Cross-border banking needs cross-border recapitalisation mechanisms. Each mechanism, however, suffers from the financial trilemma, which is that cross-border banking, national financial autonomy and financial stability are incompatible. In this paper, we study the efficiency of different burden-sharing agreements for the recapitalisation of the 30 largest banks in Europe. We consider bank bailouts for these banks in a simulation framework with stochastic country-specific bailout benefits. Among the burden sharing rules, we find that the majority and qualified-majority voting rules come close to the efficiency of a bailout mechanism with a supranational authority. Even a unanimous voting rule works better than home-country bailouts, which are very inefficient. If we assume additional systemic risk benefits, the efficiency of burden sharing rules comes close to the supranational solution.