Capacity Choice Game in a Multi-Server Queue : Existence of a Nash Equilibrium

In a multi-server, single-queue symmetric capacity choice game, Gopalakrishnan et al. (2016) characterize the existence of a Nash equilibrium under a requirement on the servers’ capacity cost functions, which excludes some highly relevant cases where servers have ample discretion over their choice of service rates. Without that requirement and when servers are free to choose any service rate, potentially leading to an unstable queueing system, the servers’ cost function is ill-behaved and standard tools for establishing the existence of an equilibrium cannot be applied. In this note, we consider a general power capacity cost function in a two-server capacity choice game with no restriction on the servers’ choice of capacity. Relying on a lesser-known result, namely Tarski's intersection theorem, we establish the existence of a Nash equilibrium, thus extending the result by Gopalakrishnan et al. (2016) to our setting. 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