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.     

The Banzhaf power index for ternary bicooperative games

In this paper we analyze ternary bicooperative games, which are a refinement of the concept of a ternary voting game introduced by Felsenthal and Machover. Furthermore, majority voting rules based on the difference of votes are simple bicooperative games. First, we define the concepts of the defender and detractor swings for a player. Next, we introduce the Banzhaf power index and the normalized Banzhaf power index. The main result of the paper is an axiomatization of the Banzhaf power index for the class of ternary bicooperative games. Moreover, we study ternary bicooperative games with two lists of weights and compute the Banzhaf power index using generating functions.     

Consistency and the graph Banzhaf value for communication graph games

The graph Banzhaf value was introduced and axiomatically characterized by Alonso-Meijide and Fiestras-Janeiro (2006). In this paper we propose the reduced game and consistency of the graph Banzhaf value for communication situations. By establishing the relationship between the Harsanyi dividends of a coalition in a communication situation and the reduced communication situation, we provide a new axiomatization of the graph Banzhaf value by means of the axioms of consistency and standardness.     

An axiomatization of the Banzhaf value for cooperative games on antimatroids

Cooperative games on antimatroids are cooperative games in which coalition formation is restricted by a combinatorial structure which generalizes permission structures. These games group several well-known families of games which have important applications in economics and politics. The current paper establishes axioms that determine the restricted Banzhaf value for cooperative games on antimatroids. The set of given axioms generalizes the axiomatizations given for the Banzhaf permission values. We also give an axomatization of the restricted Banzhaf value for the smaller class of poset antimatroids. Finally, we apply the above results to auction situations.     

Minimum norm solutions for cooperative games

A solution f for cooperative games is a minimum norm solution, if the space of games has a norm such that f(v) minimizes the distance (induced by the norm) between the game v and the set of additive games. We show that each linear solution having the inessential game property is a minimum norm solution. Conversely, if the space of games has a norm, then the minimum norm solution w.r.t. this norm is linear and has the inessential game property. Both claims remain valid also if solutions are required to be efficient. A minimum norm solution, the least square solution, is given an axiomatic characterization.      

模糊合作博弈局中人参与水平间相互作用度量

本文基于经典合作博弈局中人间相互作用现象的有关度量方法,针对具有模糊联盟的合作博弈问题,给出了模糊联盟中各局中人参与水平间相互作用的度量方法,定义了水平间相互独立性概念,建立了反映局中人各参与水平间相互作用平均程度的两指标:Shapley相互作用指标及Banzhaf相互作用指标.同时对于具有k-单调性的模糊合作博弈局中人参与水平间的边缘相互作用的有关性质作了进一步研究,得出了一些新的结论.     

An axiomatic approach to the concept of interaction among players in cooperative games

An axiomatization of the interaction between the players of any coalition is given. It is based on three axioms: linearity, dummy and symmetry. These interaction indices extend the Banzhaf and Shapley values when using in addition two equivalent recursive axioms. Lastly, we give an expression of the Banzhaf and Shapley interaction indices in terms of pseudo-Boolean functions. Received: October 1997/revised version: October 1998     

The distribution of power in the European Constitution

The aim of this paper is to analyze the distribution of voting power in the Constitution for the enlarged European Union. By using generating functions, we calculate the Banzhaf power indices for the European countries in the Council of Ministers under the decision rules prescribed by the Treaty of Nice and the new rules proposed by the European Constitution Treaty. Moreover, we analyze the power of the European citizens under the egalitarian model proposed by Felsenthal and Machover [D.S. Felsenthal, M. Machover, The measurement of voting power: Theory and practice, problems and paradoxes, Edward Elgar, Cheltenham, 1998].     

Inferior players in simple games

Power indices like those of Shapley and Shubik (1954) or Banzhaf (1965) measure the distribution of power in simple games. This paper points at a deficiency shared by all established indices: players who are inferior in the sense of having to accept (almost) no share of the spoils in return for being part of a winning coalition are assigned substantial amounts of power. A strengthened version of the dummy axiom based on a formalized notion of inferior players is a possible remedy. The axiom is illustrated first in a deterministic and then a probabilistic setting. With three axioms from the Banzhaf index, it uniquely characterizes the Strict Power Index (SPI). The SPI is shown to be a special instance of a more general family of power indices based on the inferior player axiom. Received: December 1999/Final version: June 2001     

Voting Power in the Governance of the International Monetary Fund

In general in an organisation whose system of governance involves weighted voting, a member's weight in terms of the number of votes and the formal power it represents differ. Power indices provide a means of analysing this difference. The paper uses new algorithms for computing power indices for large games. Three analyses are carried out: (1) the distribution of Banzhaf voting power among members in 1999; the results show that the United States has considerably more power over ordinary decisions than its weight of 17% but that the use of special supermajorities limits its power; (2) the effect of varying the majority requirement on the power of the IMF to act and the powers of members to prevent and initiate action (Coleman indices); the results show the effect of supermajorities severely limits the power to act and therefore renders the voting system ineffective in democratic terms, also the sovereignty of the United States within the IMF is effectively limited to just the power of veto; (3) the paper proposes the determination of the weights instrumentally by means of an iterative algorithm to give the required power distribution; this would be a useful procedure for determining appropriate changes in weights consequent on changes to individual countries' quotas; this is applied to the 1999 data. Policy recommendations are, first, that the IMF use only simple majority voting, and discontinue using special supermajorities, and, second, allocate voting weight instrumentally using power indices.     

Automorphisms and blocking in simple games

Different (proper as well as improper) simple games may share any nonseparating blocking family. An interesting question is whether all such proper games on one hand, and all improper ones on the other, are isomorphic. The answer is given here and the automorphisms of the blocking family help us to understand the structure and links of all such games.     

The Shapley–Shubik index for simple games with multiple alternatives

When analyzing mathematically decision mechanisms ruled by voting it is sometimes convenient to include abstention as a possible alternative for the voters. In classical simple games, abstention, if considered, is formally equivalent to voting against the proposal. Simple games with alternatives are useful to study voting systems where abstention does not favour any of the options. In this work, we axiomatically characterize the Shapley–Shubik index for simple games with alternatives and apply it to an example taken from real life. This work has been partially supported by Grant MTM 2006–06064 of the Education and Science Spanish Ministry and the European Regional Development Fund, and Grant SGR 2005–00651 of the Catalonia Government.     

Effectivity functions and simple games

The rather new notion of effectivity function is related to the notion of simple game. Every effectivity function is associated with a simple game. So theory about simple games may be applicable to effectivity functions. E.g. if the effectivity function is additive, then the associated simple game is weighted. Via a characterization of weighted simple games it is possible to characterize maximal additive effectivity functions.Finally we characterize additive effectivity functions and their associated simple games.I thank G.J. Otten for useful comments and stimulating conversations.     

Growth of dimension in complete simple games

Using a novel technique of Kurz and Napel (2015), we demonstrate that the worst-case growth of dimension in complete simple games is exponential in the number of players. We also prove that roughly-weighted games of arbitrary dimension exist.     

On connected coalitions in dominated simple games

In a book by Axelrod it is claimed that, in the presence of well defined policy order, only connected coalitions form. Here we investigate the compatability of Axelrod's hypothesis with several hypotheses (about coalition formation in dominated simple game) that were formulated by Peleg.     

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 Rel View after a simple translation into the tool’s programming language. In order to demonstrate the visualization facilities of Rel View we consider an example of the Catalonian Parliament after the 2003 election.     

Weighted and roughly weighted simple games

In this paper we give necessary and sufficient conditions for a simple game to have rough weights. We define two functions f(n) and g(n) that measure the deviation of a simple game from a weighted majority game and roughly weighted majority game, respectively. We formulate known results in terms of lower and upper bounds for these functions and improve those bounds. We also investigate rough weightedness of simple games with a small number of players.     

The desirability relation of simple games

We prove some properties of simple games with a complete desirability relation, and investigate the relationships between the desirability of a simple game G and that of some simple games that are derived from G. We also provide an example of a proper simple game that has a complete and acyclic desirability relation but is not a weighted majority game.     

The Banzhaf index in complete and incomplete shareholding structures: A new algorithm

In this global world, many firms present a complex shareholding structure with indirect participation, such that it may become difficult to assess a firm’s controllers. Furthermore, if there are numerous dominant shareholders, the control can be shared between them. Determining who has the most influence is often a difficult task. To measure this influence, game theory allows the modeling of voting games and the computing of the Banzhaf index. This paper firstly offers a new algorithm to compute this index in all structures and then suggests some modelisations of the floating shareholder. Then, our model is applied to a real case study: the French group Lafarge. This exemplary case demonstrates how the float’s structure and hidden coalition can impact the power relationship between dominant shareholders.     

The complexity of recursive constraint satisfaction problems

We investigate the complexity of finding solutions to infinite recursive constraint satisfaction problems. We show that, in general, the problem of finding a solution to an infinite recursive constraint satisfaction problem is equivalent to the problem of finding an infinite path through a recursive tree. We also identify natural classes of infinite recursive constraint satisfaction problems where the problem of finding a solution to the infinite recursive constraint satisfaction problem is equivalent to the problem of finding an infinite path through finitely branching recursive trees or recursive binary trees. There are a large number of results in the literature on the complexity of the problem of finding an infinite path through a recursive tree. Our main result allows us to automatically transfer such results to give equivalent results about the complexity of the problem of finding a solution to a recursive constraint satisfaction problem.