Two-Stage Asymmetric Tullock Contests with Cost Shifters and Endogenous Continuation Decision

Abstract

This paper introduces a contest-theoretic simplified model of triathlon as a sequential two-stage game. In Stage 1 (post-swim), participants decide whether to continue or withdraw from the contest, thereby generating an endogenous participation decision. In Stage 2 (bike-run), competition is represented as a Tullock contest in which swim drafting acts as a multiplicative shifter of quadratic effort costs. Closed-form equilibrium strategies are derived in the two-player case, and existence, uniqueness, and comparative statics are shown in the asymmetric n-player case. The continuation decision yields athlete-specific cutoff rules in swim drafting intensity and induces subgame-perfect equilibria (SPEs) with endogenous participation sets. The analysis relates swim drafting benefits, exposure, and group size to heterogeneous effective cost parameters and equilibrium efforts.