Axiomatizations for the Shapley-Shubik power index for games with several levels of approval in the input and output

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.Comment: 25 page

An Axiomatization of the Shapley-Shubik Index for Interval Decisions

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.Comment: 28 pages, 3 figure

The Public Good index for games with several levels of approval in the input and output

The Public Good index is a power index for simple games introduced by Holler and later axiomatized by Holler and Packel, so that some authors also speak of the Holler--Packel index. A generalization to the class of games with transferable utility was given by Holler and Li. Here we generalize the underlying ideas to games with several levels of approval in the input and output -- so-called $(j,k)$ simple games. Corresponding axiomatizations are also provided.Comment: 16 page

Measuring power and satisfaction in societies with opinion leaders: An axiomatization

URL des Documents de travail : http://centredeconomiesorbonne.univ-paris1.fr/documents-de-travail/Documents de travail du Centre d'Economie de la Sorbonne 2011.18 - ISSN : 1955-611XA well-known model in sociology and marketing is that of opinion leadership. Opinion leaders are actors who are able to affect the behavior of their followers. Hence, opinion leaders have some power over their followers, and they can exercise this power by influencing their followers choice of action. We study a two-action model for a society with opinion leaders. We assume that each member of the society has an inclination to choose one of these actions and that the collective choice is made by simple majority of the actions chosen by each member. For this model, we axiomatize satisfaction and power scores, which allow us to investigate the effects of different opinion leader-follower structures.Un modèle bien connu en sociologie et gestion est celui du leadership d'opinion. Les leaders d'opinion sont des acteurs qui peuvent influer sur les comportements de leurs disciples. En conséquence, les leaders d'opinion ont un certain pouvoir sur leurs disciples et ils peuvent exercer ce pouvoir en influençant le choix d'action de leurs disciples. Nous étudions un modèle de deux actions pour une société avec des leaders d'opinion. Nous supposons que chaque membre de la société a une inclination de choisir une des actions et que le choix collectif est fait par la majorité simple des actions choisies par chaque membre. Pour ce modèle, nous axiomatisons les scores de satisfaction et de pouvoir, ce qui nous permet d'examiner les effets des différentes structures de leader d'opinion - disciples

An axiomatization for two power indices for (3,2)-simple games

Electronic version of an article published as International Game Theory Review, Vol. 21, Issue 1, 1940001, 2019, p. 1-24. DOI: 10.1142/S0219198919400012] © World Scientific Publishing Company https://www.worldscientific.com/doi/10.1142/S0219198919400012The aim of this work is to give a characterization of the Shapley–Shubik and the Banzhaf power indices for (3,2)-simple games. We generalize to the set of (3,2)-simple games the classical axioms for power indices on simple games: transfer, anonymity, null player property and efficiency. However, these four axioms are not enough to uniquely characterize the Shapley–Shubik index for (3,2)-simple games. Thus, we introduce a new axiom to prove the uniqueness of the extension of the Shapley–Shubik power index in this context. Moreover, we provide an analogous characterization for the Banzhaf index for (3,2)-simple games, generalizing the four axioms for simple games and adding another property.Peer ReviewedPostprint (author's final draft

Measuring Power and Satisfaction in Societies with Opinion Leaders: Dictator and Opinion Leader Properties

A well known and established model in communication policy in sociology and marketing is that of opinion leadership. Opinion leaders are actors in a society who are able to affect the behavior of other members of the society called followers. Hence, opinion leaders might have a considerable impact on the behavior of markets and other social agglomerations being made up of individual actors choosing among a number of alternatives. For marketing or policy purposes it appears to be interesting to investigate the effect of different opinion leader-follower structures in markets or any other collective decision-making situations in a society. We study a two-action model in which the members of a society are to choose one action, for instance, to buy or not to buy a certain joint product, or to vote yes or no on a specific proposal. Each of the actors has an inclination to choose one of the actions. By definition opinion leaders have some power over their followers, and they exercise this power by influencing the behavior of their followers, i.e. their choice of action. After all actors have chosen their actions, a decision-making mechanism determines the collective choice resulting out of the individual choices. Making use of bipartite digraphs we introduce novel satisfaction and power scores which allow us to analyze the actors' satisfaction and power with respect to the collective choice for societies with different opinion leader-follower structures. Moreover, we study common dictator and opinion leader properties of the above scores and illustrate our findings for a society with five members.Bipartite digraph ; influence ; inclination ; collective choice ; opinion leader ; follower ; satisfaction ; power ; dictator properties ; opinion leader properties

Power in voting rules with abstention: an axiomatization of a two components power index

The final publication is available at Springer via http://dx.doi.org/10.1007/s10479-016-2124-5In order to study voting situations when voters can also abstain and the output is binary, i.e., either approval or rejection, a new extended model of voting rule was defined. Accordingly, indices of power, in particular Banzhaf’s index, were considered. In this paper we argue that in this context a power index should be a pair of real numbers, since this better highlights the power of a voter in two different cases, i.e., her being crucial when switching from being in favor to abstain, and from abstain to be contrary. We also provide an axiomatization for both indices, and from this a characterization as well of the standard Banzhaf index (the sum of the former two) is obtained. Some examples are provided to show how the indices behave.Peer ReviewedPostprint (author's final draft