Characterization of the Shapley-Shubik Power Index Without the Efficiency Axiom

We show that the Shapley-Shubik power index on the domain of simple (voting) games can be uniquely characterized without the e¢ ciency axiom. In our axiomatization, the efficiency is replaced by the following weaker require- ment that we term the gain-loss axiom: any gain in power by a player implies a loss for someone else (the axiom does not specify the extent of the loss). The rest of our axioms are standard: transfer (which is the version of additivity adapted for simple games), symmetry or equal treatment, and dummySimple Games, Shapley-Shubik Power Index, Effciency Axiom

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

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

Measuring voting power in convex policy spaces

Classical power index analysis considers the individual's ability to influence the aggregated group decision by changing its own vote, where all decisions and votes are assumed to be binary. In many practical applications we have more options than either "yes" or "no". Here we generalize three important power indices to continuous convex policy spaces. This allows the analysis of a collection of economic problems like e.g. tax rates or spending that otherwise would not be covered in binary models.Comment: 31 pages, 9 table

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

On the Felsenthal power index

The paper that introduces the Felsenthal index is titled: ‘A well-behaved index of a priori P-Power for simple n-person games.’ In 2016, Felsenthal introduced his index for simple games. His definition does not base on the axiomatic approach. Then, Felsenthal regarded some properties and proved that his index satisfies a list of six reasonable and compelling postulates. Three of the properties that he regarded refer to the weighted games, but this fact does not reduce the definition of his index to weighted games. He proves that none of seven well-known efficient power indices proposed to date satisfies the list of postulates, indicating for each of them which of the six postulates violate. In this paper we extend some of his postulates, originally defined for weighted games, to simple games. The main objective of the article is to answer three open questions motivated in his article. In particular, we prove that his index may not be the unique one fulfilling the six proposed postulates, provide an axiomatic characterization for his index and, propose an impossibility result, which is obtained by adding a new postulate to a sublist of the postulates he considered. We also remark the existence of some alternative compelling postulates which are not satisfied for the index.This research is part of the I+D+i project PID2019-104987GB-I00 supported by MCIN/AEI/10.13039/501100011033/. We thank José María Alonso-Meijide for his comments and suggestions, which helped us to improve the manuscript. We greatly appreciate the comments of two referees that have contributed to improve this work.Peer ReviewedPostprint (published version

A value for j-cooperative games: some theoretical aspects and applications

This is an Accepted Manuscript of a book chapter published by Routledge/CRC Press in Handbook of the Shapley value on December 6, 2019, available online: https://www.crcpress.com/Handbook-of-the-Shapley-Value/Algaba-Fragnelli-Sanchez-Soriano/p/book/9780815374688A value that has all the ingredients to be a generalization of the Shapley value is proposed for a large class of games called j-cooperative games which are closely related to multi-choice games. When it is restricted to cooperative games, i.e. when j equals 2, it coincides with the Shapley value. An explicit formula in terms of some marginal contributions of the characteristic function is provided for the proposed value. Different arguments support it: (1) The value can be inferred from a natural probabilistic model. (2) An axiomatic characterization uniquely determines it. (3) The value is consistent in its particularization from j-cooperative games to j-simple games. This chapter also proposes various ways of calculating the value by giving an alternative expression that does not depend on the marginal contributions. This chapter shows how the technique of generating functions can be applied to determine such a value when the game is a weighted j-simple game. The chapter concludes by presenting several applications, among them the computation of the value for a proposed reform of the UNSC voting system.Peer ReviewedPostprint (author's final draft

Balanced externalities and the Shapley value

We characterize the Shapley value using (together with standard conditions of efficiency and equal gains in two-player games) a condition of ‘undominated merge-externalities’. Similar to the well-known ‘balanced contributions’ characterization, our characterization corresponds intuitively to ‘threat points’ present in bargaining. It derives from the observation that all semivalues satisfy ‘balanced merge-externalities’. Our characterization is applicable to useful, narrow sub-classes of games (including monotonic simple games), and also extends naturally to encompass games in partition function form