Weighted Majority Games Research Articles

Complementary cooperation, minimal winning coalitions, and power indices

We introduce a new simple game, which is referred to as the complementary weighted multiple majority game (C-WMMG for short). C-WMMG models a basic cooperation rule, the complementary cooperation rule, and can be taken as a sister model of the famous weighted majority game (WMG for short). In C-WMMG, each player is characterized by a nonnegative vector with a fixed dimension, and players in the same coalition cooperate by producing a characteristic vector for this coalition (each dimension of this vector equals the maximum of the corresponding dimensions of its members). The value of a coalition is 1 if and only if the sum of its characteristic vector is larger than that of its complementary coalition, in which case the coalition is called winning. Otherwise, the coalitional value is 0. In this paper, we concentrate on the two dimensional C-WMMG. An interesting property of this case is that there are at most n+1 minimal winning coalitions (MWCs for short), and they can be enumerated in time O(nlogn), where n is the number of players. This property guarantees that the two dimensional C-WMMG is more handleable than WMG. In particular, we prove that the main power indices, i.e. the Shapley–Shubik index, the Penrose–Banzhaf index, the Holler–Packel index, and the Deegan–Packel index, are all polynomially computable. To make a comparison with WMG, we know that it may have exponentially many MWCs, and none of the four power indices is polynomially computable (unless P=NP). Still for the two dimensional case, we show that local monotonicity holds for all of the four power indices. In WMG, this property is possessed by the Shapley–Shubik index and the Penrose–Banzhaf index, but not by the Holler–Packel index or the Deegan–Packel index. Since our model fits very well the cooperation and competition in team sports, we hope that it can be potentially applied in measuring the values of players in team sports, say help people give more objective ranking of NBA players and select MVPs, and consequently bring new insights into contest theory and the more general field of sports economics. It may also provide some interesting enlightenments into the design of non-additive voting mechanisms. Last but not least, the threshold version of C-WMMG is a generalization of WMG, and natural variants of it are closely related with the famous airport game and the stable marriage/roommates problem.

A Violation of Monotonicity in a Noncooperative Setting

A power measure is monotone if a player with a larger weight is assigned at least as much power as a player with a smaller weight in the same weighted majority game. Failure of a power index to satisfy monotonicity is often considered a pathological feature. In this paper, we show that monotonicity may fail in the unique subgame perfect equilibrium of a noncooperative bargaining game. A player with a smaller weight may have a higher expected payoff than a player with a larger weight. This is possible even though coalition formation and payoff division are endogenous, all players are rational and there is no asymmetry between the players other than in the weights.

On the likelihood of dummy players in weighted majority games

When the number of players is small in a weighted majority voting game, it can occur that one of the players has no influence on the result of the vote, in spite of a strictly positive weight. Such a player is called a “dummy” player in game theory. The purpose of this paper is to investigate the conditions that give rise to such a phenomenon and to compute its likelihood. It is shown that the probability of having a dummy player is surprisingly high and some paradoxical results are observed.

Selfish bin covering

In this paper, we address the selfish bin covering problem, which is greatly related both to the bin covering problem, and to the weighted majority game. What we are mainly concerned with is how much the lack of central coordination harms social welfare. Besides the standard PoA and PoS, which are based on Nash equilibrium, we also take into account the strong Nash equilibrium, and several new equilibrium concepts. For each equilibrium concept, the corresponding PoA and PoS are given, and the problems of computing an arbitrary equilibrium, as well as approximating the best one, are also considered.

Power indices of simple games and vector-weighted majority games by means of binary decision diagrams

A simple game is a pair consisting of a finite set N of players and a set W ⊆ 2 N of winning coalitions. (Vector-) weighted majority games ((V) WMG) are a special case of simple games, in which an integer (vector) weight can be assigned to each player and there is a quota which a coalition has to achieve in order to win. Binary decision diagrams (BDDs) are used as compact representations for Boolean functions and sets of subsets. This paper shows, how a quasi-reduced and ordered BDD (QOBDD) of the winning coalitions of a (V) WMG can be build, how one can compute the minimal winning coalitions and how one can easily compute the Banzhaf, Shapley–Shubik, Holler–Packel and Deegan–Packel indices of the players. E.g. in case of weighted majority games it is shown that the Banzhaf and Holler–Packel indices of all players can be computed in expected time O ( nQ ) and in general, the Banzhaf indices can be computed in time linear in the size of the QOBDD representation of the winning coalitions. Other running times are proven as well. The algorithms were tested on some real world games, e.g. the International Monetary Fund and the EU Treaty of Nice.

A new family of regular semivalues and applications

We define a new family of values for cooperative games, including as a particular case the Shapley value. They are defined on the collection of the unanimity games, then extended by linearity. Our most relevant result shows that the family of the weighting coefficients characterizing the values so defined is an open curve on the simplex of the regular semivalues. We give an explicit formula for the values when the parameter characterizing the family is a natural number and we offer an algorithm to calculate them in weighted majority games, slightly extending previous results (see Bilbao et al., TOP, 8:191–213, 2000). The paper ends with two applications. The first one is classical, and serves to see how the indices behave with respect to the Shapley and Banzhaf values in the case of the EU parliament and in the UN Security Council. The second one is much more recent: it deals with the microarray games, introduced in Moretti et al. (TOP, 15:256–280, 2007), which are average of unanimity games. The idea is to rank genes taken from DNA of patients affected by a specific disease, with the aim of singling out a group of genes potentially responsible of the disease. In this last case we consider some microarray data available on the net and concerning some specific diseases and we show that several genes mentioned in the medical literature as potentially responsible for the onset of the disease are present in the first places according to our rankings.

Weighted and Roughly Weighted Simple Games

This paper contributes to the program of numerical characterisation and classification of simple games outlined in the classical monograph of von Neumann and Morgenstern (1944). One of the most fundamental questions of this program is what makes a simple game a weighted majority game. The necessary and sufficient conditions that guarantee weightedness were obtained by Elgot (1961) and refined by Taylor and Zwicker (1992) If a simple game does not have weights, then rough weights may serve as a reasonable substitute (see their use in Taylor and Zwicker, 1992). A simple game is roughly weighted if there exists a system of weights and a threshold such that all coalitions whose combined weight is above the threshold are winning and all coalitions whose combined weight is below the threshold are losing and a tie-breaking is needed to classify the coalitions whose combined weight is exactly the threshold. Not all simple games are roughly weighted, and the class of projective games is a prime example. In this paper we give necessary and sufficient conditions for a simple game to have rough weights. We define two functions f(n) and g(n) that measure the deviation of a simple game from a weighted majority game and roughly weighted majority game, respectively. We formulate known results in terms of lower and upper bounds for these functions and improve those bounds. We also investigate rough weightedness of simle games with a small number of players.

Weighted and roughly weighted simple games

In this paper we give necessary and sufficient conditions for a simple game to have rough weights. We define two functions f ( n ) and g ( n ) that measure the deviation of a simple game from a weighted majority game and roughly weighted majority game, respectively. We formulate known results in terms of lower and upper bounds for these functions and improve those bounds. We also investigate rough weightedness of simple games with a small number of players.

Coalition formation in weighted simple-majority games under proportional payoff allocation rules

In this paper, we introduce a simple coalition formation game in the environment of bidding, which is a special case of the weighted majority game (WMG), and is named the weighted simple-majority game (WSMG). In WSMG, payoff is allocated to the winners proportional to the players’ powers, which can be measured in various ways. We define a new kind of stability: the counteraction-stability (C-stability), where any potential deviating players will confront counteractions of the other players. We show that C-stable coalition structures in WSMG always contains a minimal winning coalition of minimum total power. For the variant where powers are measured directly by their weights, we show that it is NP-hard to find a C-stable coalition structure and design a pseudo-polynomial time algorithm. Sensitivity analysis for this variant, which shows many interesting properties, is also done. We also prove that it is NP-hard to compute the Holler-Packel indices in WSMGs, and hence in WMGs as well.

A bargaining set for monotonic simple games based on external and internal stability

A new bargaining set based on notions of both internal and external stability is developed in the context of endogenous coalition formation. It allows one to make an explicit distinction between within-group and outside-group deviation options. This type of distinction is not present in current bargaining sets. For the class of monotonic proper simple games, the outcomes in the bargaining set are characterized. Furthermore, it is shown that the bargaining set of any homogeneous weighted majority game contains an outcome for which the underlying coalition structure consists of a minimal winning coalition and its complement.

Dimension of complete simple games with minimum

Abstract Weighted majority games have the property that players are totally ordered by the desirability relation (introduced by Isbell in [J.R. Isbell, A class of majority games, Quarterly Journal of Mathematics, 7 (1956) 183–187]) because weights induce it. Games for which this relation is total are called complete simple games . Taylor and Zwicker proved in [A.D. Taylor, W.S. Zwicker, Weighted voting, multicameral representation, and power, Games and Economic Behavior 5 (1993) 170–181] that every simple game (or monotonic finite hypergraph) can be represented by an intersection of weighted majority games and consider the dimension of a game as the needed minimum number of them to get it. They provide the existence of non-complete simple games of every dimension and left open the problem for complete simple games. In this paper we prove that their result can be extended for these games and give a constructive procedure to get complete simple games of every dimension. The curious fact is that to obtain our result it is enough to consider the simplest kind of complete simple games, i.e. those that admit a minimum lattice representative. As a consequence, it is proved that there is no connection between ‘having small dimension’ and ‘being totally ordered by the desirability relation’.

Singular games in [formula omitted

Every simple monotonic game in b v ′ N A is a weighted majority game. Every game v ∈ b v ′ N A has a representation v = u + ∑ i = 1 ∞ f i ∘ μ i where u ∈ p N A , μ i ∈ N A 1 and f i is a sequence of b v ′ functions with ∑ i = 1 ∞ ‖ f i ‖ < ∞ . Moreover, the representation is unique if we require f i to be singular and that for every i ≠ j , μ i ≠ μ j .

Applications of cooperative games to business activities

We show several applications of the cooperative game theory and the Shapley value to business activities. Among them we emphasize on the sharing of costs or profits. In the theoretical part, the axiomatic setting proposed by Shapley is described and the properties of the value are studied. It is also shown the usefulness of simple games to describe and analyze collective decision systems ruled by voting. Many of these systems admit a representation as weighted majority games, a class of simple games. One of the most interesting problems in this case is to define a measure of the power distribution among the agents involved in the system. This problem receives a nice solution by means of the Shapley–Shubik index of power, which is the restriction of the Shapley value to simple games.

Proportional Payoffs in Majority Games

This paper extends the Baron-Ferejohn model of legislative bargaining to general weighted majority games with two modifications: first, payoff division can only be agreed upon after the coalition has formed (two-stage bargaining); second, negotiations in the coalition can break down, in which case a new coalition may be formed (reversible coalitions). Under the most natural bargaining protocol, both expected payoffs and actual payoff division are proportional to the voting weights provided that the set of winning coalitions of minimum weight is weakly balanced and that the breakdown probability tends to 0. Homogeneity of the voting weights is neither necessary nor sufficient for proportional payoffs. Intermediate values of the breakdown probability produce predictions consistent with the empirical evidence on portfolio allocation in Europe: a moderate propoper advantage and a linear relationship between weights and ex post payoffs for all coalition members other than the proposer.

The Aggregate and the Representation of its Parts

The main concern of this contribution is to review the approach Condorcet applied to problems of aggregation and representation. The modern viewpoint usually chosen for these topic is mainly driven by social choice theory. Here the viewpoint was altered and the problems and solutions were presented in a setting which uses cooperative game theory. After the introductory notes on Condorcet's basic ideas, we follow one logical path for an initial analysis of rules (Section 3). We present some new facts on the solutions to stand-offs. After connecting the logical approach and the social choice approach in Section 4, we discuss the probabilistic models introduced by Condorcet (Section 5). In Section 6 we provide additional structural insight, which can be derived from the game theoretic viewpoint. Based on the results of the previous sections, in Section 7 we analyse representation procedures. We develop a major tool for representation, namely the minimal representation of a weighted majority game. We discuss its construction and provide examples based on the results of the German Bundestag election in the year 2005.

On ordinal equivalence of power measures given by regular semivalues

Tomiyama [Tomiyama, Y., 1987. Simple game, voting representation and ordinal power equivalence. International Journal on Policy and Information 11, 67–75] proved that, for every weighted majority game, the preorderings induced by the classical Shapley–Shubik and Penrose–Banzhaf–Coleman indices coincide. He called this property the ordinal equivalence of these indices for weighted majority games. Diffo Lambo and Moulen [Diffo Lambo, L., Moulen, J., 2002. Ordinal equivalence of power notions in voting games. Theory and Decision 53, 313–325] extended Tomiyama's result to all linear (i.e. swap robust) simple games. Here we extend Diffo Lambo and Moulen's result to all the preorderings induced by regular semivalues (which include both classical indices) in a larger class of games that we call weakly linear simple games. We also provide a characterization of weakly linear games and use nonsymmetric transitive games to supplying examples of nonlinear but weakly linear games.

MEASURING THE POWER OF PARTIES WITHIN GOVERNMENT COALITIONS

The paper presents a coalition-structure value that is meant to capture outside options of players in a cooperative game. It deviates from the Aumann-Drèze value by violating the null-player axiom. We use this value as a power index and apply it to weighted majority games.

Control and voting power in corporate networks: Concepts and computational aspects

This paper proposes to rely on power indices to measure the amount of control held by individual shareholders in corporate networks. The value of the indices is determined by a complex voting game viewed as the composition of interlocked weighted majority games; the compound game reflects the structure of shareholdings. The paper describes an integrated algorithmic approach which allows to deal efficiently with the complexity of computing power indices in shareholding networks, irrespective of their size or structure. In particular, the approach explicitly accounts for the presence of float and of cyclic shareholding relationships. It has been successfully applied to the analysis of real-world financial networks.

On the dimension of simple monotonic games

We show that the following problem is NP-hard, and hence computationally intractable: “Given d weighted majority games, decide whether the dimension of their intersection exactly equals d”. Our result indicates that the dimension of simple monotonic games is a combinatorially complicated concept.

A law of large numbers for weighted majority

Consider an election between two candidates in which the voters' choices are random and independent and the probability of a voter choosing the first candidate is p > 1 / 2 . Condorcet's Jury Theorem which he derived from the weak law of large numbers asserts that if the number of voters tends to infinity then the probability that the first candidate will be elected tends to one. The notion of influence of a voter or its voting power is relevant for extensions of the weak law of large numbers for voting rules which are more general than simple majority. In this paper we point out two different ways to extend the classical notions of voting power and influences to arbitrary probability distributions. The extension relevant to us is the “effect” of a voter, which is a weighted version of the correlation between the voter's vote and the election's outcomes. We prove an extension of the weak law of large numbers to weighted majority games when all individual effects are small and show that this result does not apply to any voting rule which is not based on weighted majority.