*Appendix A.2. Details for Example 2*

Table A1 shows that the cycle *A*-*B*, *B*-*C*, *C*-*A* results. To illustrate the calculations, we consider the bottom of the table. The last two lines show that *A*-*C* for 69.4% and 30% of the halves of the electorate with 10:1 and 1:10 weight ratios, respectively. Thus *C*-*A* overall, by 50.3% to 49.7%.

For the last line, with a 1:10 weight ratio, (2) in the main text yields

$$\mathcal{U}(V,A) - \mathcal{U}(V,\mathbb{C}) = \left[ -(\mathbf{x} - \mathbf{1})^2 - 10(y - 7)^2 \right] - \left[ -(\mathbf{x} + 9)^2 - 10(y - 5)^2 \right] = 0,$$

or

$$20(x+2y-8) = 0\_{\prime}$$

as the applicable bisector of *PAPC*. This bisector intersects the line segment *PAPC* at (*x*, *y*) = (−4, 6), and crosses the left side of the square at *y* = 9, the right side at *y* = −1, and the *y*-axis at *y* = 4. Because the average ordinate of the bisector is *y* = 4, the proportion of the group that prefers *<sup>A</sup>* to *<sup>C</sup>* is <sup>10</sup>−<sup>4</sup> <sup>20</sup> = 30%.

**Remark.** *The same cycle of A*-*B, B*-*C, C*-*A also emerges if (at each point in the issue space) w1:w2 follows a continuous distribution over the range from (* <sup>10</sup> <sup>11</sup> *:* <sup>1</sup> <sup>11</sup> *,* <sup>1</sup> <sup>11</sup> *:* <sup>10</sup> <sup>11</sup> *) to (1:0, 0:1)—for instance, half (1* <sup>−</sup> <sup>1</sup> <sup>11</sup>*u):* <sup>1</sup> <sup>11</sup>*u and half* <sup>1</sup> <sup>11</sup>*u:(1* <sup>−</sup> <sup>1</sup> <sup>11</sup>*u) with u distributed according to the uniform (rectangular) distribution on the interval [0, 1].*


**Table A1.** Details demonstrating the existence of the Condorcet cycle.

*Appendix A.3. Details for Examples 3.1 and 3.2*

A voter *V* in the first half of the electorate has these preferences:

*V* prefers *A* to *B* if *xV* < 2, *V* prefers *B* to *C* if *xV* > 6, *V* prefers *A* to *C* if *xV* < 1, and conversely. The preferences of a voter *V* in the second half are: *V* prefers *A* to *B* if *yV* > 8, *V* prefers *B* to *C* if *yV* > −1, *V* prefers *A* to *C* if *yV* > 0, and conversely.

**Example 3.1.** *For the uncorrelated bivariate normal distribution, the proportion of voters*

that prefers *A* to *B* is [Φ(0.2) + 1 − Φ(0.8)]/2, <1 <sup>2</sup> since Φ(0.8) > Φ(0.2); that prefers *B* to *C* is [1 – Φ(0.6) + 1 − Φ(−0.1)]/2 = [1 − Φ(0.6) + Φ(0.1)]/2, <1 <sup>2</sup> since Φ(0.6) > Φ(0.1); and that prefers *A* to *C* is [Φ(0.1) + 1 − Φ(0)]/2, >1 <sup>2</sup> since Φ(0.1) > Φ(0). Thus *B*-*A*, *C*-*B*, and *A*-*C*.

**Example 3.2.** *For the uniform distribution, the proportion of the electorate*

that prefers *<sup>A</sup>* to *<sup>B</sup>* is <sup>2</sup>−(−10) <sup>20</sup> <sup>+</sup> <sup>10</sup>−<sup>8</sup> <sup>20</sup> /2 = 35%; that prefers *<sup>B</sup>* to *<sup>C</sup>* is <sup>10</sup>−<sup>6</sup> <sup>20</sup> <sup>+</sup> <sup>10</sup>−(−1) <sup>20</sup> /2 = 37.5%; and that prefers *<sup>A</sup>* to *<sup>C</sup>* is <sup>1</sup>−(−10) <sup>20</sup> <sup>+</sup> <sup>10</sup>−<sup>0</sup> <sup>20</sup> /2 = 52.5%. Again, *B*-*A*, *C*-*B*, and *A*-*C*.

*Appendix A.4. Details for Example 4*

For the third group (the one with 88% of the voters):

The perpendicular bisector of the line segment *PAPB* intersects *PAPB* at (*x*, *y*) = (2, 8) and crosses the top of the square (*y* = 10) at *x* = 2.4, the bottom at *x* = −1.6, and the *x*-axis at *x* = 0.4. The proportion of the group that prefers *A* to *B* is thus 0.4−(−10) <sup>20</sup> = 52%.

The perpendicular bisector of *PBPC* crosses *PBPC* at (*x*, *y*) = (6, −1), the left side of the square (*x* = −10) at *y* = 1, the right side at *y* = −1.5, and the *y*-axis at *y* = −0.25. Thus, the proportion of the group that prefers *B* to *C* is <sup>10</sup>−(−0.25) <sup>20</sup> = 51.25%.

The perpendicular bisector of *PAPC* crosses *PAPC* at (*x*, *y*) = (1, 0), the left side of the square at *<sup>y</sup>* <sup>=</sup> <sup>−</sup>4<sup>8</sup> <sup>9</sup> , the right side at *<sup>y</sup>* = 4, and the *<sup>y</sup>*-axis at *<sup>y</sup>* <sup>=</sup> <sup>−</sup><sup>4</sup> <sup>9</sup> . The proportion of the group that prefers *<sup>A</sup>* to *<sup>C</sup>* is, therefore, <sup>10</sup>−(<sup>−</sup> <sup>4</sup> 9 ) <sup>20</sup> = 52.22%.

Within the third group, of course, no cycle exists: *A*-*B*, *B*-*C*, and *A*-*C*.

The proportions for the first two groups (35%, 37.5%, 52.5%) are the ones obtained for Example 3.2 in Appendix A.3 just above. Thus, across all three groups, the proportion of the electorate

that prefers *A* to *B* is (0.12 × 35%) + (0.88 × 52%) = 49.96%; that prefers *B* to *C* is (0.12 × 37.5%) + (0.88 × 51.25%) = 49.6%; and that prefers *A* to *C* is (0.12 × 52.5%) + (0.88 × 52.22%) = 52.26%.

Therefore, for the entire electorate, the result is *B*-*A*, *C*-*B*, and *A*-*C*.
