Any social choice function (e.g the efficient allocation) can be implemented using different payment rules: first price, second price, all-pay, etc. All of these payment rules are guaranteed to have the same expected revenue by the revenue equivalence theorem, but have different distributions of revenue, leading to a question of which one is best. We prove that among all possible payment rules, winner-pays-bid minimizes the variance in revenue and, in fact, minimizes any convex risk measure.

翻译：任何社会选择函数（例如有效分配）均可通过不同的支付规则实现：第一价格、第二价格、全支付等。根据收益等价定理，所有这些支付规则均保证具有相同的期望收益，但其收益分布各不相同，从而引出一个问题：何种规则最优？我们证明，在所有可能的支付规则中，胜者支付竞价规则能够最小化收益方差，并且实际上可最小化任何凸风险度量。