomized Copeland rule is a supporting size SDS and satisfies therefore anonymity, neutrality, and strategyproofness.Furthermore, it satisfies also 2 -Condorcet-consistency because a Condorcet winner wins every pairwise majority comparison in .Hence, Next, let be an SDS satisfying anonymity, neutrality, strategyproofness, and 2 -Condorcet-consistency.We show that is the randomized Copeland rule.Since is anonymous, neutral, and strategyproof, we can apply Theorem 3 to represent as = point + (1 − ) sup , where ∈ [0, 1], point is a point voting SDS, and sup is a supporting size SDS.Lemma 3 states that there is a profile with Condorcet winner such that point ( , ) < 2 , and it follows from Lemma 4 that sup ( , ) ≤ 2 .Hence, ( , ) = point ( , ) + sup ( , ) < 2 if > 0. Therefore, is a supporting size SDS as it satisfies 2 -Condorcet-consistency.
Next, we show that has the same scoring vector as the randomized Copeland rule.Since is a supporting size SDS, there is a scoring vector = ( , . . ., 0 ) with is the Condorcet winner in because of 2 -Condorcet-consistency and Lemma 4. We derive from the definition of supporting size SDSs that the Condorcet winner can only achieve this probability if for every other alternatives ∈ \ { }.Moreover, observe that the Condorcet winner needs to win every majority comparison but is indifferent about the exact supporting sizes.Hence, it follows that = 2 ( −1) for all > 2 as otherwise, there is a profile in which the Condorcet winner does not receive a probability of 2 .We also know that is required by the definition of supporting size SDSs as 2 = − 2 .Hence, the scoring vector of is equivalent to the scoring vector of the randomized Copeland rule, which proves that is .
The claim is trivially true if ≤ 2 because -Condorcet consistency for > 1 is impossible.Hence, let denote a strategyproof SDS for ≥ 3 alternatives.We show in the sequel that cannot satisfy -Condorcet-consistency for > 2 .As a first step, we use Lemma 2 to construct a strategyproof SDS * that satisfies anonymity, neutrality, and -Condorcet-consistency for the same as .Since * is anonymous, neutral, and strategyproof, it follows from Theorem 3 that * can be represented as a mixture of a point voting SDS point and a supporting size SDS sup , i.e., Next, we consider point and sup separately.Lemma 3 implies for point that there is a profile with a Condorcet winner such that point ( , ) < 2 .Moreover, Lemma 4 shows that sup ( , ) ≤ 2 because supporting size SDSs never return a larger probability than 2 .Thus, we derive the following inequality, which shows that * fails -Condorcet-consistency for > 2 .Hence, no strategyproof SDS satisfies -Condorcet-consistency for > 2 when ≥ 3.
Remark 1. Lemma 2 can be applied to properties other than -Condorcet-consistency, too.For example, given a strategyproof and -ex post efficient SDS, we can construct another SDS that satisfies these axioms as well as anonymity and neutrality.
Remark 2. All axioms in the characterization of the randomized Copeland rule are independent of each other.The SDS that picks the Condorcet winner with probability 2 if one exists and distributes the remaining probability uniformly between the other alternatives only violates strategyproofness.The randomized Borda rule satisfies all axioms of Theorem 4 but 2 -Condorcetconsistency.An SDS that satisfies anonymity, strategyproofness, and 2 -Condorcet-consistency can be defined based on an arbitrary order of alternatives 0 , . . ., −1 .Then, we pick an index ∈ {0, . . ., − 1} uniformly at random and return the winner of the majority comparison between and +1 mod (if there is a majority tie, a fair coin toss decides the winner).Finally, we can use the randomized Copeland rule to construct an SDS that fails only anonymity for even : we just ignore one voter when computing the outcome of .If is even and is the Condorcet winner in , then ( ) ≥ +2 2 for all ∈ \ { }.Hence, the Condorcet winner remains a Condorcet winner after removing a single voter, which means that this SDS only fails anonymity.
Moreover, the impossibility in Theorem 4 does not hold when there are only = 2 voters because random dictatorships are strategyproof and Condorcet-consistent in this case.The reason for this is that a Condorcet winner needs to be the most preferred alternative of both voters and is therefore chosen with probability 1.
Remark 3. The randomized Copeland rule has multiple appealing interpretations.Firstly, it can be defined as a supporting size SDS as shown in Section 2.1.Alternatively, it can be defined as the SDS that picks two alternatives uniformly at random and then picks the majority winner between them; majority ties are broken by a fair coin toss.Next, Theorem 4 shows that the randomized Copeland rule is the SDS that maximizes the value of for -Condorcet-consistency among all anonymous, neutral, and strategyproof SDSs.Finally, the randomized Copeland rule is the only strategyproof SDS that satisfies anonymity, neutrality, and assigns 0 probability to a Condorcet loser whenever it exists.

-ex post Efficiency
According to Gibbard's random dictatorship theorem, random dictatorships are the only strategyproof SDSs that satisfy ex post efficiency.In this section, we show that this result is rather robust by identifying a tradeoff between -ex post efficiency and -random dictatorships.More formally, we prove that for every ∈ [0, 1], all strategyproof and 1− -ex post efficient SDSs are -randomly dictatorial for ≥ .If we set = 1, we obtain the random dictatorship theorem.On the other hand, we derive from this theorem that every 0-randomly dictatorial and strategyproof SDS is -ex post efficient for ≥ 1 , i.e., every such SDS is at least as inefficient as the uniform lottery.Moreover, we prove for every ∈ [0, 1] that mixtures of the uniform random dictatorship and the uniform lottery are the only -randomly dictatorial SDSs that satisfy anonymity, neutrality, strategyproofness, and 1− -ex post efficiency.In summary, these results demonstrate that relaxing ex post efficiency does not lead to particularly appealing strategyproof SDSs.Furthermore, we also identify a tradeoff between -Condorcet-consistency and -ex post efficiency: every -Condorcet consistent and strategyproof SDS fails -ex post efficiency for < −1 −2 .Under the additional assumption of anonymity and neutrality, we characterize the strategyproof SDSs that maximize the ratio between and : all these SDSs are mixtures of the randomized Copeland rule and the uniform random dictatorship.
For proving the tradeoff between -ex post efficiency andrandom dictatorships, we first investigate the efficiency of 0randomly dictatorial strategyproof SDSs.In more detail, we prove next that every such SDS fails -ex post efficiency for < 1 .
The proof of this result is quite similar to the one for the upper bound on -Condorcet-consistency in Theorem 4. In particular, we first show that all 0-randomly mixtures of duples and all 0-randomly dictatorial mixtures of unilaterals violate -ex post efficiency for < 1 .Next, we consider an arbitrary 0-randomly dictatorial SDS and aim to show that there are a profile and a Pareto-dominated alternative ∈ such that ( , ) ≥ .Even though Theorem 1 allows us to represent as the convex combination of a 0-randomly dictatorial mixture of unilaterals uni and a mixture of duples duple , our previous observations have unfortunately no direct consequences for the -ex post efficiency of .The reason for this is that uni and duple might violate -ex post efficiency for different profiles or alternatives.We solve this problem by transforming into a 0-randomly dictatorial SDS * that is -ex post efficient for the same as and satisfies additional properties.In particular, * can be represented as a convex combination of a 0randomly dictatorial mixture of unilaterals * uni and a 0-randomly dictatorial mixture of duples * duple such that * uni ( , ) ≥ 1 and * duple ( , ) ≥ 1 for some profile in which alternative is Paretodominated.Consequently, * fails -ex post efficiency for < 1 , which implies that also violates this axiom.
Based on Lemma 5, we can now show the tradeoff between ex post efficiency and the similarity to a random dictatorship.
Theorem 5.For every ∈ [0, 1], every strategyproof and 1− -ex post efficient SDS is -randomly dictatorial for ≥ if ≥ 3.Moreover, if = , ≥ 4, and the SDS satisfies additionally anonymity and neutrality, it is a mixture of the uniform random dictatorship and the uniform lottery.
The proof of the first claim follows easily from Lemma 5: we consider a strategyproof SDS and use the definition of -randomly dictatorial SDSs to represent as a mixture of a random dictatorship and another strategyproof SDS .Unless is a random dictatorship, the maximality of entails that is 0-randomly dictatorial.Hence, Lemma 5 implies that can only be -ex post efficient for ≥ 1 .Consequently, ≥ must be true if satisfies 1− -ex post efficiency.For the second claim, we observe first that every anonymous, neutral, and strategyproof SDS can be represented as a mixture of the uniform random dictatorship and another strategyproof, anonymous, and neutral SDS .Moreover, unless is 1randomly dictatorial, is 0-randomly dictatorial.Thus, Lemma 5 and the assumption that = require that is exactly 1 -ex post efficient.Finally, the claim follows by proving that the uniform lottery is the only 0-randomly dictatorial and strategyproof SDS that satisfies anonymity, neutrality, and 1 -ex post efficiency if ≥ 4. For = 3 the randomized Copeland rule also satisfies all required axioms and the uniform rule is thus not the unique choice.
Theorem 5 represents a continuous strengthening of Gibbard's random dictatorship theorem: the more ex post efficiency is required, the closer a strategyproof SDS gets to a random dictatorship.Conversely, our result also entails that -randomly dictatorial SDSs can only satisfy 1− -ex post efficiency for ≤ .Moreover, the second part of the theorem indicates that relaxing ex post efficiency does not allow for particularly appealing strategyproof SDSs.
The correlation between -ex post efficiency and -randomly dictatorships also suggests a tradeoff between -Condorcetconsistency and -ex post efficiency because all random dictatorships are 0-Condorcet-consistent for sufficiently large and .Perhaps surprisingly, we show next that -Condorcet consistency and -ex post efficiency are in relation with each other for strategyproof SDSs.As a consequence of this insight, two strategyproof SDSs are particularly interesting: random dictatorships because they are the most ex post efficient SDSs, and the randomized Copeland rule because it is the most Condorcet-consistent SDS.Theorem 6.Every strategyproof SDS that satisfies anonymity, neutrality, -Condorcet consistency, and -ex post efficiency with = −2 −1 is a mixture of the uniform random dictatorship and the randomized Copeland rule if ≥ 4, ≥ 5. Furthermore, there is no strategyproof SDS with < −2 −1 if ≥ 4, ≥ 5.

P
. Let be a strategyproof SDS that satisfies -Condorcet consistency for some ∈ [0, 2 ] and let ∈ [0, 1] denote the minimal value such that is -ex post efficient.We first show that ≥ −2 −1 and hence apply Lemma 2 to construct an SDS ′ that satisfies strategyproofness, anonymity, neutrality, ′ -Condorcet consistency for ′ ≥ , and ′ -ex post efficiency for ′ ≤ .In particular, if ′ is only ′ -ex post efficient for ′ ≥ −2 −1 ′ , then can only satisfy -ex post efficiency for ≥ ′ ≥ −2 −1 ′ ≥ −2 −1 .Since ′ satisfies anonymity, neutrality, and strategyproofness, we can apply Theorem 3 to represent it as a mixture of a supporting size SDS and a point voting SDS, i.e., ′ = point + (1 − ) sup for some ∈ [0, 1].Let ( 1 , . . ., ) and ( 0 , . . ., ) denote the scoring vectors describing point and sup , respectively.Next, we a derive lower bound for ′ and an upper bound for ′ by considering specific profiles.First, consider the profile in which every voter reports as his best alternative and as his second best alternative; the remaining alternatives can be ordered arbitrarily.It follows from the definition of point voting SDSs that point ( , ) = 2 and from the definition of supporting size SDS that sup ( , For the upper bound on , consider the following profile ′ where alternative is never ranked first, but it is the Condorcet winner and wins every pairwise comparison only with minimal margin.We denote for the definition of ′ the alternatives as = { , 1 , . . ., −1 }.In ′ , the voters ∈ {1, 2, 3} ranks alternatives := { ∈ \ { } : mod 3 = − 1} above and all other alternatives below.Since ≥ 4, none of them ranks first.If the number of voters is even, we duplicate voters 1, 2, and 3.As last step, we add pairs of voters with inverse preferences such that no voter prefers the most until ′ consists of voters.Since alternative is never top-ranked in ′ , it follows that point ( ′ , ) ≤ 2 .Furthermore, Using these bounds, we show next that ′ is only ′ -ex post efficiency for ′ ≥ −2 −1 ′ , which proves the second claim of the theorem.In the subsequent calculation, the first and last inequality follow from our previous analysis.The second inequality is true since −2 −1 ≤ 1 and −2 −1 ( − 1) = ( − 2).The third inequality uses the definition of supporting size SDSs.
Remark 4. All axioms of the characterization in Theorem 6 are independent of each other.Every mixture of random dictatorships other than the uniform one and the randomized Copeland rule only violates anonymity.An SDS that violates only neutrality can be constructed by using a variant of the randomized Copeland rule that does not split the probability equally if there is a majority tie.Finally, the correlation between -Condorcet-consistency andex post efficiency is required since the uniform lottery satisfies all other axioms.Moreover, all bounds on and in Theorem 6 are tight.If there are only = 2 voters, = 3 alternatives, or = 4 alternatives and = 4 voters, the uniform random dictatorship is not 0-Condorcet consistent since a Condorcet winner is always ranked first by at least one voter.Hence, the bound on does not hold in these cases.In contrast, our proof shows that Theorem 6 is also true when = 3.

CONCLUSION
In this paper, we analyzed strategyproof SDSs by considering relaxations of Condorcet-consistency and ex post efficiency.Our findings, which are summarized in Figure 2, show that two strategyproof SDSs perform particularly well with respect to these axioms: the uniform random dictatorship (and random dictatorships in general), and the randomized Copeland rule.In more detail, we prove that the randomized Copeland rule is the only strategyproof, anonymous, and neutral SDS which guarantees a probability of 2 to the Condorcet winner.Since no other strategyproof SDS can guarantee more probability to the Condorcet winner (even if we drop anonymity and neutrality), this characterization identifies the randomized Copeland rule as one of the most Condorcetconsistent strategyproof SDSs.On the other hand, Gibbard's random dictatorship theorem shows that random dictatorships are the only ex post efficient and strategyproof SDSs.We present a continuous generalization of this result: for every ∈ [0, 1], every 1− -ex post efficient and strategyproof SDS is -randomly dictatorial for ≥ .This means informally that, even if we allow that Pareto-dominated alternatives can get a small amount of probability, we end up with an SDS similar to a random dictatorship.Finally, we derive a tradeoff between -Condorcet-consistency and -ex post efficiency for strategyproof SDSs: every strategyproof and -Condorcet-consistent SDS fails -ex post efficiency for < −2 −1 .This theorem entails that it is not possible to jointly optimize these two axioms, which highlights the special role of the randomized Copeland rule and random dictatorships again.

APPENDIX: OMITTED PROOFS
Here, we discuss the missing proofs of all lemmas and of Theorem 5. Proof sketches providing intuition for the lemmas can be found in the main body.First, we discuss the proof of Lemma 1. Recall for this proof that : is the profile derived from by letting voter reinforce against .Lemma 1.A strategyproof SDS is -randomly dictatorial if and only if there are non-negative values 1 , . . ., such that: i) ii) ( : , ) − ( , ) ≥ for all alternatives , ∈ , voters ∈ , and preference profiles in which voter prefers the most and the second most.
iii) for every voter ∈ , there are alternatives , ∈ and a profile such that voter prefers the most and the second most in , and ( : , ) − ( , ) = .
P ." ⇐= " Assume that is a strategyproof SDS for which there are values 1 , . . ., such that ( : , ) − ( , ) ≥ ≥ 0 for all alternatives , ∈ , voters ∈ , and profiles such that voter prefers the most and the second most in .Furthermore, we assume that for every voter ∈ , this inequality is tight for at least one pair of alternatives , ∈ and one such profile .We show next that is -randomly dictatorial for = ∈ .As first step, note that ( , ) ≥ ∈ for every profile , alternative ∈ , and set of voters ⊆ such that all voters in report as their favorite alternative.This follows by letting the voters ∈ one after another swap with their second best alternative (note that might be a different alternative for every voter ∈ ).Using our assumption on , the probability of has to increase by at least during such a step, which means that the probability of decreases by because of localizedness.Furthermore, it holds that ( ′ , ) ≥ 0, where ′ is the profile derived by letting all voters in swap their best two alternatives.Combining these two facts then implies that ( , ) ≥ ∈ .Note that this observation implies that ≤ 1 because otherwise, cannot be a valid SDS.Moreover, is a random dictatorship if = 1.This follows from the following reasoning: for all profiles and alternatives ∈ , it holds that ( , ) ≥ ∈ , where denotes the set of voters who prefer the most in .Since the sets partition and = 1, this inequality must be tight for every alternative; otherwise, , where denotes the dictatorial SDS of voter .As next case, suppose that < 1 and define = Note that is a well-defined SDS: for all profiles and alternatives , it holds that ( , ) ≥ 0 because for all profiles .Next, we show that is strategyproof, which implies that is ′ -randomly dictatorial for ′ ≥ because = ∈ + (1 − ) .It is sufficient to show that is localized and non-perverse because then Theorem 2 implies that is strategyproof.In more detail, is localized because the SDS and all SDSs are localized.Hence, swapping two alternatives in the preferences of a voter only affects these two alternatives.For seeing that is non-perverse, consider a voter , two alternatives , ∈ and a profile such that is voter 's -th best alternative and is his + 1-th best one.We show that ( : , ) ≥ ( , ), which entails that is non-perverse.Note for this that ( : ) = ( ) for all ∈ \ { } because the preferences of these voters did not change, and ( : , ) − ( , ) ≥ 0 because is strategyproof.If and are not the two best alternatives of voter , then ( : ) = ( ) = 0. Hence, it immediately follows that ( : , ) − ( , ) = 1 1− ( : , ) − ( , ) ≥ 0 in this case.On the other hand, if and are voter 's two best alternative, we have that ( : , ) = 1 and ( , ) = 0.Moreover, our assumptions imply that ( : , ) − ( , ) ≥ because and voter 's two best alternatives.Thus, we calculate that , which shows that is non-perverse.Finally, we show that cannot be ′ -randomly dictatorial for ′ > .If this was the case, we can represent as = ∈ ′ + (1 − ′ ) ′ , where ′ ≥ 0 are values such that ∈ ′ = ′ and ′ is a strategyproof SDS.Since ′ > , there is a voter with ′ > .Furthermore, our assumptions state that there are a profile and alternatives , such that voter prefers the most and the second most in , and ( : , ) − ( , ) = .
" =⇒ " Let be a strategyproof -randomly dictatorial SDS.We show next that there are values that satisfy the requirements of the lemma.Since is -randomly dictatorial, it can be represented as = + (1 − ) , where is a random dictatorship and is another strategyproof SDS.Moreover, as is a random dictatorship, there are values 1 , . . ., such that ≥ 0 for all ∈ , ∈ = 1, and = ∈ .In the last equation, denotes the dictatorial SDS of voter .Combining these two equations, we derive that = ∈ + (1 − ) .We show in the sequel that the values = satisfy all requirements of our lemma.First, note that the conditions ≥ 0 for all ∈ and ∈ = are obviously true.
Next, consider two alternatives , ∈ , an arbitrary voter ∈ , and a profile in which voter reports as his best alternative and as his second best one.It holds that ( : , ) − ( , ) ≥ 0 because is strategyproof and therefore non-perverse, ( : , )− ( , ) = 0 for all ∈ \{ } because : = , and ( : , ) − ( , ) = 1 as is voter 's best alternative in : , but not in .Hence, it follows that ( : , ) − ( , ) ≥ = for all voters ∈ , alternatives , ∈ , and preference profiles in which voter reports as his best and as his second best alternative.
Finally, it remains to show that there is for every voter ∈ a pair of alternatives , ∈ and a profile such that voter prefers the most and the second most in and ( : , ) − ( , ) = .Assume this is not the case for some voter , i.e, that ( : , ) − ( , ) > for all alternatives , ∈ and profiles in which is voter 's best alternative and his second best one.Hence, let ′ > denote the minimal value of ( : , ) − ( , ) among all alternatives , ∈ and preference profiles in which voter reports as his best alternative and as his second best one.Moreover, define ′ = + ∈ \{ } .We can now apply the arguments for the inverse direction to derive that is ′′ -randomly dictatorial for some ′′ ≥ ′ > .This contradicts our assumption that is -randomly dictatorial as must be the maximal value such that can be represented as = + (1 − ) , where is a random dictatorship and is another strategyproof SDS.Hence, it follows that there are for every voter ∈ a profile and two alternatives , ∈ such that ( : , ) − ( , ) = and voter reports as his best alternative and as his second best one in .This means that our choice of satisfies all requirements of the lemma.

Proof of Theorem 4
Next, we show the lemmas required for the proof of Theorem 4. First, we discuss the averaging construction of Lemma 2 in detail.

P
. Let denote an arbitrary strategyproof SDS that is -Condorcet-consistent for some ∈ [0, 1].We construct in the sequel an anonymous and neutral SDS * that satisfies strategyproofness and -Condorcet-consistency for the same as .As first step, we define the SDS for arbitrary permutations : → and : → as follows.First, permutes the voters in the input profile according to and the alternatives according to .Next, we compute on the resulting profile ( ( )) and finally, we define ( , ) as the probability assigned to ( ) by in ( ( )).More formally, is defined as ( , ) = ( ( ( )), ( )), where the profile ( ( )) satisfies for all and , ∈ that ( ) ≻ ( ) ( ) in ( ( )) if and only if ≻ in .Note that is strategyproof for all permutations and because every manipulation of implies a manipulation of .Furthermore, is -Condorcet-consistent because for every preference profile with Condorcet winner , ( ) is the Condorcet winner in ( ( )).

Hence, if
violates -Condorcet-consistency in some profile , then violates this axiom in the profile ( ( )).
Finally, we define the SDS * by averaging over for all permutations and .Hence, let Π denote the set of all permutations on and let T denote the set of all permutations on .Then, * is defined as follows.* ( , Next, we show that * satisfies all axioms required by the lemma.First, * is strategyproof since all SDSs are strategyproof.The -Condorcet-consistency of * is shown by the following inequality, where denotes a profile in which is the Condorcet winner.* ( , ) = ∈Π ∈T
Next, we present the proof of Lemma 3 which demonstrates that point voting SDSs cannot satisfy -Condorcet-consistency for ≥ 2 .Note that we use additional notation for this proof.The rank ( , ) of an alternative in the preferences of a voter is the number of alternatives that are weakly preferred to by voter , i.e., ( , ) = |{ ∈ : }|.Moreover, the rank vector * ( , ) of an alternative in a preference profile is the vector that contains the rank of with respect to every voter in increasing order.An important observation for point voting SDSs is that ( , ) = ( ′ , ) if * ( , ) = * ( , ′ ).The reason for this is that a point voting SDSs assign an alternative every time probability when it is ranked -th.Finally, the proof focuses mainly on Condorcet winner candidates, which are alternatives that can be made into the Condorcet winner without changing their rank vectors.

P
. Let be a point voting SDS for ≥ 3 alternatives and ≥ 3 voters, and let = ( 1 , . . ., ) be the scoring vector that defines .Furthermore, assume for contradiction that is -Condorcet-consistent for ≥ 2 .In the sequel, we show that there can be many Condorcet winner candidates in a profile .Since we can turn Condorcet winner candidates into Condorcet winners without changing their rank vector and since ( , ) = ( ′ , ) for all profiles and ′ with * ( , ) = * ( , ′ ), it follows that each Condorcet winner candidate has at least probability in .This observation is in conflict with ∈ ( , ) = 1 if > 2 because there can be ⌈ 2 ⌉ Condorcet winner candidates.By investigating our profiles in more detail, we also deduce that = 2 is not possible.
We use a case distinction with respect to the parity of and to construct profiles with ⌈ 2 ⌉ Condorcet winner candidates.Moreover, we first focus on cases with fixed , and provide in the end an argument for generalizing the impossibility to all ≥ 3. Figure 3 illustrates our construction for all four base cases with ∈ {3, 4}.
Case 1: = 3 and is odd In this case, we choose = +1 2 alternatives which are denoted by 1 , . . ., .We construct the profile 1 with Condorcet winner candidates as follows.For every ∈ {1, . . ., }, voters 1 and 2 rank alternative at position , and voter 3 ranks it at position +2−2 .The sum of ranks of is then equal to 2 + + 2 − 2 = + 2, which means that only − 1 alternatives can be ranked above .Note for this that the sum of ranks of an alternative is the number of voters plus the number of alternatives that are ranked above .Hence, for every ∈ {1, . . ., }, we can reorder the alternatives in \ { } such that each alternative ∈ \ { } is preferred to by 2 Condorcet winner candidates and =1 ( 1 , ) ≤ 1, we derive that +1 2 ≤ 1.This is equivalent to ≤ 2 +1 < 2 , which shows that fails -Condorcet-consistency for ≥ 2 in this case.
Case 2: = 3 and is even If = 3 and is even, we construct a preference profile 2 with 2 Condorcet winner candidates similar to the last case.More precisely, we first choose an alternative , and apply the construction of the last case to the alternatives \{ }.Then, we add as the lastranked alternative of voters 1 and 2 and as first-ranked alternative of voter 3. Note that adding does not affect whether an alternative is a Condorcet winner candidate because it is last-ranked by two out of three voters.Thus, there are 2 Condorcet winner candidates in 2 and it follows analogously to the last case that ≤ 2 .Finally, we show that = 2 is also impossible.Otherwise, each of the 2 Condorcet winner candidates has a probability of 2 , which means that the other alternatives have a probability of 0. Thus, ( 2 , ) = 0 even though voter 3 reports as his best alternative.This implies for the scoring vector = ( 1 , . . ., ) of that 1 = 0.However, this is not possible because the scoring vector needs to satisfy =1 = 1 and ≥ if ≤ .Hence, we deduce also for this case that < 2 holds.
Case 3: = 4 and is odd Just as in the first case, we choose = +1 2 alternatives which are denoted by 1 , . . ., .Next, we construct a profile 3 with Condorcet winner candidates as follows.For every ∈ {1, . . ., }, voters 1 and 2 rank alternative at position , and voters 3 and 4 rank it at position +1 2 + 1 − .The sum of ranks of is then equal to 2 + 2 +1 2 + 1 − = + 3. Since the sum of ranks of an alternative is the number of voters plus the number of alternatives ranked above , we derive that only − 1 alternatives can be ranked above .Hence, for every ∈ {1, . . ., }, we can reorder the alternatives such that each alternative ∈ \ { } is ranked above once without changing the rank vector of .This entails that each alternative is a Condorcet winner candidate and thus, we derive that ≤ 2 +1 < 2 analogously to Case 1. Case 4: = 4 and is even Finally, consider the case that = 4 and is even.In this situation, we construct the profile 4 with 2 Condorcet winner candidates as follows: we choose an alternative , and apply the construction of Case 3 to the alternatives in \ { }.Then, voters 1 to 3 add as their least preferred alternative and voter 4 adds it as his best alternative.Just as in Case 2, every alternative that is a Condorcet winner candidate before adding is also a Condorcet winner candidate after adding this alternative because is the least preferred alternative of a majority of the voters.Hence, there are 2 Condorcet winner candidates in 4 , which implies that ≤ 2 .Finally, an analogous argument as in Case 2 shows that = 2 is not possible either.In particular, if = 2 , then ( 4 , ) = 0 because only Condorcet winner candidates can have positive probability.However, ( 4 , ) = 0 conflicts with the definition of point voting SDSs since voter 4 reports as his favorite choice.Therefore, it follows that fails -Condorcet-consistency for ≥ 2 .
Case 5: Generalizing the impossibility to larger Finally, we explain how to generalize the last four cases to an arbitrary number of voters ≥ 3.In this case, we also construct a profile with ⌈ 2 ⌉ Condorcet winner candidates.In more detail, we choose the suitable base case and add repeatedly pairs of voters with inverse preferences until there are voters.Note that voters with inverse preferences do not change the majority margins, and therefore they do not change whether an alternative is a Condorcet winner candidate.Hence, every alternative that is a Condorcet winner candidate in the base case is also a Condorcet winner candidate in the extended profile, which means that the arguments in the base cases also apply for larger numbers of voters.Therefore, no point voting SDS satisfies -Condorcet-consistency for ≥ 2 Next, we prove Lemma 4, which bounds the probability that can be guaranteed to Condorcet winners by supporting size SDSs.Lemma 4. No supporting size SDS can assign more than 2 probability to an alternative.

Proofs of Lemma 5 and Theorem 6
We focus next on the proofs of the lemmas that are required for Lemma 5. Hence, our goal is to derive a lower bound for the -ex post efficiency of strategyproof 0-randomly dictatorial SDSs.Since Theorem 1 allows us to represent strategyproof SDSs as a mixtures of duples and unilaterals, we focus next on these two classes.
First, we investigate the -ex post efficiency of duples.Recall therefore that a duple is a strategyproof SDS such that ( , ) = 0 for all alternatives ∈ \ { , }.Moreover, a mixture of duples is defined as ( , ) = ∈ \{ } ( , ), where = denote non-negative weights that sum up to 1.Moreover, we use in this definition that = .Finally, note that one duple for every pair is sufficient to represent every mixture of duples because two duples and ′ can be merged into one.Lemma 6.No SDS that can be represented as a convex combination of duples satisfies -ex post efficiency for < 1 if ≥ 3.

P
. Let ( , ) = ∈ \{ } ( , ) be an SDS represented as a convex combination of duples, where = is the duple SDS for the pair and and = is the weight of .Furthermore, let , denote a profile where all voters report as best alternative and as worst one.First, note that ( , , ) = ( , , ) and ( , , ) = ( , , ) for all distinct , , ∈ .Thus, we also write ,• and •, to indicate that alternative is unanimously top-ranked or bottom-ranked.
As first step, we want to bound the average probability ( , , ) + ( , , ) over all , ∈ .In more detail, the subsequent equation shows that The first equality follows from ( , , ) = ( ,• , ), ( , , ) = ( •, , ) for all alternatives , ∈ , and the observation that every alternative is both unanimously topranked and unanimously bottom-ranked in exactly ( − 1) of the considered preferences profiles.For the second equality, we replace ( ,• , ) with ∈ \{ } ( , , ) and ( •, , ) with ∈ \{ } ( , , ) according to the definition of .Furthermore, we swap the order of the sum for the second term.We derive the third equality from the fact that ( , ) + ( , ) = 1 for all profiles .Finally, the last equality uses that As a consequence of this observation, it follows that there is a pair of alternatives , ∈ such that ( , , ) + ( , , ) ≤ 2 .Otherwise, it holds that , contradicting our previous equation.Hence, ∈ \{ , } ( , , ) ≥ −2 .Since all alternatives ∈ \ { , } are Pareto-dominated by , this entails that one of these alternative receives a probability of at least We conclude therefore that the SDS fails -ex post efficient for Next, we aim to show that no 0-randomly dictatorial SDS that can be represented as a mixture of unilaterals satisfies -ex post efficiency for < 1 .Ideally, we would like to use to construct a 0-randomly dictatorial SDS * that satisfies -ex post efficiency for the same as , and that is additionally neutral and anonymous.Unfortunately, we cannot use Lemma 2 here as this lemma does not preserve that * is 0-randomly dictatorial.For demonstrating this point, let = { 1 , . . ., } denote the alternatives and assume that = ≥ 3. Furthermore, consider the unilateral which assigns probability 1 to voter 's favorite alternative in \ { }.Finally, consider the SDS + which chooses a voter ∈ uniformly at random and returns the outcome of .Lemma 1 shows that this SDS is 0-randomly dictatorial because for all ∈ , the probability of does not increase if voter reinforces it to his best alternative.Moreover, since + is a mixture of unilaterals, it is strategyproof, and its definition implies that it not anonymous.However, applying the construction of Lemma 2 to + results in the point voting SDS defined by the scoring vector ( −1 , 1 , 0, . . ., 0).It follows immediately from Lemma 1 that this SDS is not 0-randomly dictatorial as pushing an alternative from second place to first place increases its probability always by −2 > 0.
Therefore, we propose another construction in the next lemma that, given an arbitrary strategyproof and 0-randomly mixture of unilaterals, constructs a strategyproof 0-randomly dictatorial SDS that is -ex post efficient for the same as the original SDS and that has a lot of symmetries.Unfortunately, this construction does not result in a anonymous SDS.Nevertheless, the resulting SDS is significantly easier to work with and its properties are crucial for the proof of Lemma 8.Note that we require some additional terminology for the next lemma.In the sequel, we say that voter or his unilateral SDS is 0-randomly dictatorial for alternatives , if ( ) = ( : ) for all preference profiles in which is voter 's best alternative and is his second best alternative.
Lemma 7. Let be a strategyproof 0-randomly dictatorial SDS that satisfies -ex post efficiency for some ∈ [0, 1] and that can be represented as a mixture of unilaterals.Then, there is a strategyproof 0-randomly dictatorial SDS * for 2 voters that can be represented as a mixture of unilaterals and that is -ex post efficient for the same as .Moreover, * satisfies the following conditions: (i) For every voter ∈ , there is a set { , } such that voter is 0-randomly dictatorial for , and and preference profiles such that voter reports as his best alternative, as his second best one, and as his third best one.(iii) If every voter ∈ reports and as their two best alternatives, then there exists a scoring vector = ( 1 , . . ., ) and let denote a strategyproof 0randomly dictatorial SDS that is -ex post efficient and that can be represented as a mixture of unilaterals.In the sequel, we use to construct the SDS * that satisfies all requirements of the lemma.Note that this proof is quite involved and therefore, we use some auxiliary observations that are proven in the end.
We start by representing as ( ) = ∈ ( ), where denotes the unilateral SDS of voter and ≥ 0 is its weight.Note that we interpret unilaterals in this proof as SDSs that take a single preference relation as input.This is possible as unilaterals only rely on the preferences of a single voter.Observation 1 states that for every voter ∈ there are alternatives , such that is 0-randomly dictatorial for and .Even though a voter can be 0randomly dictatorial for multiple pairs of alternatives, we associate from now on every voter with exactly one such pair , .This pair can be chosen arbitrarily as it will not affect the rest of the proof.
In the next step, we merge all unilaterals in a multi-set into a single unilateral.Thus, we define the unilateral ( ) as ( ), i.e., chooses each SDS ∈ with a probability proportional to .Observe that is strategyproof because it is a mixture of strategyproof SDSs and it is 0-randomly dictatorial for , because all unilaterals in are 0-randomly dictatorial for these alternatives.Based on the SDS , we can finally define the SDS * for * = 2 voters.To this end, let * denote the electorate of * .We associate each voter ∈ * with a different pair of alternatives , ∈ and set * = .Then, the SDS * chooses one of the voters ∈ * uniformly at random and returns * ( ) = ( ), i.e., * ( ) = 1 * * =1 * ( ).Clearly, * is strategyproof because it is a mixture of strategyproof SDSs.Moreover, it is 0-randomly dictatorial because every voter ∈ * is 0-randomly dictatorial for the pair of alternatives , with which he is associated.Furthermore, Observation 3 shows that * is -ex post efficient for the same as .
It remains to show that the SDS * satisfies the properties (i), (ii), and (iii).First, note that it satisfies (i) by construction as every voter is 0-randomly dictatorial for a different pair of alternatives.For (ii) and (iii), we show first the auxiliary claim that ( , ) = ( ) ( ) ( ( ), ( )) for all permutations : → , preference profiles , and alternatives ∈ .Note that if this claim holds then the SDS * satisfies neutrality, since then for all permutations ∈ T and alternatives ∈ : * ( ( ), ( ))
Observation 1: For every voter , there exists a pair of alternatives , such that ( ) = ( : ) for all preference profiles in which voter reports as best alternative and as second best one.
First, note that is strategyproof as every manipulation of this SDS could be mapped to a manipulation of .In more detail, if voter can manipulate by switching from to ′ , he can also manipulate by switching from ( ) to ( ′ ).This is true because a manipulation requires an alternative such that ≻ ( ′ , ) > ≻ ( , ), which entails by definition of that ≻ ( ( ′ ), ( )) > ≻ ( ( ), ( )).Finally, since ≻ in if and only if ( ) ≻ ( ) in ( ), we derive that voter could manipulate by switching from ( ) to ( ′ ).
For proving this observation, we construct first another SDS + and show that this SDS is -ex post efficient for the same as .As second step, we show that * can also be derived from + by merging voters, and thus * inherits the -ex post efficiency of + .Before defining + , we introduce the SDS : just as the SDSs , it is defined as ( , ) = ( ( ), ( )).In particular, is -ex post efficient for the same as .This follows by considering an arbitrary profile in which an alternative is Pareto-dominated.It is easy to see that ( ) is then Pareto-dominated in ( ), and we derive therefore that ( , ) = ( ( ), ( )) ≤ because is -ex post efficient.Next, we define the SDS + for !voters as follows: we partition the voters {1, . . ., !} into !sets 1 , . . ., ! with | | = and associate with every set a different permutation : ), where denotes the restriction of to the voters in .Observe that + is -ex post efficient for the same as because an alternative that is Pareto-dominated in is also Pareto-dominated in all and all are -ex post efficient.Hence, it follows that Next, we show that + and * satisfy -ex post efficiency for the same .Therefore, we change the representation of + .The central observation here is that = ∈ .Hence, we can also associate every voter ∈ {1, . . ., !} with an index ∈ and a permutation such that each index-permutation pair is assigned exactly once.Thus, define + = and + = !(i.e., the weight of is the proportional to the weight of in the original SDS ).Then, we can write + as + ( ) = !=1 + + ( ).Next, note that every appears once in + ( , ) and once in the union of all .Therefore, we derive that + ( ) where * = 2 .Next, we restrict our attention to profiles such that for all { , } ⊂ 2 , all voters with ∈ submit the same preference relation.In this case, we may replace the preferences of all voters with ∈ with a single preference relation.Then, there are exactly 2 voters left, each of which is associated with a different pair of alternatives.In particular, we can use the definition of ( ) = ∈ 2( −2)! ( ) now as we apply all unilateral SDSs in to the same preference relation .Hence, + returns the same outcomes as * if for each { , } ⊂ 2 , all voters with ∈ report the same preferences.Since + is -ex post efficient, it follows therefore also that * is -ex post efficient.
Finally, we use Lemma 7 to prove that no 0-randomly dictatorial SDS that can be represented as a mixture of unilaterals is -ex post efficient for < 1 .Lemma 8.No 0-randomly dictatorial SDS that can be represented as a convex combination of unilaterals satisfies -ex post efficiency for < 1 if ≥ 3.

P
. Let the SDS denote a mixture of unilaterals.First, we apply Lemma 7 to construct the SDS * as specified by this lemma.In the sequel, we show that * is -ex post efficient for ≥ 1  and therefore is also -ex post efficient for ≥ 1 .In our proof, we construct a profile * in which every alternative must receive a probability of at most which leads to a contradiction if < 1 .Let with | | = 2 be the set of voters of * .Furthermore, Lemma 7 (i) states that every voter ∈ is associated with a different pair of alternatives { , } for which he is 0-randomly dictatorial.
We show next that * ( , ) ≤ by constructing a new preference profile ′ such that * ( , ) = * ( ′ , ) ≤ .For the construction of ′ , let all voters in the second group ∈ {2, . . ., − 1} swap and , and all voters in the third group ∈ { , . . ., 2 − 3} swap and .The resulting preference profile is shown in Figure 5 for the case that = 4.It is easy to see that Pareto-dominates in ′ and, as * is -ex post efficient, * ( ′ , ) ≤ .Alternative was moved from third to second and from second to third place by − 2 voters.It follows therefore from Lemma 7 (ii) and localizedness that the probability that alternative gains when − 2 voters swap it from third to second place is the same as the probability that looses when − 2 voters swap it from second to third place.Thus, we derive that * ( , ) = * ( ′ , ) ≤ .
Finally, note that in , all voters ∈ report the pair , for which they are 0-randomly dictatorial as their two best alternatives.Hence, Lemma 7 (iii) entails the existence of a scoring vector ( 1 , . . ., } | for all ∈ .In particular, observe that the probability of an alternative only depends on its rank vector * ( , ).Recall that the rank vector * ( , ) of an alternative in a preference profile is the vector that contains the rank of with respect to every voter in increasing order.The rank vector of alternative in is * ( , ) = ( Furthermore, observe that * ( ¯ , ) ≤ * ( , ) in every profile ¯ in which (i) each voter ∈ reports the alternatives , as his two best alternatives and (ii) * ( , ¯ ) ≥ * ( , ) for all ∈ { , . . ., 2 }.Condition (i) implies that * can be computed based on the scoring vector ( 1 , . . ., ).Furthermore, it implies that every alternative ∈ is among the two best alternatives of exactly − 1 voters, and since 1 = 2 , it follows that we can ignore these entries when comparing the probability of in with the probability of in ¯ .Finally, the claim follows as 3 ≥ • • • ≥ and * ( , ¯ ) ≥ * ( , ) for all ∈ { , . . ., 2 } entails thus that * ( , ) ≥ * ( ¯ , ).We use this fact to construct a new profile * where * ( * , ) ≤ * ( , ) ≤ for every ∈ .Let every voter ∈ report the alternatives , for which he is 0-randomly dictatorial as his two best alternatives.Furthermore, distribute all other alternatives such that no alternative is ranked third by more than − 2 voters.This is possible as there are ≥ 3 alternatives and voters.It follows from the construction that * ( , * ) ≥ * ( , ) for every ∈ { , . . ., 2 } and every ∈ .Hence, we derive that * ( * , ) ≤ * ( , ) ≤ for every ∈ .If < 1 , this entails that ∈ * ( * , ) < 1, a contradiction.Thus, * cannot satisfy -ex post efficiency for < 1 , and thus, violates this axiom, too.This show that there exists no 0randomly dictatorial SDS that can be represented as a mixture of unilaterals and that satisfies -ex post efficiency for < 1 when ≥ 3.
Finally, we use Lemma 6 and Lemma 8 to prove that there are no 0-randomly dictatorial SDSs that satisfy -ex post efficiency for < 1 .Lemma 5.No strategyproof SDS that is 0-randomly dictatorial satisfies -ex post efficiency for < 1 if ≥ 3.

P
. Let denote a strategyproof SDS for voters and ≥ 3 alternatives that is 0-randomly dictatorial.Our argument focuses mainly on the profiles , , in which all voters report as their best choice and as their second best choice.The reason for this is that if ( , ) > for some profile in which is Pareto-dominated by , then ( , , ) > .This is a direct consequence of strategyproofness as we can transform into , by reinforcing and .Hence, non-perversity implies that ( , , ) ≥ ( , ) > .Moreover, localizedness entails that the order of the alternatives ∈ \ { , } in , is not important as it does not affect the probabilities of and .
For this construction, we define as ( , ) = ( ( ), ( )) for every permutation : → .We construct the SDS * for !voters as follows: we partition the electorate in !sets with | | = and associate each of these sets with a different permutation : → .Then, we choose one of these sets uniformly at random and consider from now on only the preference profile defined by the voters in .Finally, return ( ), where denotes the permutation associated with .More formally, * ( ) = 1 !!=1 ( ).First, note that * is 0-randomly dictatorial because of Lemma 1.Since is a 0-randomly dictatorial, there is for every voter a profile and alternatives , such that voter prefers the most in and the second most, and ( , ) = ( : , ).Consequently, there are such profiles and alternatives for every voter in each SDS .Finally, we derive that such profiles and alternatives exist also for * .For a voter ∈ , the corresponding alternatives , and the preferences of the voters in are the same as for .The preferences of the remaining voters do not matter.If * does not choose in the first step, the preferences of voter do not matter, and if * chooses , it only computes ( ).Hence, if voter now swaps and , the outcome of * does not change as the outcome of does not change.Consequently, Lemma 1 implies that * is 0-randomly dictatorial.
Next, observe that * ( ) ) is strategyproof as it is a mixture of strategyproof SDSs.In particular, we can interpret each term → , where T is the set of all permutations on .
As last result, we discuss the proof of Theorem 5.
Theorem 5.For every ∈ [0, 1], every strategyproof and 1− -ex post efficient SDS is -randomly dictatorial for ≥ if ≥ 3.Moreover, if = , ≥ 4, and the SDS satisfies additionally anonymity and neutrality, it is a mixture of the uniform random dictatorship and the uniform lottery.

P
. Just as for Theorem 4, we need to show two claims: on the one hand, there is for every ∈ [0, 1] no strategyproof and 1−ex post efficient SDS that is -randomly dictatorial for < .On

1 Figure 2 :
Figure 2: Graphical summary of our results.Points in the figures correspond to SDSs and the horizontal axis indicates in both figures the value of for which the considered SDS isex post efficient.In the left figure, the vertical axis states the for which the considered SDSs are -Condorcet-consistent, and in the right figure, it shows the for which SDSs arerandomly dictatorial.Theorems 4 and 6 show that no strategyproof SDS lies in the grey area of the left figure.Theorem 5shows that no strategyproof SDS lies in the grey area below the diagonal in the right figure.Furthermore, no SDS lies in the grey area above the diagonal since a -randomly dictatorial SDS can put no more than 1 − probability on Pareto-dominated alternatives.Finally, the following SDS are marked in the figures: corresponds to all random dictatorships, to the randomized Copeland rule, to the randomized Borda rule, and to the uniform lottery.

Figure 4 : 2 2
Figure 4: The preference profile that in the proof of Lemma 8 for = 4.There are four groups of voters.The first group contains only the first voter who is 0-randomly dictatorial for and .The next two groups have both − 2 voters and are 0-randomly dictatorial for one of and .The last group contains the remaining −2 2 voters that are not 0-randomly dictatorial or .All voters have the pair for which they are 0-randomly dictatorial ranked at the top.