Probabilistic selling in vertically differentiated markets: The role of substitution

This paper studies two fundamental questions regarding probabilistic selling in vertically differentiated markets: When is it profitable and how does one design it optimally? For the first question, we identify an important but overlooked economic mechanism driving probabilistic selling in vertically differentiated markets: the convexity of consumer preferences. In stark contrast to the literature finding that probabilistic selling is {\it never} profitable except in the presence of certain capacity constraint or consumer bounded rationality, we find that with many alternative utility functions capable of representing convex preference, probabilistic selling is {\it always} profitable. For the second question, we study the optimal strategy of probabilistic selling, including the design and price of the probabilistic good and the prices of the component goods. We show that under some technical conditions, the optimal price of the high-quality component good increases while the optimal price of the low-quality component good decreases upon the introduction of probabilistic selling, thereby increasing the market coverage and the economic efficiency without launching an actual new product line. To further illustrate the design of probabilistic selling, we use an example based on the canonical utility function, which is widely used in the economics literature on vertical product differentiation. We derive a closed-form solution to the problem of optimal probabilistic selling. We also take advantage of the analytical tractability of the canonical utility to further explore the design of multiple probabilistic goods.