Abstract

The Shapley-Shubik index is a specialization of the Shapley value and is widely applied to evaluate the power distribution in committees drawing binary decisions. It was generalized to decisions with more than two levels of approval both in the input and the output. The corresponding games are called (j,k) simple games. Here we present a new axiomatization for the Shapley-Shubik index for (j,k) simple games as well as for a continuous variant, which may be considered as the limit case.

Highlights

  • In [18] Shapley introduced a function that could be interpreted as the expected utility of a game from each of its positions via the axiomatic approach the so-called Shapley value

  • A bit later, see [19], it was restricted to games with binary decisions, i.e., simple games

  • We introduce further denitions and axioms for power indices on (j, k) simple games

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Summary

Introduction

In [18] Shapley introduced a function that could be interpreted as the expected utility of a game from each of its positions via the axiomatic approach the so-called Shapley value. A bit later, see [19], it was restricted to games with binary decisions, i.e., simple games. An axiomatization of this so-called Shapley-Shubik index was given quite a few years later by Dubey [3]. A Shapley-Shubik like index for those games was motivated and introduced in [12], an axiomatization is given in [14]. We mainly focus on an axiomatic justication, see our main result in Theorem 5.1 We present another formula for the Shapley-Shubik index for (j, k) simple games which is better suited for computation issues, see Lemma 3.1 and Theorem 4.1.

Preliminaries
Axiomatization of the Shapley-Shubik index for interval simple games
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