ON CONTINUOUS SIMPLE GAMES

Résumé

In the context of collective decision-making, decision rules are typically used to aggregate the opinions of community members into a final outcome. For this purpose, several decision models have been proposed and studied in the literature. This is the case for simple games by Neumann et al. (1947), voting games with abstention by Felsenthal and Machover (1997), (j; k) simple games by Freixas and Zwicker (2003) and continuous simple games introduced by Kurz (2014). Our work focuses on the study of the continuous simple games. One of the central problems common to the previous classes of games concerns the issue of power measurement. In other words, can we formalize the ability of a member of the community to influence the final outcome? The Shapley-Shubik index, see Shapley and Shubik (1954) and the influence relation introduced by Isbell (1958) are tools that were designed to evaluate power distribution in a simple game. They were generalised to (j; k) simple games and to continuous simple games, see Freixas (2005b), Pongou et al. (2011) and Kurz (2014). For these classes of games, many mathematical challenges are still to be tackled, principally those of axiomatizing and comparing of these power measurements. In the first part of our contribution, we show that any continuous simple game viewed as a multivariate real-valued function is Riemann integrable; this result allows us to justify that the extension of the Shapley-Shubik index proposed by Kurz (2014) is well defined in the whole set of all continuous simple games. We also show that the Shapley-Shubik index for simple games as well as for (j; k) simple games appears as a special discretization of that one for continuous simple games. In the second part, we propose a rather simple and convenient formula of the Shapley-Shubik index for (j; k) simple games and provide the first axiomatic justification of this index. We also obtain two characterizations of the same index in the context of continuous simple games. The last part of our investigation leads us to the study of the properties of the influence relation introduced by Kurz (2014) on continuous simple games. We mainly characterize the class of continuous simple games on which this relation is complete, and show that it is a preordering whenever it is complete. In order to compare the influence relation and the preordering induced by the Shapley-Shubik index, we provide a sufficient condition for which these two relations coincide.

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