Equilibrium Architecture in Multi-Battle Contests with Count-Dependent Prizes

Abstract

We study how aggregating local victories shapes participation and equilibrium uniqueness. Two contestants with possibly different marginal costs compete across identical Tullock battlefields with discriminatory power at most one. A nonconstant, nondecreasing, symmetric schedule divides a fixed prize by victory count. We establish existence and show that every realized equilibrium allocation divides effort equally across battlefields. Equilibrium is unique through six battlefields, with at most two effort levels per contestant; seven first permits multiple equilibria or randomization by both contestants. The prize schedule and technology, not the cost ratio, determine whether equilibria use infinitely many effort levels; in that case, cost asymmetry changes only the higher-cost contestant's participation probability. Under majority rule, we characterize the unique equilibrium for every battlefield count and cost ratio.