**1. Introduction**

An election or vote may breed discord if there is no Condorcet winner—if every alternative faces at least one competing alternative that the electorate (strictly) prefers. That quandary reflects a Condorcet cycle, where (more than half of) the electorate prefers alternative *X* to *Y*, *Y* to *Z*, and *Z* to *X*.

We define a symmetry condition, which we refer to as radial symmetry, that is similar to that in the classic work of Plott (1967) (and is a large-electorate analog thereof) but involves an infinite (rather than finite) number of voters along with a continuous distribution of voter ideal points. Similar to Plott's condition, it is highly restrictive.

It precludes a Condorcet cycle among any three alternatives (thus guaranteeing a Condorcet winner) if all voters weight the issues alike. However, such is no longer the case if voters differ as to the relative importance they attach to the issues, as we show using counterexamples with two dimensions (issues). This interesting theoretical finding may have practical implications: It may suggest that, correspondingly, Condorcet cycles are more likely empirically if voters weight the issues disparately. Dimensions that may be afforded different relative importance by different political parties or voters may include economic left–right, social–cultural, and others (see, e.g., Polk et al. 2017). Of course, the extent to which Condorcet cycles have real-world relevance in the first place has been questioned, as in Tullock (1967). Simpson (1969) amplifies Tullock's (1967) work and mentions Plott's (1967) condition.

Our work here has several distinguishing features. (i) Its model deals with differential issue weightings among voters, a facet of spatial modeling that is not often recognized

**Citation:** Potthoff, Richard F.. 2022. Radial Symmetry Does Not Preclude Condorcet Cycles If Different Voters Weight the Issues Differently. *Economies* 10: 166. https://doi.org/ 10.3390/economies10070166

Academic Editors: Ralf Fendel, Robert Czudaj and Sajid Anwar

Received: 9 April 2022 Accepted: 9 June 2022 Published: 13 July 2022

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in prior literature. (ii) Its model uses a continuous distribution of ideal points with an infinite number of voters, in contrast to the setup with a finite number of voters that is generally specified under the Plott (1967) model. This infinite-electorate model not only bears greater resemblance to the real world of voters in large public elections but also has certain advantages of lucidity. (iii) For the case where issue weightings are the same for all voters, the continuous distribution of ideal points with an infinite electorate leads to a simpler proof of the result—akin to the classical one under the Plott model—that radial symmetry precludes Condorcet cycles. (iv) For the case where issue weightings differ among voters, we focus on showing that radial symmetry does not preclude Condorcet cycles under the infinite-electorate setup but also find that that conclusion holds under the Plott finite-electorate model as well. This primary result of ours is expressed in the title of the paper. (v) In addition to their theoretical interest, our results may have the practical implications indicated in the preceding paragraph: They may suggest a hypothesis that cycles are more apt to occur with greater weighting differences among voters.

In what follows, Section 2 covers the details of our setting and concepts. For the case where voters' weightings are homogeneous and the electorate is infinite, propositions for radial symmetry established in Section 3 have simpler proofs and stronger results than comparable ones for a finite electorate with their standard symmetry conditions.

A limited discussion of earlier writings on differential weightings is in Section 4. This review of prior results covers a short note (Hoyer and Mayer 1975) that presents a finiteelectorate example with radial symmetry where a Condorcet cycle occurs under differential weighting, though with an atypical set of utility functions.

For the case (under radial symmetry) where the voter weightings differ from one another, only one counterexample is needed to prove that Condorcet cycles are not precluded. Nonetheless, Section 5 provides several of them to illustrate different ways in which the cycles can occur for an infinite electorate; Section 6 covers the case of a finite electorate.

The degree of empirical prevalence of Condorcet cycles is examined in Section 7. In an empirical illustration based on two dimensions consisting of left–right and linguistic issues, Section 8 looks at a Condorcet preference cycle in Finland's 1931 presidential election. Section 9 summarizes.

#### **2. Framework**

We start by defining our framework, which is similar, but not identical, to that of Plott (1967; hereafter Pl); is even closer to that of Feld and Grofman (1987, especially Theorem 6; hereafter F-G) and Miller (2015, especially p. 172; hereafter Mi); and has still more common elements with Davis et al. (1972; hereafter DD&H). As we lay out our framework below, we indicate differences between it and Pl, F-G, Mi, and DD&H. (Note, though, that the second and third framework items below obviously do not apply to Section 6 below.)

**Dimensions.** We work with an issue space of two dimensions, but results are generally extendable to more than two.

**Voters.** We have an infinite number of voters (which closely approximates a large electorate). Each voter *V* has a (unique) ideal point, where *V*'s utility is maximal. By contrast, Pl, F-G, and Mi use a finite number of voters, though they can be located in varying ways. DD&H allow for either a finite or an infinite electorate.

**Distribution of voters' ideal points.** We posit that this ideal-point distribution is continuous. It is discrete for Pl, F-G, and Mi (since their number of voters is finite). DD&H provide for it to be either discrete or continuous. Use of a continuum of voters in voting applications is also found in (e.g.) Caplin and Nalebuff (1988, p. 789).

**Alternatives.** Alternatives, or options, could be political candidates, legislative proposals, or something else. Hereafter, we will just refer to the alternatives as *candidates* but with the understanding that other possibilities are not precluded. Candidates' (unique) ideal points are located in the same issue space as those of voters. Our counterexamples use three candidates, though similar ones with more than three could easily be constructed. Pl, F-G, and Mi do not consider triples of candidates (or of other alternatives) in the way that we do. DD&H have two examples that use three alternatives.

**Utility function.** We write *PV*: (*xV*, *yV*) for the ideal point of voter *V* and *PG*: (*xG*, *yG*) for the ideal point of candidate *G*. Our utility function is negative squared Euclidean distance for the case where all voters weight the two issues alike, so that the utility of voter *V* for candidate *G* is

$$\mathcal{U}(V, G) = -(\mathbf{x}\_V - \mathbf{x}\_G)^2 - (\mathcal{Y}\_V - \mathcal{Y}\_G)^2. \tag{1}$$

For F-G, Mi, and DD&H, utility is also based on Euclidean distance, but for Pl it involves the utility gradient because the Pl treatment is more general.

**Disparate weighting of issues.** For the case where voters differ in their relative weightings of the two issues, we generalize (1) and use the utility function

$$\mathcal{U}(V,G) = -w\_{1V}(\mathbf{x}\_V - \mathbf{x}\_G)^2 - w\_{2V}(y\_V - y\_G)^2. \tag{2}$$

Weight ratios *w*1*V*:*w*2*<sup>V</sup>* of 6:3, 2:1, and 1: <sup>1</sup> <sup>2</sup> (e.g.) all have the same effect. (Suitable weight ratios will vary depending on the relative scaling of *x* and *y.*) In general, the weight ratios *w*1*V*:*w*2*<sup>V</sup>* follow a distribution that can vary with (i.e., can be conditional on) the point *PV*: (*xV*, *yV*). However, for our counterexamples later, except those in Section 6, the *w*1*V*:*w*2*<sup>V</sup>* weight ratios have the same distribution for the voters at every point (*xV*, *yV*) in the issue space (e.g., a distribution that, at each point, assigns 1:9 for half the voters and 9:1 for the other half). Accordingly, with the *w*1*V*:*w*2*<sup>V</sup>* distribution independent of the ideal point *PV*, it is designated more simply as the *w*1:*w*<sup>2</sup> distribution. (We can create each type of counterexample in Section 5 without a need for the *w*1*V*:*w*2*<sup>V</sup>* distribution to differ for different *PV*.) Pl, F-G, Mi, and DD&H do not consider different weightings of the dimensions by different voters and so do not deal with anything like (2).

**Radial symmetry.** We define *radial symmetry* to be present in the distribution of voter ideal points if there exists a point *M* such that every line that passes through *M* has half the voters on each side of the line. (Because the distribution is continuous, points that are on the line itself can be disregarded.) Without loss of generality, we assume *M* to be the origin, (0, 0). Note that, with our framework, the definition of radial symmetry pertains only to the distribution of voters' ideal points and says nothing about a utility function (or about weighting of issues, or about the candidates). F-G and Mi do not use the term *radial symmetry* but, for a discrete distribution of voter ideal points, have a concept that can be deemed analogous to ours: There is a voter at *M*, and every line through *M* has the same number of voters on each side of *M*. The framework of Pl is also analogous but is more general and does involve utility. DD&H use concepts similar to ours.

**Condorcet cycles.** Under (1), our definitions that follow are consonant with standard ones and with those of Pl, F-G, Mi, and DD&H. Voter *V* at point *PV*: (*xV*, *yV*) *prefers* candidate *G* at *PG*: (*xG*, *yG*) to candidate *H* at *PH*: (*xH*, *yH*) if and only if *U*(*V*, *G*) > *U*(*V*, *H*). We write *G*-*H* to denote that more than half the electorate prefers *G* to *H*. (*G*-*H* if and only if more than half the electorate has greater utility for *G* than for *H* (i.e., is closer to *PG* than to *PH*, or, equivalently, is on the *PG* side of the perpendicular bisector of the line segment joining *PG* and *PH*).) We define a *Condorcet cycle* to exist among three candidates *A*, *B*, and *C* if *A*-*B*, *B*-*C*, and *C*-*A*; or if *B*-*A*, *C*-*B*, and *A*-*C*. We then define *preclusion of Condorcet cycles* to mean that it is not possible to choose, from anywhere in the issue space, a triple of candidates *A*, *B*, and *C* that exhibits a Condorcet cycle. If, in a group of existing candidates, *A* is a(n existing) candidate such that *G*-*A* (*A*-*G*) for no other (existing) candidate *G* located elsewhere, then *A* is a *Condorcet winner* (*Condorcet loser*). Under (2) with the weight ratio following the same distribution throughout the space of voter ideal points, definitions akin to the foregoing apply.

**Miscellaneous.** We mention three further details. First, although we use straightforward definitions of Condorcet winner and Condorcet loser, it should be noted that (rarely) there can be more than one of either (e.g., two Condorcet winners, *X* and *Y*, will exist if the preference ranking is *XYZ* for half the electorate and *YXZ* for the other half). Second, although the absence of a Condorcet winner implies the presence of a Condorcet cycle, the converse is not true (e.g., *B*-*C*, *C*-*D*, and *D*-*B*, but *A*-*B*, *A*-*C*, and *A*-*D*). Third, we refrain from dealing with concepts such as the *core* (e.g., Saari 1997; Schofield 2008) that are closely related to Condorcet winners but are not essential to our development here.
