*Appendix A.5. Details for Example 5*

Table A2 (top part) establishes the Condorcet cycle when voters' weightings are disparate. The bottom part of the table confirms that no cycle exists when all voters are alike in their weightings of the issues.

**Table A2.** Weighted <sup>a</sup> squared distance from each voter to each candidate, and preference rankings and matchups.


<sup>a</sup> Squared distances at the bottom of the table are unweighted.

### *Appendix A.6. Details for Example 6*

Table A3 (top part) establishes the Condorcet cycle when voters' weightings are disparate. The bottom part of the table confirms that no cycle exists when all voters are alike in their weightings of the issues.


**Table A3.** Weighted <sup>a</sup> squared distance from each voter to each candidate, and preference rankings and matchups.

<sup>a</sup> Squared distances at the bottom of the table are unweighted.

#### *Appendix A.7. Details for Statement about 1931 Finnish Election*

Parts (a), (b), and (c) of Table A4, for respective common weight ratios for all six party groups of 2:1, 1:1, and 1:2, establish that a Condorcet cycle still occurs only in the third case and not in the first two.

**Table A4.** Weighted squared distance from each party group to each candidate, plus preference rankings and matchups.


**(b) Weight ratio is common at 1:1 within each party group and across all party groups.**



**Table A4.** *Cont.*

<sup>a</sup> *V* is for party group: SD = Social Democrats; Pr = Progressives; Ag = Agrarians; Co = Conservatives; Sw1, Sw2 = blocs 1, 2 of Swedish People's Party. <sup>b</sup> Candidates: T = Tanner, St = Ståhlberg, K = Kallio, Sv = Svinhufvud.

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