Many queueing-based service system, such as call centers, telecommunication networks, and ride booking platforms, frequently face user abandonment due to limited service resources. Admission control thus becomes an important strategy for service providers to reject users likely to abandon after entering the system. These policies aim to minimize the long-run average system cost by balancing the costs of rejection, abandonment, and server idling. The quality of service (QoS) experienced by the user-base directly depends on the admission control policy of the service provider. In turn, users respond to QoS experienced by offering an abandonment rate at stationarity. Since the admission control policy also depends on this abandonment rate, a strategic interaction emerges between the service provider and its user-base, which can be modeled as a one-period, two-player non-cooperative game. This paper investigates the existence of pure-strategy Nash equilibrium (PSNE) in such interactions. Under a mild condition, we prove the monotonicity of PSNE with respect to key system parameters, such as arrival and service rates. In scenarios where a PSNE does not exist, we explore an equilibrium set, which captures cyclical fluctuations of the system between low and high abandonment rate regimes.