Shapley-Shubik Index Research Articles

Probabilistic Graphs and Power Indices

In this paper a method is offered for calculating the power of political parties in the Spanish parliament. This is a modification of the Shapley-Shubik index which allows the ideological affinities between parties to be taken into account. It extends Myerson's model of games with restricted communication. A version of Hart and Kurz's model is also used to study the stability of government coalitions.

A note on voting

A framework is presented in which to analyze the power of players in a cooperative game in which only certain coalitions are allowed. The allowable coalitions are characterized by a closure operator that combinatorially abstracts the notion of convexity. The Shapley-Shubik index is extended to this situation, allowing a computation of the power of each player. One example of this framework, voting on a one-dimensional spectrum, is analyzed and it is shown that the powerful players are mid-way between the middle and the extremes. This framework is also applied to an empirical study of power of the Supreme Court. Implications and generalizations are also discussed.

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.

Alternative forms of the Shapley value and the Shapley-Shubik index

We prove the validity of an alternative representation of the Shapley-Shubik (1954) index of voting power, based on the following model. Voting in an assembly consisting of n voters is conducted by roll-call. Every voter is assumed to vote "yea" or "nay" with equal probability, and all n! possible orders in which the voters may be called are also assumed to be equiprobable. Thus there are altogether 2nn! distinct roll-call patterns. Given a simple voting game (a decision rule), the pivotal voter in a roll-call is the one whose vote finally decides the outcome, so that the votes of all those called subsequently no longer make any difference. The main result, stated without proof by Mann and Shapley (1964), is that the Shapley-Shubik index of voter a in a simple voting game is equal to the probability of a being pivotal. We believe this representation of the index is much less artificial than the original one, which considers only the n! roll-calls in which all voters say "yea" (or all say "nay"). The proof of this result proceeds by generalizing the representation so that one obtains a value for each player in any coalitional game, which is easily seen to satisfy Shapley's (1953) three axioms. Thus the generalization turns out to be an alternative representation of the Shapley value. This result implies a non-trivial combinatorial theorem.

Power and Satisfaction in the United Nations Security Council

The United Nations is under pressure to make the Security Council more representative of the membership. The formal literature on voting power has discovered much that can clarify this debate, and the present analysis gives a version of the Shapley-Shubik index of voting power that considers likely alliances based on voting in the General Assembly since the breakup of the Soviet Union. Veto members occupying outlying positions, like China, have especially high voting power, whereas states without vetoes hold almost none. However, power in itself is not the goal; if another permanent member is already reliably voting as one wishes, obtaining a veto of one's own is unnecessary. Other benefits from membership are prestige and access to information. The current debate seems at an impasse, but if these alternative goals are recognized, other feasible solutions might be found.

Alternative axiomatic characterizations of the Shapley and Banzhaf values

In a paper in 1975, Dubey characterized the Shapley-Shubik index axiomatically on the class of monotonic simple games. In 1979, Dubey and Shapley characterized the Banzhaf index in a similar way. This paper extends these characterizations to axiomatic characterizations of the Shapley and Banzhaf values on the class of control games, on the class of simple games and on the class of all transferable utility games. In particular, it is shown that the additivity axiom which is usually used to characterize these values on the class of all transferable utility games can be weakened without changing the result.

Postulates and paradoxes of relative voting power ? A critical re-appraisal

This paper re-evaluates the problem of measuring thea priori relative voting power of a voter in an assembly. We propose several new intuitively compelling postualtes that any reasonable index of voting power ought to satisfy. At the same time we argue that most of the paradoxes of voting power discussed in the literature are paradoxical only in a weak sense, if at all. This leaves three crippling paradoxes — the well-known paradox ofweighted voting, and two new ones presented here: thebloc anddonation paradoxes. We evaluate the four main relative power indices discussed in the literature with respect to these three severe paradoxes. The Shapley-Shubik index is seen to be immune to all three paradoxes, while the Deegan-Packel index is vulnerable to all three. The Banzhaf and the Johnston indices are demonstrably immune to the paradox of weighted voting. However, they are shown to suffer from both the bloc and the donation paradoxes. We argue that this seriously undermines these indices in a hitherto unsuspected way. Several other theoretical issues relating to voting power are discussed.

Voting power in the EC decision making and the consequences of two different enlargements

In this paper, we study the voting power in the Council of Ministers of the EC. The purpose of this paper is twofold: first, to evaluate the voting power for the member states of today's Community and, second, to analyse the balance of power after two different enlargements of the Community. The analysis is based on the Shapley-Shubik and the Banzhaf power indices of simple cooperative games. The modified version of the Shapley-Shubik index is used to analyse how the sub-systems of the EC can change the distribution of power in the decision making process.

Designing the allocation of emergency units by using the Shapley-Shubik power index: a case study

A game theoretic approach to allocate emergency units in a set of stations is exhibited. All of the different stations are considered as separate players in a simple game context, competing and cooperating for the distribution of the resources. The aim is to allocate the resources according to the bargaining power of the players as expressed by the Shapley-Shubik (S-S) power index. This approach is different from standard operations research resource allocation techniques that divide according to some total performance criterion. The technique presented in this paper is demonstrated by a case study concerning the distribution of fire engines among the sixty eight fire stations of the Greek fire corps. The predictability properties of the model are credibly endorsed by the quite good fit of the resulting distribution with the actual distribution of fire engines. Finally, our results are compared, firstly, with the results obtained by using the Banzhaf-Coleman (B-C) power index as an alternative to the S-S index and, secondly, with the results obtained by directly apportioning according to the characteristic function of the game.

Power indices and probabilistic voting assumptions

This paper compares the theoretical bases of the Shapley-Shubik and Banzhaf indices of voting power for a legislature with weighted voting. Definitions based on probabilistic-voting assumptions, useful both as behavioral descriptions and for computation in empirical applications, are compared in terms of necessary and sufficient conditions on the choice of voting probabilities. It is shown that the Shapley-Shubik index requires stronger conditions than the Banzhaf index: the former that voting probabilities be chosen by all players from a common uniform distribution on the unit interval, the latter only that voting probabilities be selected independently from any set of distributions (on the unit interval) which have a common mean of 1/2. This result has a bearing on the theoretical criteria by which one may choose between the two indices in a voting context.

Limit properties of power indices in a class of representative systems

We study, in a class of representative systems, the limit properties of the Shapley-Shubik and the Banzhaf indices as the number of voters goes to infinity. We mainly consider the following system: 1) The representatives elected in each electoral district proportionally represent the voters' opinion; and 2) the total number of representatives is odd and a decision is made by the elected representatives according to the simple majority rule. Principal findings include the following: 1) The voters in districts with even numbers of representatives are powerless; and 2) the power is distributed among the voters in districts with odd representatives in the following manner: For each of such districts, its aggregate power is given according to an (l-1)-dimensional (l is the number of districts with odd representatives) normal distribution with mean vector 0 and covariance matrix determined by the proportions of the population in such districts in terms of the Shapley-Shubik index; and is proportional to the square root of its population in terms of the Banzhaf index. Extension of these results to more general representative systems are also examined.

Conflict and Limits of Power

Since Dahl's definition, every effort to measure power has aimed simultaneously to conceptualize and to substantiate the meaning of power. However, there is a substantial lack of rigorous and concrete models of power as a basis to discuss various characteristics of power in international politics. In this model, the “limit of power” is defined and measured. Axelrod's model of a “bully” suggests a prototype of power seeker, but Rapoport and Chammah's empirical data on intrinsic defection probability play the critical role in making it measurable in Dahl's sense. The distribution of power as related to Axelrod's conflict of interest is neither directly nor inversely proportional to C.I.. Three phases of C.I.—mild, moderate, and severe—are shown to characterize three important types of international political and political economy systems: alliance, interdependence, and bipolarity, respectively. Other important characteristics of power are formulated, as the rate of collective goods consumed, credibility, the acquisitive and punitive efficacies of power, and characteristics of the limit of power are reviewed in comparison with the Shapley-Shubik power index.

The Relationship Between Shareholding Concentration and Shareholder Voting Power in British Companies: A Study of the Application of Power Indices for Simple Games

This paper reports an analysis of the relationships between shareholding and voting power distributions in a sample of British companies. It applies two standard approaches to the measurement of power in simple games: the Shapley-Shubik and the Banzhaf power indices. The results indicate that power is more concentrated than ownership in every case. A comparison of the two indices reveals that typically the Banzhaf index gives a more concentrated power distribution. For the Shapley-Shubik index the power ratio for the largest shareholder is accurately described in terms of the size of holding and the concentration of the remainder. The corresponding Banzhaf power ratio is less dependent on these variables. There is no association between power concentration and company size.

The shapley value on some lattices of monotonic games

Abstract It is shown that the restriction of the Shapley value to the lattice of all monotonic games whose range is contained in an arbitrary set of non-negative real numbers (which contains 0) can be uniquely characterized by the axioms that were given by Dubey (in order to characterize the Shapley value on the class of monotonic simple games). We also derive formulas for the Shapley-Shubik and Banzhaf power indices in terms of the minimal winning coalitions of the game.

On Ordinal Equivalence of Power Indices in Simple Games

The ordinal equivalence of the Shapley-Shubik and Banzhaf-Coleman power indices in a simple game would be important to many applications, where consideration of “how much” more power one member has than another is irrelevant. The reason is that if the ordinal equivalence holds, then another question of which index to use goes out. To the ordinal equivalence problem, this paper gives the following answers. The ordinal equivalence holds true either in the weighted majority games as a subclass of simple games or in the five- or less-member simple games. However, it does not necessarily hold true in the six- or more-member simple games.

Values of Voting Games

The Shapley-Shubik index of political power measures the probability that a given player in a simple game is pivotal, with th2 power to turn a losing coalition into a winning coalition. Until now, this index has been applied only to games with transferable utility. This paper considers the case where utility is not transferable. A voting game is a nontransferable utility game with a simple gane structure. The solution of a voting game is its nontransferable utility value. This paper studies the values of such games. In particular, the Shapley-Shubik index is the value for a certain class of voting games. This shows that the measurement of political power does not depend on the transferability of utility.

Assessment of Political Power in the Israeli Knesset

Immediately after the election to the tenth Israeli parliament (Knesset), 21 students of political science, 24 Knesset members, and seven parliamentary correspondents were each asked (a) to assess the political power ratios of the 10 parties represented in the Knesset and (b) to judge the ideological similarity between them. As ascertained by Saaty's analytic hierarchy scaling technique, the power ratio judgments proved sufficiently consistent to justify the construction of individual ratio scales of perceived political power. The ideological proximities were adequately represented by two-dimensional ideological spaces. Analyses of the derived power measures showed that the higher the political sophistication of the subject, the higher the combined power attributed to the religious parties and the lower the combined power assigned to the two largest parties Likud and Labor. The derived power measures were then compared to the predictions of six power indices, three of which only consider the ideological space. Of the six models, the generalized Banzhaf power index best accounted for the perceived power of 62% of the subjects, whereas the classical Shapley-Shubik index provided the best fit for 31% of the subjects. The generalized power indices were found only partly satisfactory with a need for further revision.

The structure of voting-power indices

The difference between the Shapley-Shubik index and the Banzhaf index refers to one assumption, the assignment of a priori probabilities to how voting units choose between “aye” and “no” in voting sequences. This behavior assumption is specified differently in the two indices, which accounts for the differences between their numerical values. Moreover, it has been shown that other probability assignments may be made corresponding to different assumptions about voting behavior—indeed, there are as many power indices as there are possible probability assignments. This result opens up the possibility of employing the idea of a power index to study real voting power and not simply formal voting power. The probability of a voter choosing a certain type of voting behavior reflects his or her capacity to enter coalitions in various voting situations. These probabilities may be estimated empirically, which means that a priori assumptions about equiprobability may be replaced by a posteriori probability assignments. These empirically derived probabilities would reflect the constraints on voting behavior (deriving from party affiliation, ideology and coalition norms) typical of real-life social choice.

The Predictive Power of Voting Power Indices: FASB Voting on Statements of Financial Accounting Standards Nos. 45-69

Previous papers on the voting activities of the FASB (Newman [1981a; 1981b] and Selto and Grove [1982] have examined whether some members or coalitions of members of the FASB wielded voting power in excess of their votes. Each of these studies was conducted without the benefit of a general theory of voting on the FASB-a theory which would predict (explain) members' voting behavior(s). Instead, each provided descriptive evidence on the relative ability of different voting indices to ascertain the existence of voting coalitions from prior voting behavior. To illustrate, Newman [1981] suggested that such voting power indices as the Shapley-Shubik or the Banzhaf indices could describe how former Big Eight members of the FASB appeared to control the outcomes of the FASB. Selto and Grove (SG) [1982] extended Newman's work to confirm that while former Big Eight members apparently did not control the FASB on SFAS Nos. 26-44, three other coalitions seemed likely. SG tested the descriptive power of the Shapley-Shubik and the Banzhaf indices for SFAS Nos. 26-44 for these coalitions. For two of the three the best descriptive index was the Shapley-Shubik index. However, the best descriptor for one of the coalitions was the nominal voting power or simply the proportion of votes held. Since we have no theory to support these results, we attempted to

The Banzhaf power index for political games

The Banzhaf index of a voting game is a measure of a priori power of the voters. The model on which the index is based treats the voters symmetrically, i.e. the ideology, outlook, etc., of the voters influencing their voting behavior is ignored. Here we present a nonsymmetric generalization of the Banzhaf index in which the ideology of the voters affecting their voting behavior is taken into account. A model of ideologies and issues is presented. The conditions under which our model gives the Shapley-Shubik index (another index of a priori power of the voters) are given. Finally several examples are presented and some qualitative results are given for straight majority and pure bargaining games.