Summary: The following game is played on a weighted graph: Alice selects a matching \(M\) and Bob selects a number \(k\). Alice’s payoff is the ratio of the weight of the \(k\) heaviest edges of \(M\) to the maximum weight of a matching of size at most \(k\). If \(M\) guarantees a payoff of at least \(\alpha\) then it is called \(\alpha \)-robust. In [SIAM J. Discrete Math. 15, No. 4, 530–537 (2002; Zbl 1006.05051)] R. Hassin and S. Rubinstein gave an algorithm that returns a \(1 / \sqrt{ 2} \)-robust matching, which is best possible. We show that Alice can improve her payoff to 1/ln(4) by playing a randomized strategy. This result extends to a very general class of independence systems that includes matroid intersection, b-matchings, and strong 2-exchange systems. It also implies an improved approximation factor for a stochastic optimization variant known as the maximum priority matching problem and translates to an asymptotic robustness guarantee for deterministic matchings, in which Bob can only select numbers larger than a given constant. Moreover, we give a new LP-based proof of Hassin and Rubinstein’s bound.