Symmetry-in-Asymmetry: On the Existence and Structure of Asymmetric Pure-Strategy Equilibria in Symmetric Tullock Two-Group Contests

When information is complete, symmetric group-contest games `a la Tullock are known to display within-group-symmetric (WGS) pure-strategy equilibria. Far less is known about the possible existence and structure of their asymmetric counterparts—pure-strategy equilibria where, symmetry notwithstanding, members of the same group exert different effort levels. This note fills the gap by proving that within-group-asymmetric (WGA) pure-strategy equilibria indeed exist, but only for a specific subset of incentivisation schemes. All such equilibria display a peculiar ‘group-splitting’ structure: a subgroup of members free-ride exerting null effort, whereas the other members provide the same positive effort level. Because of this symmetry-in-asymmetry, the game displays the same aggregative structure identified in the context of WGS equilibria. Accordingly, an explicit characterisation is feasible also for their WGA counterparts.