# Voting Paradoxes

**DOI:**https://doi.org/10.1057/978-1-349-95189-5_1920

## Abstract

After using an example to motivate why voting theory is so central to the social sciences, this survey describes some of the more recent (and, surprisingly, benign) interpretations of Arrow’s Impossibility Theorem as well as explanations of the wide selection of voting paradoxes that drive this academic area. As described, it now is possible to explain all positional voting paradoxes while creating any number of illustrating examples.

### Keywords

Anti-plurality system Approval voting Arrow, K Axiomatic approach to decision rule choice Borda Count Condorcet, M Cumulative voting Elections Gibbard–Satterthwaite theorem Impossibility theorem Independence of irrelevant alternatives Kruskal–Wallis test Luce, D Plurality vote Sonnenshein–Mantel–Debreu theorem Transitivity Voting paradoxes Voting rules Walras’s LawAlmost daily, news articles describe important elections being held somewhere in the world. The newsworthiness of these events is obvious: election outcomes can change the political, societal and economic directions of a city, a state, or even a country. Elections, in fact, are everywhere; their use ranges from legislative bodies busily determining laws to a kindergarten class selecting a recess treat ‘with a show of hands’. As elections are important, we impose safeguards such as the secret ballot. But a strong message coming from voting theory is that the choice of a voting rule can do more to frustrate the ‘will of the voters’ than any scheming, cigar-smoking political boss.

*plurality vote*, or ‘vote for one’,*A wins*with the A > B > C ranking;*Borda Count*, where 2, 1, 0 points are assigned, respectively, to a voter’s first, second and third ranked candidate,*B wins*with the B > C > A ranking;*anti-plurality*, or ‘vote for two’, system, which is equivalent to voting against a candidate,*C wins*where its C > B > A ranking happens to reverse the plurality ranking.

Not all candidates reflect the ‘will of these voters’, yet each ‘wins’ by selecting an appropriate voting rule. Pairwise majority votes offer no help with their A > B, B > C, C > A cycle. The message is that, rather than capturing the views of the voters, an election outcome may more accurately reflect the *choice of the voting rule*.

More general rules include *n*-candidate positional methods defined by *n* weights *w*_{1}, *w*_{2}, …, *w*_{n} = 0; *w*_{1} > 0 and *w*_{j} ≥ *w*_{j+1} where *w*_{j} points are assigned to a voter’s *j*th ranked candidate; candidates are ranked by the sums of assigned points. While (1, 0, 0), (2, 1, 0) and (1, 1, 0) represent the above rules, (8, 3, 0) is still another choice. Different weights, however, may generate other election outcomes. Indeed, the above example allows *seven* different positional election rankings. For instance, the (8, 3, 0) outcome is a fourth strict ranking B > A > C; the three remaining rankings involve ties.

One probable reason for the many different election rules is that inventing new ones is limited only by one’s imagination; for example, positional methods define run-off rules whereby, after the bottom-ranked candidates are dropped, the remaining two are reordered. With our example, the plurality, Borda, and anti-plurality run-offs elect, respectively, A, B and B. Other approaches allow each *voter* to select a positional method to tally his ballot. With *cumulative voting*, for instance, a voter splits, say, three points in any integer manner; for example, she may use (3, 0, 0), or (2, 1, 0). *Approval voting* (AV) allows a voter to vote for (approve) any number of candidates; for example, he could select (1, 0, 0) or (1, 1, 0). But, whenever voters can determine how to tally their own ballots, we must anticipate that a single profile (that is, listing of voters’ preferences) can admit many different outcomes. Indeed, while changing positional methods generates seven different rankings for our example, *all 13* ways to rank three candidates are admissible cumulative or AV outcomes. Some theorists view this flexibility as a virtue (for example, Brams et al. 1988); others treat this extreme indeterminacy as a serious failing (for example, Saari and Van Newenhizen 1988).

As our example demonstrates, selecting an inappropriate voting or decision rule could inadvertently cause inferior outcomes – with negative concomitant consequences. This is not an isolated phenomenon: with conservative assumptions, about 69 % of contested three-candidate elections allow election rankings to change with different positional methods (Saari and Tataru 1999). The percentage significantly increases with more candidates.

Further underscoring the complexity is Arrow’s (1951) seminal impossibility theorem. He first requires voters to have complete (all pairs are ranked), transitive (a voter preferring A > B and B > C prefers A > C) preferences without restrictions, and the societal outcomes to be complete transitive rankings. Then Arrow characterizes all rules satisfying two basic properties. The first (Pareto) is a unanimity condition whereby, if everyone ranks a pair of candidates in the same manner, that is the societal ranking.

To motivate the second, ‘independence of irrelevant alternatives’ (IIA), condition with a reoccurring phenomenon in the judging of figure-skating, suppose a committee’s ranking is Susan > Barb > Jeannie. Imagine Barb’s anguish if, told that had more judges liked Jeannie, Barb would have ranked over Susan. Why should the judges’ opinion of Jeannie affect the (Susan, Barb) ranking? Arrow’s ‘independence of irrelevant alternatives’ (IIA) condition prohibits this difficulty. Essentially, IIA requires each pair’s ranking to depend only on each voter’s relative ranking of this pair.

With these minimal conditions, Arrow proves that, for three or more candidates, the only admissible rule is a *dictator* – a specified voter whereby the societal outcome *always* agrees with her preferences independent of what other voters want. Understandably, Arrow’s result is often interpreted to mean ‘no voting rule is fair’. An alternative, significantly more benign explanation is given below.

The overpowering message is that the choice of a decision rule is crucial. Indeed, determining which rules are ‘optimal’ is the primary concern of voting theory, where finding axiomatic characterizations of rules, or discovering paradoxical examples, seems to dominate. Another approach (Luce 1959) imposes structure on the outcomes; this structure determines what voting rules are admitted and what restrictions must be imposed on voter choices. A third, recent emphasis examines the data structure – voter preferences – to determine what the voters want and then which voting rules deliver the appropriate outcome (Saari 2000).

For a template, treat a voting rule as a mapping from the domain (space of individual preferences) to the range (space of societal outcomes). The axiomatic approach emphasizes properties of the mapping, Luce’s approach emphasizes the structure of the range, and my recent approach emphasizes the structure of the domain. All three approaches are briefly described.

## Axiomatic Approach and Paradoxes

Borrowed from mathematics, a standard justification for the ‘axiomatic approach’ is that ‘it tells us what we are getting’. After all, axioms are intended to form the fundamental building blocks of a theory, so axiomatic characterizations should specify what to expect from different voting rules. But this expectation requires the conditions to be true axioms; most often they are not. Instead, many results *uniquely identify* a rule in terms of special, perhaps idiosyncratic, properties rather than characterizing the rule. As an analogy, it is easy to envision settings where certain properties uniquely identify ‘John’ as a studious, well-behaved student, while different properties uniquely identify ‘John’ as a street-wise juvenile delinquent. By concentrating on particular traits, both sets of properties uniquely identify John, but neither completely describes nor characterizes him.

Similarly, many so-called ‘axiomatic characterizations’ of voting rules are, in reality, properties that inadvertently emphasize *special profiles*, so while they uniquely *identify* certain rules, they do not characterize them. As an example, certain technical assumptions plus the condition ‘a candidate top-ranked by most voters wins’ uniquely *identifies* the plurality vote. Alternatively, the same technical conditions accompanied with the ‘with *n*-candidates, a candidate may win even if bottom-ranked by all but one more than 1/*n* of the voters’ property also uniquely *identifies* the plurality vote. Neither is an axiomatic characterization: by depending on special profiles, neither really ‘tells us what we are getting’.

This literature, however, identifies valued voting rule properties. Another widely used approach with the same objective is to find ‘voting paradoxes’, that is, unexpected outcomes. Indeed, the origin of this field derives from a 1770 example (published in Borda 1781) that Borda constructed to question the plurality vote: with his example the C > B > A plurality outcome conflicts with the pairwise rankings that are consistent with A > B > C; his (2, 1, 0) Borda Count conclusion agrees with the pairwise rankings.

In contrast, Condorcet (1785) believed we should decide via pairwise comparisons: a Condorcet winner (loser) is the candidate who beats (loses to) all other candidates in majority pairwise votes. To distinguish his approach from Borda’s, he constructed an example whereby the Condorcet winner is not top-ranked by the Borda Count – or any positional rule. The controversy over whether Borda’s or Condorcet’s method is superior continues: comments on this debate are given below.

With examples, Condorcet illustrated that his method can fail; for example, the *Condorcet triplet* A > B > C, B > C > A, C > A > B defines the pairwise cycle A > B, B > C, C > A where neither a Condorcet winner nor loser exists. Later I explain why Condorcet’s example remains central to voting theory. Others continuing Condorcet’s philosophy explored ways to handle cyclic outcomes; for example, Dodgson’s (1876) (Lewis Carroll from ‘Alice in Wonderland’) method finds the ‘closest’ Condorcet winner (that is, over all possible lists of pairwise rankings, find the list with a Condorcet winner that is ‘closest’ to the actual election tallies), while Kemeny’s Rule finds the ‘closest’ transitive ranking. Surprisingly, as Ratliff (2001) proved, the Dodgson winner need not be Kemeny top-ranked; it can be anywhere within the Kemeny ranking. As Ratliff (2003) also proved with examples, if Dodgson’s method is extended to select the top two, or top three, candidates, the outcomes need not be consistent; that is, examples exist where the Dodgson winner is not a Dodgson top-two candidate, and none of them is in the Dodgson top three. Voting behaviour is very complex.

‘Paradoxes’, then, identify new properties of voting rules. Nurmi (1999, 2002), for instance, creates several examples illustrating how major voting rules disagree over a wide selection of desirable properties. His work suggests it may be futile to select voting rules based on specified properties because no rule may satisfy all of them, and most surely there are other valued properties that we have yet to recognize. Fishburn creates many fascinating examples; one (1981) has a plurality ranking of A > B > C > D, but, if D drops out, the same voters have the plurality ranking of C > B > A; Fishburn’s example illustrates an unexpected reversal property of the plurality vote.

Examples disclose subtle properties of voting rules, so a way to find all such properties is to find *everything* that can happen: that is, a profile defines a list – an election ranking for each possible subset of candidates. The goal is to find *all* lists that can be created with all possible choices of positional rules and all possible profiles. Call this collection of lists a ‘dictionary’. Entries in a dictionary, then, describe all possible ranking properties for all positional rules and even for methods, such as AV and run-offs, based on positional and pairwise rules. Even entries outside the dictionary describe properties; for example, lists of the (A > B > C, B > A, C > A, C > B) type, where some profile allows the pairwise rankings to reverse the positional ranking, never are in the Borda Dictionary, so, by being a missing listing, it describes a Borda consistency property.

Such dictionaries exist (for example, Saari 1989; Saari and Merlin 2000) showing, for instance, that most positional rules allow *anything* to happen. For instance, rank seven candidates in any desired manner. Next, re-rank the seven six-candidate subsets (created by dropping someone) in any desired manner; for example, if you wish, reverse the original ranking, or select them randomly. Continue doing so with each subset of five, four, three and two candidates. While the choices could be chaotic, a profile exists where the voters’ plurality ranking for each subset is the selected one. (The same conclusion holds for most choices of positional rules over the different subsets.) What provides hope from these dictionaries is that the Borda Count – defined by (*n* − 1, *n* − 2, …, 1, 0) – is the unique rule (when used with every subset of candidates) that significantly minimizes the number and kinds of allowed paradoxes. Thus, the Borda Count enjoys the maximum number of positive properties; for example, only Borda always ranks a Condorcet winner over a Condorcet loser.

A related ‘dictionary’ result (Saari 1992a) proves that a ten-candidate profile exists where 9(9!) (recall, 9! = (9)(8)(7) … (2)(1), so 9(9!) is over three million) different election rankings without ties result from changing the positional method; each candidate is top ranked with some rules and bottom ranked with others. (For *n*-candidates, up to (*n* − 1)[(*n* − 1)!] different strict election rankings can emerge from changes in positional methods.)

## Luce’s Approach

Arrow (1951) proved that with three or more candidates no voting rule satisfying his conditions always has transitive outcomes. Luce (1959) adopted a different approach; he imposed constraints on admissible election outcomes. His conditions, which are described in terms of probabilities to reflect his interest in individual decisions, are stricter than Arrow’s. Expressed in terms of voting, Luce requires a candidate’s vote percentage to remain consistent over all subsets of candidates. For instance, if A, B, and C receive, respectively, 1/3, 1/2, and 1/6 of the vote, then in a pairwise comparison B beats A by receiving (1/2)/[(1/3) + (1/2) = 3/5 of the vote. Luce’s conditions, then, capture settings where a candidate’s support is intrinsic; relative to other candidates, the support remains fixed over all sets of candidates even should new ones join.

The accompanying voting rule and admissible profiles are not specified; they are selected to be consistent with Luce’s conditions. But, even with his strong conditions, the accompanying profile restriction with the plurality vote is surprisingly relaxed. Only limited extensions of this approach have been explored for voting theory, but more is possible for settings where candidates have intrinsic support.

## Emphasizing the Data

So far I have sampled ways to analyse voting rules through properties of the rules and by imposing restrictions on admissible election outcomes. It remains to explore how the domain structure – the individual preferences – sheds light on these rules. The approach mimics how we might determine whether an election outcome reflects the ‘will of the voters’: one way is to compare the outcome with what the voters say they want. To develop methodology, reverse the order: first determine what the voters want, and then determine which voting rules respect these outcomes.

To indicate how to determine what the voters want, consider tallying an Alice > Barb ‘22:20’ election outcome. One tallying approach combines an Alice and a Barb vote – a tie. After counting the 20 ties, Alice breaks the tie as she has two extra supporters. For more candidates, the approach is to determine configurations of preferences that arguably constitute ties. This provides a filter; if a voting rule fails to deliver a tie, expect it to introduce a bias in election outcomes. While this is the motivation, the technical objective is to find a coordinate system for the space of profiles. Different coordinates represent how portions of profiles influence different voting rules.

profiles that cause

*all possible*positional method problems, but with no effect on pairwise rankings;profiles that cause

*all*problems with pairwise majority votes, but with no effect on positional rankings; andprofiles where no problems arise with any positional or majority vote rule.

The coordinates allow us to explain properties of election rules. For instance, positional rules failing to have a tie for the first class of profiles can seriously disagree with pairwise majority vote outcomes.

The second class of profiles explains problems dating to the 1780s about conflicts between pairwise and positional methods as well as agendas, tournaments and so forth.

Conflicts associated with any profile, such as our initial one, can be explained; for example, finding the portions of a profile in each of these directions identifies why different rules have different election outcomes.

Examples illustrating any possible paradox can be constructed. Start with a profile in the last class where there is complete agreement among all rules. To introduce a conflict with positional methods, add a profile portion from the first class; to create conflict with pairwise outcomes, add a profile portion from the second class.

To determine the first coordinate direction, we must find all profiles affecting only positional outcomes. While this is done mathematically, for an intuitive explanation combine a ranking with its reversal, for example, (A > B > C, C > B > A): it is arguable that the outcome should be a tie. It is a tie for majority votes over pairs. But with positional rules (w_{1}, w_{2}, 0), the A:B:C tallies are w_{1}:2w_{2}:w_{1}.where a tie occurs if and only if (iff) *w*_{1} = 2*w*_{2}; that is, the desired tie occurs iff the Borda Count is used. If this configuration is used as a filter, then beware of a non-Borda rule. This is because, instead of a tie, rules with *w*_{1} > 2*w*_{2} (for example, the plurality vote) have an A = C > B outcome, while rules with *w*_{1} < 2*w*_{2} (for example, the anti-plurality vote) have a B > A = C outcome. Consequently, profiles exist where non-Borda positional rankings must differ from majority vote outcomes.

Surprisingly, *all possible differences* among three-candidate positional election rankings reflect how different rules handle these *reversal* profile components. Indeed, to create the initial example, I started with one voter with the B > C > A preference. To generate differences in positional outcomes, add x reversal units of (A > B > C, C > B > A) and y of (A > C > B, B > C > A). As the plurality and anti-plurality tallies for A:B:C are, respectively, *x* + *y*:*y*:*x* and *x* + *y*:2*x* + *y*:*x* + 2*y*, algebra yields my *x* = 2, *y* = 3 choices creating the desired positional outcomes – and conflicts. (Borda is not affected by reversal terms, so its ranking remains the starting B > C > A.) As all possible positional differences are generated by reversal terms, any justification for one positional rule (for example, properties that uniquely identify one rule over others) must reduce to analysing the reversal component (A > B > C, C > B > A) tally.

The second coordinate direction, capturing all conflict among pairwise majority votes, is the Condorcet triplet with its resulting cycle. This component is responsible for all pairwise voting mysteries, including the majority vote cycles, differences in Dodgson’s and Kemeny’s methods, problems with agendas, tournaments and so forth. This assertion holds for any number of candidates. To create a Condorcet *n*-tuple, start with an *n*-candidate ranking, say A > B > C > D > E. For the next ranking, place the top candidate on the bottom, creating B > C > D > E > A. Continue until each candidate is in first, second, …, last place precisely once. This configuration should define a tie, and it does for all positional methods. But the profile also creates majority vote cycles. Surprisingly, these profile coordinate components cause all possible pairwise problems.

To illustrate with our initial example, start with the B > C > A preference. Adding *z* units of (A > B > C, B > C > A, C > B > A) results in A:B, B:C, C:A pairwise votes of, respectively, 2*z*:1 + *z*, 2*z* + 1:*z*, 2*z*:1:*z*. So *z* = 2 creates the desired cycle. Adding these reversal and Condorcet terms to the starting ranking yields the initial example.

The remaining coordinate directions, where nothing goes wrong, are called *Basic* directions. For candidate A, it consists of two preferring A > B > C, two preferring A > C > B, one preferring B > A > C, one preferring C > A > B; that is, two for each ranking where A is top-ranked, one for each where A is second-ranked. More generally with *n*-candidates, candidate X’s Basic direction has (*n*–*j*) voters with each ranking where X is *j*th ranked. While not intuitive, these coordinate directions come from mathematics. The important point is that no conflict occurs in this profile space; for example, the tallies for *any* voting rule for all candidates identifies the tally for *all* voting rules over any subset of candidates. Nothing goes wrong. These three kinds of directions span the six dimensions of profile space, so they complete the three-alternative analysis. (A profile, of course, normally has only parts in each direction.)

## Explaining All Differences

All possible differences among three-candidate standard voting rules, then, reflect how voting rules react to reversal and Condorcet profile components. The many desirable properties of the Borda Count, for instance, arise because it is the only rule based on positional and majority votes that always delivers a tie for these components.

I indicated how all positional differences reflect how positional rules treat reversal terms, so it remains to describe the Condorcet components. For motivation, suppose three voters must vote for one of two candidates from each of three schools. Suppose the candidates are [Anne, Bob], [Connie, Dave], [Ellen, Fred]. Does a [Bob, Dave, Fred] outcome, each by 2:1, reflect the voters’ views? To answer this question without knowing the actual preferences, all supporting preferences must be listed.

Four of the five profiles have two voters selecting different candidates from each school; this causes a tie. Breaking the tie is the last voter’s [Bob, Dave, Fred] preference. The fifth profile has the preferences [Anne, Dave, Fred], [Bob, Connie, Fred], [Bob, Dave, Ellen]. It is difficult to argue against the outcome for the first four profiles as a tie is broken. At least statistically, then, the outcome respects most supporting profiles. But it is difficult to justify the fifth ‘outlier’ profile other than pointing to the 2:1 votes.

While most profiles justify the conclusion, suppose the fifth ‘outlier’ profile is the actual one where each voter wanted to elect a woman and a man. The profile reflects their wishes; the outcome does not. The reason is clear: the majority vote strictly emphasizes information about specific pairs; it ignores information – even intended relationships – among pairs. Consequently, rather than recognizing the added ‘balanced gender’ condition, the majority vote must ignore it.

To connect this example with the Condorcet triplet, identify Anne = B > A, Bob = A > B; Connie = C > B, Dave = B > C; Ellen = A > C, Fred = C > A: the Condorcet triplet becomes the outlier ‘fifth profile’, and the ‘balanced gender condition’ is equivalent to ‘transitivity’. Because any argument applied to one setting transfers to the other, it follows that the cyclic outcome for the Condorcet triplet (the ‘paradox of voting’) occurs because (*a*) this outcome reflects most supporting profiles (even though, by involving cyclic preferences, they are not admitted), and (*b*) the majority vote strips all connecting information, *including transitivity*, from the profile. (*c*) While majority pairwise voting may suffice if candidates have ‘intrinsic support’, it can distort outcomes for usual cases.

Pairwise outcomes reflect the average over

*all possible*supporting profiles; paradoxes, such as with the Condorcet triplet, indicate that the actual profile is an outlier relative to the average.Majority votes strip away all intended relationships, including transitivity, from the profile.

Whenever intended relations are dropped, they come from profile portions based on Condorcet

*n*-tuples.

## Explaining Mysteries

The above structure explains several mysteries. The ones described here compare the Borda and Condorcet rules, briefly discuss all rules based on pairwise outcomes, and explain Arrow’s Impossibility Theorem.

As indicated, for any number of candidates all possible differences between the Borda and pairwise rankings manifest the majority vote’s reaction to Condorcet *n*-tuples, which introduce cyclic affects. As an illustrating example, with two preferring A > B > C, and one preferring B > A > C, both the Borda and pairwise rankings reflect A > B > C. Adding x units of the Condorcet [B > A > C, A > C > B, C > B > A] never affects the Borda ranking, but its cyclic effect changes the A:B, B:C, C:A pairwise tallies to 2 + *x*:2*x* + 1, *x* + 3:2*x*, 2:2*x* + 3 where *x* = 2 makes B the Condorcet winner, *x* ≥ 4 creates a cycle.

*Any* difference between the Borda and Condorcet winners, then, reflects Condorcet profile components. Thus, any argument supporting Condorcet over Borda must justify something other than a tie for a Condorcet triplet or *n*-tuple.

Voting rules relying on majority vote pairwise rankings, such as Kemeny’s and Dodgson’s rules, inherit the majority vote difficulties caused by Condorcet *n*-tuples. As these rules are primarily intended to handle cyclic behaviour, their value presumably emerges when the Condorcet component is dominant. But the stripping action of the majority vote over these components means that, unexpectedly, the rule cannot use information about the voters’ transitive preferences. Consequently, if the transitivity of voter preferences is valued, such rules should not be used. If transitivity is not valued, we must question using rules that impose transitivity on the outcomes.

A similar analysis holds for Arrow’s Theorem (Saari 2001). An unexpected feature of IIA, as with the majority vote, is to strip from the decision rule all information that individuals have transitive preferences. But, if the rule cannot use the transitivity of individual preferences, then transitive societal outcomes cannot be expected unless profiles are severely restricted; that is, the societal outcome reflects the imposed data structure rather than properties of the rule. One severe restriction is to use the preferences of a single voter; this explains Arrow’s dictator.

As Arrow’s negative result is strictly caused by IIA unintentionally stripping away valued information about individual preferences, resolutions must modify IIA to allow the rule to use this information. To illustrate, a transitive ranking, say A > B > C, separates some alternatives from others. Listing these separations as [A > B, 0], [B > C, 0], [A > C, 1] provides information about the transitive individual preferences. Let IIIA (*Intensity IIA*) be where a pair’s societal ranking is determined by how each voter ranks the pair *and the number of separating alternatives*. By replacing IIA with IIIA in Arrow’s conditions, Arrow’s dictator is replaced with the Borda Count, and rules based on the Borda Count.

## Strategic Behaviour

Beyond the above ‘single-profile’ problems, multiple-profile concerns catalogue interesting changes in outcomes by changing a profile. They include the seminal Gibbard (1973)–Satterthwaite (1975) theorem asserting that, with three or more alternatives, no decision rule is immune from strategic behaviour: that is, with any rule, situations exist where some voter ensures a personally better outcome by voting according to other than her true preferences. There is, in fact, a host of related behaviour; see, for example, Nurmi (1999, 2002). Some rules, for instance, can cause a winning candidate to *lose* by attracting more supporting voters. Similarly, Fishburn and Brams (1983) discovered the ‘no-show’ paradox where, with the plurality run-off, a voter obtains a personally better outcome by *not* voting.

These results reflect the higher dimensionality of profiles that accompanies added alternatives. With two candidates, a voter can vote for, or against, her favorite. With more alternatives, beyond her top and bottom choice, a voter can consider intermediate options. As suggested by the ‘don’t waste your vote’ cry for strategic voting, situations exist where, by voting strategically, some voters can block personally lower-ranked candidates from winning. The Gibbard–Satterthewaite result proves this happens for all realistic rules.

A common source of problems, such as the no-show paradox, or where two subcommittees elect ‘A’ but the combined committee does not, and so forth, is when the rule loses monotonicity. Positional methods are monotonic; that is, with added support a candidate has higher tallies. But difficulties occur with rules involving several subsets of candidates; for example, a run-off involves {all *n*-candidates} and {top two}. What causes problems is that the first election determines who is advanced to the second. Consequently, added support for a winning candidate could also advance a stronger opponent to the run-off.

## Implications for Economics

Voting rules are aggregation methods: voters’ preference rankings are aggregated into a societal ranking. But as much of economics, and the social sciences, also involves aggregation rules, we must anticipate that the behaviour of voting rules predicts behaviour elsewhere in economics and other disciplines. This happens. As illustrations, the above result allowing 9(9!) different positional election rankings for a single ten-candidate profile, where almost any specified outcome can occur, has a parallel with the Sonnenshein (1972)–Mantel (1972)–Debreu (1974) Theorem asserting that any continuous function satisfying Walras’s Laws can be (up to minor technical conditions on prices) the aggregate excess demand function for some exchange economy. As another example, recall the voting result stating that, even if the rankings for the different subsets of candidates are selected in an arbitrary manner, a supporting profile can be found. The same behaviour arises in economics. The voting result allowing a ranking to be selected for each subset of candidates, and a profile can be found so that the selected ranking is the actual election ranking also has an economic parallel: that is, the Sonnenshein–Mantel–Debreu Theorem extends to where a different function can be selected for each subset of commodities, and an economy (initial endowment and utility function for each agent) can be found so that (with the same technical condition) the aggregate excess demand for each subset is the selected one (Saari 1992b).

Voting results have parallels in non-parametric statistics, namely, select rankings for each subset of alternatives: for most non-parametric rules, a data-set can be found so that each set’s actual ranking is the selected one. In voting, the positional rule most immune from the ‘anything can happen’ difficulty is the Borda Count. In nonparametric statistics, the Kruskal–Wallis test has similar properties (Haunsperger 1992).

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