Axiomatisation of the Shapley value and power index for bi-cooperative games

URL des Cahiers : https://halshs.archives-ouvertes.fr/CAHIERS-MSECahiers de la Maison des Sciences Economiques 2006.23 - ISSN 1624-0340Bi-cooperative games have been introduced by Bilbao as a generalization of classical cooperative games, where each player can participate positively to the game (defender), negatively (defeater), or do not participate (abstentionist). In a voting situation (simple games), they coincide with ternary voting game of Felsenthal and Mochover, where each voter can vote in favor, against or abstain. In this paper, we propose a definition of value or solution concept for bi-cooperative games, close to the Shapley value, and we give an interpretation of this value in the framework of (ternary) simple games, in the spirit of Shapley-Shubik, using the notion of swing. Lastly, we compare our definition with the one of Felsenthal and Machover, based on the notion of ternary roll-call, and the Shapley value of multi-choice games proposed by Hsiao and Ragahavan.Les jeux bi-coopératifs ont été introduits par Bilbao comme une généralisation des jeux coopératifs classiques. Dans ces jeux, chaque joueur peut participer positivement au jeu (pour l'objectif), ou négativement (contre l'objectif), ou ne pas participer du tout. Dans une situation de vote, ces jeux coïncident avec les jeux de vote ternaires proposés par Felsenthal et Machover, dans lequels chaque votant peut voter en faveur, contre, ou s'abstenir. Dans ce papier, on propose une définition d'une valeur ou solution pour les jeux bi-coopératifs, dans l'esprit de la valeur de Shapley, et nous donnons une interprétation de cette valeur dans le cadre des jeux de vote ternaires, à la manière de Shapley-Shubik. Dans une dernière partie, nous comparons notre approche avec celle de Felsenthal et Machover, et celle de Hsiao et Raghavan qui ont proposé une valeur de Shapley pour les jeux multi-choix

Generalized roll-call model for the Shapley-Shubik index

In 1996 Dan Felsenthal and Mosh\'e Machover considered the following model. An assembly consisting of $n$ voters exercises roll-call. All $n!$ possible orders in which the voters may be called are assumed to be equiprobable. The votes of each voter are independent with expectation $0<p<1$ for an individual vote {\lq\lq}yea{\rq\rq}. For a given decision rule $v$ the \emph{pivotal} voter in a roll-call is the one whose vote finally decides the aggregated outcome. It turned out that the probability to be pivotal is equivalent to the Shapley-Shubik index. Here we give an easy combinatorial proof of this coincidence, further weaken the assumptions of the underlying model, and study generalizations to the case of more than two alternatives.Comment: 19 pages; we added a reference to an earlier proof of our main resul

Power indices expressed in terms of minimal winning coalitions

A voting situation is given by a set of voters and the rules of legislation that determine minimal requirements for a group of voters to pass a motion. A priori measures of voting power, such as the Shapley-Shubik index and the Banzhaf value, show the influence of the individual players. We used to calculate them by looking at marginal contributions in a simple game consisting of winning and losing coalitions derived from the rules of the legislation. We introduce a new way to calculate these measures directly from the set of minimal winning coalitions. This new approach logically appealing as it writes measures as functions of the rules of the legislation. For certain classes of games that arise naturally in applications the logical shortcut drastically simplifies calculations. The technique generalises directly to all semivalues. Keywords. Shapley-Shubik index, Banzhaf index, semivalue, minimal winning coalition, Möbius transform.Shapley-Shubik index, Banzhaf index, semivalue, minimal winning coalition, Möbius transform.

The men who weren't even there: Legislative voting with absentees

Voting power in voting situations is measured by the probability of changing decisions by altering the cast `yes' or `no' votes. Recently this analysis has been extended by strategic abstention. Abstention, just as `yes' or `no' votes can change decisions. This theory is often applied to weighted voting situations, where voters can cast multiple votes. Measuring the power of a party in a national assembly seems to fit this model, but in fact its power comprises of votes of individual representatives each having a single vote. These representatives may vote yes or no, or may abstain, but in some cases they are not even there to vote. We look at absentees not due to a conscious decision, but due to illness, for instance. Formally voters will be absent, say, ill, with a certain probability and only present otherwise. As in general not all voters will be present, a thin majority may quickly melt away making a coalition that is winning in theory a losing one in practice. A simple model allows us to differentiate between winning and more winning and losing and less losing coalitions reflected by a voting game that is not any more simple. We use data from Scotland, Hungary and a number of other countries both to illustrate the relation of theoretical and effective power and show our results working in the practice.a priori voting power; power index; being absent from voting; minority; Shapley-Shubik index; Shapley valuea priori voting power; power index; being absent from voting; minority; Shapley-Shubik index; Shapley value

The men who weren't even there: Legislative voting with absentees

Voting power in voting situations is measured by the probability of changing decisions by altering the cast 'yes' or 'no' votes. Recently this analysis has been extended by strategic abstention. Abstention, just as 'yes' or 'no' votes can change decisions. This theory is often applied to weighted voting situations, where voters can cast multiple votes. Measuring the power of a party in a national assembly seems to fit this model, but in fact its power comprises of votes of individual representatives each having a single vote. These representatives may vote yes or no, or may abstain, but in some cases they are not even there to vote. We look at absentees not due to a conscious decision, but due to illness, for instance. Formally voters will be absent, say, ill, with a certain probability and only present otherwise. As in general not all voters will be present, a thin majority may quickly melt away making a coalition that is winning in theory a losing one in practice. A simple model allows us to differentiate between winning and more winning and losing and less losing coalitions reected by a voting game that is not any more simple. We use data from Scotland, Hungary and a number of other countries both to illustrate the relation of theoretical and effective power and show our results working in the practice.a priori voting power; power index; being absent from voting; minority; Shapley-Shubik index; Shapley value

The Shapley value analyzed under the Felsenthal and Machover bargaining model

This is a post-peer-review, pre-copyedit version of an article published in Public choice. The final authenticated version is available online at: http://dx.doi.org/10.1007/s11127-018-0560-2.In 1996, Felsenthal and Machover proposed a bargaining procedure for a valuable payoff in cooperative and simple games. They proved that the value underlying their bargaining scheme was the Shapley value by showing that it verifies the axioms that Shapley proposed for characterizing his value. They remarked that a direct proof of the result involves rather formidable combinatorial difficulties, but that it has some independent interest. In this paper, we prove such a combinatorial result and obtain a formula for the Shapley value that has a great potential to be extended to more general classes of games.Peer ReviewedPostprint (author's final draft

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