Dimension of complete simple games with minimum (original) (raw)
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Minimal winning coalitions in weighted-majority voting games
Social Choice and Welfare, 1996
Riker's size principle for n-person zero-sum games predicts that winning coalitions that form will be minimal in that any player's defection will negate the coalition's winning status. Brains and Fishburn (1995) applied Riker's principle to weighted-majority voting games in which players have voting weights wl > w2 > ... > w,, and a coalition is winning if its members' weights sum to more than half the total weight. We showed that players' bargaining power tends to decrease as their weights decrease when minimal winning coalitions obtain, but that the opposite trend occurs when the minimal winning coalitions that form are "weight-minimal", referred to as least winning coalitions. In such coalitions, large size may be more harmful than helpful.
Annals of Operations Research, 2013
This paper is a twofold contribution. First, it contributes to the problem of enumerating some classes of simple games and in particular provides the number of weighted games with minimum and the number of weighted games for the dual class as well. Second, we focus on the special case of bipartite complete games with minimum, and we compare and rank these games according to the behavior of some efficient power indices of players of type 1 (or of type 2). The main result of this second part establishes all allowable rankings of these games when the Shapley-Shubik power index is used on players of type 1.
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.
On the dimensionality of voting games
2008
In a yes/no voting game, a set of voters must determine whether to accept or reject a given alternative. Weighted voting games are a well-studied subclass of yes/no voting games, in which each voter has a weight, and an alternative is accepted if the total weight of its supporters exceeds a certain threshold. Weighted voting games are naturally extended to k-vector weighted voting games, which are intersections of k different weighted voting games: a coalition wins if it wins in every component game. The dimensionality, k, of a kvector weighted voting game can be understood as a measure of the complexity of the game. In this paper, we analyse the dimensionality of such games from the point of view of complexity theory. We consider the problems of equivalence, (checking whether two given voting games have the same set of winning coalitions), and minimality, (checking whether a given k-vector voting game can be simplified by deleting one of the component games, or, more generally, is equivalent to a k -weighted voting game with k < k). We show that these problems are computationally hard, even if k = 1 or all weights are 0 or 1. However, we provide efficient algorithms for cases where both k is small and the weights are polynomially bounded. We also study the notion of monotonicity in voting games, and show that monotone yes/no voting games are essentially as hard to represent and work with as general games.
Bargaining Sets of Voting Games
2004
Let A be a finite set of m <FONT FACE="Symbol">³</FONT> 3 alternatives, let N be a finite set of n <FONT FACE="Symbol">³</FONT> 3 players and let R<SUP>n</SUP> be a profile of linear preference orderings on A of the players. Throughout most of the paper the considered voting system is the majority rule. Let u<SUP>N</SUP> be a profile of utility functions
Mathematical Social Sciences, 2000
In a weighted majority game each player has a positive integer weight and there is a positive integer quota. A coalition of players is winning (losing) if the sum of the weights of its members exceeds (does not exceed) the quota. A player is pivotal for a coalition if her omission changes it from a winning to a losing one. Most game theoretic measures of the power of a player involve the computation of the number of coalitions for which that player is pivotal. Prasad and Kelly [Prasad, K., Kelly, J.S., 1990. NP-completeness of some problems concerning voting games. International Journal of Game Theory 19, 1-9] prove that the problem of determining whether or not there exists a coalition for which a given player is pivotal is NP-complete. They also prove that counting the number of coalitions for which a given player is pivotal is [P-complete. In the present paper we exhibit classes of weighted majority games for which these problems are easy.
NP-completeness of some problems concerning voting games
International Journal of Game Theory, 1990
The problem of confirming lower bounds on the number of coalitions for which an individual is pivoting is NP-complete. Consequently, the problem of confirming non-zero values of power indices is NP-complete. The problem of computing the Absolute Banzhaf index is #P-complete. Related problems for power indices are discussed.
Bargaining Sets of Majority Voting Games
Mathematics of Operations Research, 2007
Let A be a finite set of m alternatives, let N be a finite set of n players, and let R N be a profile of linear orders on A of the players. Let u N be a profile of utility functions for R N . We define the nontransferable utility (NTU) game V u N that corresponds to simple majority voting, and investigate its Aumann-Davis-Maschler and Mas-Colell bargaining sets. The first bargaining set is nonempty for m ≤ 3, and it may be empty for m ≥ 4. However, in a simple probabilistic model, for fixed m, the probability that the Aumann-Davis-Maschler bargaining set is nonempty tends to one if n tends to infinity. The Mas-Colell bargaining set is nonempty for m ≤ 5, and it may be empty for m ≥ 6. Furthermore, it may be empty even if we insist that n be odd, provided that m is sufficiently large. Nevertheless, we show that the Mas-Colell bargaining set of any simple majority voting game derived from the k-fold replication of R N is nonempty, provided that k ≥ n + 2.