**Appendix A**

*Appendix A.1. Details for Examples 1.1 and 1.2*

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

*V* prefers *A* to *B* if *xV* < 2 [since <sup>1</sup> <sup>2</sup> (*xA* <sup>+</sup> *xB*) = <sup>1</sup> <sup>2</sup> (1 + 3) = 2],

*V* prefers *B* to *C* if *xV* > −3,

*V* prefers *A* to *C* if *xV* > −4,

and conversely (e.g., *V* prefers *B* to *A* if *xV* > 2). The preferences of a voter *V* who is in the second half are:

*V* prefers *A* to *B* if *yV* > 0 [since <sup>1</sup> <sup>2</sup> (*yA* + *yB*) = 0], *V* prefers *B* to *C* if *yV* < −1, *V* prefers *A* to *C* if *yV* > 6, and conversely.

**Example 1.1.** *The ideal points follow the uncorrelated bivariate normal distribution (means of 0, standard deviations equal to 10). Let Φ(*•*) denote the cumulative distribution function of the standard (univariate) normal distribution (with mean 0 and variance 1). Then, the proportion of the electorate*

```
that prefers A to B is [Φ(0.2) + 1 − Φ(0)]/2,
    >1
      2 since Φ(0.2) > Φ(0);
that prefers B to C is [1 − Φ(−0.3) + Φ(−0.1)]/2,
    >1
      2 since Φ(−0.1) > Φ(−0.3); and
that prefers A to C is [1 − Φ(−0.4) + 1 − Φ(0.6)]/2 = [Φ(0.4) + 1 − Φ(0.6)]/2,
    <1
      2 since Φ(0.6) > Φ(0.4).
     Thus A-
              B, B-
                    C, and C-
                               A.
```
**Example 1.2.** *The distribution of ideal points is uniform rather than bivariate normal. Then 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>0</sup> <sup>20</sup> /2 = 55%; that prefers *<sup>B</sup>* to *<sup>C</sup>* is <sup>10</sup>−(−3) <sup>20</sup> <sup>+</sup> <sup>−</sup>1−(−10) <sup>20</sup> /2 = 55%; and that prefers *<sup>A</sup>* to *<sup>C</sup>* is <sup>10</sup>−(−4) <sup>20</sup> <sup>+</sup> <sup>10</sup>−<sup>6</sup> <sup>20</sup> /2 = 45%.

Thus *A*-*B*, *B*-*C*, and *C*-*A* with 55%-to-45% preference margins in all three cases.
