A Shapley Value Index for Market Basket Analysis: Efficient Computation Using an Harsanyi Dividend Representation

Abstract

Market basket analysis (MBA) aims to discover purchasing patterns and item associations from customer transaction data. A major drawback of current techniques for MBA is a lack of quantitative metrics to measure the real value associated with basket items. This paper addresses this gap by deriving a practical game-theoretic measure for MBA based on the Shapley value of cooperative games, which we call Shapley value index for MBA (SIMBA). The SIMBA of an item represents the average revenue it earns, including its influence on the revenue earned from sales of other items. A significant challenge when applying Shapley value-inspired approaches in practical domains is the exponential complexity of Shapley value computation. However, for the MBA domain, we show that SIMBA admits a scalable exact computation method that does not require sampling or other approximations. Specifically, a characteristic function for the MBA game is constructed so that the transaction dataset input corresponds to the game’s Harsanyi dividends. The relationship between Harsanyi dividends and the Shapley value is then exploited to efficiently compute SIMBA. This approach scales linearly in the number of transactions, making SIMBA a feasible approach for quantitative MBA. SIMBA can be used to screen conventional MBA techniques, such as association rules, to identify significant rules based on the items’ cross-selling capacity. This combination of existing MBA methods and SIMBA will generate rules based not only on frequency of co-occurrence, but also on the significance of the items. We demonstrate the working of the algorithm by analyzing openly available transaction data from an online retail store. To the best of our knowledge, this is the first time Shapley value is used in this way to solve market basket analyses of a practical size.