On Wars of Attrition Without Assessment

Abstract

Contests in nature are frequently won by the animal with the higher resource holding potential (RHP), consistent with animals assessing opponents' RHP accurately. Nevertheless RHP asymmetry can determine a contest without any assessment of opponents' RHP. To establish this result, we develop an analytical model of the war of attrition for an arbitrary distribution of initial RHP. If its coefficient of variation, κ, is sufficiently high or if cost of persistence per unit time is sufficiently small compared with the rate at which a victor translate its remaining reserves into fitness, then there is a unique evolutionarily stable strategy (ESS) at which—despite no assessment—the victor is always the animal with higher RHP. Thus victory by the contestant with higher RHP does not by itself imply that an animal assesses its opponent's RHP accurately. Data from a damselfly, Calopteryx maculata, suggest that κ is large enough for the ESS to exist when RHP is determined by energy reserves. We identify characteristics of the ESS that may help to clarify whether animals “observe” (via any sensory modality) a difference in physical magnitudes by comparing separate perceptions of each magnitude (Hypothesis A) or by perceiving the difference itself (Hypothesis B). First, a critical value of κ, above which the ESS invariably exists, exceeds a half under Hypothesis A but is less than a third under Hypothesis B; its precise value depends weakly on the initial RHP distribution. Second, the ESS distributions of final RHP among winners and losers both have a lower mean and higher variance under Hypothesis A than under Hypothesis B. Third, the ESS distribution of contest duration times has a lower mean and higher variance under Hypothesis B than under Hypothesis A. All derived distributions are determined analytically for arbitrary initial RHP.