Decisions with a continuum of options

Title data

Kurz, Sascha ; Moyouwou, Issofa ; Touyem, Hilaire:
Decisions with a continuum of options.
2019
Event: Application-Oriented Computational Social Choice , 15.-20.09.2019 , Wadern, Deutschland.
(Conference item: Workshop , Speech )

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Abstract in another language

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. You may think of e.g. tax rates. 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. Also for the Penrose-Banzhaf index there exists a variant for games with interval decisions in the literature on aggregation function. The general framework of games with a continuum of options deserves to be explored more. We collect a list of some open problems in that direction.