Multi-dimensional Virtual Values and Second-degree Price Discrimination
summary
Claims that posting a single price for the favorite outcome is revenue-optimal whenever the high-value and discount factor are positively correlated (first-order stochastic dominance), and shows a counterexample when the discount factor is non‑monotone. Proves this by building multi-dimensional virtual values via an extended integration-by-parts lemma, choosing integration-by-parts paths to reverse-engineer a virtual value vector so revenue = expected virtual surplus and the posted-price allocation maximizes it pointwise; extends to multiple agents, outcomes, costs and ironing.
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