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.