María Albina Puente

Reliability Importance Measures of the Components in a System Based on Semivalues and Probabilistic Values

The main contribution of this paper consists in providing different ways to value importance measures for components in a given reliability system or in an electronic circuit. The main tool used is a certain type of semivalues and probabilistic values. One of the results given here extends the indices given by Birnbaum [3] and Barlow and Proschan [2], which respectively coincide with the Banzhaf [1] and the Shapley and Shubik [15] indices so well-known in game theory.

Semivalues as power indices

Abstract A restricted notion of semivalue as a power index, i.e. as a value on the lattice of simple games, is axiomatically introduced by using the symmetry, positivity and dummy player standard properties together with the transfer property. The main theorem, that parallels the existing statement for semivalues on general cooperative games, provides a combinatorial definition of each semivalue on simple games in terms of weighting coefficients, and shows the crucial role of the transfer property in this class of games. A similar characterization is also given that refers to unanimity coefficients, which describe the action of the semivalue on unanimity games. We then combine the notion of induced semivalue on lower cardinalities with regularity and obtain a series of characteristic properties of regular semivalues on simple games, that concern null and nonnull players, subgames, quotients, and weighted majority games.

A note about games-composition dimension

Abstract The main contribution of this paper is the calculation of the dimension of simple games that are a composition of unanimity games via individualism. We also provide a constructive procedure that represent this type of game as an intersection of a number of weighted majority games equal to its dimension. This provides a better way to achieve exponential dimension (in a monotonic setting) than that of Taylor and Zwicker (Simple Games, Desirability Relations, Trading and Pseudoweightings, Princeton University Press, Princeton, NJ, 1999).

Complete games with minimum

Some real-world examples of simple games, like the procedure to amend the Canadian Constitution, are complete simple games with minimum. Using characteristic invariants for this class of games, we study different types of solution concepts. For an arbitrary number of players we get the nucleolus by means of a determinate compatible system of equations, characterize the maximality of the kernel and give a method to calculate semivalues. Several applications are found at the end of the paper.

Axiomatic characterizations of the symmetric coalitional binomial semivalues

The symmetric coalitional binomial semivalues extend the notion of binomial semivalue to games with a coalition structure, in such a way that they generalize the symmetric coalitional Banzhaf value. By considering the property of balanced contributions within unions, two axiomatic characterizations for each one of these values are provided.

A Parametric Family of Mixed Coalitional Values

We introduce here a family of mixed coalitional values. They extend the binomial semivalues to games endowed with a coalition structure, satisfy the property of symmetry in the quotient game and the quotient game property, generalize the symmetric coalitional Banzhaf value introduced by Alonso and Fiestras and link and merge the Shapley value and the binomial semivalues. A computational procedure in terms of the multilinear extension of the original game is also provided and an application to political science is sketched.

Partnership formation and binomial semivalues

Partnership formation in cooperative games is studied, and binomial semivalues are used to measure the effects of such a type of coalition arising from an agreement between (a group of) players. The joint effect on the set of involved players is also compared with that of the alternative alliance formation. The simple game case is especially considered, and the application to a real life example illustrates the use of coalitional values closely related to the binomial semivalues when dealing with partnership formation and coalitional bargaining simultaneously.

Coalitional multinomial probabilistic values

We introduce a new family of coalitional values designed to take into account players’ attitudes with regard to cooperation. This new family of values applies to cooperative games with a coalition structure by combining the Shapley value and the multinomial probabilistic values, thus generalizing the symmetric coalitional binomial semivalues. Besides an axiomatic characterization, a computational procedure is provided in terms of the multilinear extension of the game and an application to the Catalonia Parliament, Legislature 2003–2007, is shown.

Partnership formation and multinomial values

We use multinomial values to study the effects of the partnership formation in cooperative games, comparing the joint effect on the involved players with the alternative alliance formation. The simple game case is especially considered and the application to the Catalonia Parliament (Legislature 2003-2007) is also studied.

Consecutive expansions of k-out-of-n systems

We introduce consecutive expansions of k-out-of-n systems, which have the property that components are totally ordered by the node criticality relation and with respect to well-known structural importance measures. We propose some formulae to easily compute these measures and study the hierarchies induced for them for large systems.