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We propose a decentralized solution for a pursuit-evasion game involving a heterogeneous group of rational (selfish) pursuers and a single evader based on the framework of potential games. In the proposed game, the evader aims to delay (or, if possible, avoid) capture by any of the pursuers whereas each pursuer tries to capture the latter only if this is to his best interest. Our approach resembles in principle the so-called relay pursuit strategy introduced in [1], in which only the pursuer that can capture the evader faster than the others is active. In sharp contrast with the latter approach, the active pursuer herein is not determined by a reactive ad-hoc rule but from the solution of a corresponding potential game. We assume that each pursuer has different capabilities and his decision whether to go after the evader or not is based on the maximization of his individual utility (conditional on the choices and actions of the other pursuers). The pursuers' utilities depend on both the rewards that they will receive by capturing the evader and the time of capture (cost of capturing the evader) so that a pursuer should only seek capture when the incurred cost is relatively small. The determination of the active pursuer-evader assignments (in other words, which pursuers should be active) is done iteratively by having the pursuers exchange information and updating their own actions by executing a learning algorithm for games known as Spatial Adaptive Play (SAP). We illustrate the performance of our algorithm by means of extensive numerical simulations.