Gianfranco Gambarelli

Indirect control of corporations

We consider situations in which shares ofn firms are held by the samen firms as well as by individual investors. The probelm is to determine the actual control of the firms by the individual investors.In part 1, we develop the mathematical model, and explain the basic notions of clutter and of effective reduction. In part 2 we introduce the idea of a consistent reduction, prove existence of such, and show the relation between consistent and effective reductions. In part 3 we introduce multilinear extensions and show how these can be used to calculate the effective and consistent reductions. Several examples are worked out in detail.

Common behaviour of power indices

The variations ensuing in a weighted majority game are studied when a player increases his weight in prejudice of others or decreases in favor, or trades shares outside the game (in particular when an-person game becomes an (n+1)-person one). An invariant behaviour for different game values is found for all these cases. Possible applications to politics, shareholdings and large games are pointed out.

Power indices for political and financial decision making: A review

The most important power indices are presented. The effectiveness of these indicators is discussed with reference to description of political and financial events. Some recent studies and applications are shown.

COMPARING POWER INDICES

This paper aims to give a global vision concerning the state of the art of studies on 13 power indices and to establish which of them are more suitable for describing the real situations which are, from time to time, taken into consideration. In such contexts, different comparisons have been developed in terms of properties, axiomatic grounds and so on. This analysis points out various open problems.

Minimax Multi-District Apportionments

The problem of seat apportionment in electoral systems turns out to be quite complex, since no apportionment method exists which succeeds in verifying all the principal fairness criteria. Gambarelli (1999) introduced an apportionment technique which is custom made for each case, respects Hare minimum, Hare maximum and Monotonicity and satisfies other criteria in order of preference. In this chapter a generalization of that method is proposed, in order to extend it to the multi-district election case, where criteria should be respected at a global as well as at a local level. An existence theorem and a generating algorithm are supplied.

A Public Help Index

As far as we known, the first concept of power index dates back to 1780s and is due to Luther Martin (see Felsenthal and Machover (2005), Gambarelli and Owen (2004) and Riker (1986)). Lionel S. Penrose (1946) gave, probably, the first scientific discussion of voting power where he introduced the concept of a priori voting power (a similar analysis was independently carried out by John F. Banzhaf (1965)). Lloyd S. Shapley, in cooperation with Martin Shubik (Shapley and Shubik 1954), came up with a specialization of the Shapley (1953) value as a power index. Other power indices were introduced later; some derived from existing values, others built exclusively for simple games. The Public Good Index introduced by Manfred Holler in (1978) belongs to the latter category.

The “coherent majority average” for juries’ evaluation processes

Abstract I propose a method to synthesize the performance scores for artistic sports such as rhythmic gymnastics, figure skating, synchronized swimming, and diving by taking into account inter-judge variability, while maintaining all the reliable scores. This procedure is based on the assumption that the majority of the scores in each event are reliable and they relate well to those scores that are closest to them. The method consists of putting scores in order and considering clusters of m consecutive scores, where m is the number of judges making up the simple majority. For each cluster, the difference between the highest and the lowest score is calculated. In cases where the minimum difference is positive, the arithmetic mean of those scores that belong to clusters where the difference is minimal is computed. In cases where the minimum difference is zero (i.e. if the majority of judges unanimously assign the same score), then the set of the scores to consider within the mean is extended to those scores that are very near to those of the majority of the judges. A comparison between the actual evaluation procedures and the proposed model is provided.

Political and Financial Applications of the Power Indices

Some theoretical and algorithmical studies on common properties of the power indices, with applications to Finance and Politics, are presented together with some ideas for further developments.

Discontinuous Solutions in n-Person Games

Publisher Summary This chapter discusses the discontinuous solutions in n -person games. One of the basic concepts of the theory of n -person games is the so-called Shapley value, which allows the power of each player to be defined, taking into account the payoff of all possible coalitions that can be joined by the player. The Shapley value has a number of practical applications, for instance in voting schemes, ranging from the case of political parties to that of shareholders of companies. The drawback of the original formulation of the Shapley value is the complexity of its computation which is based on set-theoretical operations; thus, only in certain cases, suitable numerical algorithms have been developed. Gambarelli was able to produce a simple geometric rule and a corresponding algorithm for computation of the Shapley value for all possible games. The application of this formula to a variety of problems led to some interesting results, especially if the actual buying and selling operations performed by a player, which induce a set of discontinuities in the Shapley value, are included in the game to increase its value. Common to all these new solutions is the concept of the set of minimal winning coalitions, that is, winning coalitions that become losing coalitions when any one of the participants abandons it.

Multicameral voting cohesion games

This work links two research branches regarding cooperative games: multi-cameral games and a priori coalitions. Regarding the former, many parliaments are based on a bicameral system where bills have to be approved by both chambers. The parties’ power indices are affected by it. In fact one certain political party for one certain coalition can be crucial in one chamber and not in the other. Usually the problem can be solved by building a unified game, related to two or more chambers, where the global power indices can be calculated. Concerning cohesion games, some coalitions among parties might be possible, but have a low probability of materializing if the parties involved have very different ideologies. It is necessary to take this into consideration when we compute the relative power indices. In this work, a model is built that takes into consideration both topics. An automatic calculation algorithm is elaborated and is applied to the current situations in Belgium, the Czech Republic, France, Italy, the Netherlands, Poland and Romania, considering the location of the parties on the left-right axis. This model is then applied globally to the European Union. In this work the model is applied to Politics, but its applications are not limited to that field. For instance, in Finance it can be applied to describe agreements for debt restructuring and corporate mergers.