Cesarino Bertini

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.

Discrete time portfolio selection with Lévy processes

This paper analyzes discrete time portfolio selection models with Levy processes. We first implement portfolio models under the hypotheses the vector of log-returns follow or a multivariate Variance Gamma model or a Multivariate Normal Inverse Gaussian model or a Brownian Motion. In particular, we propose an ex-ante and an ex-post empirical comparisons by the point of view of different investors. Thus, we compare portfolio strategies considering different term structure scenarios and different distributional assumptions when unlimited short sales are allowed.

QUANTUM COMPUTATIONAL FINITE-VALUED LOGICS

The theory of logical gates in quantum computation has suggested new forms of quantum logic, called quantum computational logics. The basic semantic idea is the following: the meaning of a sentence is identified with a quantum information quantity, represented by a quregister (a system of qudits) or, more generally, by a mixture of quregisters (called qumix), whose dimension depends on the logical complexity of the sentence. At the same time, the logical connectives are interpreted as logical operations defined in terms of quantum logical gates. Physical models of quantum computational logics can be built by means of Mach-Zehnder interferometers.

Logics from quantum computation with bounded additive operators

The theory of gates in quantum computation has suggested new forms of quantum logic, called quantum computational logics, where the meaning of a sentence is identified with a system of qubits in a pure or, more generally, mixed state. In this framework, any formula of the language gives rise to a quantum circuit that transforms the state associated to the atomic subformulas into the state associated to the formula and vice versa. On this basis, some holistic semantic situations can be described, where the meaning of whole determines the meaning of the parts, by non-linear and anti-unitary operators. We prove that the semantics with such operators and the semantics with unitary operators turn out to characterize the same logic.

Indices of Collusion among Judges and an Anti-collusion Average

We propose two indices of collusion among Judges of objects or events in a context of subjective evaluation, and an average based on these indices. The aim is manifold: to serve as a reference point for appeals against the results of voting already undertaken, to improve the quality of scores summarized for awards by eliminating those that are less certain, and, indirectly, to provide an incentive for reliable evaluations. An algorithm for automatic computation is supplied. The possible uses of this technique in various fields of application are pointed out: from Economics to Finance, Insurance, Arts, artistic sports and so on.

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.

Quantum structures in qudit spaces

Abstract The Toffoli gate and the Hadamard gate give rise to an approximately universal set of quantum computational gates. We study the algebraic properties of extensions of this system by introducing the notion of the extended Shi–Aharonov structure. The quotient of this structure is isomorphic to a structure based on a particular set of real numbers (the Bloch hypersphere). The aim of this paper is to bring together researchers working in different areas (quantum circuits, algebra, geometry) opening up new perspectives. The algebraic approach may be useful in circuit analysis.

A fuzzy approach to quantum logical computation

The theory of logical gates in quantum computation has inspired the development of new forms of quantum logic, called quantum computational logics. The basic semantic idea is the following: the meaning of a formula is identified with a quantum information quantity, represented by a density operator, whose dimension depends on the logical complexity of the formula. At the same time, the logical connectives are interpreted as operations defined in terms of quantum gates. In this framework, some possible relations between fuzzy representations based on continuous t-norms for quantum gates and the probabilistic behavior of quantum computational finite-valued connectives are investigated.