Posted Pricing versus Bargaining in Sequential Selling Process

Abstract

We study the role of bargaining in a firm’s sequential selling process. The seller firm under consideration sequentially sells a fixed amount of stock to a random arrival stream of potential buyers who are heterogeneous in product valuation. Based on the stock level and the time to the end of the selling season, the seller may dynamically choose either to post a take-it-or-leave-it price or to engage in bargaining with an arriving buyer. We introduce a stochastic order, called the scaled pricing order, for the buyer’s valuation distribution. The scaled pricing order measures the average portion of trade surplus that the seller can obtain by posting a price. For distributions with increasing length-biased hazard rate, checking the property of increasing (decreasing) scaled pricing order boils down to verifying the convexity (concavity) of the Mills’ ratio. We fully characterize the family of distributions that are invariant in the scaled pricing order, which includes uniform and exponential distributions. When the buyer’s valuation distribution belongs to the invariant family, the seller should stick to either pricing or bargaining throughout the selling season, depending on her bargaining power vis-a-vis the buyer. When the buyer’s valuation is increasing (decreasing) in the scaled pricing order, the seller should choose pricing (bargaining) when she has a small (large) stock to sell over a long (short) horizon. We also discuss several model variations to demonstrate the application of the scaled pricing order in analyzing sequential trading processes.