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We study the design of core-selecting payment rules for combinatorial auctions, a challenging setting where no strategyproof rules exist. We show that the rule most commonly used in practice, the Quadratic rule, can be improved on in terms of efficiency, incentives, and revenue. We present a new computational search framework for finding good mechanisms, and we apply it toward a search for good core-selecting rules. Within our framework, we use an algorithmic Bayes–Nash equilibrium solver to evaluate 366 rules across 31 settings to identify rules that outperform the Quadratic rule. Our main finding is that our best-performing rules are large-style rules—that is, they provide bidders with large values with better incentives than does the Quadratic rule. Finally, we identify two particularly well-performing rules and suggest that they may be considered for practical implementation in place of the Quadratic rule.

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