Conjugate information disclosure in an auction with learning

We consider a single-item, independent private value auction environment with two bidders: a leader, who knows his valuation, and a follower, who privately chooses how much to learn about his valuation. We show that, under some conditions, an ex-post efficient revenue-maximizing auction—which solicits bids sequentially—partially discloses the leader's bid to the follower, to influence his learning. The disclosure rule that emerges is novel; it may reveal to the follower only a pair of bids to which the leader's actual bid belongs. The identified disclosure rule, relative to the first-best, induces the follower to learn less when the leader's valuation is low and more when the leader's valuation is high