9 - The potential of the Shapley value

Summary

Introduction

We study multiperson games in characteristic function form with transferable utility. The problem is to solve such a game (i.e., to associate to it payoffs to all the players).

Three main solution concepts are as follows. The first was introduced by von Neumann and Morgenstern: A “stable set” of a given game is a set of payoff vectors; such a set, if it exists, need not be unique. Next came the “core,” due to Shapley and Gillies, which is a unique set of payoff vectors. Finally, the Shapley “value” consists of just one payoff vector. There is thus an apparent historical trend from “complexity” to “simplicity” in the structure of the solution.

We propose now an even simpler construction: Associate to each game just one number! How would the payoffs to all players then be determined? by using the “marginal contribution” principle, an approach with a long tradition (especially in economics). Thus, we assign to each player his or her marginal contribution according to the numbers defined earlier. The surprising fact is that only one requirement, that the resulting payoff vector be “efficient” (i.e., that the payoffs add up to the worth of the grand coalition), determines this procedure uniquely.