Axioms for Defeat in Democratic Elections
aa r X i v : . [ ec on . T H ] A ug Axioms for Defeat in Democratic Elections
Wesley H. Holliday † and Eric Pacuit ‡ † University of California, Berkeley ([email protected]) ‡ University of Maryland ([email protected])
August 15, 2020
Abstract
We propose six axioms concerning when one candidate should defeat another in a democratic electioninvolving two or more candidates. Five of the axioms are widely satisfied by known voting procedures.The sixth axiom is a weakening of Kenneth Arrow’s famous condition of the Independence of IrrelevantAlternatives (IIA). We call this weakening Coherent IIA. We prove that the five axioms plus CoherentIIA single out a voting procedure studied in our recent work: Split Cycle. In particular, Split Cycle is themost resolute voting procedure satisfying the six axioms for democratic defeat. In addition, we analyzehow Split Cycle escapes Arrow’s Impossibility Theorem and related impossibility results.
Contents
Conclusion 32A Proofs for Section 3 33B Arrow’s alleged confusion and VIIA 33
In the abstract for his lecture at a 2017 Lindau Nobel Laureate Meeting, Eric Maskin (2017) claimed that“The systems that most countries use to elect presidents are deeply flawed,” a claim defended in writing byMaskin and Sen (2016; 2017a; 2017b). In fact, the issue goes far beyond presidential elections: the samevoting systems are used in elections ranging from national elections to elections in small committees andclubs. In our view, the key issue highlighted by Maskin and Sen can be stated in terms of the followingnormative principle (closely related to what voting theorists call Condorcet consistency, defined below ).Majority Defeat: if a candidate does not win in an election, they must have been defeated by someother candidate in the election, and a candidate should defeat another only if a majority of votersprefer the first candidate to the second.As is well known, widely used voting systems can violate the principle of Majority Defeat. Example 1.1.
In the 2000 U.S. presidential election in Florida, George W. Bush defeated Al Gore andRalph Nader according to Plurality voting, which only allows voters to vote for one candidate. Yet basedon the plausible inference that most Nader voters preferred Gore to Bush (see Magee 2003), it follows thata majority of all voters preferred Gore to Bush.
Example 1.2.
In the 2009 mayoral election in Burlington, Vermont, the Progressive candidate Bob Kissdefeated the Democratic candidate Andy Montroll according to Ranked Choice voting (defined in Example2.11 below), but Montroll was preferred to each of the other candidates including Kiss by majorities of voters,according to the ranked ballots collected.
Example 1.3.
During the 2016 U.S. presidential primary season, an NBC News/Wall Street Journal poll(March 3-6) asked respondents both for their top choice and their preference between Donald Trump and eachof Ted Cruz, John Kasich, and Marco Rubio. Trump was the Plurality winner, receiving 30% of first placevotes, but Cruz, Kasich, and Rubio were each preferred to Trump by 57%, 57%, and 56% of respondents,respectively (see Kurrild-Klitgaard 2018 concerning statistical significance). For further discussion of whetheranother Republican might have been majority preferred to Trump, see Maskin and Sen 2016, Maskin 2017,Kurrild-Klitgaard 2018, and Woon et al. 2020.For related examples outside the U.S., see, e.g., Kaminski 2015, § 20.3.2 and Feizi et al. 2020.The above failures of Majority Defeat involve spoiler effects . In Example 1.1, although it is likely thata majority of voters preferred Gore to Bush and also preferred Gore to Nader, Nader’s inclusion in therace spoiled the election for Gore, handing victory to Bush. In Example 1.2, although a majority of voterspreferred Montroll to Kiss and a majority preferred Montroll to the Republican candidate, Kurt Wright, For the relation between Majority Defeat and Condorcet consistency, see Remark 4.14. Due to the possibility of strategic voting, we cannot guarantee that voters’ rankings ofthe candidates always reflect their sincere preferences (see Taylor 2005), but one can try to choose votingprocedures that provide fewer incentives for strategic voting (see, e.g., Chamberlin 1985, Nitzan 1985, Bassi2015, Holliday and Pacuit 2019). Assuming we collect ranked ballots, a wide variety of voting proceduresbecome available (see, e.g., Brams and Fishburn 2002, Pacuit 2019, and Examples 2.11 and 2.16 below).One obvious idea for satisfying the principle of Majority Defeat is to say that one candidate defeatsanother if and only if a majority of voters prefer the first candidate to the second. Notoriously, however,this can result in every candidate being defeated, leaving no candidate who wins. In particular, there canbe a majority cycle , wherein a majority of voters prefer a to b , a majority of voters prefer b to c , anda majority of voters prefer c to a (Condorcet 1785). Majority cycles may also involve more than threecandidates. This so-called Paradox of Voting is perhaps the main theoretical obstacle to the possibility ofrational democratic decision making with more than two candidates. Riker (1982) has famously arguedthat the Paradox of Voting, along with the related Arrow Impossibility Theorem (Arrow 1963), destroysthe notion of a coherent “will of the people” in a democracy. Although there is not yet enough empiricalresearch to know how prevalent majority cycles are in real elections of various scales, majority cycles havebeen found in some large elections (see Deemen 2014). In a typical election, we expect (or at least hope) thatthere will be a Condorcet winner—a candidate a such that for every candidate b , a majority of voters prefer a to b —in which case some voting theorists believe that the choice is clear: elect the Condorcet winner (see,e.g., Felsenthal and Machover 1992; Maskin and Sen 2017a,b). A voting method is Condorcet consistent ifit chooses as the unique winner of an election the Condorcet winner, whenever a Condorcet winner exists. One could collect even more information from each voter than a rankings of the candidates: e.g., a ranking plus a distin-guished set of “approved” candidates (cf. Brams and Sanver 2009) or a grading of each candidate (cf. Balinski and Laraki 2010)from which a ranking can be derived. In this paper, we assume that only rankings of the candidates are collected from voters. Wolf (1970) takes these results to show that “majority rule is fatally flawed by an internal inconsistency” (p. 59), and Hardin(1990) takes them to cast “doubt on the conceptual coherence of majoritarian democracy” (p. 184). For further discussion, seeRisse 2001, 2009. As Van Deemen (2014) remarks, “it is remarkable to see that the empirical research on the paradox has been conductedmainly for large elections. Collective decision making processes in relatively small committees, such as corporate boards ofdirectors, management teams in organizations, government cabinets, councils of political parties and so on, have hardly beenstudied” (p. 325). Concerning the relevance of such empirical research, Ingham (2019) argues that “Arrow’s theorem and relatedresults threaten the populist’s principle of democratic legitimacy even if majority preference cycles never occur” (p. 97). if there is a Condorcet winner, elect that person, and if not, do something else with a differentjustification , Split Cycle provides a unified rule for cases with or without Condorcet winners:In an election with candidates x and y , say that x wins by n over y when there are n more voters whoprefer x to y than who prefer y to x . Then x defeats y according to Split Cycle if x wins by more than n over y for the smallest number n such that there is no majority cycle, containing x and y , in whicheach candidate wins by more than n over the next candidate in the cycle.As we show, Split Cycle mitigates spoiler effects (see Section 4.3) and has several other virtues, includingavoiding the so-called Strong No Show Paradox (see Holliday and Pacuit 2020a).In this paper, we arrive at Split Cycle (defined formally in Section 3) by another route. We proposesix general axioms concerning when one candidate should defeat another in a democratic election involvingtwo or more candidates (Section 4). Five of the axioms are widely satisfied by proposed voting procedures.The sixth axiom is a weakening of Kenneth Arrow’s famous condition of the Independence of IrrelevantAlternatives (IIA) (Arrow 1963). We call this weakening Coherent IIA. We prove that the five axioms plusCoherent IIA single out Split Cycle (Section 5). In particular, our main result is that Split Cycle is the mostresolute voting procedure satisfying the six axioms for democratic defeat.In the second half of the paper, we analyze how Split Cycle manages to escapes Arrow’s ImpossibilityTheorem and related impossibility results in social choice theory (Section 6). The answer is twofold: weweaken IIA to Coherent IIA, and we relax Arrow’s assumptions about the properties of the defeat relationbetween candidates. We explain how neither of these moves is sufficient by itself to escape Arrow-likeimpossibility theorems. But by doing both, Split Cycle provides a compelling response, we think, to boththe Paradox of Voting and Arrow’s Impossibility Theorem.A key aspect of our characterization of Split Cycle using the six axioms for defeat is that we work witha model in which elections can have different sets of voters and different sets of candidates, just as they doin reality. Given the importance of this variable-election setting to our characterization, we consider howstandard impossibility results for a fixed set of candidates/voters can be adapted to and even strengthenedin the variable-election setting, and yet how Split Cycle still escapes them (Sections 6.2 and 6.3). One of themethodological lessons of the paper, in our view, is the value of working in a variable-election framework.We start in Section 2 by reviewing the formal framework we will use to conduct our analysis. As suggested in Section 1, we work in a variable voter and variable candidate setting. This means that ourgroup decision method can input elections (formalized as profiles below) with different sets of voters anddifferent sets of candidates. To allow sets of voters and candidates of arbitrary (but finite) size in elections, By contrast, in voting theory and social choice theory, one often studies group decision methods all of whose inputs havethe same set of voters and same set of candidates. See Section 6.1.
4e first fix infinite sets V and X of voters and candidates , respectively. A given election will use only finitesubsets V ⊆ V and X ⊆ X . We consider elections in which each voter in the election submits a ranking ofall the candidates in the election, which we assume (for simplicity) is a strict linear order on the set X ofcandidates, i.e., a binary relation P on X satisfying the following conditions for all x, y, z ∈ X : • asymmetry: if xP y then not yP x ; • transitivity: if xP y and yP z , then xP z ; • connectedness: if x = y , then xP y or yP x .We take xP y to mean that the voter strictly prefers candidate x to candidate y . For a set X , let L ( X ) bethe set of all strict linear orders on X . Then we formalize the notion of an election as follows. Definition 2.1. A profile is a function P : V → L ( X ) for some nonempty finite V ⊆ V and nonempty finite X ⊆ X , which we denote by V ( P ) (called the set of voters in P ) and X ( P ) (called the set of candidates in P ), respectively. We call P ( i ) voter i ’s ballot , and we write ‘ x P i y ’ for ( x, y ) ∈ P ( i ) .When we display profiles, we show their “anonymized form” that records only the number of candidateswith each type of ballot, rather than the identities of the voters. For example: a b cb c ac a b The above diagram indicates that two voters rank b above c above a (notation: bca ), etc.It will be important later to consider the restriction of a profile to a subset of the candidates. Definition 2.2.
Given a binary relation P on X and Y ⊆ X , let P | Y be the restriction of P to the set Y ,i.e., P | Y = P ∩ ( Y × Y ) . Given a profile P , let P | Y be the profile with X ( P | Y ) = Y and V ( P | Y ) = V ( P ) obtained from P by restricting each voter’s ballot to the set Y . We now consider two different kinds of group decision methods, differing in what they output. The firstkind outputs a set of winners for the election.
Definition 2.3. A voting method is a function F on the domain of all profiles such that for any profile P , ∅ = F ( P ) ⊆ X ( P ) . We call F ( P ) the set of winners for P under F .As usual, if F ( P ) contains multiple candidates, we assume that some further tie-breaking process would thenapply, though we do not fix the nature of this process (see Schwartz 1986, pp. 14-5 for further discussion).If x F ( P ) , this means that x is excluded from the rest of the process that leads to the ultimate winner.The second kind of group decision method outputs an asymmetric binary relation on the set of candidates.In social choice theory, this relation is typically called the “strict social preference” relation. We interpretthis binary relation as a defeat relation for the election in the sense of Section 1. I.e., for all i ∈ V ( P ) , P | Y ( i ) = P ( i ) | Y . efinition 2.4. A variable-election collective choice rule (VCCR) is a function f on the domain of allprofiles such that for any profile P , f ( P ) is an asymmetric binary relation on X ( P ) , which we call the defeatrelation for P under f . For x, y ∈ X ( P ) , we say that x defeats y in P according to f when ( x, y ) ∈ f ( P ) .A well-known special case of a collective choice rule is what Arrow called a social welfare function (SWF).The output of an SWF is a strict weak order , i.e., a binary relation P on X satisfying asymmetry and thecondition that for all x, y, z ∈ X : • if xP y , then xP z or zP y .Note that these conditions imply that P is transitive. In the variable-election setting, we define the following. Definition 2.5. A variable-election social welfare function (VSWF) is a VCCR f such that for any profile P , f ( P ) is a strict weak order.For readers familiar with the standard setup in social choice theory, we note some subtleties about ourdefinitions. Remark 2.6.
1. We add the modifier ‘variable-election’ because the term ‘collective choice rule’ due to Sen (2017,Ch. 2 ∗ ) appears in a fixed voter and fixed candidate setting.2. In social choice theory, one often defines the output of a CCR (as Sen does) to be a “weak socialpreference” relation R that is reflexive instead of asymmetric. For social welfare functions, the choicedoes not matter, because strict weak orders P are in one-to-one correspondence with complete andtransitive relations R . However, since we aim to study the concept of defeat , an asymmetric relation,we have defined VCCRs accordingly.3. For simplicity, we build the axiom of Universal Domain into the definition of a VCCR, but one could ofcourse define a notion of VCCR where only certain profiles are in the domain of f (cf. Gaertner 2001).Any VSWF f induces a voting method f such that for any profile P , f ( P ) is the set of candidates whoare not defeated by any candidates in P according to f . That f ( P ) is a strict weak order implies that f ( P ) is nonempty, but in fact a much weaker condition is sufficient—namely, acyclicity. Definition 2.7.
Let P be an asymmetric binary relation on a set X . A cycle in P is a sequence x , . . . , x n of elements of X such that x P x , . . . , x n − P x n , x n = x , and all elements are distinct except x and x n . The relation P is acyclic if there is no cycle in P . A VCCR f is acyclic if for all profiles P , f ( P ) is acyclic.Any acyclic VCCR induces a voting method that outputs for a given profile the set of undefeated candi-dates. All defeated candidates are excluded from the rest of the process that leads to the ultimate winner. Lemma 2.8.
Given any acyclic VCCR f , the function f on the set of profiles defined by f ( P ) = { x ∈ X ( P ) | there is no y ∈ X ( P ) : y defeats x in P according to f } is a voting method, as ∅ = f ( P ) ⊆ X ( P ) . A relation R on X is complete if for all x, y ∈ X , we have xRy or yRx . Thus, we are use the term ‘cycle’ for what is usually called a simple cycle . F , we can consider the acyclic VCCRs from which F arises as in Lemma 2.8. Definition 2.9.
Let F be a voting method and f a VCCR. Then F is defeat rationalized by f if F = f .Let us now review some standard VCCRs. Several of the VCCRs are based on the majority preferencerelation, defined as follows. Definition 2.10.
Given a profile P and x, y ∈ X ( P ) , we say that x is majority preferred to y in P (and y is majority dispreferred to x in P ) if more voters rank x above y in P than rank y above x in P . We write x → P y (or x → y if P is clear from context) to indicate that x is majority preferred to y in P .A majority cycle in P is a cycle in the relation → P . Example 2.11. Simple Majority . For x, y ∈ X ( P ) , x defeats y in P if and only if x → y .2. Covering (Gillies 1959; Fishburn 1977; Miller 1980). For x, y ∈ X ( P ) , say that x left-covers y in P if for all z ∈ X ( P ) , if z → x , then z → y ; and x right-covers y in P if for all z ∈ X ( P ) , if y → z ,then x → z . (Left-covering and right-covering are equivalent if P has an odd number of voters but notfor an even number of voters.) We say that x defeats y in P according to the Left Covering VCCR(resp. Right Covering VCCR) if x → y and x left-covers y (resp. x → y and x right-covers y ). We saythat x defeats y in P according to the Fishburn VCCR if x left-covers y but y does not left-cover x .3. Copeland (Copeland 1951). The Copeland score of a candidate x in profile P is the number ofcandidates to whom x is majority preferred in P minus the number of candidates who are majoritypreferred to x in P : |{ z ∈ X ( P ) | x → z }| − |{ z ∈ X ( P ) | z → x }| . Then for x, y ∈ X ( P ) , x defeats y in P if and only if the Copeland score of x is greater than the Copeland score of y .4. Borda . The Borda score of a candidate x in profile P is calculated as follows: for every voter whoranks x in last place, x receives 0 points, and for every voter who ranks x in second to last place, x receives 1 point, and so on. That is, for every voter who ranks x in k places above last place, x receives k points. The sum of the points that x receives is x ’s Borda score in P . Then for x, y ∈ X ( P ) , x defeats y in P if and only if the Borda score of x is greater than the Borda score of y .5. Plurality . The plurality score of a candidate x in profile P is the number of voters who rank x in firstplace. Then for x, y ∈ X ( P ) , x defeats y in P if and only if the plurality score of x is greater than theplurality score of y .6. Ranked Choice ( Hare ). Given a profile P , define a sequence P , . . . , P n of profiles as follows. First, P = P . Second, given a profile P k in the sequence, if all candidates in P k have the same pluralityscore, set n = k to end the sequence; otherwise, where A k is the set of candidates whose pluralityscore in P k is above the lowest plurality score of a candidate in P k , let P k +1 be obtained from P k byrestricting the set of candidates to A k , i.e., P k +1 = ( P k ) | A k . The Hare score of candidate x in P isthe number of rounds of elimination that x survives, i.e., the greatest k such that x ∈ A k . Then x defeats y in P if and only if the Hare score of x is greater than the Hare score of y . When there is more than one candidate with lowest plurality score, this definition of Ranked Choice eliminates all suchcandidates. For discussion of another way of dealing with ties for the lowest plurality score, see Freeman et al. 2015, § 3. margins betweencandidates in the given profile.
Definition 2.12.
Let P be a profile and x, y ∈ X ( P ) . The margin of x over y in P is the number of voterswho rank x above y in P minus the number of voters who rank y above x in P . Let
M argin P ( x, y ) be themargin of x over y in P .The margin graph of P , M ( P ) , is the directed graph with weighted edges whose set of nodes is X ( P ) with an edge from x to y when x is majority preferred to y , weighted by the margin of x over y in P . Example 2.13.
For a profile P shown in anonymized form on the left, its margin graph M ( P ) is shown onthe right: a b cb c ac a b a cb Now the idea that the output of a VCCR depends only on margins can be formalized as follows. Definition 2.14.
A VCCR f is margin based if for any profiles P and P ′ , if M ( P ) = M ( P ′ ) , then f ( P ) = f ( P ′ ) .It is obvious that VCCRs 1-3 in Example 2.11 are margin based, but this is less obvious for Borda. Lemma 2.15.
For any profile P and x, y ∈ X ( P ) , x defeats y according to the Borda VCCR if and only ifthe sum of the margins of x over other all other candidates is greater than the sum of the margins of y overall other candidates. Other examples of margin based VCCR include the following.
Example 2.16. Weighted Covering (Dutta and Laslier 1999). Given a profile P and x, y ∈ X ( P ) , x defeats y in P if x → y and for all z ∈ X ( P ) , M argin P ( x, z ) ≥ M argin P ( y, z ) .2. Beat Path (Schulze 2011). Given a profile P and x, y ∈ X ( P ) a path from x to y is a sequence z , . . . , z n of candidates with z = x and z n = y such that each candidate is majority preferred to thenext candidate in the sequence. The strength of a path is the smallest margin between consecutivecandidates in the path. Then x defeats y in P according to the Beat Path VCCR if the strength ofthe strongest path from x to y is greater than the strength of the strongest path from y to x . Note that the margin of x over y is negative when y is majority preferred to x . Cf. De Donder et al.’s (2000) notion of C1.5 functions. Note that Fishburn’s (1977) C2 functions can use not only the difference between the number of voters who prefer x to y and the number of voters who prefer y to x , but also those twonumbers themselves. The Pareto VCCR in Example 4.6 is C2 but not margin based (not C1.5). Remember that the margin of x over z is negative when z is majority preferred to x . emark 2.17. Within the family of margin based VCCRs, we can make a useful three-way distinction.1. A majority graph is any directed graph M whose edge relation is asymmetric. Given a profile P , the majority graph of P , M ( P ) , is the directed graph whose set of nodes is X ( P ) with an edge from x to y when x is majority preferred to y in P . We say that a VCCR f is majority based if for any profiles P and P ′ , if M ( P ) = M ( P ′ ) , then f ( P ) = f ( P ′ ) . The Simple Majority, Covering, and CopelandVCCRs in Example 2.11 are majority based in this sense.2. A qualitative margin graph is a pair M = ( M, ≺ ) where M is a majority graph and ≺ is a strict weakorder on the set of edges of M . The qualitative margin graph of P is the pair M ( P ) = ( M ( P ) , ≺ P ) suchthat for any edges ( a, b ) and ( c, d ) in M ( P ) , we have ( a, b ) ≺ P ( c, d ) if M argin P ( a, b ) < M argin P ( c, d ) .We say that a VCCR f is qualitative-margin based if for any profiles P and P ′ , if M ( P ) = M ( P ′ ) ,then f ( P ) = f ( P ′ ) . The Weighted Covering and Beat Path VCCRs in Example 2.16 are qualitative-margin based in this sense, as is the Split Cycle VCCR defined in Section 3, but none of these VCCRsare majority based.3. We already defined margin based
VCCRs in Definition 2.14. Note that the Borda VCCR in Example2.11 is margin based but not qualitative-margin based.This concludes our review of basic notions. In the next section we turn to our preferred voting methodand VCCR.
In Holliday and Pacuit 2020a, we studied a voting method that we call Split Cycle. Here we formulate SplitCycle as a VCCR, which defeat rationalizes the Split Cycle voting method (recall Definition 2.9). We givetwo formulations in Definition 3.1 and Lemma 3.5, respectively. The first definition of Split Cycle formalizesthe definition given in Section 1. For a profile P , candidates x, y ∈ X ( P ) , and natural number n , say that x wins by n over y if the margin of x over y in P is n (recall Definition 2.12). Definition 3.1.
Given a profile P and candidates x, y ∈ X ( P ) , x defeats y in P according to Split Cycle if x wins by more than n over y for the smallest number n such that there is no majority cycle, containing x and y , in which each candidate wins by more than n over the next candidate in the cycle.The basic idea is that when the electorate’s majority preference relation is incoherent , in the sense thatthere is a majority cycle, this raises the threshold required for one candidate a in the cycle to defeat another b —but not infinitely. If we raise the threshold n sufficiently, then there will be no incoherence involving a and b with respect to the higher threshold, i.e., no cycles in the win by more than n relation that contain a and b . If the margin of a over b is greater than this sufficiently large n , Split Cycle says that a defeats b . Example 3.2.
Consider a profile P with the following margin graph: Cf. Fishburn’s (1977) C1 functions. In terms of the C1, C1.5, and C2 classifications in Footnotes 11 and 13, qualitative-margin based methods could becalled C1.25. Another example of a qualitative-margin based VCCR defeat rationalizes the Simpson-Kramer Minimax method(Simpson 1969, Kramer 1977): x defeats y if x ’s largest majority loss is smaller than y ’s largest majority loss. cbd The only majority cycle is a, b, c, a . Note that each candidate wins by more than over the next candidatein the sequence. However, it is not the case that each candidate wins by more than over the next candidatein the sequence. Thus, a threshold of win by more than splits the a, b, c, a cycle: a cb Hence there is no incoherence involving a and b with respect to the win by more than relation. Then since a wins by more than over b , Split Cycle says that a defeats b . Similarly, since b wins by more than over c , Split Cycle says that b defeats c . However, since c does not win by more than over a , Split Cycle saysthat c does not defeat a . Crucially, though, since c and d are not involved in any cycles together, and c winsby more than 0 over d , Split Cycle says that c defeats d . The key point is that incoherence can be localized : c and a belong to a cycle together, but c and d do not. By the same reasoning, a defeats d , and b defeats d .Thus, we obtain the following defeat relation: a cbdD DDDD Note that just as in a sporting tournament, it can happen that while team a defeats team b and team b defeats team c , team a does not defeat team c , the same phenomenon occurs in the defeat relation above.Finally, since a is the only undefeated candidate, a is the winner according to Split Cycle.Additional examples of determining the Split Cycle defeat relation will be given below (Example 4.8,Remark 4.10, and Example 4.12). For still more examples, see Holliday and Pacuit 2020a. Remark 3.3.
Where f is the Split Cycle VCCR as in Definition 3.1, the induced voting method f , whichpicks as winners the undefeated candidates, is the Split Cycle voting method. As a voting method, SplitCycle is Condorcet consistent: if x is majority preferred to every other candidate y —if x is a Condorcetwinner—then x is the unique winner of the election. For if x is the Condorcet winner, then there are nocycles involving x , so x defeats all other candidates according to Split Cycle.An equivalent definition of Split Cycle can be given in terms of the following concept.10 efinition 3.4. Let P be a profile and ρ a majority cycle in P . The splitting number of ρ in P is thesmallest margin between consecutive candidates in ρ . Let Split P ( ρ ) be the splitting number of ρ in P .For example, the splitting number of the cycle a, b, c, a in the profile in Example 3.2 is 3. In Holliday and Pacuit2020a, we took the following formulation of Split Cycle to be the official definition (for a proof of Lemma3.5, see Appendix A). Lemma 3.5.
Let P be a profile and x, y ∈ X ( P ) . Then x defeats y in P according to Split Cycle if andonly if M argin P ( x, y ) > and M argin P ( x, y ) > Split P ( ρ ) for every majority cycle ρ in P containing x and y. Thus, in Example 3.2, a defeats b because M argin P ( a, b ) = 5 , the only majority cycle is a, b, c, a , and itssplitting number is . Note that since we are only comparing the sizes of margins, Split Cycle is qualitative-margin based in the sense of Remark 2.17.A useful fact, proved in Holliday and Pacuit 2020a, is that it suffices to only look at majority cycles inwhich y directly follows x . We include the proof in Appendix A to keep the paper self-contained.
Lemma 3.6.
Let P be a profile and x, y ∈ X ( P ) . Then x defeats y in P according to Split Cycle if andonly if M argin P ( x, y ) > and M argin P ( x, y ) > Split P ( ρ ) for every majority cycle ρ in P of the form x → y → z → · · · → z n → x. In Holliday and Pacuit 2020a, we show that Split Cycle—understood as a voting method, i.e., as f forthe Split Cycle VCCR f —satisfies a number of desirable axioms for voting methods, and we systematicallycompare Split Cycle to other margin based voting methods, including Beat Path and Ranked Pairs (Tideman1987). In particular, we show that Split Cycle is the only known voting method that simultaneously satisfiesa Pareto axiom, an anti-spoiler axiom, and an axiom preventing the so-called Strong No Show Paradox. Inthe next two sections, we take a different approach: we characterize Split Cycle as a VCCR, rather than avoting method, and we characterize the Split Cycle VCCR relative to all VCCRs.
In this section, we propose six axioms concerning when one candidate should defeat another in a democraticelection involving two or more candidates. Four axiom are standard (Section 4.1); one is less well known butalso from the previous literature (Section 4.2); and the key axiom is new (Section 4.3).
The first four axioms are ubiquitous in social choice and voting theory. The first axiom appears already inMay’s (1952) characterization of majority rule for two-candidate elections: This fact relates Split Cycle to concepts in Heitzig 2002, as discussed in Holliday and Pacuit 2020a. x defeats y in P , and P ′ is obtained from P by swapping the ballotsassigned to two voters, then x still defeats y in P ′ (Anonymity); and if x defeats y in P , and P ′ isobtained from P by swapping x and y on each voter’s ballot, then y defeats x in P ′ (Neutrality). It is clear that all VCCRs defined so far in this paper satisfy Anonymity and Neutrality.The second axiom is definitive of the problem of choosing winners that we aim to solve:A2. Availability: for every P , there is some undefeated candidate in P .To say that in some profiles all candidates are defeated and hence excluded from further consideration—sono candidate is available to become the ultimate winner—is to give up on solving the problem. Unlike theSimple Majority VCCR (Example 2.11), Split Cycle satisfies Availability. Proposition 4.1.
Split Cycle satisfies Availability.
Proof.
Suppose there is a profile P in which every candidate is defeated by some other. Since X ( P ) is finite,it follows that there is a sequence ρ = x , . . . , x n of candidates with x = x n such that each candidate defeatsthe next candidate in the sequence. It follows by Lemma 3.5 that ρ is a majority cycle in which the marginof each candidate over the next is greater than the splitting number of ρ , which is impossible.Note that Availability is strictly weaker than the assumption that a VCCR is acyclic. There being no defeatcycles implies that some candidate is undefeated (given that the set of candidates in a profile is finite), butsome candidate being undefeated does not imply that there are no defeat cycles (e.g., a could be undefeatedwhile there is a defeat cycle involving b , c , and d ). Nonetheless, the proof of Proposition 4.1 (starting in thesecond sentence) shows that Split Cycle is an acyclic VCCR as well.For the third axiom, given any profile P and natural number m , the profile m P is obtained from P byreplacing each voter by m copies of that voter. For example, if P has three voters i, j, k , then P has sixvoters i , i , j , j , k , k such that the ballots of i and i in P are the same as those of i in P , etc.A3. (Upward) Homogeneity: for every P , if x defeat y in P , then x defeats y in P .Homogeneity is usually stated as the condition that for any m ≥ , x defeats y in P if and only if x defeats y in m P . As Smith (1973, p. 1029) remarks, “Homogeneity seems an extremely natural requirement; if eachvoter suddenly splits into m voters, each of whom has the same preferences as the original, it would be hardto imagine how the “collective preference” would change.” Nonetheless, we use the weaker version statedabove since it is sufficient for our main result. Almost all standard voting procedures satisfy (the usualstatement of) Homogeneity. That Split Cycle satisfies Homogeneity follows from the fact that Split Cycleis qualitative-margin based as in Remark 2.17, and P and P have the same qualitative margin graphs.The fourth axiom is another one of the most basic principles of voting theory. The term ‘Monotonicity’is used for a number of different conditions, but our formulation is equivalent (for profiles of linear ballots)to Arrow’s (1963) axiom of Positive Association of Social and Individual Values: These versions of Anonymity and Neutrality stated in terms of the transposition of two ballots/candidates are equivalentto the usual versions stated in terms of a permutation of the ballots/candidates, since any permutation can be obtained by asequence of transpositions. The usual version of, e.g., Neutrality states that if σ is a permutation of X , and σ P is the profileobtained from P by setting xσ P i y if and only if σ ( x ) P i σ ( y ) , then x defeats y in σ P if and only if σ ( x ) defeats σ ( y ) in P . One exception is the Dodgson voting procedure (see Fishburn 1977, Brandt 2009). Positive Association states that if x defeats y in P according to f , and P ′ is a profile such that for all x ′ , y ′ ∈ X \ { x } ,(i) P |{ x ′ ,y ′ } = P ′|{ x ′ ,y ′ } , (ii) for all i ∈ V , x P i y ′ implies x P ′ i y ′ , and (iii) for all i ∈ V , y ′ P ′ i x implies y ′ P i x , then x defeats y in P ′ according to f . x defeats y in a profile (resp. two-candidate profile) P , and P ′ is obtained from P by some voter i moving x above the candidate that i ranked immediately above x in P , then x defeats y in P ′ .We think Monotonicity should hold for profiles with any number of candidates, but Monotonicity for two-candidate profiles is sufficient for the proof of our main result. This is noteworthy because the Ranked ChoiceVCCR in Example 2.11 does not satisfy Monotonicity for arbitrary profiles (see, e.g., Felsenthal 2012) butdoes for two-candidate profiles. All other VCCRs defined above satisfy Monotonicity for arbitrary profiles.The axioms proposed so far imply the principle of Majority Defeat for two-candidates profiles. The proofis essentially part of the proof of May’s (1952) characterization of majority rule. Lemma 4.2. If f satisfies Anonymity, Neutrality, and Monotonicity with respect to two-candidate profiles,then f satisfies Special Majority Defeat: for any two-candidate profile P , x defeats y in P according to f only if x is majority preferred to y . Proof.
Suppose x defeats y in P . It follows by Anonymity, Neutrality, and the asymmetry of defeat that thenumber of voters who rank x above y is not equal to the number who rank y above x . Now we claim that x is majority preferred to y . Suppose instead that y is majority preferred to x by a margin of m . Flip voterswith y P i x to x P ′ i y until we obtain a profile P ′ in which x is majority preferred to y by a margin of m . Since x defeats y in P , x still defeats y in P ′ by Monotonicity. But P ′ can also be obtained from P by the voterand candidate swaps described in the statements of Anonymity and Neutrality. Thus, since x defeats y in P , y defeats x in P ′ . Hence in P ′ , x defeats y and y defeats x , contradicting the asymmetry of defeat. Remark 4.3.
Monotonicity is weaker than May’s (1952) condition of Positive Responsiveness, which inaddition requires that if y does not defeat x in P , then changing a single voter from y P i x to x P ′ i y results in x defeating y in P ′ . We find imposing Positive Responsiveness in general, i.e., for elections with any numberof candidates, much too strong, so we prefer to motivate majority rule in two-candidate elections usingaxioms that we find plausible for any number of candidates, as in Proposition 4.7 below. Like the axiom of Homogeneity, the next axiom is a variable voter axiom. Say that two voters i and j have reversed ballots in a profile Q if for all x, y ∈ X ( Q ) , we have x Q i y if and only if y Q j x . For example, if i has abcd and j has dcba , then i and j have reversed ballots. Adding a pair of voters with reversed ballotsto a profile does not change the margins between any candidates. A natural thought is that such votersbalance each other out, so adding such a pair to an election should not change the defeat relations betweencandidates. This leads to what Saari (2003) calls the Neutral Reversal Requirement.A5. Neutral Reversal: if P ′ is obtained from P by adding two voters with reversed ballots, then x defeats y in P if and only if x defeats y in P ′ .Not only Split Cycle but all other margin based VCCRs (recall Example 2.11) satisfy Neutral Reversal.However, Neutral Reversal is weaker than the assumption that a VCCR is margin based. Cf. Woeginger (2002), who notes that while Anonymity and Neutrality “are natural and fairly weak, the positive respon-siveness axiom is usually criticized for being too strong” (p. 89). xample 4.4. We define the Positive/Negative VCCR (cf. Heckelman and Ragan Forthcoming) as follows.In a profile P , a candidate x receives point for every voter who ranks x first and − point for every voterwho ranks x last. The score of x in P is the sum of the points x receives from voters. Then x defeats y in P if the score of x is greater than the score of y . This Positive/Negative VCCR satisfies Neutral Reversal,as a pair of reversed ballots adds a net score of 0 to each candidate. Yet it is easy to construct profiles withthe same margin graphs that have different defeat relations.Not all common VCCRs satisfy Neutral Reversal, as Examples 4.5-4.6 below show. To analyze violationsof Neutral Reversal, we distinguish its two directions: • Upward Neutral Reversal: if P ′ is obtained from P by adding two voters with reversed ballots, then if x defeats y in P , x defeats y in P ′ . • Downward Neutral Reversal: if P ′ is obtained from P by adding two voters with reversed ballots, thenif x defeats y in P ′ , x defeats y in P . Example 4.5.
Consider the Plurality VCCR from Example 2.11. Let P be any profile for candidates a, b, c .Adding to P a pair of voters with the reversed ballots abc and cba to obtain a profile P ′ increases the pluralityscores of a and c by one but does not increase the plurality score of b . From here it is easy to see that thePlurality VCCR violates both Upward and Downward Neutral Reversal. Example 4.6.
The Pareto VCCR f is defined as follows: for any profile P and x, y ∈ X ( P ) , x defeats y in P if and only if all voters in P rank x above y . Clearly adding two voters with reversed ballots to a profilein which x is unanimously ranked above y results in a profile in which x is not unanimously ranked above y .Thus, the Pareto VCCR violates Upward Neutral Reversal. However, it trivially satisfies Downward NeutralReversal, because if P ′ has a pair of voters with reversed ballots, then no candidates defeats any other in P ′ .The Pareto VCCR seems reasonable for certain special purposes, e.g., in a small club, unanimity may bevalued and often possible. However, in elections where disagreement is expected, the Pareto VCCR wouldbe of little help in narrowing down the range of possible winners, as so few candidates would defeat others.Of course, we agree that it is a sufficient condition for x to defeat y that x is unanimously ranked above y . A VCCR f is said to satisfy the Pareto axiom if for all profiles P and x, y ∈ X ( P ) , if x P i y for all i ∈ V ( P ) , then x defeats y in P according to f . Split Cycle clearly satisfies the Pareto axiom. Moreover,we can use Pareto and Upward Neutral Reversal to derive the converse of Special Majority Defeat (Lemma4.2), thereby obtaining a characterization of majority rule for two-candidate elections that differs from May’s(1952) famous characterization (cf. Aşan and Sanver 2002, Woeginger 2002, and Llamazares 2006, Cor. 12). Proposition 4.7.
For any VCCR f , the following are equivalent:1. f coincides with majority rule on two-candidate profiles;2. f satisfies the following axioms with respect to two-candidate profiles: Anonymity, Neutrality, Mono-tonicity, Pareto, and Upward Neutral Reversal. Proof.
The implication from 1 to 2 is easy to check. From 2 to 1, we already proved in Lemma 4.2 that if f satisfies Anonymity, Neutrality, and Monotonicity for two-candidate profiles, then in such a profile, x defeats14 only if x is majority preferred to y . We now use Pareto and Upward Neutral Reversal to show that if x ismajority preferred to y , then x defeats y . For suppose P is a two-candidate profile in which x is majoritypreferred to y . Consecutively remove pairs of voters ( i, j ) with x P i y and y P j x until we obtain a profile P in which all voters rank x over y . By Pareto, x defeats y in P . Then by repeated application of UpwardNeutral Reversal, adding back the removed pairs of voters, x defeats y in P .We prefer this characterization of majority rule to that of May (1952) for the reason given in Remark 4.3. Suppose x defeats y in a profile P , and a profile P ′ is exactly like P with respect to how every voter ranks x vs. y . Should it follow that x defeats y in P ′ ? Arrow’s (1963) famous axiom of the Independence of IrrelevantAlternatives says ‘yes’ (see Section 6.1). But we say ‘no’ if P ′ is more incoherent than P , in terms of cycles.If P ′ is sufficiently incoherent, we may need to suspend judgment on many defeat relations that we couldcoherently accept in P . To overlook this point is to commit what we call The Fallacy of IIA. Although there is a perfectly reasonable notion of the advantage of x over y that only depends on howvoters rank x vs. y , whether that intrinsic advantage is sufficient for x to defeat y may depend on a standardthat takes into account the whole election, e.g., that takes into account whether the electorate is incoherentwith respect to a set of candidates including x, y (see Holliday and Kelley 2020 for a formalization of theadvantage-standard idea). That standards may be context dependent should be no surprise: just as whethera person counts as “tall” depends on who else is being assessed for tallness in the context of our judgment,whether one candidate’s performance against another counts as “a defeat” depends on which other pairwisecandidate performances are also being assessed as potential defeats in the context of our judgment. Example 4.8.
In the profiles P and P ′ below, we have P |{ a,b } = P ′|{ a,b } . In the context of the perfectlycoherent profile P , the margin of n for a over b should be sufficient for a to defeat b . But in the context ofthe incoherent profile P ′ , it is not sufficient: no one can be judged to defeat anyone else (this follows fromAnonymity, Neutrality, and Availability). Thus, this is a counterexample to IIA. n n n a b c b a a c c b a cbn nnn n n a b c b c a c a b a cbn nn However, it is not as if the standard for defeat in every case of a majority cycle is unattainable. In the profile Q below, we believe that the advantage of a over b is sufficient for a to defeat b : We take this criticism of IIA to differ from some other criticisms of IIA, such as those in Mackie 2003, Ch. 6. For a related proposal in the setting of judgment aggregation to set supermajority thresholds in a local, context sensitiveway, see Cariani 2016. a b cb c ac a b a cb According to Split Cycle, the standard for a to defeat b in a profile, which the margin of a over b must surpass,is the maximum of the splitting numbers of the cycles containing a and b . Since the splitting number of thecycle in the profile Q is 1, the margin of a over b surpasses the standard, so a defeats b . Remark 4.9.
In an illuminating result, Patty and Penn (2014) prove that Arrow’s IIA is equivalent to thecondition of unilateral flip independence , which states that if two profiles are alike except that one voterflips one pair of adjacent candidates on her ballot, then the defeat relations for the two profiles can differat most on the flipped candidates. They write that this theorem “demonstrates a fundamental basis of thenormative appeal of IIA” (p. 52) (cf. Patty and Penn 2019, p. 155). However, observe that for the profiles P and P ′ in Example 4.8, if n = 1 , then a single voter (the middle voter) flipping adjacent candidates onher ballot ( ac to ca ) takes us from the coherent profile P , in which there is no difficulty in judging that a defeats b , to the incoherent profile P ′ , in which no one can be judged to defeat anyone else. Hence unilateralflip independence makes the same mistake as IIA in ignoring how context can affect the standard for defeat. Remark 4.10.
Maskin (2020) proposes a weakening of IIA called Modified IIA, which states that if profiles P and P ′ are alike in how every voter ranks x vs. y , and for each voter i and candidate z , i ranks z inbetween x and y in P if and only if i ranks z in between x and y in P ′ , then x defeats y in P if and only if x defeats y in P ′ . Saari (1994; 1995; 1998) proposed a stronger axiom, though still weaker than IIA, calledIntensity IIA: if profiles P and P ′ are alike in how every voter ranks x vs. y , and for each voter i , the numberof candidates that i ranks in between x and y in P is the same as the number of candidates that i ranks inbetween x and y in P ′ , then x defeats y in P if and only if x defeats y in P ′ . Modified IIA and Intensity IIAare problematic for the same reason that IIA is, only we now need four candidates to see why. In the profiles P and P ′ below, we have P |{ a,b } = P ′|{ a,b } , and for each voter i and candidate z , i ranks z in between a and b in P if and only if i ranks z in between a and b in P ′ . In the context of the perfectly coherent profile P , themargin of n for a over b should be sufficient for a to defeat b . But in the context of the incoherent profile P ′ , it is not sufficient: no one can be judged to defeat anyone else (this follows from Anonymity, Neutrality,and Availability). Thus, this is a counterexample to Modified IIA and Intensity IIA. n n n n a b a ab c b b c d c dd a d c a bcd n n n n Patty and Penn (2019) do not think that IIA is compelling for voting procedures in elections (see their Section 3.1), butthey do find it compelling in contexts of multicriterial decision making. n n n a b c d b c d a c d a b d a b c a bcd n n n n Maskin (2020) suggests the benefit of Modified IIA is that it rules out vote-splitting, which he illustrates usingspoiler effects in Plurality voting as in Examples 1.1 and 1.3. However, Modified IIA is neither necessary norsufficient for a voting procedure to have good anti-spoiler properties. Split Cycle does not satisfy ModifiedIIA—it correctly says that a defeats b in P but not P ′ above—yet Split Cycle satisfies strong anti-spoilerproperties: not only Independence of Clones (Tideman 1987), as shown in Holliday and Pacuit 2020a, butalso a condition of Immunity to Spoilers, as shown below. On the other hand, Borda satisfies Modified IIAyet satisfies neither Independence of Clones nor Immunity to Spoilers, as shown in Example 4.15 below.Avoiding The Fallacy of IIA does not mean abandoning the idea behind IIA entirely. We need onlydepart from its local evaluation of x vs. y when increasing incoherence demands that we be more conservativein locking in relations of defeat. If there is no increase in incoherence from profile P to P ′ , then if the intrinsicadvantage of x over y is sufficient for x to defeat y in P , we think it should still be sufficient for x to defeat y in P ′ . Moreover, a clearly sufficient condition for there to be no increase in incoherence from P to P ′ isthe following: the margin graph of P ′ is obtained from that of P by deleting zero or more candidates otherthan x and y and deleting or reducing the margins on zero or more edges not connecting x and y . For suchdeletions or reductions can only reduce incoherence. For example, in the profile Q in Example 4.8, deletingcandidate c or deleting or reducing the margins on the c → a or b → c edges only reduces incoherence, so a ’s defeat of b should be preserved. Thus, we arrive at our proposed axiom of Coherent IIA.A6. Coherent IIA: if x defeats y in P , and P ′ is a profile such that P |{ x,y } = P ′|{ x,y } and the margin graphof P ′ is obtained from that of P by deleting zero or more candidates other than x and y and deletingor reducing the margins on zero or more edges not connecting x and y , then x still defeats y in P ′ .In Section 6.1.2, we show that Coherent IIA implies Weak IIA (Baigent 1987): if two profiles are exactlyalike with respect to how every voter ranks x vs. y , it cannot be that in one profile x defeats y while in theother profile y defeats x . At most, a defeat that holds in one can be withdrawn in the other. Proposition 4.11.
Split Cycle satisfies Coherent IIA.
Proof.
Suppose x defeats y in P , so by Lemma 3.5, M argin P ( x, y ) > Split P ( ρ ) for any majority cycle ρ in P containing x and y . Since P |{ x,y } = P ′|{ x,y } , M argin P ( x, y ) = M argin P ′ ( x, y ) . Since M ( P ′ ) isobtained from M ( P ) by deleting zero or more candidates other than x and y and zero or more edges notconnecting x and y , every majority cycle ρ in P ′ containing x and y is already a majority cycle in P containing x and y , and as no margins have increased from P to P ′ , Split P ( ρ ) ≥ Split P ′ ( ρ ) . It followsthat M argin P ′ ( x, y ) > Split P ′ ( ρ ) for any majority cycle ρ in P ′ containing x and y . Thus, x defeats y in P ′ by Lemma 3.5. For an intellectual history of the ideas behind IIA, going back to Condorcet and Daunou, see McLean 1995. By allowing for the deletion of candidates, Coherent IIA is a weakening of a variable-candidate version of IIA that we callVIIA, defined in Section 6.2.
Example 4.12.
To see that Borda fails Coherent IIA, consider the following profiles P and P ′ : x y y a x x b a cc b b y c a xa ycb
24 4 4 2 2 21 1 2 a y y b a x c b c x c b y x a xa ycb According to the Borda VCCR, x defeats y in P : despite the fact that only one person prefers x to y , whereas three prefer y to x , the proponent of Borda ascribes some significance to the fact that the first voter places a, b, c between x and y (perhaps strategically, of course). Now although P |{ x,y } = P ′|{ x,y } and the margingraph of P ′ is obtained from that of P by deleting some edges not connecting x and y , Borda changes theverdict for P ′ and says that y defeats x in P ′ . Thus, Borda violates Coherent IIA. The mistake, in our view,was to judge that x defeats y in P in the first place. According to Split Cycle, by contrast, y defeats x in P because y is majority preferred to x and there are no majority cycles in P .Example 4.12 shows that Borda fails to satisfy the principle of Majority Defeat from Section 1: if x defeats y in P , then x is majority preferred to y in P . Lemma 4.13.
Anonymity, Neutrality, Monotonicity (for two-candidate profiles), and Coherent IIA togetherimply Majority Defeat.
Proof.
Suppose x defeats y in P . Then by Coherent IIA, x defeats y in P |{ x,y } . Given Anonymity, Neutrality,and Monotonicity, it follows by Lemma 4.2 that x is majority preferred to y in P |{ x,y } and hence in P . Remark 4.14.
Whenever an election has a Condorcet winner, Majority Defeat implies that the Condorcetwinner is undefeated, but it does not imply that the Condorcet winner is the only undefeated candidate.Thus, Majority Defeat does not by itself imply Condorcet consistency.Finally, we will show that Coherent IIA (together with Anonymity, Neutrality, and Monotonicity) rulesout the kind of spoiler effects shown in Examples 1.1-1.3. For this, we must consider an election with andwithout a potential spoiler. Given a profile P and b ∈ X ( P ) , let P − b be the profile obtained from P bydeleting b from all ballots. In Holliday and Pacuit 2020a, we define the following axiom: I.e., P − b = P | X ( P ) \{ b } , using the notation of Definition 2.2. Immunity to Spoilers: if a is undefeated in P − b , and a is majority preferred to b in P , and b is defeatedin P , then a is still undefeated in P .Examples 1.1 and 1.3 show how Plurality voting can violate Immunity to Spoilers. In the first example,assume Gore would win in the two-candidate profile P − Nader and that Gore is majority preferred to Naderin the full election P . Then since Nader is defeated in P , Immunity to Spoilers requires that Nader not spoilthe election for Gore, i.e., that Gore is still a winner in P . The analysis of Example 1.3 is similar, makingsome assumptions about voters’ rankings of Cruz, Kasich, and Rubio. Example 1.2 shows how RankedChoice voting can violate Immunity to Spoilers. Montroll wins in the two-candidate profile P − Wright , andMontroll is majority preferred to Wright in the full election P . Then since Wright is defeated in P , Immunityto Spoilers requires that Wright not spoil the election for Montroll, i.e., that Montroll is still a winner in P .To see how Borda violates Immunity to Spoilers, consider the following. Example 4.15.
Let P − b (left) and P (right) be the following profiles: c aa c a c c ab ca b a c b According to Borda, a defeats c in P − b . Note that a is majority preferred to b in P , and b is defeated in P by both a and c according to Borda. But the addition of the loser b spoils the election for a , as c defeats a in P according to Borda. This violates Immunity to Spoilers. It is also violates Independence of Clones(Tideman 1987), as b is a clone of c (no candidates appear in between b and c on any voter’s ballot). Finally, c defeating a in P but not in P − b is another violation of Coherent IIA (see the proof of Proposition 4.16).In fact, Coherent IIA (together with the other mentioned axioms) implies an even strong anti-spoileraxiom from Holliday and Pacuit 2020a: • Strong Stability for Winners: if a in undefeated in P − b , and b is not majority preferred to a in P , then a is still undefeated in P . Proposition 4.16.
Anonymity, Neutrality, Monotonicity (for two-candidate profiles), and Coherent IIAtogether imply Strong Stability for Winners.
Proof.
Suppose a is undefeated in P − b according to f and that b is not majority preferred to a in P . Supposefor contradiction that a is defeated in P according to f . Since b is not majority preferred to a in P , it followsby Lemma 4.13 that b does not defeat a in P according to f . Hence there is some c ∈ X ( P ) \ { b } thatdefeats a in P according to f . Then since P |{ a,c } = ( P − b ) |{ a,c } and the margin graph of P − b is obtainedfrom that of P by deleting a candidate other than a and c , it follows by Coherent IIA that c defeats a in P − b according to f , contradicting our initial assumption.Thus, contrary to Maskin 2020, it is Coherent IIA rather than Modified IIA that mitigates spoiler effects.19 Characterization
In this section, we prove our main result using the axioms proposed in Section 4. Given VCCRs f and g , wesay that g is at least as resolute as f if for every profile P and x, y ∈ X ( P ) , if x defeats y in P accordingto f , then x defeats y in P according to g . It follows that the set of winners according to g (i.e., the set ofundefeated candidates according to g ) is always a subset of the set of winners according to f (i.e., the set ofundefeated candidates according to f ). Roughly speaking, more resolute means smaller sets of winners .Given a class C of VCCRs and g ∈ C , we say that g is the most resolute VCCR in C if g is at leastas resolute as every VCCR in C , or equivalently, if for every profile P , x, y ∈ X ( P ) , and f ∈ C , if x defeats y in P according to f , then x defeats y in P according to g . A number of voting procedures can becharacterized as “the most resolute procedure satisfying such and such properties” (see, e.g., Brandt et al.2013; Brandt and Seedig 2014). We now give such a characterization of Split Cycle. Theorem 5.1.
Split Cycle is the most resolute of all VCCRs satisfying the six axioms for defeat:A1. Anonymity and Neutrality: if x defeats y in P , and P ′ is obtained from P by swapping the ballotsassigned to two voters, then x still defeats y in P ′ (Anonymity); and if x defeats y in P , and P ′ isobtained from P by swapping x and y on each voter’s ballot, then y defeats x in P ′ (Neutrality).A2. Availability: for every P , there is some undefeated candidate in P .A3. (Upward) Homogeneity: for every P , if x defeats y in P , then x defeats y in P .A4. Monotonicity (for two-candidate profiles): if x defeats y in P (a two-candidate profile), and P ′ isobtained from P by some voter i moving x above the candidate that i ranked immediately above x in P , then x defeats y in P ′ .A5. Neutral Reversal: if P ′ is obtained from P by adding two voters with reversed ballots, then x defeats y in P if and only if x defeats y in P ′ .A6. Coherent IIA: if x defeats y in P , and P ′ is a profile such that P |{ x,y } = P ′|{ x,y } and the margin graphof P ′ is obtained from that of P by deleting zero or more candidates other than x and y and deletingor reducing the margins on zero or more edges not connecting x and y , then x still defeats y in P ′ . Proof.
We have already observed that Split Cycle satisfies the axioms. Next, we show that for any VCCR f satisfying the axioms for defeat and any profile P , if x defeats y in P according to f , then x defeats y in P according to Split Cycle. Toward a contradiction, suppose x defeats y in P according to f but notaccording to Split Cycle. Since P may have an odd number of voters, consider P . It follows by (Upward)Homogeneity that x defeats y in P according to f . But according to Split Cycle, x does not defeat y in P (whose qualitative margin graph is the same as P ). Since x defeats y according to f , by MajorityDefeat (Lemma 4.13), we have M argin P ( x, y ) > . Moreover, since P has an even number of voters, M argin P ( x, y ) is even. Now since x does not defeat y in P according to Split Cycle, by Lemma 3.6there is a majority cycle x → y → z → · · · → z n → x in P such that M argin P ( x, y ) is less than orequal to every margin along the cycle. Let Q be obtained from by adding zero or more reversal pairsof voters so that | V ( Q ) | ≥ ( n + 2) M argin P ( x, y ) . Then by (Upward) Neutral Reversal, x defeats y in Q according to f . Let M ′ be the weighted directed graph obtained from M (2 P ) by deleting all candidates20xcept x, y, z , . . . , z n and all edges except the edges in the cycle and reducing the weights on all remainingedges so they are equal to M argin P ( x, y ) . Let k = M argin P ( x, y ) / . We now construct a profile P ′ whosemargin graph is M ′ . For each edge a → b in x → y → z → · · · → z n → x , where c , . . . , c n are theelements from { x, y, z , . . . , z n } \ { a, b } such that b → c → · · · → c n → a , we add voters to P ′ as follows: • if a = x (and hence b = y ), then for k voters from V ( Q ) who rank x above y in Q , add them to P ′ with the ballot a , b , c , . . . , c n (which in this case is x , y , z , . . . , z n ), and for k other voters from V ( Q ) who rank x above y in Q , add them to P ′ with the ballot c n , . . . , c , a , b (which in this case is z , . . . , z n , x , y ). See the first and second columns in the profile in Figure 1. • if a = x and y occurs before x in the sequence b, c , . . . , c n , a , then for k voters from V ( Q ) who rank y above x in Q , add them to P ′ with the ballot a , b , c , . . . , c n , and for k voters from V ( Q ) who rank x above y in Q , add them to P ′ with the ballot c n , . . . , c , a , b . See, e.g., the third and fourth columnsin the profile in Figure 1. • if a = x and x occurs before y in the sequence b, c , . . . , c n , a , then for k voters from V ( Q ) who rank x above y in Q , add them to P ′ with the ballot a , b , c , . . . , c n , and for k voters from V ( Q ) who rank y above x in Q , add them to P ′ with the ballot c n , . . . , c , a , b . See, e.g., the fifth and sixth columns inthe profile in Figure 1.This construction uses k + ( n + 1) k voters from V ( Q ) who rank x over y and ( n + 1) k voters from V ( Q ) whorank y over x , for a total of ( n + 2) M argin P ( x, y ) voters from V ( Q ) . Then P ′ has the form in Figure 1.Observe that M ( P ′ ) = M ′ (e.g., M argin P ′ ( x, y ) = 2 k = M argin P ( x, y ) , M argin P ′ ( x, z ) = 0 , etc.). k k k k k k · · · k k x z n y x z y · · · z n z n − y ... z z n z x · · · x ... z ... z ... ... z n · · · y z ... z ... z z n ... · · · z y ... x z n y x z · · · ... z n z n y x z y z · · · z n − x Figure 1: the profile P ′ .Now we claim that x does not defeat y in P ′ according to f . Toward a contradiction, suppose x doesdefeat y in P ′ according to f . Let σ ( P ′ ) be the profile obtained from P ′ by the permutation σ that mapseach a ∈ { x, y, z , . . . , z n } to the unique b ∈ { x, y, z , . . . , z n } such that a → b in P ′ , as shown in Figure 2.By Neutrality, since x defeats y in P ′ , it follows that y defeats z in σ ( P ′ ) . But P ′ can obviously beobtained from σ ( P ′ ) by a permutation of the voters (e.g., in Figures 1-2, the first column in P ′ is the sameas the second to last column in σ ( P ′ ) ). Thus, by Anonymity, since y defeats z in σ ( P ′ ) , it follows that y defeats z in P ′ . By similar reasoning using σ ( σ ( P ′ )) , we have that z defeats z in P ′ , etc., until we conclude This standard kind of construction is used in the proof of McGarvey’s theorem (McGarvey 1953). Here we use the permutation version of Neutrality in Footnote 16. k k k k k · · · k k y x z y z z · · · x z n z z n z x z y · · · y ... z ... ... z n ... x · · · z ...... z z n ... z n ... · · · ... z z n y x z y z · · · ... x x z y z z z · · · z n y Figure 2: the profile σ ( P ′ ) .that xDyDz D . . . Dz n Dx where D is the defeat relation in P ′ . This contradicts Availability. Hence x doesnot defeat y in P ′ according to f .Since V ( P ′ ) ⊆ V ( Q ) and M argin Q ( x, y ) = M argin P ′ ( x, y ) , it follows that half of the voters in V ( Q ) \ V ( P ′ ) (which may be empty) rank x above y and half of the voters in V ( Q ) \ V ( P ′ ) rank y above x . Let P ′′ be obtained from P ′ as follows: for each voter in V ( Q ) \ V ( P ′ ) who ranks x above y , add themto P ′′ with the ballot x, y, z , . . . , z n , and for each voter V ( Q ) \ V ( P ′ ) who ranks y above x , add them to P ′′ with the ballot z n , . . . , z , y, x . Thus, P ′′ is obtained from P ′ by adding zero or more reversal pairs ofvoters, so by (Downward) Neutral Reversal, since x does not defeat y in P ′ according to f , it follows that x does not defeat y in P ′′ according to f . Finally, we have: • V ( Q ) = V ( P ′′ ) and Q |{ x,y } = P ′′|{ x,y } ; • M ( P ′′ ) is obtained from M ( Q ) by deleting zero or more candidates other than x and y and deletingor reducing the margins on zero or more edges other than the x → y edge.Thus, by Coherent IIA, since x defeats y in Q according to f , we have that x defeats y in P ′′ according to f , contradicting what we derived above.Of course Split Cycle is not the only VCCR that satisfies the six axioms for defeat. For example, thenull VCCR according to which no one ever defeats anyone else satisfies all six axioms—although it can easilybe ruled out by other axioms that Split Cycle satisfies, such as the Pareto axiom. Another example is theVCCR according to which x defeats y if M argin P ( x, y ) is greater than the splitting number of every majoritycycle in P , not only those majority cycles containing x and y . Since these VCCRs are not as resolute asSplit Cycle, they do not satisfy the seventh “axiom” that the VCCR should be the most resolute VCCRamong those satisfying the first six axioms. A natural next step would be to obtain another axiomaticcharacterization of Split Cycle as the only VCCR satisfying some axioms without reference to resoluteness.Theorem 5.1 shows that using a VCCR other than Split Cycle requires either violating one of the sixaxioms for defeat or sacrificing resoluteness. For these and other reasons (see Holliday and Pacuit 2020a),we settle on Split Cycle as our preferred VCCR and hence as our preferred answer to the question of whenone candidate should defeat another in a democratic election using ranked ballots. E.g., in the profile P in Example 3.2, neither of the mentioned VCCRs judges that a defeats d . Escaping impossibility
In this section, we address the question: how does Split Cycle escape Arrow’s Impossibility Theorem andrelated impossibility results? In Section 6.1, we recall the standard formulation of Arrow’s theorem andrelated results, and we explain how Split Cycle escapes these results. In Section 6.2, we reformulate theseresults in the variable candidate setting in which we characterized Split Cycle. Finally, in Section 6.3, weconsider some simple impossibility results based not on IIA but instead on a choice-consistency principlesometimes conflated with IIA, allegedly even by Arrow himself (see Appendix B).
Arrow (1963) worked in a fixed voter and fixed candidate setting (see Campbell and Kelly 2002, Penn 2015for modern presentations). Fix a set V of voters and a set X of candidates. A ( V, X ) -profile is a profile P asin Definition 2.1 in which V ( P ) = V and X ( P ) = X . A ( V, X ) -collective choice rule (or ( V, X ) -CCR) is afunction on the set of ( V, X ) -profiles such that for any ( V, X ) -profile P , f ( P ) is an asymmetric binary relationon X . If f is a VCCR as in Section 2, then for any V ⊆ V and X ⊆ X , the restriction f | V,X of f to the setof ( V, X ) -profiles is a ( V, X ) -CCR. In particular, for any such ( V, X ) , the Split Cycle VCCR restricts to theSplit Cycle ( V, X ) -CCR. Our question is: how does the Split Cycle ( V, X ) -CCR escape Arrow’s Theorem?Let f be a ( V, X ) -CCR. To state Arrow’s Theorem, we recall the following key notions: • f is a ( V, X ) - social welfare function (or ( V, X ) -SWF) if for any ( V, X ) -profile P , f ( P ) is a strict weakorder (recall Section 2); • f satisfies Independence of Irrelevant Alternatives (IIA) if for any ( V, X ) -profiles P and P ′ and x, y ∈ X ,if P |{ x,y } = P ′|{ x,y } , then x defeats y in P according to f if and only if x defeats y in P ′ according to f ; • f satisfies Pareto if for any ( V, X ) -profile P and x, y ∈ X , if x P i y for all i ∈ V , then x defeats y in P according to f ; • an i ∈ V is a dictator for f if for all ( V, X ) -profiles P and x, y ∈ X , if x P i y , then x defeats y in P according to f .Then Arrow’s famous Impossibility Theorem can be stated as follows. Theorem 6.1 (Arrow 1963) . Assume V is finite and | X | ≥ . Any ( V, X ) -SWF satisfying IIA and Paretohas a dictator. Remark 6.2.
Since our profiles are profiles of linear ballots, the conclusion of Arrow’s theorem can bestrengthened to say that f has a strong dictator , i.e., an i ∈ V such that for all ( V, X ) -profiles P and x, y ∈ X , we have that x defeats y in P according to f if and only if x P i y , i.e., f ( P ) = P i . Note that we have built the condition of Universal Domain with respect to ( V, X ) into the definition of a ( V, X ) -CCR. Arrow considered profiles where each voter’s strict preference relation is a strict weak order, whereas we have assumed strictlinear orders. However, it is well known that Arrow’s Theorem can be proved for strict linear order profiles (see, e.g., Fishburn1973, p. 208). In fact, Arrow’s Theorem for strict linear order profiles is a corollary of the statement of Arrow’s Theorem forstrict weak orders, by applying Lemma 3.4 of Holliday and Pacuit 2020b. ( V, X ) -CCR avoids Arrow’s theorem due to the following facts:1. We weaken IIA to Coherent IIA.
2. We weaken Arrow’s assumption that the defeat relation is a strict weak order to it being acyclic.
Neither of these moves by itself is sufficient to escape Arrow-style impossibility theorems, as we show below.
To see that weakening IIA to Coherent IIA is not sufficient, we first observe that Coherent IIA implies aweakening of IIA known as Weak IIA, which states that if P |{ x,y } = P ′|{ x,y } and x defeats y in P ′ accordingto f , then y does not defeat x in P according to f . Lemma 6.3. If f is a VCCR satisfying Coherent IIA, then for any V ⊆ V and X ⊆ X , f | V,X satisfies WeakIIA.
Proof.
Suppose that P |{ x,y } = P ′|{ x,y } and x defeats y in P . Then by Coherent IIA, x defeats y in P |{ x,y } ,so y does not defeat x in P |{ x,y } . Now if y defeats x in P ′ , then by Coherent IIA, y defeats x in P ′|{ x,y } andhence in P |{ x,y } , since P |{ x,y } = P ′|{ x,y } , which is a contradiction. Therefore, y does not defeat x in P ′ .Under Weak IIA, Baigent (1987) proved an Arrow-style impossibility theorem asserting the existence of avetoer instead of a dictator. Given a ( V, X ) -CCR f , a voter i ∈ V is a vetoer for f if for all ( V, X ) -profiles P and x, y ∈ X ( P ) , if x P i y , then y does not defeat x in P according to f . Theorem 6.4 (Baigent 1987) . Assume V is finite and | X | ≥ . Any ( V, X ) -SWF satisfying Weak IIA andPareto has a vetoer.The existence of a vetoer for an SWF is inconsistent with the SWF satisfying both Pareto and Anonymity. Proposition 6.5.
Suppose | V | ≥ and | X | ≥ . Let f be a ( V, X ) -SWF satisfying Pareto. If f has avetoer, then f has a unique vetoer and hence violates Anonymity. Proof.
Suppose there are two vetoers j and k . Consider a profile P in which (i) x P i y for all i ∈ V , (ii) y P j z ,and (iii) z P k x . By (i) and Pareto, x defeats y in P according to f . By (ii), z does not defeat y , since j is avetoer. By (iii), x does not defeat z , since k is a vetoer. But since f is an SWF, f ( P ) is a strict weak order,so if x defeats y , then either z defeats y or x defeats z . Thus, we have a contradiction.As a corollary of Theorem 6.4 and Proposition 6.5, we have the following. Corollary 6.6.
Assume V is finite and | X | ≥ . There is no ( V, X ) -SWF satisfying Weak IIA, Pareto, andAnonymity.In light of Lemma 6.3, Theorem 6.4, and Corollary 6.6, only weakening IIA to Coherent IIA is not sufficientto escape Arrow-style impossibility results. Strictly speaking, we have stated Coherent IIA in a variable candidate setting and IIA in a fixed candidate setting, so theyare not comparable in strength, but see Proposition 6.11 in Section 6.2. Cf. Campbell and Kelly (2000), who observe that at least four candidates are required for Baigent’s result. .1.3 Blau-Deb Theorem Weakening Arrow’s assumption that the defeat relation is a strict weak order to it being acyclic is also notsufficient by itself. Blau and Deb (1977) prove a vetoer theorem for acyclic CCRs under IIA together withNeutrality and Monotonicity (recall Section 4.1). Let f be a ( V, X ) -CCR. A coalition C ⊆ V of voters has veto power for f if for any ( V, X ) -profile P and x, y ∈ X , if x P i y for all i ∈ C , then y does not defeat x in P according to f . Theorem 6.7 (Blau and Deb 1977) . Let f be an acyclic ( V, X ) -CCR satisfying IIA, Neutrality, and Mono-tonicity.1. For any partition of V into at least | X | -many coalitions, at least one of the coalitions has veto power.2. If | X | ≥ | V | , then f has a vetoer. Remark 6.8.
Part 2 is an immediate consequence of part 1 by considering the finest partition.
Remark 6.9.
Inspection of the proof of the Veto Theorem in Blau and Deb 1977 shows that the assumptionof acyclicity may be replaced by the weaker axiom of Availability (recall Section 4.1).As an example of applying Theorem 6.7.1, if there are five candidates, then for any partition of theelectorate into five coalitions—say, five coalitions of equal size—one of the five coalitions has veto power(and hence, assuming Anonymity, all coalitions of the same size would have veto power). Moreover, inthe variable candidate setting, we can use Theorem 6.7 to prove the existence of a single vetoer under avariable candidate version of IIA (see Proposition 6.15 below), without the assumption that | X | ≥ | V | . Thus,weakening Arrow’s strict weak order assumption to the assumption of acyclicity (or even Availability) is notenough by itself to escape Arrow-style impossibility theorems, as we would like to retain such appealingproperties as Neutrality and Monotonicity.It is the combination of weakening IIA to Coherent IIA and weakening Arrow’s strict weak order as-sumption to acyclicity that allows Split Cycle to escape Arrow-style impossibility theorems. Since we have analyzed Split Cycle as a VCCR in this paper, to properly make claims about how Split Cyclerelates to Arrow-style impossibility theorems, we should recast these results in the variable-election setting.In this setting, there are two versions of Arrow’s Independence of Irrelevant Alternatives (IIA).
Definition 6.10.
Let f be a VCCR.1. f satisfies fixed-candidate Independence of Irrelevant Alternatives (FIIA) if for any profiles P and P ′ with X ( P ) = X ( P ′ ) , if P |{ x,y } = P ′|{ x,y } , then x defeats y in P according to f if and only if x defeats y in P ′ according to f ;2. f satisfies variable-candidate Independence of Irrelevant Alternatives (VIIA) if for any profiles P and P with x, y ∈ X ( P ) ∩ X ( P ′ ) , if P |{ x,y } = P ′|{ x,y } , then x defeats y in P according to f if and only if x defeats y in P ′ according to f . 25e suggest in Appendix B that if asked to formulate his axioms for VCCRs, Arrow would formulateIIA as VIIA. Our Coherent IIA is a weakening of VIIA, as Coherent IIA strengthens the assumption from P |{ x,y } = P ′|{ x,y } to the assumption that not only P |{ x,y } = P ′|{ x,y } but also that the margin graph of P ′ isobtained from that of P in a certain way. Proposition 6.11.
Any VCCR satisfying VIIA also satisfies Coherent IIA.We reject VIIA in favor of Coherent IIA for the reasons explained in Section 4.3.Arrow’s Impossibility Theorem can be stated in the variable-election setting using some additional no-tions. First, given V ⊆ V , a V -profile is a profile P as in Definition 2.1 in which V ( P ) = V . Second, given i ∈ V ⊆ V and X ⊆ X , we say that i is ( V, X ) -dictator (resp. V -dictator ) for f if for any ( V, X ) -profile(resp. V -profile) P and x, y ∈ X ( P ) , x P i y implies that x defeats y in P according to f . Theorem 6.12 (Arrow’s Theorem for VSWFs) . Suppose f is a VSWF satisfying the Pareto principle.1. If f satisfies FIIA, then for any finite sets V ⊆ V and X ⊆ X with | X | ≥ , there is a ( V, X ) -dictatorfor f .2. If f satisfies VIIA, then for any finite set V ⊆ V , there is a V -dictator for f . Proof.
For part 1, let f | V,X be the restriction of f to ( V, X ) -profiles. Then f | V,X is a ( V, X ) -SWF as inSection 6 satisfying IIA and Pareto. Since V is finite and | X | ≥ , Arrow’s Theorem (Theorem 6.1) gives usthe desired ( V, X ) -dictator for f .For part 2, consider any finite V ⊆ V . Pick some finite X ⊆ X such that | X | ≥ . Then as in part 1,Arrow’s Theorem applied to f | V,X gives us an i V,X ∈ V who is a ( V, X ) -dictator for f . We claim that i V,X is a V -dictator for f . Let Q be a V -profile. We must show that for all x, y ∈ X ( Q ) , x Q i V,X y implies that x defeats y in Q according to f . Suppose x Q i V,X y . Let X = X ∪ X ( Q ) . Since | X | ≥ , Arrow’s Theoremapplied to f | V,X gives us an i V,X ∈ V who is a ( V, X ) -dictator for f . We claim that i V,X = i V,X . Supposenot. There is a ( V, X ) -profile P ⋆ such that for some a, b ∈ X , voter i V,X ranks a above b in P ⋆ while i V,X ranks b above a in P ⋆ . Then since i V,X is a ( V, X ) -dictator, b defeats a in P ⋆ according to f . Hence byVIIA, b defeats a in P ⋆ | X according to f . But this contradicts the fact that i V,X is a ( V, X ) -dictator, giventhat X ( P ⋆ | X ) = X and i V,X ranks a above b in P ⋆ | X . Hence i V,X = i V,X , so i V,X is a ( V, X ) -dictator for f . Now let Q + be any ( V, X ) -profile such that Q + | X = Q . Then x Q i V,X y implies x Q + i V,X y , so x defeats y in Q + according to f because i V,X is a ( V, X ) -dictator. Hence by VIIA, x defeats y in Q according to f ,which completes the proof, as diagrammed in Figure 3 with Y = X ( Q ) . ( V, X ) -dictator ( V, Y ) -dictator ( V, X ∪ Y ) -dictatorFigure 3: To show that any ( V, X ) -dictator is also a ( V, Y ) -dictator, we first show that any ( V, X ) -dictatoris also a ( V, X ∪ Y ) -dictator and then show that any ( V, X ∪ Y ) -dictator is also a ( V, Y ) -dictator.26 emark 6.13. As in Remark 6.2, since our profiles are profiles of linear ballots, the conclusions of parts 1and 2 of Theorem 6.12 can be strengthened with ‘strong dictator’ in place of ‘dictator’.
Remark 6.14.
There are VSWFs satisfying Pareto and VIIA for which there is no i ∈ V who is a V -dictatorwith respect to all V ⊆ V with i ∈ V . For example, let V be the set of natural numbers, and for any profile P ,let f ( P ) = P max( V ( P )) , where max( V ( P )) is the greatest number in the set V ( P ) . Thus, in the variable-votersetting Arrow’s axioms are consistent with different electorates having different dictators.Just as Arrow’s Theorem can be adapted to the variable-election setting, so can Baigent’s Theorem(Theorem 6.4), which we leave as an exercise to the reader (hint: use Proposition 6.5 to obtain the analogueof Theorem 6.12.2). More interesting is the reformulation of the Blau-Deb Theorem (Theorem 6.7) in thevariable-election setting—in particular, the variable-candidate setting—as VIIA allow us to strengthen theconclusion of the theorem to state the existence of a vetoer without the restriction that | X | ≥ | V | .To state the variable-candidate version of the Blau-Deb Theorem, we need the following notions. Givenfinite V ⊆ V , i ∈ V , finite X ⊆ X , and a, b ∈ X , we say that: • i is a ( V, X ) -vetoer for f on ( a, b ) if for all ( V, X ) -profiles P , if a P i b , then b does not defeat a in P according to f ; • i is a ( V, X ) -vetoer for f if for every a, b ∈ X , i is a ( V, X ) -vetoer for f on ( a, b ) ; • i is a V -vetoer for f if for every finite X ⊆ X , i is a ( V, X ) -vetoer for f . Theorem 6.15. If f is a VCCR satisfying VIIA, Availability, Neutrality, and Monotonicity, then for anyfinite V ⊆ V , f has a V -vetoer. Proof.
Consider any finite V ⊆ V . Pick some finite X ⊆ X such that | X | ≥ | V | . Then Theorem 6.7.2 (andRemark 6.9) applied to f | V,X gives us an i V,X ∈ V who is a ( V, X ) -vetoer for f . We claim that i V,X isa V -vetoer for f . Let Q be a V -profile. We must show that for all x, y ∈ X ( Q ) , x Q i V,X y implies that y does not defeat x in Q according to f . Suppose x Q i V,X y . Let X = X ∪ X ( Q ) . We claim that for any a, b ∈ X , voter i V,X is a ( V, X ) -vetoer on ( a, b ) . Suppose P is a ( V, X ) -profile such that a P i V,X b . Then i V,X ranks a above b in the restricted profile P | X , and i V,X is a ( V, X ) -vetoer, so b does not defeat a in P | X , which by VIIA implies that b does not defeat a in P . Thus, i V,X is a ( V, X ) -vetoer on ( a, b ) , whichby Neutrality implies that i V,X is a ( V, X ) -vetoer. Now let Q + be any ( V, X ) -profile extending Q . Then x Q i V,X y implies x Q + i V,X y , so y does not defeat x in Q + according to f because i V,X is a ( V, X ) -vetoer.Hence by VIIA, y does not defeat x in Q according to f , which completes the proof.Theorem 6.15 shows how moving to the variable-candidate setting and interpreting IIA as VIIA can strengthenimpossibility theorems. But by weakening VIIA to Coherent IIA, impossibility results like Theorem 6.15disappear. Split Cycle satisfies Coherent IIA, Availability, Neutrality, and Monotonicity but has no vetoer. For suppose i V,X is not a ( V, X ) -vetoer, so there are a ′ , b ′ ∈ X and a ( V, X ) -profile Q such that i V,X ranks a ′ above b ′ in Q but b ′ defeats a ′ in Q . Consider any permutation σ of X such that σ ( a ) = a ′ and σ ( b ) = b ′ . Applying this permutation to Q as in Footnote 16 yields a profile σ Q in which i V,X ranks a above b . By the permutation version of Neutrality in Footnote16, since b ′ defeats a ′ in Q , b defeats a in σ Q . This contradicts the fact that i V,X is a ( V, X ) -vetoer on ( a, b ) . .3 Impossibility, independence, and choice consistency Our rejection of VIIA in favor of Coherent IIA also leads us to reject another well-known principle thatis related to VIIA, at least under one interpretation. In particular, the term ‘Independence of IrrelevantAlternatives’ is sometimes used in the theory of rational choice for a condition that differs from Arrow’s butalso leads to impossibility theorems when applied in a certain way to voting. A choice function on a set X of candidates is a function C such that for any nonempty subset Y of X , C ( Y ) is a nonempty subset of Y .Such a choice function satisfies Sen’s (1971) condition α iffor all nonempty Z ⊆ Y ⊆ X , we have Z ∩ C ( Y ) ⊆ C ( Z ) . As a famous illustration of this condition in the context of individual choice, attributed to Sidney Mor-genbesser, imagine that when offered a choice between apple pie and blueberry pie, you choose apple pie; butwhen offered a choice between apple, blueberry, and cherry, you switch to blueberry. This violates α where Z = { apple, blueberry } , Y = { apple, blueberry, cherry } , C ( Y ) = { blueberry } , and C ( Z ) = { apple } . Sen’s α is also known as ‘Chernoff’s axiom’ (Chernoff 1954) and sometimes ‘Independence of Irrelevant Alternatives’(cf. Radner and Marschak 1954). But as Suzumura warns (1983, p. 66), “[C]are should be taken concerningthe occasional unfortunate confusions in the literature between condition I [IIA] and Chernoff’s axiom ofchoice consistency, despite rather obvious contextual differences between them.” In Appendix B, we discussthe common allegation that Arrow himself was guilty of this confusion, as it relates to VIIA.How can a choice-consistency axiom such as Sen’s α be applied to voting to compare it with IIA? Givenan acyclic VCCR f and profile P , there are two ways to use f and P to define a choice function on X ( P ) :
1. the global choice function G f ( P , · ) induced by f, P : for any nonempty Y ⊆ X ( P ) , G f ( P , Y ) = { y ∈ Y | there is no z ∈ Y that defeats y in P according to f } .
2. the local choice function L f ( P , · ) induced by f, P : for any nonempty Y ⊆ X ( P ) , L f ( P , Y ) = { y ∈ Y | there is no z ∈ Y that defeats y in P | Y according to f } . Example 6.16.
The distinction between the global choice function and local choice function can be illus-trated by the well-known distinction between global Borda count and local Borda count. Let f be theVCCR according to which x defeats y in a profile P according to f just in case the Borda score of x in P is greater than that of y . Then G f ( P , Y ) , the elements of Y chosen according to global Borda count, arethe elements of Y whose Borda scores are maximal among elements of Y , where Borda scores are calculatedwith respect to the full profile P . By contrast, L f ( P , Y ) , the element of Y chosen according to local Bordacount, are the elements of Y whose Borda scores are maximal among elements of Y , where Borda scores arecalculated with respect to the restricted profile P | Y . For example, consider the following profiles P (left)and P | Y (right) where Y = { x, y, a } : For G f ( P , Y ) to be a choice function, i.e., for ∅ = Y ⊆ X ( P ) to imply G f ( P , Y ) = ∅ , f must be acyclic. But for L f ( P , · ) to be a choice function, it suffices that f satisfies the weaker axiom of Availability. This terminology is due to Kelly (1988, p. 71, 74). Sen (1987, pp. 78-9) uses the terms ‘broad’ and ‘narrow’. x y ya x xb a cc b by c a x y ya x xy a a Global Borda count yields G f ( P , Y ) = { x } , as x has the highest Borda score in P , while local Borda countyields L f ( P , Y ) = { y } , as y has the highest Borda score in the restricted profile P | Y .Before using the global and local choice functions to define two senses of Sen’s α for voting, we must notethat the distinction between global and local is lost under the assumption of VIIA. Proposition 6.17.
Let f be an acyclic VCCR. The following are equivalent:1. f satisfies VIIA;2. for any profile P and Y ⊆ X ( P ) , G f ( P , Y ) = L f ( P , Y ) . Proof.
Suppose f satisfies VIIA. Consider any profile P and Y ⊆ X ( P ) . Then the following are equivalentfor any y, z ∈ Y : • z defeats y in P according to f ; • z defeats y in P |{ y,z } according to f (by VIIA); • z defeats y in ( P | Y ) |{ y,z } according to f (since P |{ y,z } = ( P | Y ) |{ y,z } ); • z defeats y in P | Y according to f (by VIIA).Hence G f ( P , Y ) = L f ( P , Y ) by the definitions of G f ( P , · ) and L f ( P , · ) .Suppose condition 2 holds. Consider profiles P and P ′ with x, y ∈ X ( P ) ∩ X ( P ′ ) and P |{ x,y } = P ′|{ x,y } .We must show that x defeats y in P according to f if and only if x defeats y in P ′ according to f . This isequivalent to the claim that G f ( P , { x, y } ) = G f ( P ′ , { x, y } ) . We claim that the following equations hold: G f ( P , { x, y } ) G f ( P ′ , { x, y } ) = = L f ( P , { x, y } ) = L f ( P ′ , { x, y } ) The vertical equations hold by condition 2, while the horizontal equation holds since P |{ x,y } = P ′|{ x,y } .Hence G f ( P , { x, y } ) = G f ( P ′ , { x, y } ) .Assuming we weaken VIIA, we can make the local-global distinction and hence distinguish two senses ofSen’s α in the context of voting. Definition 6.18.
Let f be an acyclic VCCR.1. f satisfies Global- α if G f ( P , · ) satisfies α for all profiles P ;2. f satisfies Local- α if L f ( P , · ) satisfies α for all profiles P .29t is well known that Global- α imposes no constraint on an acyclic VCCR. Proposition 6.19. If f is an acyclic VCCR, then f satisfies Global- α . Proof.
The claim that f satisfies Global- α is the claim that for any profile P and nonempty Z ⊆ Y ⊆ X ( P ) ,we have Z ∩ G f ( P , Y ) ⊆ G f ( P , Z ) . Indeed, if y ∈ Z ∩ G f ( P , Y ) , so by definition there is no z ∈ Y thatdefeats y in P according to f , then since Z ⊆ Y , there is no z ∈ Z that defeats y in P according to f , whichby definition implies y ∈ G f ( P , Z ) .Let us now consider Local- α as a constraint on VCCRs. First, we note that it is a weakening of VIIA. Proposition 6.20.
1. If f is an acyclic VCCR satisfying VIIA, then f satisfies Local- α ;2. There are acyclic VCCRs satisfying Local- α but not FIIA and hence not VIIA. Proof.
For part 1, assuming that f satisfies VIIA, we show that for any profile P and nonempty Z ⊆ Y ⊆ X ( P ) , we have Z ∩ L f ( P , Y ) ⊆ L f ( P , Z ) . Suppose y ∈ Z but y
6∈ L f ( P , Z ) , so there is an x ∈ Z thatdefeats y in P | Z according to f . Then since ( P | Z ) |{ x,y } = ( P | Y ) |{ x,y } , it follows by VIIA that x defeats y in P | Y according to f , which with x ∈ Z ⊆ Y implies y
6∈ L f ( P , Y ) .For part 2, let f be a VCCR such that x defeats y in P according to f if and only if (i) x is unanimouslypreferred to y and (ii) there is a z ∈ X ( P ) \ { x, y } such that x → z but y z . Then f is acyclic invirtue of (i). To see that f violates FIIA, consider two profiles P and P ′ with X ( P ) = X ( P ′ ) = { x, y, z } and V ( P ) = V ( P ′ ) such that all i ∈ V ( P ) have x P i z P i y while all i ∈ V ( P ′ ) have x P ′ i y P ′ i z . Then P |{ x,y } = P ′|{ x,y } , and x defeats y in P but not in P ′ , violating FIIA. To see that f satisfies Local- α , wemust show that for any profile P and nonempty Z ⊆ Y ⊆ X ( P ) , we have Z ∩ L f ( P , Y ) ⊆ L f ( P , Z ) . Suppose y ∈ Z but y
6∈ L f ( P , Z ) , so there is an x ∈ Z that defeats y in P | Z according to f . Hence x is unanimouslypreferred to y in P | Z and there is a z ∈ Z such that x → z but y z . Then since Z ⊆ Y , we have that x ∈ Y , that x is unanimously preferred to y in P | Y according to f , and that there is a z ∈ Y such that x → z but y z . Therefore, x defeats y in P | Y according to f , which with x ∈ Y implies y
6∈ L f ( P , Y ) .Although weaker than VIIA, Local- α is still a significant restriction on an acyclic VCCR, as it rules outthat the VCCR coincides with majority rule on two-candidates profiles. Definition 6.21.
A VCCR f satisfies Binary Majoritarianism if for any profile P with X ( P ) = { x, y } , x defeats y in P according to f if and only if x is majority preferred to y in P .We have the following easy impossibility result. Proposition 6.22.
There is no VCCR satisfying Local- α , Availability, and Binary Majoritarianism. Proof.
Consider a profile P with X ( P ) = { x, y, z } and a majority cycle x → y → z → x . By Availability,there is some a ∈ { x, y, z } who is undefeated in P according to f . Since there is a majority cycle, there issome b ∈ { x, y, z } such that b → a . Hence by Binary Majoritarianism, b defeats a in P |{ a,b } according to f .Thus, we have a ∈ { a, b } ∩ L f ( P , { x, y, z } ) but a
6∈ L f ( P |{ a,b } , { a, b } ) , so f violates Local- α .Combining Propositions 6.20.1 and 6.22, we have the analogue of Proposition 6.22 under VIIA.30 orollary 6.23. There is no VCCR satisfying VIIA, Availability, and Binary Majoritarianism.Finally, we can show how an analogue of Proposition 6.22 applies to voting methods (recall Definition2.3). First, we adapt the definition of α to voting methods. Definition 6.24.
A voting method F satisfies α if for all nonempty Z ⊆ X ( P ) , Z ∩ F ( P ) ⊆ F ( P | Z ) . Given a VCCR f , its induced voting method f (recall Lemma 2.8) is related to its local choice functionin the following obvious way. Lemma 6.25.
Let f be a VCCR satisfying Availability. Then for any profile P , f ( P ) = L f ( P , X ( P )) . In light of Lemma 6.25 and the definitions of Local- α and α , the following is immediate. Lemma 6.26. If f is a VCCR satisfying Availability and Local- α , then f satisfies α .The proof of Proposition 6.22 is then easily adapted to yield the following impossibility result. Proposition 6.27.
There is no voting method satisfying α and Binary Majoritarianism.As a voting method (resp. VCCR) Split Cycle satisfies Binary Majoritarianism but not α (resp. Local- α ).The mistake of insisting on α for voting is essentially the same as the mistake of insisting on IIA, which canbe seen by reformulating α as follows: for all nonempty Z ⊆ X ( P ) , if x ∈ Z but x F ( P | Z ) , then x F ( P ) .If F is defeat rationalizable, this means that if x is defeated in the smaller profile P | Z then x must also bedefeated in the larger profile P . But this should not follow if the larger profile P is more incoherent than P | Z . If P is sufficiently incoherent, we may need to suspend judgment on many defeat relations that wecould coherently accept in P | Z . Example 6.28.
The same example used against IIA in Example 4.8 can be adapted to argue against Local- α for VCCRs or α for voting methods. Consider the following profiles P and P ′ : n n na b ab a b a bnn n n a b c b c a c a b a cbn nn In the context of the perfectly coherent profile P , the margin of n for a over b should be sufficient for a to defeat b , so a should be the uniquely chosen winner. But in the context of the incoherent profile P ′ ,it is not sufficient: no one can be judged to defeat anyone else—this follows from Anonymity, Neutrality,and Availability—so all three must be included in the choice set to which a further tie-breaking process isapplied. This is a counterexample to Local- α and α : b is undefeated in P ′ but not in P ′|{ a,b } = P . Note that this is equivalent to the more direct translation of α to voting methods: for all nonempty Z ⊆ Y ⊆ X ( P ) , Z ∩ F ( P | Y ) ⊆ F ( P | Z ) . α applied to voting is in the spirit of Sen’s (1993) view that “Violations ofproperty α . . . can be related to various different types of reasons—easily understandable when the externalcontext is spelled out” (p. 501). Sen (pp. 500-502) focuses on rationalizing violations of α in individual choiceby reference to features of the context of choice. Here we have focused on rationalizing violations in votingby reference to features of the context given by the profile—namely, an increase in incoherence from oneprofile to another. To overlook this context would be to commit The Fallacy of IIA from Section 4.3. The pessimistic conclusions about democracy that some have drawn from the Paradox of Voting and Arrow’sImpossibility Theorem are not justified. Like most voting theorists, we are more optimistic. In particular, webelieve that many majority cycles can be resolved in a rational way, while respecting the principle of MajorityDefeat, as shown by Split Cycle. Of course there remain some cycles, such as a perfect cycle a → b → c → a in which each candidate is majority preferred to the next by exactly the same margin (and there are no othercandidates), which must lead to a tie between all candidates. But to think that democracy is devastated bythe possibility of such ties seems almost as unreasonable as thinking that democracy is devastated by thepossibility that in an election with only two candidates and an even number of voters, it could happen thathalf of the voters vote for x over y while half vote for y over x . The fact that the set of winners cannotalways be a singleton—that some further tie-breaking mechanism must be in place—hardly warrants verypessimistic conclusions, especially if non-singleton sets of winners are not likely, as we expect when there aremany voters compared to candidates (see Holliday and Pacuit 2020a).Far from justifying pessimism about democracy, social choice theory leads the way to voting proceduresthat can improve democratic decision making. We agree with Maskin and Sen (2017a; 2017b) that a majorimprovement would come in replacing Plurality voting with a voting procedure using ranked ballots thatelects a Condorcet winner whenever there is one. In this paper, we have arrived at a unique voting procedureof this kind, Split Cycle, via six axioms concerning when one candidate should defeat another in a democraticelection—with the key axiom being the axiom of Coherent IIA that weakens Arrow’s IIA and explains why thelatter is too strong. As theorists, we sleep well at night knowing that we have a solid theoretical justificationfor handling majority cycles in a certain way should they arise. As citizens and committee members, wehope that in practice our elections will have Condorcet winners and that we will elect them. Acknowledgements
We thank Mikayla Kelley for helpful comments. We are also grateful for useful feedback received at theWork in Progress Seminar and Logic Seminar at the University of Maryland in July 2020 and at the FERCreading group at UC Berkeley in August 2020. 32
Proofs for Section 3
Lemma 3.5.
Let P be a profile and x, y ∈ X ( P ) . Then x defeats y in P according to Split Cycle if andonly if M argin P ( x, y ) > and M argin P ( x, y ) > Split P ( ρ ) for every majority cycle ρ in P containing x and y. Proof.
Suppose that in P , x wins by more than n over y for the smallest natural number n such that there isno majority cycle, containing x and y , in which each candidate wins by more than n over the next candidatein the cycle. Then M argin P ( x, y ) > n ≥ . Now consider some majority cycle ρ in P containing x and y .By our choice of n , we have n ≥ Split P ( ρ ) , so M argin P ( x, y ) > n implies M argin P ( x, y ) > Split P ( ρ ) .Conversely, suppose M argin P ( x, y ) > and M argin P ( x, y ) > Split P ( ρ ) for every majority cycle ρ in P containing x and y . If there exist such cycles, let n be the maximum of their splitting numbers, andotherwise let n = 0 . It follows that there is no majority cycle containing x and y in which each candidatewins by more than n over the next candidate in the cycle; moreover, n is the smallest natural number forwhich this holds. By our initial supposition, M argin P ( x, y ) > n , so we are done. Lemma 3.6.
Let P be a profile and x, y ∈ X ( P ) . Then x defeats y in P according to Split Cycle if andonly if M argin P ( x, y ) > and M argin P ( x, y ) > Split P ( ρ ) for every majority cycle ρ in P of the form x → y → z → · · · → z n → x. Proof.
We use the formulation of Split Cycle in Lemma 3.5. If
M argin P ( x, y ) > Split P ( ρ ) for everymajority cycle ρ in P containing x and y , then in particular M argin P ( x, y ) > Split P ( ρ ) for every majoritycycle ρ in P of the form x → y → z → · · · → z n → x . Conversely, suppose M argin P ( x, y ) > Split P ( ρ ) for every majority cycle ρ in P of the form x → y → z → · · · → z n → x . Now consider a majority cycle ρ in P containing x and y , whose splitting number is maximal among all such majority cycles. We must show M argin P ( x, y ) > Split P ( ρ ) . If y occurs immediately after x in ρ , then by “rotating the cycle” we obtaina cycle ρ ′ of the form x → y → z → · · · → z n → x with the same splitting number as ρ , in which case M argin P ( x, y ) > Split P ( ρ ′ ) by our initial supposition and hence M argin P ( x, y ) > Split P ( ρ ) . Thus,suppose y does not occur immediately after x in ρ . Then without loss of generality, we may assume ρ is ofthe form y → z → · · · → z n → x → w → · · · → w m → y . Let ρ ′ be the cycle x → y → z → · · · → z n → x .Then M argin P ( x, y ) > Split P ( ρ ′ ) by our initial supposition, so the splitting number of ρ ′ is the marginassociated with some successive candidates in the sequence y, z , . . . z n , x . Since y, z , . . . z n , x is a subsequenceof ρ , and the splitting number is defined as a minimum, it follows that Split P ( ρ ′ ) ≥ Split P ( ρ ) . Thensince M argin P ( x, y ) > Split P ( ρ ′ ) , we have M argin P ( x, y ) > Split P ( ρ ) , and since ρ was chosen to havemaximal splitting number among all majority cycles containing x and y , we are done. B Arrow’s alleged confusion and VIIA
Arrow has been accused of confusing his own condition of IIA, an interprofile condition, with a choice-consistency condition such as Sen’s α , defined in Section 6.3 (see, e.g., Hansson 1973, § 3, Ray 1973, p. 989,and Suzumura 1983, p. 97, endnote 16, p. 250). To clarify this matter, which is relevant to our distinction33etween FIIA and VIIA, we first note that Arrow did not state IIA in what is now its most common form,given in Definition 6.10.1. Instead, he stated it in the equivalent form (assuming acylicity) in Definition B.1.1. Definition B.1.
Let f be an acyclic VCCR.1. f satisfies global choice FIIA if for any profiles P and P ′ with V ( P ) = V ( P ′ ) and X ( P ) = X ( P ′ ) and Y ⊆ X ( P ) , if P | Y = P ′| Y , then G f ( P , Y ) = G f ( P ′ , Y ) .2. f satisfies local choice FIIA if for any profiles P and P ′ with V ( P ) = V ( P ′ ) and X ( P ) = X ( P ′ ) and Y ⊆ X ( P ) , if P | Y = P ′| Y , then L f ( P , Y ) = L f ( P ′ , Y ) . Proposition B.2.
Let f be an acyclic VCCR. Then f satisfies global choice FIIA if and only if f satisfiesFIIA. Proof.
Assume f satisfies global choice FIIA. To show that f satisfies FIIA, suppose P |{ x,y } = P ′|{ x,y } . Thenby global choice FIIA, G f ( P , { x, y } ) = G f ( P ′ , { x, y } ) . It follows by definition of G f that x defeats y in P according to f if and only if x defeats y in P ′ according to f . Hence f satisfies FIIA.Now assume f satisfies FIIA. To show that f satisfies global choice FIIA, suppose P | Y = P ′| Y . Then forany two x, y ∈ Y , P |{ x,y } = P ′|{ x,y } . Hence by FIIA, x defeats y in P according to f if and only if x defeats y in P ′ according to f . It follows by definition of G f that G f ( P , Y ) = G f ( P ′ , Y ) .In contrast to global choice FIIA, which is a significant restriction on an acyclic VCCR, local choice FIIAis no restriction. Proposition B.3. If f is an acyclic VCCR, then f satisfies local choice FIIA. Proof.
By definition, we have L f ( P , Y ) = { y ∈ Y | there is no z ∈ Y that defeats y in P | Y according to f }L f ( P ′ , Y ) = { y ∈ Y | there is no z ∈ Y that defeats y in P ′| Y according to f } , which with P | Y = P ′| Y implies L f ( P , Y ) = L f ( P ′ , Y ) .Now consider one of Arrow’s (1963, p. 26) supposed arguments for IIA:Suppose that an election is held, with a certain number of candidates in the field, each individualfiling his list of preferences, and then one of the candidates dies. Surely the social choice shouldbe made by taking each individual’s preference lists, blotting out completely the dead candidate’sname, and considering only the orderings of the remaining names in going through the procedureof determining the winner. That is, the choice to be made among the set S of surviving candidatesshould be independent of the preferences of individuals for candidates not in S . To assumeotherwise would be to make the result of the election dependent on the obviously accidentalcircumstance of whether a candidate died before or after the date of polling.We agree with the literature cited above (Hansson 1973; Ray 1973; Suzumura 1983) that this is not anargument that one’s VCCR should satisfy IIA. But neither is it an argument that one’s VCCR should satisfyLocal- α . In our view, the argument above is at most an argument for the thesis that if a candidate who34ppeared on the ballots in P dies after the ballots are collected, then one should choose among the survivingcandidates using the local choice function L f ( P , · ) . As long as one chooses using the local choice function,one follows all of Arrow’s recommendations above, regardless of whether f satisfies IIA or Local- α .However, Arrow does not officially make the distinction between the global and local choice functions.He only officially defines the global choice function induced by a CCR. But if in the example above,Arrow wants the global choice function to act like the local choice function, this leads to VIIA accordingto Proposition 6.17. Thus, one can understand the otherwise puzzling example of the dead candidate aspossibly related to Arrow’s implicit commitment to VIIA.Arrow (1963, p. 27) gives another supposed argument for IIA, based on Borda count:[S]uppose that there are three voters and four candidates, x , y , z , and w . Let the weights for thefirst, second, third, and fourth choices be 4, 3, 2, and 1, respectively. Suppose that individuals1 and 2 rank candidates in the order x , y , z , and w , while individual 3 ranks them in the order z , w , x , and y . Under the given electoral system, x is chosen. Then, certainly, if y is deletedfrom the ranks of the candidates, the system applied to the remaining candidates should yieldthe same result, especially since, in this case, y is inferior to x according to the tastes of everyindividual; but, if y is in fact deleted, the indicated electoral system would yield a tie between x and z .Let P be the initial profile described by Arrow with X ( P ) = { x, y, z, w } . When Arrow says “if y is deletedfrom the ranks of the candidates, the system applied to the resulting candidates should yield the same result”which of the following did he mean?1. since G f ( P , X ( P )) = { x } , we should have G f ( P , { x, z, w } ) = { x } ;2. since G f ( P , X ( P )) = { x } , we should have G f ( P |{ x,z,w } , { x, z, w } ) = { x } ;3. since L f ( P , X ( P )) = { x } , we should have L f ( P , { x, z, w } ) = { x } ;4. since L f ( P , X ( P )) = { x } , we should have L f ( P |{ x,z,w } , { x, z, w } ) = { x } .In fact, options 2, 3, and 4 are equivalent. Since Arrow only officially discusses the global choice functioninduced by a CCR, he could not have officially meant 3 or 4. Moreover, since Arrow assumes that all of theprofiles in the domain of a given SWF have the same set of candidates, he could not have officially meant 2,which requires that both P and P |{ x,z,w } be in the domain of f . Thus, only option 1 officially makes sensein his framework. 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