**5. With an Infinite Electorate, Radial Symmetry, and Unlike Weighting, Cycles Can Occur**

For an infinite electorate, we next provide counterexamples to establish that, even with (unlikely) radial symmetry of the distribution of voter ideal points, preclusion of Condorcet cycles is not guaranteed if voters differ (in their utility functions) as to the comparative importance they attach to the issues in the policy space. Even given the tight radial-symmetry restrictions, some critics might still find this result unsurprising. Here, we are proving the result by counterexample to provide broad insights, although other methods of proof might be used. Even though with equal weighting slight perturbances from radial symmetry are sufficient to create a Condorcet cycle, our counterexamples with unequal weighting (and radial symmetry) were not easy to construct, rest on thin margins for some candidate pairings, and (in most cases) place all or almost all of a given voter's weight on just one of the two issues.

We consider two bivariate (infinite-electorate) ideal-point distributions that observe radial symmetry. The first is a bivariate normal distribution with a mean of 0 and standard deviation of 10 for both *x* and *y*; we use a correlation coefficient of 0 (although, even if it is not 0, radial symmetry still holds). The second is a uniform (rectangular) distribution with probability density function *f*(*x*, *y*) = 1/400 over the square with vertices (*x*, *y*)=(−10, 10), (10, 10), (−10, −10), and (10, −10).

A single counterexample would suffice to show that Condorcet cycles are no longer precluded for voter ideal-point distributions with radial symmetry in an infinite electorate if voters differ as to how they judge the relative importance of the issues. We provide several counterexamples, however, in order to illustrate particular features.

**Examples 1.1 and 1.2.** The respective spatial locations of candidates *A*, *B*, and *C* are

$$P\_A \colon (\mathbf{x}\_A, \mathcal{Y}\_A) = (1, \mathcal{T}); \\ P\_B \colon (\mathbf{x}\_B, \mathcal{Y}\_B) = (\mathcal{S}, -\mathcal{T}); \\ \text{and } P\_C \colon (\mathbf{x}\_C, \mathcal{Y}\_C) = (-\mathcal{Y}, \mathcal{S}).$$

If all voters have the same utility function, as in (1), then *A*-*B*, *B*-*C*, and *A*-*C* since *D*2 *<sup>A</sup>* = 50, *<sup>D</sup>*<sup>2</sup> *<sup>B</sup>* = 58, and *<sup>D</sup>*<sup>2</sup> *<sup>C</sup>* = 106 (no cycle).

However, now suppose instead that (at each point in the issue space) half the electorate attaches sole importance to the issue on the *x*-axis whereas the other half cares only about the *y*-dimension. The utility functions (of voter *V* for candidate *G*) can then be represented by (2) with respective weight ratios *w*1:*w*<sup>2</sup> of 1:0 and 0:1.

**Example 1.1.** *If the ideal points follow the uncorrelated bivariate normal distribution indicated above, then A*-*B, B*-*C, and C*-*A, so a Condorcet cycle exists. See Appendix A.1 for calculation details.*

**Example 1.2.** *Suppose now that the distribution of ideal points is uniform (as described above) rather than bivariate normal. Then a Condorcet cycle exists again, once more with A*-*B, B*-*C, and C*-*A. Here, the margins by which the electorate prefers A over B, B over C, and C over A are identical: 55% to 45% in all three cases. See Appendix A.1 for calculation details.*

In Examples 1.1–1.2, as well as below in Examples 3.1–3.2 and in parts of Example 4, each voter is concerned about *only one* of the two issues. Those examples were constructed in that way to provide ease of exposition and calculation.

It might be claimed, though, that all these counterexamples are problematic because the concern of voters for only one dimension renders the setting unidimensional rather than truly two-dimensional. Although we find it hard to grant that this is a valid objection, we nonetheless counter by pointing out that the proportion of the electorate with *G*-*H* (for two candidates *G* and *H*) can be a continuous—not discontinuous—function of the weight ratios. Thus, a change in the weight ratios from 1:0 and 0:1 to 1:ε and ε:1, for tiny enough ε, will not break the cycle. In fact, ε does not (in general) even need to be tiny, as evidenced by the next example.

**Example 2.** *The same cycle as in Examples 1.1–1.2 (A*-*B, B*-*C, C*-*A) also occurs if (e.g.) Example 1.2 is unchanged except that all voters care (somewhat) about both dimensions with weight ratios w1:w2 of 10:1 and 1:10 in place of 1:0 and 0:1. The (complicated) calculation details are in Appendix A.2.*

**Examples 3.1 and 3.2.** For candidates *A*, *B*, and *C*, respective locations are

*PA*: (*xA*, *yA*)=(−3, 9); *PB*: (*xB*, *yB*) = (7, 7); and *PC*: (*xC*, *yC*) = (5, −9).

With the same utility function for all voters as in (1), no cycle can exist: *A*-*B*, *B*-*C*, and *A*-*C*, with *D*<sup>2</sup> *<sup>A</sup>* = 90, *<sup>D</sup>*<sup>2</sup> *<sup>B</sup>* = 98, and *<sup>D</sup>*<sup>2</sup> *<sup>C</sup>* = 106. However, suppose now that, as in Examples 1.1–1.2, weightings differ across voters according to (2) with weight ratios *w*1:*w*<sup>2</sup> of 1:0 for half the electorate and 0:1 for the other half.

**Example 3.1.** *If the ideal points follow the uncorrelated bivariate normal distribution, a Condorcet cycle occurs with B*-*A, C*-*B, and A*-*C. See Appendix A.3 for calculation details.*

**Example 3.2.** *If the ideal points follow the uniform rather than the bivariate normal distribution, a Condorcet cycle exists again, also with B*-*A, C*-*B, and A*-*C. See Appendix A.3 for calculation details.*

Note that, even though Examples 1.1–1.2 and 3.1–3.2 all have *A*-*B*, *B*-*C*, and *A*-*C* if all voters have the same utility function, with the change to 1:0 and 0:1 Examples 1.1–1.2 have the cycle *A*-*B*, *B*-*C*, and *C*-*A* whereas the cycle in Examples 3.1–3.2 is the opposite— *B*-*A*, *C*-*B*, and *A*-*C*. Thus, interestingly, only *A*-*C* is reversed in the former case, but both *A*-*B* and *B*-*C* are reversed in Examples 3.1–3.2—a salient difference between the two cases.

**Example 4.** *We extend Example 3.2, still with the same uniform distribution for the ideal points and the same three candidates, but now we suppose that the weight ratios w1:w2 for (2) are 1:0 for only 6% of the voters, 0:1 for another 6%, and 1:1 (equal weighting) for the remaining 88%. Even though an overwhelming proportion of the voters (88%) are identical with one another as to how they weight the issues, it turns out that the Condorcet cycle with B*-*A, C*-*B, and A*-*C nonetheless emerges again. See Appendix A.4 for calculation details. Therefore, here, even with radial symmetry, the generation of a cycle does not require a very large fraction of voters with non-conforming weightings.*

Obviously, where radial symmetry holds, many possible cases of unlike voter weightings of issues will *not* serve to trigger Condorcet cycles. The above counterexamples serve simply to establish the possibility of those cycles.

In all of those counterexamples, the distribution of the weight ratios *w*1*V*:*w*2*<sup>V</sup>* is the same (set at *w*1:*w*2) for voters at every point in the issue space (e.g., 10:1 for half the voters at each point and 1:10 for the other half). This allows for easier presentation but also shows that counterexamples need not be made more elaborate through use of different distributions at different points. Because of similar considerations, all of our chosen weight ratio distributions are discrete (in fact, mostly dichotomous) rather than continuous, although a possible continuous weight-ratio distribution is described briefly just before Table A1 in Appendix A.2 below.

#### **6. Counterexamples for the Case of a Finite Electorate**

Although this paper concentrates on establishing that Condorcet cycles are not precluded with differential weighting under radial symmetry if the number of voters is infinite, a comparable result holds for a finite number of voters. Thus, our final two examples demonstrate that, even under the customary symmetry conditions for a finite electorate, Condorcet cycles can occur if voters are disparate in their weightings of the issues based on the utility function (2). Similar to the examples in Section 5, which each have a common

weight-ratio distribution at all points, Example 5 observes this condition (with a 7:1 ratio for one voter and 1:7 for the other) at each voter point except the origin. Example 6, though, has only one voter at each point, with weight-ratios that differ among points.

These two examples differ from the one in Hoyer and Mayer (1975, p. 806) in that both are based on the simple utility function (2). That earlier example uses a more complex utility function for some voters, as noted above in Section 4.

**Example 5.** *This counterexample has three candidates at the same locations as in Examples 1.1–1.2, and nine voters as follows:*

*V*1–*V*4 are at (−6, 2), (6, 2), (−6, −2), (6, −2); *V*5–*V*8 are (respectively) at these same four locations; *V*9 is at (0, 0).

These nine voters satisfy the symmetry conditions of F-G, Mi, and Pl: A voter exists at the origin, and any line through the origin has the same number of voters on each side. Under (1), there is no Condorcet cycle because *A*-*B*, *B*-*C*, and *A*-*C* with 5-to-4 preference margins for each matchup.

Now let (2) apply with weight ratios of 7:1 for *V*1–*V*4, 1:7 for *V*5–*V*8, and 1:1 for *V*9. A Condorcet cycle—*A*-*B*, *B*-*C*, *C*-*A*, with 5-to-4 preference margins for each—then results. See Appendix A.5 for calculation details.

**Example 6.** *This example differs from Example 5 by having no more than one voter at the same point in the issue space—a condition that may sometimes be invoked but (unsurprisingly) does not avoid the possibility of a cycle, as Example 6 shows. Here, the three candidates have the same locations as in Example 5 (and Examples 1.1–1.2). The voters are five of the nine voters of Example 5: V5, V2, V3, V8, and V9. This set of five voters satisfies the radial-symmetry requirements. Again, there is no Condorcet cycle under (1), with a 3-to-2 margin for each of A*-*B, B*-*C, and A*-*C.*

However, under (2), with weight ratios as in Example 5 for the five voters, the cycle *A*-*B*, *B*-*C*, *C*-*A* results again in Example 6, with 3-to-2 margins this time. For calculation details, see Appendix A.6.

#### **7. Empirical Frequency of Condorcet Cycles**

The importance of Condorcet winners stems from the concept that, in a singlewinner contest or election, the selectee or electee should be a Condorcet candidate (Condorcet winner)—assuming that one exists. Most voting theorists (e.g., Abramson et al. 2002; Dasgupta and Maskin 2004; Gehrlein 2006; Maskin and Sen 2018; Merrill 1988; Regenwetter et al. 2006) embrace this principle, though exceptions can be found (e.g., Saari 1995). The absence of a Condorcet winner reflects a Condorcet cycle.

Our results for radial symmetry may have empirical ramifications even though the symmetry condition itself would not be expected to exist (probably not even approximately) in practical situations. Thus, in broader, real-world milieus, it is reasonable to conjecture that Condorcet cycles—and the muddlement that they can create where no Condorcet winner emerges (e.g., Van Deemen 2014)—will be more frequent if voters weight the issues differently (rather than alike), in line with what happens where radial symmetry does exist.

The impact, though, may be greater or less depending both on the extent to which unlike versus like weightings can more often trigger Condorcet cycles and on the general frequency of conditions that are ripe for Condorcet cycles in the first place. As for the latter, judgments may differ as to how often Condorcet cycles occur empirically. Table 4 of Van Deemen (2014) shows that a Condorcet paradox (considered there to be the absence of a candidate against whom no rival is strictly preferred by the electorate) existed in 25, or 9.4%, of 265 elections or votes. Although that percentage is quite important and meaningful, an alternative percentage, calculated as the number of triples with a Condorcet cycle divided by total number of triples rather than as the number of elections with a Condorcet paradox

divided by total number of elections, might be appreciably lower. It also appears that Condorcet paradoxes in the table occur more frequently in legislative votes or the like than in elections that involve political candidates or political parties; the former may have been winnowed more selectively, with votes perhaps likelier to reflect contrived outcomes rather than true preferences.

A few results do not appear in Table 4 of Van Deemen (2014). They include the 2009 mayoral election in Burlington, Vermont (USA), in which voters ranked five candidates and none of the (5 × 4 × 3)/6 = 10 candidate triples were cyclical (Laatu and Smith 2009; Olson 2009); a 1952 poll of 562 college students that asked for ranking of 10 candidates for U.S. president and showed no Condorcet cycles among the (10 × 9 × 8)/6 = 120 triples (Potthoff 1970); a 2019 poll of "1002 likely Democratic presidential primary voters" that requested rankings of 20 Democratic candidates for U.S. president and produced no cycles in any of the (20 × 19 × 18)/6 = 2280 triples (FairVote 2019); a similar 2020 poll of 825 Democratic voters covering eight presidential candidates that also produced no cycles, in any of the (8 × 7 × 6)/6 = 56 triples (FairVote 2020); and some of the items listed by Lagerspetz (2016, pp. 384–85). In any event, although the degree of empirical incidence of Condorcet cycles, and of the nonexistence of Condorcet winners, can be difficult to determine, their impact cannot be dismissed given their disruptive potential, and so any effect of differential issue weights on generation of greater cyclicity is deserving of attention.

#### **8. The 1931 Election for President of Finland**

Finding a good example of an actual election that is in tune with the work of this paper is challenging. The real-world example that we present here is necessarily imperfect. It does have a Condorcet cycle that is well-documented, which is a rarity. It has a two-dimensional issue space but does not have radial symmetry, of course. Numerical values associated with the voters and candidates have to be assigned somewhat arbitrarily. The distribution of voter ideal points cannot be portrayed as continuous, in part because the voters are electors representing political parties (rather than typical citizens) and are thus subject to party pressure for homogeneity. In contrast to our main counterexamples above, the weighting of the two issues is the same for all voters with a given ideal point but differs from one point to another. Although some features of our example are thus not in accord with our structures set forth previously, it may still provide some useful insight.

The example is from the 1931 presidential election in Finland, in which 300 electors, chosen through proportional representation and representing six political parties or factions thereof, voted among four candidates. The preference orderings of all 300 electors were generally known (Lagerspetz 2016, p. 397) and are shown, along with the number of electors, in Table 1 for each of the six party groups. The Swedish People's Party had no candidate and had two blocs (which we call Sw1 and Sw2). Otherwise, the four candidates and their parties were: Tanner, Social Democrats; Ståhlberg, Progressives; Kallio, Agrarians; and Svinhufvud, Conservatives. Tanner was the Condorcet loser. A preference cycle, however, encompassed the three at the top: Ståhlberg over Kallio, Kallio over Svinhufvud, and Svinhufvud over Ståhlberg, as shown on the right side of Table 1.

Lagerspetz (2016, pp. 393–94) identifies three applicable dimensions or issues: traditional left–right; linguistic—Swedish-speaking versus Finnish-speaking; and degree of acceptance of democracy. Because the third dimension seems to be closely aligned with the first, we treat it as combined with the first and consider just the left–right (*x*) and linguistic (*y*) issues. The latter concerns mainly a sharp discord between Kallio and the Swedish blocs: "the Agrarian candidate Kallio—a Finnish nationalist who, unlike most leading politicians, could not even speak Swedish—was totally unacceptable to the Swedish-speaking group" (Lagerspetz 2016, p. 396).


**Table 1.** For each party or faction, preference rankings among the four candidates.

<sup>a</sup> T = Tanner, St = Ståhlberg, K = Kallio, Sv = Svinhufvud. <sup>b</sup> Pairings of T against St, K, and Sv are not shown; T loses to each of the other three by 210 to 90.

Shown in Table 2, and also graphically in Figure 1, are our (*x*, *y*) spatial-location numerical assignments to the six party groups and the four candidates, selected so as to try to reflect the essence of the political situation for the two issues. For all 24 combinations of party groups with candidates, the table provides the weighted squared distance corresponding to (2) above, using our chosen weight ratios *w*1*V*:*w*2*<sup>V</sup>* for the first to second issue. *V* now refers to a party group rather than an individual voter. There is one ratio for each party group *V*.

**Table 2.** Weighted squared distance from each party group to each candidate.


<sup>a</sup> Same for all voters in the party group. <sup>b</sup> T = Tanner, St = Ståhlberg, K = Kallio, Sv = Svinhufvud.

Our weight ratios (Table 2) are taken as 2:1 except for the two Swedish blocs, which each receive 1:2 to reflect their strong concern for the linguistic dimension, and the Agrarians, with 1:1. As an illustration, the weighted squared distance between Sw1 and Svinhufvud is 1(0 − 2)2 + 2(−<sup>1</sup> − 0)2 = 1·4+2·1 = 6. From low to high, the four values for Sw1 are 3 (Ståhlberg), 6 (Svinhufvud), 9 (Kallio), and 27 (Tanner), thus producing the preference ordering on the right side of Table 2.


**Figure 1.** Two-dimensional issue space for party groups and candidates. Traditional left-right dimension is on the *x*-axis; linguistic dimension (Swedish versus Finnish) is on the *y*-axis. Party groups: *SD* = Social Democrats; *Pr* = Progressives; *Ag* = Agrarians; *Co* = Conservatives; *Sw*1, *Sw*2 = blocs 1, 2 of Swedish People's Party. Candidates: **T** = Tanner, **St** = Ståhlberg, **K** = Kallio, **Sv** = Svinhufvud.

All six of the preference orderings in Table 2—obtained using our (*x*, *y*) points and through the weight ratios *w*1*V*:*w*2*<sup>V</sup>* that differ by party group *V*—agree with the documented orderings in Table 1. They, thus, beget the same Condorcet cycle.

If for all six party groups *w*1*V*:*w*2*<sup>V</sup>* were set alike (to *w*1:*w*2) as either 2:1 or 1:1 (though not 1:2), one can show that no cycle would arise. See Appendix A.7 for details.

The 1931 Finnish presidential election provides a far-from-perfect illustration for our work. However, it does show how an election with a Condorcet cycle can be fitted by a model that has a two-dimensional issue space together with differential weights for the voters.

#### **9. Summary**

Different aspects of our work may be of significance or have interesting implications. Our work brings out some benefits of continuous (versus discrete) ideal-point distributions and infinite (versus finite) electorates. It calls attention to the topic, mostly neglected heretofore, of voters' differential weightings of issues. The findings on the failure of symmetry to preclude cycles under differential weighting may be of some theoretical interest, and our approach to proofs in Section 3 for the case of equal weighting and an infinite electorate may have some pedagogical advantages over alternative methods.

We showed that, even under the stringent condition of radial symmetry (which we define for an infinite electorate), Condorcet cycles can occur if voters differ on their weightings of issues (though not if they do not differ). It is reasonable to hypothesize that this theoretical result has a real-world counterpart: that, in empirical settings (necessarily without radial symmetry), the absence of Condorcet winners stemming from Condorcet cycles will occur more often where voters weight the issues heterogeneously. Such a hypothesis might be examined. If it holds, the question may arise, in some cases, as to whether measures to try to lessen voters' differences in issue weightings could (or should) be taken to try to reduce cycles and associated instability.

**Funding:** This research received no external funding.

**Data Availability Statement:** All data used in this paper are hypothetical and are shown in the paper itself.

**Acknowledgments:** The author thanks the three anonymous reviewers for their useful comments.

**Conflicts of Interest:** The author declares no conflict of interest.
