**3. Radial Symmetry as a Sufficient Condition to Preclude Condorcet Cycles under (1)**

For (1) for our framework above, we state and prove three propositions.

**Proposition 1.** *Condorcet cycles are precluded if radial symmetry holds.*

**Proof.** For two candidates *G* at (*xG*, *yG*) and *H* at (*xH*, *yH*), define

$$D\_G^2 = \mathfrak{x}\_G^2 + \mathfrak{y}\_G^2 \quad \text{and} \quad D\_H^2 = \mathfrak{x}\_H^2 + \mathfrak{y}\_H^2 \downarrow$$

which are the respective squares of their distances from the origin (i.e., from *M*). We start by showing that *G*-*H* if and only if *D*<sup>2</sup> *<sup>G</sup>* < *<sup>D</sup>*<sup>2</sup> *H*.

A voter *V* at (*xV*, *yV*) prefers *G* to *H* if and only if *U*(*V*, *G*) > *U*(*V*, *H*), or if and only if

$$-\left(\mathbf{x}\_V - \mathbf{x}\_G\right)^2 - \left(\mathbf{y}\_V - \mathbf{y}\_G\right)^2 > -\left(\mathbf{x}\_V - \mathbf{x}\_H\right)^2 - \left(\mathbf{y}\_V - \mathbf{y}\_H\right)^2,$$

or if and only if

$$2(\mathbf{x}\_{\rm G} - \mathbf{x}\_{H})\mathbf{x}\_{V} + 2(\mathbf{y}\_{\rm G} - \mathbf{y}\_{H})\mathbf{y}\_{V} > D\_{\rm G}^{2} - D\_{\rm H}^{2} \,. \tag{3}$$

The line that bounds the region (3) is parallel to the line that bounds

$$2(\mathbf{x}\_G - \mathbf{x}\_H)\mathbf{x}\mathbf{v} + 2(y\_G - y\_H)y\mathbf{v} \ge 0 \,. \tag{4}$$

However, the latter line passes through the origin and so, by the assumption of radial symmetry, has half the electorate on each side of it. The region defined by (3) (which is the set of voters *V* who prefer *G* to *H*) is larger than the region defined by (4) if and only if *D*<sup>2</sup> *<sup>G</sup>* < *<sup>D</sup>*<sup>2</sup> *<sup>H</sup>*. Thus, *G*-*H* if and only if *D*<sup>2</sup> *<sup>G</sup>* < *<sup>D</sup>*<sup>2</sup> *<sup>H</sup>*, since the latter region contains half the electorate.

To complete the proof, consider any three candidates *A*, *B*, and *C* whose respective squared distances from the origin are *D*<sup>2</sup> *<sup>A</sup>*, *<sup>D</sup>*<sup>2</sup> *<sup>B</sup>*, and *<sup>D</sup>*<sup>2</sup> *<sup>C</sup>*. It is, of course, not possible for these three values to exhibit intransitivity, so a Condorcet cycle among *A*, *B*, and *C* cannot occur.

Although Proposition 1 and its proof are only for two dimensions, we remark that extension to more than two dimensions is immediate. Note also that it is easy to extend the proof to preclude any cyclicity among any group of more than three candidates.

**Proposition 2.** *If (given radial symmetry) any set of candidates includes a candidate K located at M (the origin), then K is a Condorcet winner.*

**Proof.** Follows at once from the foregoing, by noting that *D*<sup>2</sup> *<sup>K</sup>* = 0.

Proposition 2 can be considered an analog, for continuous ideal-point distributions, of the basic result of Pl, F-G, and Mi (for discrete distributions) that proves that a Condorcet winner, located at the origin, must exist under their symmetry conditions. Not only is Proposition 2 unusual in this setting in that it pertains to continuous rather than discrete ideal-point distributions, but also its proof has the benefit of being comparatively simple.

**Proposition 3.** *Radial symmetry guarantees a Condorcet winner even if there is no candidate located at the origin.*

**Proof.** This result, stronger than Proposition 2, follows from Proposition 1 because the impossibility of any Condorcet cycles implies the existence of a Condorcet winner.

Curiously, as a matter of theoretical interest, and in contrast to the finite-electorate setup, Proposition 3 functions even if no *voter* is located at the origin. For example, in the distribution of voter ideal points there could be a doughnut hole centered at the origin.

The possible pedagogical value of the above results should not be overlooked. In particular, the proof of Proposition 1, under a continuous distribution of ideal points, is shorter and more straightforward than corresponding proofs under discrete distributions. Related results and proofs in DD&H, however, do apply to a continuous ideal-point distribution and an infinite electorate and are similar (though not identical) to ours above. Our development may be less comprehensive, but also more understandable, than that of DD&H.

We note that Propositions 1–3 still hold under a special case of (2) in which *w*1*V*, *w*2*<sup>V</sup>* are replaced by *w*1, *w*<sup>2</sup> such that the weights are the same for all voters at a given point (as well as from one point to another) but can still differ between the two issues. This is just a transformation of scale: If we substitute *x* = *x*0/*w*1/2 <sup>1</sup> and *<sup>y</sup>* <sup>=</sup> *<sup>y</sup>*0/*w*1/2 <sup>2</sup> into the modified (2), then *x*<sup>0</sup> and *y*<sup>0</sup> satisfy (1), and radial symmetry still holds after the transformation.

#### **4. Previous Work That Has Considered Unequal Weighting**

Thus far, we have concentrated on the case where all voters weight the issues alike. For utility functions, whether with or without radial or other symmetry in the ideal-point distributions that apply, most authors do not consider cases where voters differ as to how they weight the dimensions. However, such differences in weighting are central to the present paper.

In earlier work, Davis and Hinich (1968, p. 68) treat a situation where some voters care only about a single issue (which may vary from one voter to another). Although Davis et al. (1970, p. 434) refer to but eschew a model with different weights for different voters, they do briefly mention that a voter may care about only one issue (p. 433), that farmers may care especially about farm price supports and petroleum-industry people about oil import quotas (p. 440), and that an assumption of equal weighting is not generally satisfied (p. 446). Hoyer and Mayer (1974) provide extensive comparisons of a model where all voters have the same weighting versus a model where voter weightings can differ. Hinich and Pollard (1981) use a model that allows voters to differ not only in their weightings of issues but also in their perceptions of candidates' positions on issues. Each candidate is located along a single "predictive dimension" (though more than one such dimension may be possible). Rabinowitz et al. (1982), Niemi and Bartels (1985), and Erikson and Romero (1990) consider weighting differences that are associated with differences in the saliences that voters attach to issues. Rivers (1988, p. 740) disapproves that voter heterogeneity in issue weighting is ignored "in common practice," before developing an approach that handles it. Although the utility function in the model of Adams (1997, p. 1254) allows for differences among voters as to how they weight the dimensions, that flexibility is not put to use. Adams and Adams (2000, p. 140) mention the possibility that "different policy dimensions matter to different voters" but judge that such a point does not affect their conclusions. Feld et al. (2014, pp. 480–81) briefly allude to a possible model "that would allow variation in the salience different voters attach to different issues."

Unlike the references just cited, Hoyer and Mayer (1975, p. 806) provide an example, for a finite electorate, where a Condorcet cycle occurs with radial symmetry and weighting differences among voters. That example has three candidates and nine voters. Though it is presented geometrically rather than algebraically, its utility functions do not all conform to (2) above. They do so for three of the nine voters. For the other six, however, the utility function is the same as (2) but with an added cross-product term of the form −2*w*12*V*(*xV* − *xG*)(*yV* − *yG*).
