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.
International audienceGehrlein et al. (Math Soc Sci 66:352–365, 2013) have shown that an increase of the voters’ preference diversity, as measured by the number kkk of preference types in a voting situation, implies a decrease in the probability of having a Condorcet Winner. The results offered in this paper indicate that this relationship is far from being so clear when we consider instead the proximity of voting situations to having kk distinct preference types. This measure of agreement is compared to other measures of group mutual coherence previously analyzed in Gehrlein (Condorcet’s paradox, Springer Publishing, Berlin, 2006). It turns out that our results are completely consistent with the theory introduced by List (Good Soc 11:72–79, 2002) that is based on an important distinction between two different concepts of agreement
International audienceA Condorcet social choice procedure elects the candidate that beats every other candidate under simple majority when such a candidate exists. The reinforcement axiom roughly states that given two groups of individuals, if these two groups select the same alternative, then this alternative must also be selected by their union. Condorcet social choice procedures are known to violate this axiom. Our goal in this paper is to put this important voting theory result into perspective. We then proceed by evaluating how frequently this phenomenon is susceptible to occur
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