Strategic Pricing and Ranking in Recommendation Systems with Seller Competition

We study a recommendation system where sellers compete for visibility by strategically offering commissions to a platform that optimally curates a ranked menu of items and their respective prices for each customer. Customers interact sequentially with the menu following a cascade click model, and their purchase decisions are influenced by price sensitivity and positions of various items in the menu. We model the seller-platform interaction as a Stackelberg game with sellers as leaders and consider two different games depending on whether the prices are set by the platform or prefixed by the sellers. It is complicated to find the optimal policy of the platform in complete generality; hence, we solve the problem in an important asymptotic regime. The core contribution of this paper lies in characterizing the equilibrium structure of the limit game. We show that when sellers are of different strengths, the standard Nash equilibrium does not exist due to discontinuities in utilities. We instead establish the existence of a novel equilibrium solution, namely `$\mu$-connected equilibrium cycle' ($\mu$-EC), which captures oscillatory strategic responses at the equilibrium. Unlike the (pure) Nash equilibrium, which defines a fixed point of mutual best responses, this is a set-valued solution concept of connected components. This novel equilibrium concept identifies a Cartesian product set of connected action profiles in the continuous action space that satisfies four important properties: stability against external deviations, no external chains, instability against internal deviations, and minimality. We extend a recently introduced solution concept equilibrium cycle to include stability against measure-zero violations and, by avoiding topological difficulties to propose $\mu$-EC.