The paper focuses on a particular polling system known as the cyclic Bernoulli polling (CBP) system, where a server moves cyclically between the stations and serves the queue at a station with a certain probability when polled. Each station follows either a gated or partially exhaustive service discipline. In the steady state of such a system, we study a new game-theoretic aspect, where, the stations strategically choose the probability of accepting or rejecting the service from the server when polled. We examine three variants of non-cooperative games among stations: (i) each station selfishly minimizes its expected waiting time, (ii) a team game where each station minimizes the expected workload of the system, and (iii) stations act with partial cooperation, incurring an additional linear cost. We begin by presenting a new result for the CBP system regarding the continuity of expected waiting times in relation to the probabilities selected by the stations. For each game, we then investigate the existence and uniqueness of the Nash equilibrium (NE). In some cases, the NE is explicitly derived, while in others, characterizing the NE remains challenging due to the complex dependence of waiting times on the non-trivial buffer occupancy equations. Nonetheless, we analyze the NE and its properties through numerical experiments. Notably, in many instances, stations opt to accept service with a probability less than 1—a trend observed even among selfish stations.