Assortment Competition in a Two-Tier Supply Chain Under Cardinality Constraints

We study a two-tier supply chain in which multiple manufacturers sell their products through a single retailer, and consumer choices follow a Multinomial Logit model. Each manufacturer selects from its portfolio an assortment of products to sell, subject to a cardinality constraint on the size of the assortment. We consider both the case where manufacturers take wholesale prices as given and the case where they set their own wholesale prices. The retailer determines the retail prices of the products chosen. We develop efficient methods for computing profit-maximizing assortment and pricing strategies for manufacturers in a competitive environment. Numerical experiments show that our algorithms can reduce computational complexity by orders of magnitude in identifying pure strategy Nash equilibria, which we prove always exist. We also conduct a numerical case study to compare different wholesale pricing regimes, and find that the retailer may favor a cost-plus pricing with a uniform markup percentage, not because the rule induces assortment outcomes that maximize the total profit of the supply chain, but because it grants the retailer a larger share of that profit.