Monotonicity of power in games with a priori unions (original) (raw)

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.

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.

Strategic power indices: Quarrelling in coalitions

2006

Abstract While they use the language of game theory known measures of a priory voting power are hardly more than statistical expectations assuming voters behave randomly. 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.

ANNEXATIONS AND MERGING IN WEIGHTED VOTING GAMES - The Extent of Susceptibility of Power Indices

Proceedings of the 3rd International Conference on Agents and Artificial Intelligence, 2011

This paper discusses weighted voting games and two methods of manipulating those games, called annexation and merging. These manipulations allow either an agent, called an annexer to take over the voting weights of some other agents in the game, or the coming together of some agents to form a bloc of manipulators to have more power over the outcomes of the games. We evaluate the extent of susceptibility to these manipulations in weighted voting games of the following prominent power indices: Shapley-Shubik, Banzhaf, and Deegan-Packel indices. We found that for unanimity weighted voting games of n agents and for the three indices: the manipulability, (i.e., the extent of susceptibility to manipulation) via annexation of any one index does not dominate that of other indices, and the upper bound on the extent to which an annexer may gain while annexing other agents is at most n times the power of the agent in the original game. Experiments on non unanimity weighted voting games suggest that the three indices are highly susceptible to manipulation via annexation while they are less susceptible to manipulation via merging. In both annexation and merging, the Shapley-Shubik index is the most susceptible to manipulation among the indices.

On the Present and Future of Power Measures

Homo Oeconomicus, 2002

Power indices are commonly required to assign at least as much power to a player endowed with some given voting weight as to any player of the same game with smaller weight. This local monotonicity and a related global property however are frequently and for good reasons violated when indices take account of a priori unions amongst subsets of players (reflecting, e.g., ideological proximity). This paper introduces adaptations of the conventional monotonicity notions that are suitable for voting games with an exogenous coalition structure. A taxonomy of old and new monotonicity concepts is provided, and different coalitional versions of the Banzhaf and Shapley-Shubik power indices are compared accordingly.

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.