Shapley Value and Other Axiomatic Extensions to Shapley Value

Abstract

If the Pythagoras theorem is central to Euclidean Geometry, so is the Shapley value to cooperative game theory. Its applications permeate several areas of social, biological, and physical sciences. It has a large impact on machine learning models [8]. It identifies the relevant genes in a co-expressive biological network [3]. It is an index of unfairness in tax and revenue [2]. It helps health networks to cooperate and share health care equipment, facilities, and personnel and share these costs in an equitable manner [13]. It allocates cleaning costs of industrial wastes dumped into a common river by multiple industries [5]. It identifies key drivers in a customer satisfaction analysis [6]. It apportions appropriate penalties among multiple causation in tort law [7]. The applications are endless. Here we use simple models to motivate the intuition behind the Shapley value. We formally discuss the mathematical axioms leading to the derivation of the Shapley value for cooperative TU games. In any cooperative venture, individuals can exert various effort levels. The notion of Multi-choice Shapley value measures the worth of a participant at various effort levels. Since computing the Shapley value is NP hard, one looks for alternative approaches to approximating the Shapley value. The notion of a potential function is a great promise in this direction.