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We consider an online assortment problem with $[n]:=\{1,2,\ldots,n\}$sellers, each holding exactly one item $i\in[n]$ with initial inventory $c_i\in\mathbb{Z}_+$, and a sequence of homogeneous buyers arriving over a finite timehorizon $t=1,2,\ldots,m$. There is an online platform whose goal is to offer asubset $S_t\subseteq [n]$ of sellers to the arriving buyer at time $t$ tomaximize the expected revenue derived over the entire horizon while respectingthe inventory constraints. Given an assortment $S_t$ at time $t$, it is assumedthat the buyer will select an item from $S_t$ based on the well-knownmultinomial logit model, a well-justified choice model from the economicliterature. In this model, the revenue obtained from selling an item $i$ at agiven time $t$ critically depends on the assortment $S_t$ offered at that timeand is given by the Nash equilibrium of a Bertrand game among the sellers in$S_t$. This imposes a strong dependence/externality among the offeredassortments, sellers' revenues, and inventory levels. Despite that challenge,we devise a constant competitive algorithm for the online assortment problemwith homogeneous buyers. We also show that the online assortment problem withheterogeneous buyers does not admit a constant competitive algorithm. Tocompensate for that issue, we then consider the assortment problem under anoffline setting with heterogeneous buyers. Under a mild market consistencyassumption, we show that the generalized Bertrand game admits a pure Nashequilibrium over general buyer-seller bipartite graphs. Finally, we develop an$O(\ln m)$-approximation algorithm for optimal market segmentation of thegeneralized Bertrand game which allows the platform to derive higher revenuesby partitioning the market into smaller pools.

Online Assortment and Market Segmentation under Bertrand Competition with Set-Dependent Revenues