Symmetric zero-sum games with only asymmetric equilibria

We know that a) two-player symmetric zero-sum games with non-empty equilibrium sets always admit symmetric equilibria and that b) two-player and multiplayer symmetric non-zero-sum games might have only asymmetric equilibria (Fey, 2012). But what about multiplayer symmetric zero-sum games? This paper shows that these games might also have only asymmetric equilibria. One of the examples employed to illustrate this point is the three-candidate version of the popular Hotelling–Downs model of electoral competition. This demonstrates that symmetric games with only asymmetric equilibria are not technical paradoxes but are integrated in economics and political science literature for quite a while.