Constrained Monotonicity and the Measurement of Power (original) (raw)

On the Present and Future of Power Measures

Homo Oeconomicus, 2002

Power indices are commonly required to assign at least as much power to a player endowed with some given voting weight as to any player of the same game with smaller weight. This local monotonicity and a related global property however are frequently and for good reasons violated when indices take account of a priori unions amongst subsets of players (reflecting, e.g., ideological proximity). This paper introduces adaptations of the conventional monotonicity notions that are suitable for voting games with an exogenous coalition structure. A taxonomy of old and new monotonicity concepts is provided, and different coalitional versions of the Banzhaf and Shapley-Shubik power indices are compared accordingly.

Monotonicity of power and power measures

Theory and Decision, 2004

Monotonicity is commonly considered an essential requirement for power measures; violation of local monotonicity or related postulates supposedly disqualifies an index as a valid yardstick for measuring power. This paper questions if such claims are really warranted. In the light of features of real-world collective decision making such as coalition formation processes, ideological affinities, a priori unions, and strategic interaction, standard notions of monotonicity are too narrowly defined. A power measure should be able to indicate that power is non-monotonic in a given dimension of players' resources if -given a decision environment and plausible assumptions about behaviour -it is non-monotonic.

The Impossibility of a Preference-Based Power Index

Journal of Theoretical Politics, 2005

This paper examines a recent debate in the literature on power indices in which classical measures such as the Banzhaf, Shapley-Shubik, and Public Good indices have been criticized on the grounds that they do not take into account player preferences. It has been argued that an index that is blind to preferences misses a vital component of power, namely strategic interaction. In this vein, there has been an attempt to develop so-called strategic power indices on the basis of non-cooperative game theory. We argue that the criticism is unfounded and that a preference-based power index is incompatible with the definition of power as a generic ability: 'the ability to affect outcomes'. We claim that power resides in, and only in, a game form and not in a game itself. KEY WORDS . ability . game theory . power indices . strategic power Journal of Theoretical Politics 17(1): 137-157

Power Indices in Politics: Some Results and Open Problems

2009

We present an overview of the political applications of power indices, carried out at the University of Bergamo with partners in Europe and United States. New designs, explanations and examples are added so as to better illustrate the results obtained. Additionally, certain open problems are described.

FREEDOM OF CHOICE AND WEIGHTED MONOTONICITY OF POWER

Metroeconomica, 2009

The paper discusses the substitution of mergeability by the weighted monotonicity property in the definition of the Public Good Index. The cardinality of sets, implicit to the measurement of power, can thus be related to comparing sets and relations of counting and set inclusion. This allows for the application of results from the measurement of power to the specification of freedom of choice and thereby to connect the 'world of agents' with the 'world of opportunities'. The relationship between weighted monotonicity and constrained monotonicity is specified.

Monotonicity of power in games with a priori unions

Theory and Decision, 2009

The Bicameral Postulates and Indices of a Priori Voting Power

1998

is an index of relative voting power for simple voting games, the bicameral postulate requires that the distribution of K-power within a voting assembly, as measured by the ratios of the powers of the voters, be independent of whether the assembly is viewed as a separate legislature or as one chamber of a bicameral system, provided that there are no voters common to both chambers. We argue that a reasonable index-if it is to be used as a tool for analysing abstract, 'uninhabited' decision rules-should satisfy this postulate. We show that, among known indices, only the Banzhaf measure does so. Moreover, the Shapley-Shubik, Deegan-Packel and Johnston indices sometimes witness a reversal under these circumstances, with voter x 'less powerful' than y when measured in the simple voting game G 1 , but 'more powerful' than y when G 1 is 'bicamerally joined' with a second chamber G 2. Thus these three indices violate a weaker, and correspondingly more compelling, form of the bicameral postulate. It is also shown that these indices are not always co-monotonic with the Banzhaf index and that as a result they infringe another intuitively plausible condition-the price monotonicity condition. We discuss implications of these findings, in light of recent work showing that only the Shapley-Shubik index, among known measures, satisfies another compelling principle known as the bloc postulate. We also propose a distinction between two separate aspects of voting power: power as share in a fixed purse (P-power) and power as influence (I-power). KEY WORDS: Banzhaf, Deegan-Packel, index of voting power, Johnston, paradoxes of voting power, Penrose, postulates for index of voting power, Shapley value, Shapley-Shubik, simple voting game, weighted voting game.

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. We used to calculate them by looking at marginal contributions in a simple game consisting of winning and losing coalitions derived from the rules of the legislation. We introduce a new way to calculate these measures directly from the set of minimal winning coalitions. 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 calculations. The technique generalises directly to all semivalues.

Local Monotonicity of Voting Power: A Conceptual Analysis

2002

This paper examines a fundamental and on-going debate in the literature on voting power about what constitutes a reasonable measure of a priori voting power. We focus on one of the central axioms or postulates known as local montonicity which says that voting power should be ranked in the same order as the order of voting weights. By examining a general violation of local montonicity under Straffins partial homogeneity approach we show that this postulate lacks convincing justification. However, and somewhat paradoxically, we argue that the previous arguments against local montonicity are flawed, and the intuition behind the postulate is essentially correct. The problem lies with the definition of a prioricity and the nature of the voting game.

On Public Values and Power Indices

Decision Making in Manufacturing and Services, 2015

In this paper, we analyze some values and power indices from a different point of view that are well-defined in the social context where the goods are public. In particular, we consider the Public Good index (Holler, 1982), the Public Good value (Holler and Li, 1995), the Public Help index (Bertini et al., 2008), the König and Bräuninger index (1998) also called the Zipke index (Nevison et al., 1978), and the Rae index (1969). The aims of this paper are: to propose an extension of the Public Help index to cooperative games; to introduce a new power index with its extension to a game value; and to provide some characterizations of the new index and values.

Power Indices Taking into Account Agents’ Preferences

Studies in Choice and Welfare, 2006

A set of new power indices is introduced extending Banzhaf power index and taking into account agents? preferences to coalesce. An axiomatic characterization of intensity functions representing a desire of agents to coalesce is given. A set of axioms for new power indices is presented and discussed. An example of use of these indices for Russian parliament is given.

The Voting Power Approach: A Theory of Measurement

European Union Politics, 2003

Max Albert has argued that the theory of power indices “should not ... be considered as part of political science” and that “[v]iewed as a scientific theory, it is a branch of probability theory and can safely be ignored by political scientists”. Albert’s argument rests on a particular claim concerning the theoretical status of power indices, namely that the theory of power indices is not a positive theory, i.e., not one that has falsifiable implications. I re-examine the theoretical status of power indices and argue that it would be unwise for political scientists to ignore such indices. Although I agree with Albert that the theory of power indices is not a positive theory, I suggest that it is a theory of measurement that can usefully supplement other positive and normative social- scientific theories. Preprint: http://personal.lse.ac.uk/LIST/PDF-files/List-EUPolitics.pdf

On the performance of the Shapley Shubik and Banzhaf power indices for the allocations of mandates

THEMA Working Papers, 2007

A classical problem in the power index literature is to design a voting mechanism such as the distribution of power of the different players is equal (or closer) to a pre established target. This tradition is especially popular when considering two tiers voting mechanisms: each player votes in his own jurisdiction to designate a delegate for the upper tier; and the question is to assign a certain number of mandates for each delegate according the population of the jurisdiction he or she represents. Unfortunately, there exist several measures of power, which in turn imply different distributions of the mandates for the same pre established target. The purposes of this paper are twofold: first, we calculate the probability that the two most important power indices, the Banzhaf index and the Shapley-Shubik index, lead to the same voting rule when the target is the same. Secondly, we determine which index on average comes closer to the pre established target.

Compound Voting and the Banzhaf Power Index

RePEc: Research Papers in Economics, 2003

We present three axioms for a power index defined on the domain of simple (voting) games. Positivity requires that no voter has negative power, and at least one has positive power. Transfer requires that, when winning coalitions are enhanced in a game, the change in voting power depends only on the change in the game, i.e., on the set of new winning coalitions. The most crucial axiom is composition: the value of a player in a compound voting game is the product of his power in the relevant first-tier game and the power of his delegate in the second-tier game. We prove that these three axioms categorically determine the Banzhaf index. JEL Classification Numbers: C71, D72.

Measuring voting power: The paradox of new members vs. the null player axiom

2009

Qualified majority voting is used when decisions are made by voters of different sizes. In such situations the voters' influence on decision making is far from obvious; power measures are used for an indication of the decision making ability. Several power measures have been proposed and characterised by simple axioms to help the choice between them. Unfortunately the power measures also feature a number of so-called paradoxes of voting power.