Endogenous entry and investment in an all-pay auction

Contests are ubiquitous but do not simply arise in a vacuum. Competitors make conscious decisions before the fighting stage. This paper looks at the interplay between the decision to enter and then undertake a pre-contest investment to enhance the chance of winning the prize. We use an all-pay auction to model the contest stage; investment cost is private information and its return is stochastic. We characterize equilibrium in terms of threshold strategies on the cost parameter, both for the entry and the subsequent investment decision. We show that the stochastic nature of the investment outcome has a non-monotonic effect on players decisions in equilibrium. A contest designer can use our results to directly achieve goals related to the maximization of contest effort, entry fee revenue and investment propensity. In all cases, we demonstrate that limiting entry may be optimal. • We consider the interplay between entry and investment decisions in a contest. • We derive threshold strategy equilibrium for entry and investment. • Conditions for full entry and investment, and limited entry/investment. • Stochastic return to investment has a non-monotonic effect on equilibrium actions. • An entry fee can be set to achieve design goals. Full entry is not always optimal.