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When rational, utility-maximizing agents encounter an opportunity to collaborate on a group activity, they must determine whether to commit to that activity. We refer to this problem as the initial-commitment decision problem (ICDP). This paper describes a mechanism that agents may use to solve the ICDP. The mechanism is based on a combinatorial auction in which agents bid on sets of roles in the group activity, each role comprising constituent subtasks that must be done by the same agent. Each bid may specify constraints on the execution times of the subtasks it covers. This mechanism permits agents to keep most details of their individual schedules of prior commitments private. The paper reports the results of several experiments testing the performance of the mechanism. These results demonstrate a significant improvement in performance when constituent subtasks are grouped into roles. They also show that as the number of time constraints in bids increases, the probability that there is a solution decreases, the cost of an optimal solution (if one exists) increases, and the time required finding an optimal solution (if one exists) decreases. The paper also describes several strategies that agents might employ when using this mechanism.