The war of attrition: two players choose when to concede a contested prize. Each prefers the other to concede first but pays a cost per unit time of waiting. With complete information, the equilibrium is mixed — each player randomizes their concession time. With two-sided incomplete information (each player's private valuation is unknown to the other), the equilibrium structure becomes richer and multiple equilibria emerge.
The paper classifies the multiplicity. Two distinct sources: when the hazard potential diverges at its lower limit (types near the boundary have extreme behavior), the multiplicity is in aggressiveness — how quickly types concede, shifted uniformly. When the hazard potential is finite at its lower limit, the multiplicity is in immediate concession mass — some types concede instantly, and the fraction doing so is indeterminate.
The refinement equivalence is the technical result. Two standard approaches to selecting among equilibria — perturbing payoffs slightly and introducing “behavioral types” (irrational players who never concede) — are mathematically identical. They impose the same restrictions on the equilibrium set. Neither is stronger or weaker; they're the same refinement in different clothes.
But both fail for unbounded type supports. When the private values can be arbitrarily large, neither perturbation nor behavioral types eliminate the multiplicity. The tail of the distribution creates equilibrium slack that no finite perturbation can resolve. For bounded supports, both methods select uniquely.
The distribution's tail determines whether the game has a sharp prediction or an irreducible indeterminacy. Bounded types, bounded uncertainty; unbounded types, unbounded multiplicity.