We introduce a new solution for tournaments called the unsurpassed set. This solution lies between the uncovered set and the Copeland winner set. We show that this solution is more decisive than the uncovered set in discriminating among alternatives, and avoids a deficiency of the Copeland winner set. Moreover, the unsurpassed set is more sensitive than the uncovered set but less sensitive than the Copeland winner set to the reinforcement of the chosen alternatives. Besides, it turns out that this solution (...) violates the other standard properties including independence of unchosen alternatives, stability, composition consistency and indempotency. (shrink)