Izabella Stach

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.

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.

Some Open Problems In Simple Games

This paper presents a review of literature on simple games and highlights various open problems concerning such games; in particular, weighted games and power indices.

Sub-Coalitional Approach to Values

The behavioral models of classical values (like the Shapley and Banzhaf values) consider the contributions to coalition S as contributions delivered by the players individually joining such a coalition as it is being formed; i.e., v(S) – v(S \ {i}). In this paper, we propose another approach to values where these contributions are considered as given by sets of players: (v(S) – v(S \ R)), where S, R are subsets of the set of all players involved in cooperative game v. Based on this new approach, several sub-coalitional values are proposed, and some properties of these values are shown.

Power Indices for Finance

The weight of the share stock of a company may be described by power indices that quantify the possibility for each shareholder to get majority positions by coalitions with other shareholders. To study such indices allows us to build efficient models for forecasting, simulating, and regulating financial, political, and economic fields. An overview of financial applications of power indices is presented; this was carried out at the University of Bergamo along with partners in Europe and United States. New explanations and examples are added so as to better illustrate the results obtained. Additionally, certain open problems are described.

Power Measures and Public Goods

In this paper, we analyze some power indices that are well-defined in the social context where goods are public. We consider the following indices: Public Help index i¾? [1], Public Help index i¾? [2], the Konig and Brauninger index [3, 4], the Nevison index [5], and the Rae index [6]. This paper continues the earlier work on public good indices see [2]. The aim of this paper is to compare several power indices, taking into account the various properties, rankings amongst players, and ranges of the power indices.

Guest editorial (to Special Issue on Game Theory and Applications)

This special issue of Decision Making in Manufacturing and Services is devoted to Game Theory and Applications and related topics. The origin of the issue is the 10th Spain-Italy-Netherlands Meeting on Game Theory (SING10), which took place on 7th-9th July, 2014. The conference was hosted by the Faculty of Management at AGH University of Science and Technology in Krakow, Poland (main organizer was Izabella Stach). The history of the SING meetings started at the beginning of the 1980s with the first meetings held in Italy. Then, subsequently, meetings were added in Spain, the Netherlands and Poland. Nowadays SING is one of the most important international meetings on game theory organized each year in a European country. The SING10 meeting in 2014 attracted more than one hundred and ninety scientist from five continents. More about the SING meetings and in particular about SING10 can be funded in Gambarelli (2011) and Bertini et al. (2014). The submitted papers (139 presentations, 135 in parallel sessions and 4 in plenary sessions) covered a variety of topics on game theory and its applications. This special issue collects some surveys on recent results in different fields, presented in the conference.