Efficient mixtures of priority rules for assigning objects

Equity motivates randomization, but often comes at the cost of efficiency. We study the tradeoff within the strategy-proof family of priority rules. Although randomization over all priority orders is incompatible with efficiency, we characterize the maximal subsets of priority for which randomization preserves efficiency: free-agent and adjacent-three families. Introducing equity measures for asymmetric rules, we show that mixtures within these subfamilies admit probabilistic guarantees that priority rules cannot deliver.