Wars of Attrition with Unrecoverable Upfront Investments

Abstract

This paper models a war of attrition where unrecoverable upfront investments decide for how long a participant can compete. In a two-stage setting, participants competing for a single prize first privately choose a resource investment to utilize through the contest, then second choose for how long they are willing to compete given those resources. Cases where unused resources from the contest have zero, partial, and full repurpose value (e.g., for future contests) are considered, which allows for a direct comparison to the first price all-pay auction (zero repurpose value) and the standard war of attrition (full repurpose value). The theoretical contribution of this model has direct implications for wars of attrition where upfront costs are a prerequisite to competing, such as for R&D endeavors, market exit scenarios, and contract fulfillments. In addition to these applications, the model is illustrated in a charity auction setting where the revenue equivalence theorem is violated and the effect of varying repurpose values on expected revenue is discussed.