A buyer suggests prices to N sellers in a time period and buys from the seller who accepts the bid first. The number of bidding rounds is determined by how frequently the buyer can make an offer. We show that with no limit on the frequency and without discounting, the price path is either kept flat initially with large jumps at the end or increasing steadily over time. Which class of path occurs in equilibrium depends on the buyer's trade-off between committing to a price ceiling versus finely screening the sellers' costs. With discounting, limiting the number of rounds mitigates the delay caused by the reluctance to raise bids in the first class of equilibrium and therefore can benefit the buyer. This result suggests why, in reality, bargaining parties often take measures to make their offers rigid and consequently force themselves to make fewer offers.

This paper provides a general analysis of signaling under double‐crossing preferences with a continuum of types. There are natural economic environments where the indifference curves of two types cross twice, such that the celebrated single‐crossing property fails to hold. Equilibrium exhibits a threshold type below which types choose actions that are fully revealing and above which they pool in a pairwise fashion, with a gap separating the actions chosen by these two sets of types. The resulting signaling action is quasi‐concave in type. We also provide an algorithm to establish equilibrium existence by construction.