Sequential second-price auctions with private budgets

We study an auction game in which two units of a good are sold via two second-price auctions sequentially to three or more bidders who have private budgets that limit their spending in the auctions. We characterize the unique symmetric equilibrium. In equilibrium, while bidders bid the minimum of their valuations and budgets in the second auction, their aggression in the first round depends on two factors: the competition from other bidders, as measured by the probability of facing high budget bidders, and, the option value from participating the second auction, as measured by the budget difference between various bidders. The former factor makes bidders more aggressive and the latter makes them less aggressive. In particular, when the competition factor dominates, the sequential auctions generate higher revenue in expectation than the uniform price auction.