# An Axiomatization of the Shapley-Shubik Index for Interval Decisions

## TL;DR

This paper extends the Shapley-Shubik index to interval decision games, providing an axiomatization and showing its relation to classical indices, thus broadening its applicability.

## Contribution

It introduces an axiomatization for the Shapley-Shubik index in interval decision games and links it to classical discrete cases.

## Key findings

- Axiomatization of the index for interval decisions
- Connection between interval and classical Shapley-Shubik index
- Potential for generalization to a value measure

## Abstract

The Shapley-Shubik index was designed to evaluate the power distribution in committee systems drawing binary decisions and is one of the most established power indices. It was generalized to decisions with more than two levels of approval in the input and output. In the limit we have a continuum of options. For these games with interval decisions we prove an axiomatization of a power measure and show that the Shapley-Shubik index for simple games, as well as for $(j,k)$ simple games, occurs as a special discretization. This relation and the closeness of the stated axiomatization to the classical case suggests to speak of the Shapley-Shubik index for games with interval decisions, that can also be generalized to a value.