This paper investigates first-price and Dutch auctions when bidders have preferences exhibiting the Allais paradox. We characterize an equilibrium for both auctions, paying particular attention to the dynamic inconsistency problems which can arise with such preferences. We show that the Dutch auction systematically yields a higher revenue than the first-price auction. This stands in sharp contrast with the presumption that these auctions are strategically equivalent, which is indeed valid in the expected utility case. We also show that introducing a "buy-it-now-price" to the first-price auction increases seller's expected revenue when bidders have Allais paradox preference while it does not for expected-utility maximizers.