We identify a new monotonicity condition (called cover monotonicity) for tournament solutions which allows a discrimination among main tournament solutions: The top-cycle, the iterated uncovered set, the minimal covering set, and the bipartisan set are cover monotonic while the uncovered set, Banks set, the Copeland rule, and the Slater rule fail to be so. As cover monotonic tournament solutions induce social choice rules which are Nash implementable in certain non-standard frameworks (such as those set by Bochet and Maniquet (CORE Discussion (...) Paper No. 2006/84, 2006) or Özkal-Sanver and Sanver (Social Choice and Welfare, 26(3), 607–623, 2006), the discrimination generated by cover monotonicity becomes particularly notable when implementability is a concern. (shrink)