Rent-Seeking and Loot-Seeking when Stalemate is a Possible Outcome

Standard contest models assume that some contestant always wins the prize. This paper studies rent-seeking and loot-seeking contests that may instead end in a stalemate: nobody wins, yet every party forfeits its expenditure. A stalemate probability function (SPF) maps aggregate effort into stalemate risk; conditional on no stalemate, a Tullock contest success function allocates the prize. The third-party SPF coincides with the contest success function with a draw axiomatized by Blavatskyy (2010). Four results emerge. First, stalemate risk deflates the expected prize but need not reduce effort: when effort lowers the stalemate probability, equilibrium spending can exceed the Tullock level—up to full dissipation—so stalemate risk cannot resolve the Tullock paradox. Second, when a third party of strength w generates the stalemate, weak intervention (4vw < 1, where v is the larger stake) cannot stop a loot-seeking war; moderate intervention (4vw > 1 but w < v) produces a one-sided cease-fire by the side with more to lose, favoring the side with less to lose; only intervention exceeding the larger stake produces peace. Third, for any SPF, which side fights harder depends on its slope: both equally under a constant SPF, the side with less to lose under a decreasing SPF—echoing Hirshleifer’s paradox of power—and the side with more to lose under an increasing SPF. Fourth, a player who alone knows the stalemate probability gains from revealing it when it is positive but from concealing it when it is zero. Implications for contest design and peacekeeping are discussed.