Date of Submission


Date of Award


Institute Name (Publisher)

Indian Statistical Institute

Document Type

Doctoral Thesis

Degree Name

Doctor of Philosophy

Subject Name

Quantitative Economics


Economic Research Unit (ERU-Kolkata)


Chakravarty, Satya Ranjan (ERU-Kolkata; ISI)

Abstract (Summary of the Work)

MotivationThe issue of measurement of voting power is a very important topic of discussion in social science these days. The concept of voting power concerns any collective decision making body (or, equivalently, a collectivity) which makes ‘yes’ or ‘no’ decisions on any issue, by the process of voting. Examples of such bodies abound in today’s world. The United Nations Security Council, The Council of Ministers in the European Union, the Parliament of the republic of India, the board room of any corporate house etc., are all examples of such decision making bodies.The voting process of each of these bodies is governed by its own constitution, which lays down the decision making rule for the collectivity. This decision rule in turn aggregates individual votes to determine the decision of the voting body as whole. Typically, when a proposal suggesting a certain course of action is presented before such a body, its members are asked to vote either for the bill (‘yes’) or against it (‘no’). The decision rule then transforms these individual votes into a collective decision of the voting body. As an example, consider a board of directors of a company consisting of five members. Let the decision rule, as laid down by the constitution of the board be ‘simple majority’, i.e., at least three members of the board have to vote ‘yes’ in order that the board collectively passes the bill. So in a situation in which only two members of the board vote ‘yes’ and the remaining three vote ‘no’, the decision rule spells out that the bill is rejected and the course of action as suggested by the bill cannot be taken by the board (in spite of two members wanting it).In this framework, by individual voting power we mean an individual voter’s ability to change the outcome of the voting procedure by changing his stand on the bill. It is a rough measure of the extent of control that an individual voter has over the collective action of the voting body. As the ability of a voter to influence the outcome of the voting process by changing his vote is determined by the decision rule, it can be said that the decision rule determines how the formal control over the actions of the collectivity is shared among its members. Often there arises the need to assess decision rules for its capacity in ensuring fairness in sharing of control over the collectivity’s action among the members (according to some given definition of fairness, which might seem relevant in the given context). For this purpose, the use of some kind of a measure of individual power becomes imperative.To understand this point better let us consider a real life example, which is the topic of much research these days, the European Union (EU). Of all the decision making bodies of the European Union, the Council of Ministers is by far the most important. The direct voters in the council are themselves representatives of the electorate of the respective EU states. Thus the electorate of the individual EU states exercise indirect influence over the council’s decisions. If the accepted notion of fairness is that of equitability (i.e., one person one vote), then the indirect influence of electorate in various constituent countries ought to be equal, irrespective of the difference in their population size. In other words, a citizen of Germany ought, in principle, to have just as much influence over a decision of the council as a citizen of (say) Greece1 . Thus in order to evaluate whether the decision rule of the council is equitable in this sense, we need to first quantify this amount of influence (see Felsenthal and Machover (2000)).There could be many other reasons why the evaluation of a decision rule is necessary. Consider a voting body, which requires unanimity among all the members to pass a resolution, i.e., every member has to vote for the resolution (‘yes’) in order that the voting body passes it. Then it obvious that here, the power of the decision making body to act is very small. In fact, even in a situation where only one member votes ‘no’, and all the remaining members vote for the bill, the body cannot translate the wishes of the majority of the members into actual collective action. Thus it might sometimes be important to evaluate the degree to which the decision making body as a whole, is empowered as a decision maker. Here it is obvious that what concerns us is not the individual voting power but collective voting power. Hence the need for an index that gives us a quantification of the extent to which the body is able to control the outcome of a division of it.Having thus stated the need for a quantitative measure of both individual and collective voting power, we proceed to the remaining part of the chapter. In different subsections of section 1.2, we discuss in details the issue of individual voting power. In section 1.2.1, we introduce some preliminary definitions and in section 1.2.2, we formally define what we mean by an index of individual voting power, and discuss some well-known indices. Then in section 1.2.3, some postulates which an index of individual power are expected to satisfy (following Felsenthal and Machover (1995, 1998)) and the associated paradoxes are presented. In section 1.2.4, we discuss some characterizations of the well-known indices of power and in section 1.2.5 some alternative approaches for measuring voting power are introduced. Section 1.2.6 deals with voting power when voters have more than two alternatives to choose from. Section 1.3 presents some characteristics of the voting body as a whole and finally in section 1.4 we list some applications where these indices have been used.


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