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We consider decentralized medium-access control in which many pairwise interactions occur between randomly selected users that belong to a large population. In each local interaction, the users involved compete over an access opportunity. A given user has a fixed number of access attempts and a fixed budget for buying different priority levels. In each time-slot, the access is attributed to the user with the largest priority level. We analyze this problem under both cooperative as well as competitive frameworks. We show that unlike many standard team problems, optimal pure policies do not exist in the team framework, but both an optimal solution as well as equilibria exist within the class of mixed policies. We establish structural properties as well as explicit characterization of these: We show that the optimal policy requires only three priority levels, whereas the noncooperative game possesses a unique symmetric equilibrium point that uses at most two priority levels. Our analysis is applied to power control over wireless capture channels, where the budget constraint corresponds to the battery lifetime.