Paying for Quality vs. Paying for Rank: When Purifying a Metric Backfires

Matching markets allocate scarce opportunities using performance metrics that agents can game. When does making a metric less gameable actually reduce gaming? We show that the answer hinges on how the market converts evaluations into stakes. Where evaluations assign prices — wages, credit terms, score-indexed payments — partially purifying the metric raises gaming precisely when gameable variation dominates it: the market's trust in the metric then rises faster than its exposure to gaming falls. By contrast, where evaluations assign positions — fixed seats allocated by rank — purification never backfires: ranking cancels trust from the incentive, leaving a rank price that deflates with the metric's dispersion, so mean gaming falls monotonically. We solve both markets in closed form under a first-order equilibrium concept; transfers microfound the first regime, rank tournaments the second. A prize-curvature index then determines when gaming improves rather than harms sorting, and when perfecting the metric is itself sorting-suboptimal.