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. Keywords. Shapley-Shubik index, Banzhaf index, semivalue, minimal winning coalition, Möbius transform.Shapley-Shubik index, Banzhaf index, semivalue, minimal winning coalition, Möbius transform.

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

Strategic power indices: Quarrelling in coalitions

While they use the language of game theory known measures of a priory voting power are hardly more than statistical expectations assuming voters behave randomly. Focusing on normalised indices we show that rational players would behave differently from the indices predictions and propose a model that captures such strategic behaviour.Banzhaf index, Shapley-Shubik index, a priori voting power, rational players

Strategic power indices: Quarrelling in coalitions

While they use the language of game theory known measures of a priory voting power are hardly more than statistical expectations assuming voters behave randomly. Focusing on normalised indices we show that rational players would behave differently from the in-dices predictions and propose a model that captures such strategic behaviour.Banzhaf index, Shapley-Shubik index, a priori voting power, rational players

A Relation-algebraic Approach to Simple Games

Simple games are a powerful tool to analyze decision-making and coalition formation in social and political life. In this paper, we present relation-algebraic models of simple games and develop relational algorithms for solving some basic problems of them. In particular, we test certain fundamental properties of simple games (being monotone, proper, respectively strong) and compute specific players (dummies, dictators, vetoers, null players) and coalitions (minimal winning coalitions and vulnerable winning coalitions). We also apply relation-algebra to determine central and dominant players, swingers and power indices (the Banzhaf, Holler-Packel and Deegan-Packel indices). This leads to relation-algebraic specifications, which can be executed with the help of the BDD-based tool RelView after a simple translation into the tool's programming language. In order to demonstrate the visualization facilities of RelView we consider an example of the Catalonian Parliament after the 2003 election.relation algebra; RelView; simple game; winning coalition; swinger; dominant player; central player; power index

Connected Coalition Formation and Voting Power in the Council of the European Union: An Endogenous Policy Approach. EIPA Working Paper 99/W/05

Resorting to political economy approaches, this paper attempts to associate the industrial structure in the European Union (EU) to the coalition formation process between European member states. Using a well-known measure of relative voting power, the (normalized) Banzhaf power index, we relax the common assumption that coalitions form randomly. Instead, we adopt the standard interest group model and look at the structure of European industry, mainly in terms of industrial concentration in the EU, as an indicator of its lobbying influence on domestic politics and governments’ preferences. This, in turn, influences the political stance, and thus the coalition building process, of the different member states in the Council. We derive estimates on members’ relative influence within the Council for different policy areas in the broader framework of industry and trade, on the basis of both weighted votes and likely patterns of coalition-formation in the Council

Proximity and linkages among coalition participants: a new voting power measure applied to the International Monetary Fund

Voting power methodology offers insights to understand coalition building in collective decision making. Using cooperative game theory, Banzhaf (1965) developed an index to capture the numerical importance of voters in coalition building. This voting power index is still widely used today in applications to international politics. Yet, it assumes that voters are symmetric and focuses on particular voters only. This paper proposes a new measure of voting power which account for the numerical proximity between voters by capturing how often they appear in winning coalitions together. The index is also developed to account for the relative importance of coalitions and the relative linkages among coalition participants. We present an application to the governance structure of the International Monetary Fund, with linkages being represented by bilateral trade between voters. The results are able to explain several important features of the functioning of this particular voting body, and may be useful for other applications of international politics. JEL Classification: C71, F33coalition building, International Monetary Fund, voting power

A relation-algebraic approach to simple games

Simple games are a powerful tool to analyze decision - making and coalition formation in social and political life. In this paper, we present relation-algebraic models of simple games and develop relational specifications for solving some basic problems of them. In particular, we test certain fundamental properties of simple games and compute specific players and coalitions. We also apply relation algebra to determine power indices. This leads to relation-algebraic specifications, which can be evaluated with the help of the BDD-based tool RelView after a simple translation into the tool's programming language. In order to demonstrate the visualization facilities of RelView, we consider an example of the Catalonian Parliament after the 2003 election.Relation algebra ; RelView ; simple game ; winning coalition ; swinger ; dominant player ; central player ; power index