Competitive and Cooperative Assortment Games under Markov Chain Choice Model

In this work, we study the assortment planning games in which multiple retailers interact in the market. Each retailer owns some of the products and their goal is to select a subset of products, that is, an assortment to offer to the customers so as to maximize their expected revenue. The purchase behavior of the customer is assumed to follow the Markov chain choice model. We consider two types of assortment games under the Markov chain choice model—a competitive game and a cooperative game. In the assortment competition game, we show that there always exists a pure‐strategy Nash equilibrium and such equilibrium can be found in polynomial time. We also identify an easy‐to‐check condition for the uniqueness of the Nash equilibrium. Then we analyze the equilibrium outcome of this competition game, and compare the outcome with that in a monopolistic setting and a central planner setting. We show that under the assortment competition game, each retailer will offer a broader assortment in the equilibrium, which could include products that are not profitable in the monopolistic or the central planner setting, and it will eventually lead to a decrease in revenue for each player. Furthermore, we show that the price‐of‐anarchy and the price‐of‐stability of the game can be arbitrarily large. Motivated by these results, we further consider the assortment cooperation game under the Markov chain choice model, in which retailers are allowed to form coalitions. We consider two settings of cooperative games distinguished by the way we assume other players’ behaviors outside a coalition. Interestingly, we find that when the players are assumed to be intrinsically competitive (meaning that players outside a coalition are assumed to be a collection of competing players), then there is incentive for all the players to form a grand coalition and there exists an allocation of the total revenue that makes the coalition stable (exists a core to the game). However, when the players are assumed to be intrinsically collaborative (meaning that players outside a coalition are assumed to form another coalition), then a stable coalition may not exist.