Inverse optimization of integer programming games for parameter estimation arising from competitive retail location selection

Abstract

When determining store locations, competing retailers must take customers’ store choice into consideration. Customers predominantly select which store to visit based on price, accessibility, and convenience. Incumbent retailers can estimate the weight of these factors (customer attraction parameters) using granular historical data. Their location decision under full information and simultaneous competition translates into an integer programming game. Unlike incumbents, new entrants lack this detailed information; however, they can observe the resulting location structure of incumbents. Assuming the observed location structure is (near-)optimal for all incumbent retailers, a new entrant can use these observations to estimate customer attraction parameters. To facilitate this estimation, we propose an “inverse optimization approach” for integer programming games (IPGs), enabling a new entrant to identify parameters that lead to the observed equilibrium solutions. We solve this “inverse IPG” via decomposition by solving a master problem and a subproblem. The master problem identifies parameter combinations for which the observations represent (approximate) Nash equilibria compared with optimal solutions enumerated in the subproblem. This row-generation approach extends prior methods for inverse integer optimization to competitive settings with (approximate) equilibria.We compare the decision-making of new entrants selecting locations based on scenarios, or information about the underlying distribution of customer attraction parameters (expected values), with new entrants using inversely estimated parameters for their location decisions. New entrants who rely on inversely optimized parameters can improve their profits. On average over a large set of synthetic numerical experiments, we observe improvements of 4–11%. This benefit can be realized with as little as one or two observations, yet additional observations help to increase prediction reliability significantly.