The present paper analyzes multi-player contests where participants compete for a valuable prize and their probability of victory depends on the difference between their effective efforts. These difference-form contests have appealing properties but remain largely understudied due to the non-existence of pure-strategy equilibria and the preemption effect they display (e.g. Che and Gale, 2000). We show that these features rest critically on the assumption of full linearity. Pure strategy equilibria with multiple active contestants exist under mild conditions as soon as full linearity is assumed away. In addition, we show that symmetric difference-form contests are equilibrium equivalent to rank-order tournaments à la Lazear and Rosen (1981) and characterize the level of total expenditures as a function of the heterogeneity in participants’ valuations of victory
