*Article* **Radial Symmetry Does Not Preclude Condorcet Cycles If Different Voters Weight the Issues Differently**

**Richard F. Potthoff**

Department of Political Science and Social Science Research Institute, Duke University, Box 90420, Durham, NC 27708, USA; potthoff@duke.edu

**Abstract:** Radial symmetry, by our definition, is a precise condition on continuous ideal-point distributions, rarely if ever found exactly in practice, that is similar to the classical 1967 symmetry condition of Plott but pertains to an infinite electorate; the bivariate normal distribution provides an example. A Condorcet cycle exists if the electorate prefers alternative *X* to *Y*, *Y* to *Z*, and *Z* to *X*. An alternative *K* is a Condorcet winner if there is no alternative that the electorate prefers to *K*. Lack of a Condorcet winner may engender turmoil. The nonexistence of a Condorcet winner implies that a Condorcet cycle exists. Radial symmetry precludes the existence of Condorcet cycles and thus guarantees a Condorcet winner; but this result assumes that all voters weight the dimensions alike. Our counterexamples show that a Condorcet cycle can arise, even under radial symmetry, if the weighting of issues varies across voters. This finding may be of more than theoretical value: It may suggest that in an empirical setting (without radial symmetry), a Condorcet cycle may be more frequent if voters differ as to how they weight the dimensions. We examine, for illustration based on two dimensions (left–right, linguistic), a Condorcet preference cycle in Finland's 1931 presidential election.

**Keywords:** Condorcet cycle; Condorcet winner; Condorcet paradox; multidimensional issue space; radial symmetry; spatial modeling
