The Semi-Random Satisfaction of Voting Axioms

1 Introduction . The “ widespread presence of impossibility results ” [ 48 ] is one of the most fundamental and significant challenges in social choice theory . These impossibility results often assert that no “ perfect ” voting rule exists for three or more alternatives [ 1 , 28 , 45 ] . Nevertheless , an ( imperfect ) voting rule must be designed and used in practice for agents to make a collective decision . In the social choice literature , the dominant paradigm of doing so has been the axiomatic approach , i.e. , voting rules are designed , evaluated , and compared to each other w.r.t . their satisfaction of desirable normative properties , known as ( voting ) axioms . Most definitions of dissatisfaction of voting axioms are based on worst-case analysis . For example , a voting rule r does not satisfy CONDORCET CRITERION ( CC for short ) , if there exists a collection of votes , called a profile , where the Condorcet winner exists but is not chosen by r as a winner . The Condorcet winner is the alternative who beats all other alternatives in their head-to-head competitions . As another example , a voting rule r does not satisfy PARTICIPATION ( PAR for short ) , if there exists a profile and a voter who has incentive to abstain from voting . An instance of dissatisfaction of PAR is also known as the no-show paradox [ 19 ] . Unfortunately , when the number of alternatives m is at least four , no irresolute voting rule satisfies CC and PAR simultaneously [ 39 ] . While the classical worst-case analysis of ( dis ) satisfaction of axioms can be desirable in high-stakes applications such as political elections , it is often too coarse to serve as practical criteria for comparing different voting rules in more frequent , low-stakes applications of social choice , such as business decision-making [ 5 ] , crowdsourcing [ 34 ] , informational retrieval [ 33 ] , meta-search engines [ 14 ] , recommender systems [ 51 ] , etc . A decision maker who desires both axioms would find it hard to choose between a voting rule that satisfies CC but not PAR , such as Copeland , and a voting rule that satisfies PAR but not CC , such as plurality . A finer and more quantitative measure of satisfaction of axioms is therefore called for . 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . One natural and classical approach is to measure the likelihood of satisfaction of axioms under a probabilistic model of agents ’ preferences , in particular the independent and identically distributed ( i.i.d . ) uniform distribution over all rankings , known as Impartial Culture ( IC ) in social choice . This line of research was initiated and established by Gehrlein and Fishburn in a series of work in the 1970 ’ s [ 24 , 26 , 25 ] , and has become a “ new sub-domain of the theory of social choice ” [ 13 ] . Some classical results were summarized in the 2011 book by Gehrlein and Lepelley [ 27 ] , and recent progresses can be found in the 2021 book edited by Diss and Merlin [ 13 ] . While this line of work is highly significant and interesting from a theoretical point of view , its practical implications may not be as strong , because most previous work focused on a few specific distributions , especially IC , which has been widely criticized to be unrealistic ( see , e.g. , [ 42 , p. 30 ] , [ 23 , p. 104 ] , and [ 30 ] ) . Indeed , conclusions drawn under any specific distribution may not hold in practice , as “ all models are wrong ” [ 9 ] . Technically , characterizing the likelihood of satisfaction of CC and of PAR are already highly challenging w.r.t . IC , and despite that Berg and Lepelley [ 3 ] explicitly posed them as open questions in 1994 , not much is known beyond a few voting rules . Therefore , the following question largely remains open . How likely are voting axioms satisfied under realistic models ? The importance of successfully answering this question is two-fold . First , it tells us whether the worst-case violation of an axiom is a significant concern in practice . Second , it provides a finer and more quantitative foundation for comparing voting rules . We believe that the semi-random models [ 15 ] , originally investigated by Blum [ 7 ] and [ 8 ] and includes the models employed in the celebrated smoothed analysis proposed by Spielman and Teng [ 49 ] , provides a promising framework for addressing the question . In this paper , we adopt the semirandom model by Xia [ 54 ] , which models the satisfaction of a per-profile voting axiom X by a function X ( r , P ) ∈ { 0 , 1 } , where r is a voting rule and P is a profile , such that r satisfies X if minP X ( r , P ) = 1 . Let Π denote a set of distributions over all rankings over the m alternatives ( denoted by L ( A ) ) , which represents the “ ground truth ” preferences for a single agent that the adversary can choose from . Let n denote the number of agents . Because a higher value of X ( r , P ) is more desirable to the decision maker , the adversary aims at minimizing expected X ( r , P ) by choosing ~π ∈ Πn—the profile P is generated from ~π . The semi-random satisfaction of X under r with n agents , denoted by X̃minΠ ( r , n ) , is defined as follows [ 54 ] : X̃minΠ ( r , n ) , inf~π∈Πn PrP∼~πX ( r , P ) ( 1 ) Notice that agents ’ ground truth preferences can be arbitrarily correlated , while the noises are independent , which is a standard assumption in the literature and in practice [ 54 ] . Example 1 ( Semi-Random CC under plurality ) . Let X = CC and r = Plu denote the irresolute plurality rule , which chooses all alternatives that are ranked at the top most often as the ( co- ) winners . Suppose there are three alternatives , denoted by A = { 1 , 2 , 3 } , and suppose Π = { π1 , π2 } , where π1 and π2 are distributions shown in Table 1 . Then , we have C̃C min Π ( Plu , n ) = inf~π∈ { π1 , π2 } n PrP∼~π CC ( Plu , P ) . When n = 2 , the adversary has four choices of ~π , i.e. , { ( π1 , π1 ) , ( π1 , π2 ) , ( π2 , π1 ) , ( π2 , π2 ) } . Each ~π leads to a distribution over the set of all profiles of two agents , i.e. , L ( A ) 2 . We have C̃C min Π ( Plu , 2 ) = 1 , because CC is satisfied at all profiles of two agents . As we will see later in Example 3 , for all sufficiently large n , C̃C min Π ( Plu , n ) = exp ( −Θ ( n ) ) . 1.1 Our Contributions . We initiate the work towards a comprehensive picture of semi-random satisfaction of voting axioms under commonly-studied voting rules , by focusing on CC and PAR in this paper due to their importance , popularity , and incompatibility [ 39 ] . Recall that m is the number of alternatives and n is the number of agents . Our technical contributions are two-fold . First , semi-random satisfaction of CC ( Theorem 1 and 2 ) . We prove that , under mild assumptions , for any fixed m ≥ 3 and any sufficiently large n , the semi-random satisfaction of CC under a wide range of voting rules is 1 , 1 − exp ( −Θ ( n ) ) , Θ ( n−0.5 ) , exp ( −Θ ( n ) ) , or being Θ ( 1 ) and 1 −Θ ( 1 ) at the same time ( denoted by Θ ( 1 ) ∧ ( 1−Θ ( 1 ) ) ) . The 1 − exp ( −Θ ( n ) ) case is positive news , because it states that CC is satisfied almost surely when n is large , regardless of the adversary ’ s choice . The remaining three cases are negative news , because they state that the adversary can make CC to be violated with non-negligible probability , no matter how large n is . Second , semi-random satisfaction of PAR ( Theorems 3 , 4 , 5 , 6 ) . We prove that , under mild assumptions , for any fixed m ≥ 3 and any sufficiently large n , the semi-random satisfaction of PAR under a wide range of voting rules is 1−Θ ( n−0.5 ) . These are positive news , because they state that PAR is satisfied almost surely for large n , regardless of the adversary ’ s choice . While this message may not be surprising at a high level , as the probability for a single agent to change the winner vanishes as n → ∞ , the theorems are useful and non-trivial , as they provide asymptotically tight rates . In particular , straightforward corollaries of our theorems to IC address open questions posed by Berg and Lepelley [ 3 ] in 1994 , and also provides a mathematical justification of two common beliefs related to PAR : first , IC exaggerates the likelihood for paradoxes to happen , and second , the dissatisfaction of PAR is not a significant concern in practice [ 31 ] , especially when it is compared to our results on semi-random CC . Table 2 summarizes corollaries of our results under some commonlystudied voting rules w.r.t . IC as well as the satisfaction of CC and PAR on Preflib data [ 35 ] , where Θ ( 1 ) ∧ ( 1−Θ ( 1 ) ) means that there exists constants 0 < α < β ≤ 1 that do not depend on n , such that the likelihood is in [ α , β ] . Table 2 provides a more quantitative way of comparing voting rules . Suppose the decision maker puts 50 % weight ( or any fixed non-zero ratio ) on both CC and PAR , and assume that the preferences are generated from IC . Then , when n is sufficiently large , the last five voting rules in the table ( that satisfy CC ) outperform the first five voting rules in the table ( the first four satisfies PAR ) . Beyond CC and PAR . Theorems 1–6 are proved by ( non-trivial ) applications of a categorization lemma ( Lemma 1 ) , which characterizes semi-random satisfaction of a large class of axioms that can be represented by unions of finitely many polyhedra , including CC and PAR . We believe that Lemma 1 is a promising tool for analyzing other axioms in future work . 1.2 Related Work and Discussions . The Condorcet criterion ( CC ) was proposed by Condorcet in 1785 [ 11 ] , has been one of the most classical and well-studied axioms , and has “ nearly universal acceptance ” [ 44 , p. 46 ] . CC is satisfied by many commonly-studied voting rules , except positional scoring rules [ 18 ] and multiround-score-based elimination rules , such as STV . Most previous work focused on characterizing the Condorcet efficiency , which is the probability for the Condorcet winner to win conditioned on its existence [ 17 , 16 , 43 , 25 , 40 ] . Beyond positional scoring rules , the study was mostly based on computer simulations , see , e.g. , [ 20 , 21 , 37 , 41 ] . The participation axiom ( PAR ) was motivated by the no-show paradox [ 19 ] and was proved to be incompatible with CC for every m ≥ 4 [ 39 ] . The likelihood of PAR under commonly studied voting rules w.r.t . IC was posed as an open question by Berg and Lepelley [ 3 ] in 1994 , and has been investigated in a series of works including [ 32 , 31 , 52 ] , see [ 27 , Chapter 4.2.2 ] . In particular , Lepelley and Merlin [ 31 ] analyzed the likelihood of various no-show paradoxes for three alternatives under scoring runoff rules , which includes STV , w.r.t . IC and other distributions , and “ strongly believe that the no-show paradox is not an important flaw of the scoring run-off voting systems ” . Our work vs. previous work on CC and PAR . Our results address open questions by Berg and Lepelley [ 3 ] about the likelihood of satisfaction of CC and PAR in two dimensions : first , we conduct semi-random analysis , which extends i.i.d . models and is believed to be significantly more general and realistic . Second , our results cover a wide range of voting rules whose likelihood of satisfaction under CC or PAR even w.r.t . IC were not mathematically characterized before , including CC under STV , and PAR under maximin , Copeland , ranked pairs , Schulze , and Black ’ s rule . While all results in this paper assume that the number of alternatives m is fixed , they are already more general than many previous work that focused on m = 3 . Semi-random analysis . There is a large body of literature on the applications of semi-random analysis , especially smoothed analysis , to computational problems [ 50 ] . Its main idea , i.e. , the worst average-case analysis , has been proposed and investigated in other disciplines as well . For example , it is the central idea in frequentist statistics ( as in the frequentist expected loss and minimax decision rules [ 4 ] ) and is also closely related to the min of means criteria in decision theory [ 29 ] . Recently , Baumeister et al . [ 2 ] and Xia [ 54 ] independently proposed to conduct smoothed analysis in social choice . We adopt the framework in the latter work , though our motivation and goal are quite different . We aim at providing a comprehensive picture of semi-random satisfaction of voting axioms , while [ 54 ] focused on analyzing semi-random likelihood of Condorcet ’ s voting paradox and an impossibility on anonymity and neutrality . On the technical level , while Lemma 1 is a straightforward corollary of [ 55 , Theorem 2 ] , applications of results like Lemma 1 can be highly non-trivial and problem dependent as commented in [ 55 ] , which is the case of this paper . We believe that Lemma 1 ’ s main merit is conceptual , as it provides a general categorization of semi-random satisfaction of a large class of per-profile axioms beyond CC and PAR for future work .

Summary: This paper studies the smoothed satisfaction of voting axioms, which is a novel notion to quantify the dissatisfaction of voting axioms based on smoothed analysis. This paper aims to address the following open question: "How likely are voting axioms satisfied under realistic models?"  The main contribution of this work is that they propose the framework of smoothed satisfaction and give pretty general quantitative results of dissatisfaction of various voting axioms for different voting rules. In particular, their results imply two important but contradicting axioms (under worst-case analysis): Condorcet criterion (CC) and Participation both can be actually satisfied in many realistic settings.