On Measuring Influence in Non-Binary Voting Games

In this note, we demonstrate using two simple examples that generalization of the Banzhaf measure of voter influence to non-binary voting games that requires as starting position a voter’s membership in a winning coalition is likely to incompletely reflect the influence a voter has on the outcome of a game. Generalization of the Banzhaf measure that takes into consideration all possible pivot moves of a voter including those moves originating from a losing coalition will, on the other hand, result in a measure that is proportional to the Penrose measure only in the ternary case.Penrose measure, Banzhaf index, ternary games, multicandidate weighted voting games

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

Power Indices and Minimal Winning Coalitions in Simple Games with Externalities

We propose a generalization of simple games to sit uations with coalitional externalities. The main novelty of our generalization is a monotonicity property that we define for games in partition function form. This property allows us to properly speak about minimal winning embedded coalitions. We propose and characterize two power indices based on these kind of coalitions. We provide methods based on the multilinear extension of the game to compute the indices. Finally, the new indices are used to study the distribution of power in the current Parliament of Andalusia

Power indices and minimal winning coalitions for simple games in partition function form

We propose a generalization of simple games to partition function form games based on a monotonicity property that we define in this context. This property allows us to properly speak about minimal winning embedded coalitions. We propose and characterize two power indices based on such coalitions. Finally, the new indices are used to study the distribution of power in the Parliament of Andalusia that emerged after the elections of March 22, 2015

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

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

Weighted Committee Games

Many binary collective choice situations can be described as weighted simple voting games. We introduce weighted committee games to model decisions on an arbitrary number of alternatives in analogous fashion. We compare the effect of different voting weights (share-holdings, party seats, etc.) under plurality, Borda, Copeland, and antiplurality rule. The number and geometry of weight equivalence classes differ widely across the rules. Decisions can be much more sensitive to weights in Borda committees than (anti-)plurality or Copeland ones.Comment: 26 pages, 9 tables, 4 figure