Abstract

Shapley's combinatorial representation of the Shapley value is embodied in a formula that gives each player his expected marginal contribution to the set of players that precede him, where the expectation is taken with respect to the uniform distribution over the set of all orders of the players. We obtain alternative combinatorial representations that are based on allocating to each player the average relative payoff of coalitions that contain him, where one averages first over the sets of fixed cardinality that contain the player and then averages over the different cardinalities. Different base levels in comparison to which relative payoffs are evaluated yield different combinatorial formulas.

Introduction

The familiar representation of the Shapley value gives each player his “average marginal contribution to the players that precede him,” where averages are taken with respect to all potential orders of the players; see Shapley (1953). This chapter looks at three alternative representations of the Shapley value, each expressing the idea that a player gets the “average relative payoff to coalitions that contain him.” The common feature of the three representations we obtain is the way averages are taken, whereas the distinctive feature is the base level in comparison to which relative payoffs are evaluated.