Capacity choice game in a multiserver queue: Existence of a Nash equilibrium

Abstract

AbstractIn many congestion‐prone services, front‐line employees have discretion over the rate at which they serve customers. To evaluate the impact of queue pooling on their decisions, we model the situation as a two‐server, single‐queue symmetric capacity choice game. Gopalakrishnan et al. (2016) characterize the existence of a Nash equilibrium in this game under a requirement on the servers' capacity cost functions, that is, where servers have limited discretion. Without that requirement and when servers are free to choose any service rate, the servers' cost function is ill‐behaved and standard tools for establishing the existence of an equilibrium cannot be applied. We consider a general power capacity cost function with no restriction on the servers' choice of capacity, and rely on a lesser‐known result, namely Tarski's intersection theorem, to establish the existence of a symmetric pure‐strategy Nash equilibrium. Comparing settings where queue stability is enforceable versus not, we show that there always exists a Nash equilibrium in the former case, unlike in the latter, and that some of the capacity choices that are equilibria in the former case are no longer equilibria in the latter. Our analysis highlights the criticality of the enforceability of system stability on equilibrium outcomes.