Shapley-Shubik Index Research Articles

Coalition formation in general apex games under monotonic power indices

An apex game consists of one apex player and a set of minor players. We identify two key properties of apex games and use them to introduce the class of general apex games. We derive players' preferences over winning coalitions by applying strongly monotonic power indices on such a game and all its subgames and investigate whether there are core stable coalitions in the induced hedonic coalition formation game. Besides several general results, in particular, we develop conditions on the game for the Shapley–Shubik index, the Banzhaf index, and the normalized Banzhaf index.

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.

Voting power and proportional representation of voters

We prove that for the proportional representative election system if parties’ sizes are uniformly distributed on the simplex, the expected ratio of a party size to its political power, measured by the Shapley–Shubik index, converges to $$1$$ , as the number $$n$$ of parties increases indefinitely. The rate of convergence is high and it is of the magnitude of $$\frac{1}{n}$$ . Empirical evidence from the Netherlands elections supports our result. A comparison with the Banzhaf index is provided.

Power measures derived from the sequential query process

We study a basic sequential model for the formation of winning coalitions in a simple game, well known from its use in defining the Shapley–Shubik power index. We derive in a uniform way a family of measures of collective and individual decisiveness in simple games, and show that, as for the Shapley–Shubik index, they extend naturally to measures for TU-games. These individual measures, which we call weighted semivalues, form a class whose intersection with that of the class of weak semivalues yields the class of all semivalues.We single out the simplest measure in this family for more investigation, as it is new to the literature as far as we know. Although it is very different from the Shapley value, it is closely related in several ways, and is the natural analogue of the Shapley value under a nonstandard, but natural, definition of simple game. We illustrate this new measure by calculating its values on some standard examples.

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.

Analyse spatiale du pouvoir de vote : application au cas de l’intercommunalité dans le département du Val d’Oise

L’analyse spatiale des indices de pouvoir est generalement employee dans la litterature afin d’etudier l’importance des criteres ideologiques. Notre propos est ici de montrer, a travers l’exemple de structures intercommunales francaises, que la situation geographique des joueurs peut avoir un impact important en terme de pouvoir. En effet, prendre en compte la situation geographique (en utilisant l’indice de Shapley-Owen) change parfois radicalement les mesures usuelles du pouvoir (en utilisant l’indice de Shapley-Shubik).

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

The Isbell desirability relation (I), the Shapley–Shubik index (SS) and the Banzhaf–Coleman index (BC) are power theories that grasp the notion of individual influence in a yes–no voting rule. Also, a yes–no 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–no 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.

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 in a voting situation and are calculated by looking at marginal contributions in a simple game consisting of winning and losing coalitions derived from the legislative rules. We introduce a new way to calculate these measures directly from the set of minimal winning coalitions and derive explicit formulae for the Shapley–Shubik and Banzhaf values. 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 the numerical calculations to obtain the indices. The technique generalises directly to all semivalues.

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 in a voting situation and are calculated by looking at marginal contributions in a simple game consisting of winning and losing coalitions derived from the legislative rules.We introduce a new way to calculate these measures directly from the set of minimal winning coalitions and derive explicit formulae for the Shapley-Shubik and Banzhaf values. 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 the numerical calculations to obtain the indices. The technique generalises directly to all semivalues.

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 in a voting situation and are calculated by looking at marginal contributions in a simple game consisting of winning and losing coalitions derived from the legislative rules. We introduce a new way to calculate these measures directly from the set of minimal winning coalitions and derive explicit formulae for the Shapley-Shubik and Banzhaf values. 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 the numerical calculations to obtain the indices. The technique generalises directly to all semivalues.

The influence relation for ternary voting games

Although simple games are very useful in modeling decision-making bodies, they allow each voter only two choices: to support or oppose a measure. This restriction ignores that voters often can abstain from voting, which is effectively different from the other two options. Following the approach of Felsenthal and Machover (1997), for modeling voting with abstentions, we will look at the extension of the influence relation for simple games to the Ternary Voting Game given in Tchantcho et al. (2008). That paper showed that the influence relation is ordinally equivalent to the classical Banzhaf and Shapley–Shubik indices in a class of games called weakly equitable. In this paper, we will show that this result does hold true for all Ternary Voting Games. Also we will show that adding a third voting option allows for asymmetric distribution of power that cannot be achieved by any simple game.

Manipulating the quota in weighted voting games

Weighted voting games provide a simple model of decision-making in human societies and multi-agent systems. Such games are described by a set of players, a list of playersʼ weights, and a quota; a coalition of the players is said to be winning if the total weight of its members meets or exceeds the quota. The power of a player in a weighted voting game is traditionally identified with her Shapley–Shubik index or her Banzhaf index, two classic power measures that reflect the playerʼs marginal contribution under different coalition formation scenarios. In this paper, we investigate by how much one can change a playerʼs power, as measured by these indices, by modifying the quota. We give tight bounds on the changes in the individual playerʼs power that can result from a change in quota. We then describe an efficient algorithm for determining whether there is a value of the quota that makes a given player a dummy, i.e., reduces her power (as measured by both indices) to 0. We also study how the choice of quota can affect the relative power of the players. Finally, we investigate scenarios where oneʼs choice in setting the quota is constrained. We show that optimally choosing between two values of the quota is complete for the complexity class PP, which is believed to be significantly more powerful than NP. On the other hand, we empirically demonstrate that even small changes in quota can have a significant effect on a playerʼs power.

Voting power in the EU council of ministers and fair decision making in distributive politics

We analyze and evaluate the different decision rules describing the Council of Ministers of the EU starting from 1958 up to now. Most of the existing studies use the Banzhaf index (for binary voting) or the Shapley–Shubik index (for distributive politics). We argue in favor of the nucleolus as a power measure in distributive situations and an alternative to the Shapley–Shubik index. We then calculate the nucleolus and compare the results of our calculations with the conventional measures. In the second part, we analyze the power of the European citizens as measured by the nucleolus under the egalitarian criterion proposed by Felsenthal and Machover (1998), and characterize the first best situation. Based on these results we propose a methodology for the design of the optimal (fair) decision rules. We perform the optimization exercise for the earlier stages of the EU within a restricted domain of voting rules, and conclude that Germany should receive more than the other three large countries under the optimalvoting rule.

Egalitarian property for power indices

In this study, we introduce and examine the Egalitarian property for some power indices on the class of simple games. This property means that after intersecting a game with a symmetric or anonymous game the difference between the values of two comparable players does not increase. We prove that the Shapley–Shubik index, the absolute Banzhaf index, and the Johnston score satisfy this property. We also give counterexamples for Holler, Deegan–Packel, normalized Banzhaf and Johnston indices. We prove that the Egalitarian property is a stronger condition for efficient power indices than the Lorentz domination.

Power indices in large voting bodies

There is no consensus on the properties of voting power indices when there is a large number of voters in a weighted-voting body. On the one hand, in some real-world cases that have been studied the power indices have been found to be nearly proportional to the weights (e.g., the EUCM, US Electoral College); this is true for both the Penrose-Banzhaf and the Shapley-Shubik indices. It has been suggested that this is a manifestation of a conjecture by Penrose (known subsequently as the Penrose limit theorem, that has been shown to hold under certain conditions). On the other hand, we have the older literature from cooperative game theory, due to Shapley and his collaborators, showing that, where there is a finite number of voters whose weights remain constant in relative terms, and where the quota remains constant in relative terms, while the total number of voters increases without limit, the powers of the voters with finite weight tend to limiting values that are, in general, not proportional to the weights. These results, too, are supported by empirical studies of large voting bodies (e.g., the IMF/WB boards, corporate shareholder meetings). This paper proposes a restatement of the Penrose limit theorem and shows that, in general, both the ‘classical’ power indices converge in the limit to proportionality with weights as the Laakso-Taagepera index of political fragmentation increases. This new version reconciles the different theoretical and empirical results that have been found for large voting bodies.

Voting Power in the EU Council of Ministers and Fair Decision Making in Distributive Politics

We analyze and evaluate the different decision rules describing the Council of Ministers of the EU starting from 1958 up to date. Most of the existing studies use the Banzhaf index (for binary voting) or the Shapley-Shubik index (for distributive politics). We argue in favor of the nucleolus as a power measure in distributive situations and an alternative to the Shapley-Shubik index. We then calculate the nucleolus and compare the results of our calculations with the conventional measures. In the second part, we analyze the power of the European citizens as measured by the nucleolus under the egalitarian criterion proposed by Felsenthal and Machover (1998), and characterize the first best situation. Based on these results we propose a methodology for the design of the optimal (fair) decision rules. We perform the optimization exercise for the earlier stages of the EU within a restricted domain of voting rules, and conclude that Germany should receive more than the other three large countries under the optimal voting rule.

False-Name Manipulations in Weighted Voting Games

Weighted voting is a classic model of cooperation among agents in decision-making domains. In such games, each player has a weight, and a coalition of players wins the game if its total weight meets or exceeds a given quota. A player's power in such games is usually not directly proportional to his weight, and is measured by a power index, the most prominent among which are the Shapley-Shubik index and the Banzhaf index.In this paper, we investigate by how much a player can change his power, as measured by the Shapley-Shubik index or the Banzhaf index, by means of a false-name manipulation, i.e., splitting his weight among two or more identities. For both indices, we provide upper and lower bounds on the effect of weight-splitting. We then show that checking whether a beneficial split exists is NP-hard, and discuss efficient algorithms for restricted cases of this problem, as well as randomized algorithms for the general case. We also provide an experimental evaluation of these algorithms. Finally, we examine related forms of manipulative behavior, such as annexation, where a player subsumes other players, or merging, where several players unite into one. We characterize the computational complexity of such manipulations and provide limits on their effects. For the Banzhaf index, we describe a new paradox, which we term the Annexation Non-monotonicity Paradox.

The Shapley–Shubik index for multi-criteria simple games

In this paper we address multi-criteria simple games which constitute an extension of the basic framework of voting systems and related social-choice situations. For these games, we propose the extended Shapley–Shubik index as the natural generalization of the Shapley–Shubik index in conventional simple games, and establish an axiomatic characterization of this power index.

Hierarchy of players in swap robust voting games

Ordinarily, the process of decision making by a committee through voting is modeled by a monotonic game the range of whose characteristic function is restricted to {0, 1}. The decision rule that governs the collective action of a voting body induces a hierarchy in the set of players in terms of the a-priori influence that the players have over the decision making process. In order to determine this hierarchy in a swap robust game, one has to either evaluate a power index (e.g., the Shapley–Shubik index, the Banzhaf–Coleman index) for each player or conduct a pairwise comparison between players, whereby a player i is ranked higher than another player j if there exists a coalition in which i is more desirable as a coalition partner than j. In this paper, we outline an alternative mechanism to determine the ranking of players in terms of their a-priori power. This simple and elegant method uses only minimal winning coalitions, rather than the entire set of winning coalitions.

Axiomatics for power indices that account for player preferences

We offer a general approach to describing power indices that account for preferences as suggested by F. Aleskerov. We construct two axiomatizations of these indices. Our construction generalizes the Laruelle-Valenciano axioms for Banzhaf (Penrose) and Shapley-Shubik indices. We obtain new sets of axioms for these indices, in particular, sets without the anonymity axiom.