A note on voting (original) (raw)

Coalitional Power Indices Applied to Voting Systems

Proceedings of the 9th International Conference on Operations Research and Enterprise Systems, 2020

We describe voting mechanisms to study voting systems. The classical power indices applied to simple games just consider parties, players or voters. Here, we also consider games with a priori unions, i.e., coalitions among parties, players or voters. We measure the power of each party, player or voter when there are coalitions among them. In particular, we study real situations of voting systems using extended Shapley-Shubik and Banzhaf indices, the so-called coalitional power indices. We also introduce a dynamic programming to compute them.

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.

Spectrum value for coalitional games

Games and Economic Behavior, 2013

Assuming a 'spectrum' or ordering of the players of a coalitional game, as in a political spectrum in a parliamentary situation, we consider a variation of the Shapley value in which coalitions may only be formed if they are connected with respect to the spectrum. This results in a naturally asymmetric power index in which positioning along the spectrum is critical. We present both a characterisation of this value by means of properties and combinatoric formulae for calculating it. In simple majority games, the greatest power accrues to 'moderate' players who are located neither at the extremes of the spectrum nor in its centre. In supermajority games, power increasingly accrues towards the extremes, and in unaninimity games all power is held by the players at the extreme of the spectrum.

A Note on Positions and Power of Players in Multicameral Voting Games

Transactions on Computational Collective Intelligence XXVII, 2017

A multicameral simple game is an intersection of a number of simple games played by the same set of players: a coalition is winning in the multicameral game if and only if it is winning in all the individual games played. Examples include decision rules in multicameral parliaments where a bill must be passed in all the houses of the parliament, and voting rules in the European Union Council where a winning coalition of countries must satisfy two or three independent criteria. This paper is a preliminary study of relations between the positions and power indices of players in the "chamber" games and in the multicameral game obtained as the intersection. We demonstrate that for any power index satisfying a number of standard properties, the index of a player in the multicameral game can be smaller (or greater) than in all the chamber games; this can occur even when the players are ordered the same way by desirability relations in all the chamber games. We also observe some counterintuitive effects when comparing the positions and decisiveness of players. However, as expected, introducing an additional chamber with all the players equal (a one man-one vote majority game) to a complete simple game reduces all the differences between the Shapley-Shubik indices of players. Keywords: Simple games • Multicameral voting • Complete games • Power indices • Reducing power inequalities 1 Of course, if some independent MPs are present, they also are voters.

A power analysis of linear games with consensus

Mathematical Social Sciences, 2004

Linear simple games with consensus are considered. These games are obtained by intersecting a linear simple game (a game where the desirability order for players is total) and a symmetric (k-outof-n) game. We investigate the behavior of the Shapley -Shubik power index when passing from a linear game with consensus level q 1 to the same game with consensus level q 2 >q 1 . We also introduce a ''range notion'', which intuitively represents the egalitarianism of the Shapley -Shubik index, obtain an upper bound on this measure, and characterize when it is achieved. Finally, the theory developed here is illustrated with several real-world voting systems. D

Quarrelling in coalitions to increase p-voting power

Abstract While they use the language of game theory known measures of a priory voting power are hardly more than statistical expectations assuming the random behaviour of the players. Focusing on normalised indices we show that rational players would behave differently from the indices predictions and propose a model that captures such strategic behaviour. Keywords and phrases: Banzhaf index, Shapley-Shubik index, a priori voting power, rational players.

The Bicameral Postulates and Indices of a Priori Voting Power

1998

is an index of relative voting power for simple voting games, the bicameral postulate requires that the distribution of K-power within a voting assembly, as measured by the ratios of the powers of the voters, be independent of whether the assembly is viewed as a separate legislature or as one chamber of a bicameral system, provided that there are no voters common to both chambers. We argue that a reasonable index-if it is to be used as a tool for analysing abstract, 'uninhabited' decision rules-should satisfy this postulate. We show that, among known indices, only the Banzhaf measure does so. Moreover, the Shapley-Shubik, Deegan-Packel and Johnston indices sometimes witness a reversal under these circumstances, with voter x 'less powerful' than y when measured in the simple voting game G 1 , but 'more powerful' than y when G 1 is 'bicamerally joined' with a second chamber G 2. Thus these three indices violate a weaker, and correspondingly more compelling, form of the bicameral postulate. It is also shown that these indices are not always co-monotonic with the Banzhaf index and that as a result they infringe another intuitively plausible condition-the price monotonicity condition. We discuss implications of these findings, in light of recent work showing that only the Shapley-Shubik index, among known measures, satisfies another compelling principle known as the bloc postulate. We also propose a distinction between two separate aspects of voting power: power as share in a fixed purse (P-power) and power as influence (I-power). KEY WORDS: Banzhaf, Deegan-Packel, index of voting power, Johnston, paradoxes of voting power, Penrose, postulates for index of voting power, Shapley value, Shapley-Shubik, simple voting game, weighted voting game.