A class of power indices for voting games

A class of power indices is discussed which includes not only the usual power indices but also the class of semivalues and the indices introduced byDeegan/Packel [1978]. Sufficient conditions for members of this class to be symmetric, support-independent, and self-dual are given. We then discuss various conditions under which members of this class coincide with the ShapleyShubik and Banzhaf indices.     

Ternary voting games

We defineternary voting games (TVGs), a generalization ofsimple voting games (SVGs). In a play of an SVG each voter has just two options: voting yes or no. In a TVG a third option is added: abstention. Every SVG can be regarded as a (somewhat degenerate) TVG; but the converse is false. We define appropriate generalizations of the Shapley-Shubik and Banzhaf indices for TVGs. We define also theresponsiveness (ordegree of democratic participation) of a TVG and determine, for eachn, the most responsive TVGs withn voters. We show that these maximally responsive TVGs are more responsive than the corresponding SVGs.We wish to thank Robert Aumann and an anonymous referee for their helpful comments, and Ralph Amelan and Hazem Ghobarah for the information they provided us regarding voting rules in the US Congress.     

Information aggregation in multicandidate elections under plurality rule and runoff voting

I consider a model in which imperfectly informed voters with common interests participate in a multicandidate election decided by either plurality rule or a runoff. Prior to the election, each voter receives a private signal corresponding to the candidate the voter thinks is best. Voters are relatively more likely to think a given candidate is best if the candidate is a relatively better candidate. I show that there is a sequence of equilibrium strategies for the voters such that, as the number of voters goes to infinity, the probability that the best candidate is elected goes to 1. I further show that all candidates receive significant vote shares in any equilibrium in which information fully aggregates under plurality rule and that voters do at least as well when the election is decided by a runoff as they do when the election is decided by plurality rule.     

Multicameral voting cohesion games

This work links two research branches regarding cooperative games: multi-cameral games and a priori coalitions. Regarding the former, many parliaments are based on a bicameral system where bills have to be approved by both chambers. The parties’ power indices are affected by it. In fact one certain political party for one certain coalition can be crucial in one chamber and not in the other. Usually the problem can be solved by building a unified game, related to two or more chambers, where the global power indices can be calculated. Concerning cohesion games, some coalitions among parties might be possible, but have a low probability of materializing if the parties involved have very different ideologies. It is necessary to take this into consideration when we compute the relative power indices. In this work, a model is built that takes into consideration both topics. An automatic calculation algorithm is elaborated and is applied to the current situations in Belgium, the Czech Republic, France, Italy, the Netherlands, Poland and Romania, considering the location of the parties on the left-right axis. This model is then applied globally to the European Union. In this work the model is applied to Politics, but its applications are not limited to that field. For instance, in Finance it can be applied to describe agreements for debt restructuring and corporate mergers.     

On minimum sum representations for weighted voting games

A proposal in a weighted voting game is accepted if the sum of the (non-negative) weights of the ??yea?? voters is at least as large as a given quota. Several authors have considered representations of weighted voting games with minimum sum, where the weights and the quota are restricted to be integers. In Freixas and Molinero (Ann. Oper. Res. 166:243?C260, 2009) the authors have classified all weighted voting games without a unique minimum sum representation for up to 8 voters. Here we exhaustively classify all weighted voting games consisting of 9?voters which do not admit a unique minimum sum integer weight representation.     

Mathematical structures of simple voting games

We address simple voting games (SVGs) as mathematical objects in their own right, and study structures made up of these objects, rather than focusing on SVGs primarily as co-operative games. To this end it is convenient to employ the conceptual framework and language of category theory. This enables us to uncover the underlying unity of the basic operations involving SVGs.     

On the design of voting games

By focusing on the protectionist tendency found in the design of voting games, a thorough analysis is provided for the role of blocking coalitions in a simple game. We characterize those blocking families that univocally determine the game, and show that otherwise at least three games share a given nonempty blocking family, also giving an upper bound for the number of such games. Some examples illustrate the application of these ideas to political science.     

Achievable hierarchies in voting games with abstention

It is well known that he influence relation orders the voters the same way as the classical Banzhaf and Shapley–Shubik indices do when they are extended to the voting games with abstention (VGA) in the class of complete games. Moreover, all hierarchies for the influence relation are achievable in the class of complete VGA. The aim of this paper is twofold. Firstly, we show that all hierarchies are achievable in a subclass of weighted VGA, the class of weighted games for which a single weight is assigned to voters. Secondly, we conduct a partial study of achievable hierarchies within the subclass of H-complete games, that is, complete games under stronger versions of influence relation.     

A core of voting games with improved foresight

This paper considers voting situations in which the vote takes place iteratively. If a coalition replaces the status quo a with a contestant b, then b becomes the new status quo, and the vote goes on until a candidate is reached that no winning coalition is willing to replace. It is well known that the core, that is, the set of undominated alternatives, may be empty. To alleviate this problem, Rubinstein [Rubinstein, A., 1980. Stability of decision systems under majority rule. Journal of Economic Theory 23, 150–159] assumes that voters look forward one vote before deciding to replace an alternative by a new one. They will not do so if the new status quo is going to be replaced by a third that is less interesting than the first. The stability set, that is, the set of undominated alternatives under this behavior, is always non-empty when preferences are strict. However, this is not necessarily the case when voters’ indifference is allowed. Le Breton and Salles [Le Breton, M., Salles, M., 1990. The stability set of voting games: Classification and generecity results. International Journal of Game Theory 19, 111–127], Li [Li, S., 1993. Stability of voting games. Social Choice and Welfare 10, 51–56] and Martin [Martin, M., 1998. Quota games and stability set of order d. Economic Letters 59, 145–151] extend the sophistication of the voters by having them look d votes forward along the iterative process. For d sufficiently large, the resulting set of undominated alternatives is always non-empty even if indifference is allowed. We show that it may be unduly large. Next, by assuming that other voters along a chain of votes are also rational, that is, they also look forward to make sure that the votes taking place later on will not lead to a worst issue for them, we are able to reduce the size of this set while insuring its non-emptiness. Finally, we show that a vote with sufficient foresight satisfies a no-regret property, contrarily to the classical core and the stability set.     

A further result on the core of voting games

LetN be a set of individuals,A a closed bounded interval ofRe. Using a former result ofNakamura, it is shown that if every individual inN has a quasi-concave utility function overA, then a proper simple game has a non-empty core which is a convex set. In particular, the majority core is explicitly characterized. When every individual inN has a strictly quasi-concave utility function overA, then it is shown that the local core in a proper simple game is precisely the core.     

A class of non-normalized power indices for simple games

Weights are attached to the coalitions of a finite set of players in order to reflect the fact that some coalitions are more likely to be formed than others. A new power index for simple games which depends only on these weights and the minimal winning coalitions of a game is introduced. Two axiomatic characterizations of this index are given. Symmetry, dummy player and quarrel properties of the index are treated. Two special forms of the index arising from certain assumptions about the weights are described.     

Monotonicity of power in weighted voting games with restricted communication

Indices that evaluate the distribution of power in simple games are commonly required to be monotonic in voting weights when the game represents a voting body such as a shareholder meeting, parliament, etc. The standard notions of local or global monotonicity are bound to be violated, however, if cooperation is restricted to coalitions that are connected by a communication graph. This paper proposes new monotonicity concepts for power in games with communication structure and investigates the monotonicity properties of the Myerson value, the restricted Banzhaf value, the position value, and the average tree solution.     

Computing the nucleolus of weighted voting games in pseudo-polynomial time

Mathematical Programming - In cooperative games, players have a possibility to form different coalitions. This leads to the questions about ways to motivate all players to collaborate, i.e. to...     

A note on hierarchies in weighted voting games and partitions

Weighted voting games (WVGs) model situations where voters, possibly controlling different numbers of votes, vote yes or no on a proposition. A proposition passes if and only if the number of yes votes meets or exceeds a quota \(q\). Each winning coalition is a partition of an integer greater than or equal to \(q\), with parts taken from the set of all weights for that game. Results about WVGs are here interpreted as results about partitions.     

Interaction indices for games on combinatorial structures with forbidden coalitions

The notion of interaction among a set of players has been defined on the Boolean lattice and Cartesian products of lattices. The aim of this paper is to extend this concept to combinatorial structures with forbidden coalitions. The set of feasible coalitions is supposed to fulfil some general conditions. This general representation encompasses convex geometries, antimatroids, augmenting systems and distributive lattices. Two axiomatic characterizations are obtained. They both assume that the Shapley value is already defined on the combinatorial structures. The first one is restricted to pairs of players and is based on a generalization of a recursivity axiom that uniquely specifies the interaction index from the Shapley value when all coalitions are permitted. This unique correspondence cannot be maintained when some coalitions are forbidden. From this, a weak recursivity axiom is defined. We show that this axiom together with linearity and dummy player are sufficient to specify the interaction index. The second axiomatic characterization is obtained from the linearity, dummy player and partnership axioms. An interpretation of the interaction index in the context of surplus sharing is also proposed. Finally, our interaction index is instantiated to the case of games under precedence constraints.     

The stability set of voting games: Classification and genericity results

In 1980 Rubinstein introduced a new solution concept for voting games called the stability set which incorporates the idea that before entering into a possibly winning coalition with respect to some pair of alternatives, a voter will consider what might happen in the future. He showed that if the voters' preferences are given by linear orders the stability set is non-empty for a large class of voting games with finite sets of alternatives. We consider the case where indifference is allowed (preferences are then complete preorders) and show that the picture is then quite different. First, in the finite case, we obtain classification results for the non-emptiness of the stability set which are based on Nakamura's number. When preferences are continuous, we prove a general non-emptiness theorem and show that the set of profiles for which the stability set is non-empty is dense in the set of profiles.We are most grateful to the French Commissariat Général du Plan for financial support and to Georges Bordes, Prasanta K. Pattanaik, Hervé Moulin, Bezalel Peleg, Ariel Rubinstein, Norman Schofield and two anonymous referees of this journal for helpful comments. In particular, Bezalel Peleg has called our attention to the similarities of the stability set with the Gillies set and the uncovered set. A previous draft of this paper has been presented at the conference on Economic Models and Distributive Justice in Brussels, at the European Meeting of the Econometric Society in Copenhagen and at seminars in Europe and the United States. We thank the participants for their remarks.     

Comparing influence theories in voting games under locally generated measures of dissatisfaction

The Isbell desirability relation (I), the Shapley?CShubik index (SS) and the Banzhaf?CColeman index (BC) are power theories that grasp the notion of individual influence in a yes?Cno voting rule. Also, a yes?Cno voting rule is often used as a tool for aggregating individual preferences over any given finite set of alternatives into a collective preference. In this second context, Diffo Lambo and Moulen (DM) have introduced a power relation which ranks the voters with respect to how ably they influence the collective preference. However, DM relies on the metric d that measures closeness between preference relations. Our concern in this work is: do I, SS, BC and DM agree when the same yes?Cno voting rule is the basis for collective decision making? We provide a concrete and intuitive class of metrics called locally generated (LG). We give a characterization of the LG metrics d for which I, SS, BC and DM agree on ranking the voters.     

A note on balancedness and nonemptiness of the core in voting games

Consider a society with a finite number,n, of individuals who have to choose an alternative from a setA in them-dimensional Euclidean space IR ^m . Assuming that the preference relation overA of every individual is convex and continuous, Greenberg (1979) showed some that if the set of winning coalitions (i.e. those that have the veto power) consists of all coalitions with more thanmn/m + 1 individuals the core of the induced game is nonempty. Greenberg and Weber (1984) have strengthened this result by showing that the induced game is in fact balanced. On the other hand Le Breton (1987), Schofield (1984a) and Strnad (1985) have generalized Greenberg's theorem to arbitrary voting games. The purpose of this note is to show that however the induced game is not generally balanced.