https://www.overleaf.com/read/rtyhfnfxcgrd#727801

This paper develops a two-stage Tullock conflict model with two agents and a violence parameter that determines the onset of open conflict. Agents invest costly effort to compete for a valuable prize, awarded probabilistically following the standard Tullock formulation. The winning probability is the ratio of a player’s effort (raised to a positive power) to the sum of both players’ efforts raised to the same power. When this power equals one, the contest reduces to a lottery. In the canonical two-stage rent-seeking model with linear and symmetric effort costs, the Subgame Perfect Nash Equilibrium (SPNE) predicts that each player expends one-fourth of the prize value in case of open conflict, resulting in equal winning probabilities of one-half. We extend this framework by assuming quadratic effort costs. Under symmetry, equilibrium efforts remain positive and winning probabilities still equalize at one-half. However, we identify multiple equilibria depending on the prize value and the violence parameter. In some parametric settings, unilateral positive effort by one player is sufficient to win without triggering violent conflict. We then introduce cost asymmetry, allowing one player to face lower marginal effort costs. Our analysis shows that even an infinitesimal advantage leads the lower-cost agent to exert strictly higher effort, significantly increasing total effort in open conflict. This results in greater overall waste relative to the symmetric benchmark.