A Multiplicative Variant of the Shapley Value for Factorizing the Risk of Disease

Abstract

This article sets out to build the bridge between the mathematical philosophy of axiomatic approaches to cooperative game theory and practical concerns of statistical applications in epidemiology. It deals with the methodological task of appropriately assessing the impact of multiple characteristics of human behaviour, human constitution or environmental agents (the so called risk factors) on the disease load in the population. It is shown that this epidemiologic problem of multifactorial risk attribution can be formalized in a way that is comparable to the game-theoretic description of several players that act together in a grand coalition and are faced with the problem of dividing their profit fairly among them. In particular, epidemiologic methods of partitioning the risk of disease that are equivalent to the Shapley value are reviewed briefly. Moreover, a new parameter based on a multiplicative analogue of the Shapley value is introduced. Its application and the interpretation of its results are illustrated by epidemiologic data.