Montserrat Pons

On the ordinal equivalence of the Johnston, Banzhaf and Shapley power indices

In this paper, we characterize the games in which Johnston, Shapley–Shubik and Penrose–Banzhaf–Coleman indices are ordinally equivalent, meaning that they rank players in the same way. We prove that these three indices are ordinally equivalent in semicomplete simple games, which is a newly defined class that contains complete games and includes most of the real–world examples of binary voting systems. This result constitutes a twofold extension of Diffo Lambo and Moulen’s result (Diffo Lambo and Moulen, 2002) in the sense that ordinal equivalence emerges for three power indices (not just for the Shapley–Shubik and Penrose–Banzhaf–Coleman indices), and it holds for a class of games strictly larger than the class of complete games.

Identifying Optimal Components in a Reliability System

The first step in a reliability optimization process is to make a reliability assessment for each component in the system. If this assessment is made in a qualitative way, by grouping together components with the same reliability, and establishing a prevalence order among groups, is there a way to decide which components have the greatest Birnbaum measure without computing the exact value of this measure? In this paper, three relations between components are introduced and studied, and it is proved that they are useful for selecting the components that have the biggest effect on the system reliability in the sense of Birnbaum. An algorithm that uses the results in the paper to select these important components is also provided.

The influence of the node criticality relation on some measures of component importance

For different reliability importance measures we prove that the criticality relation between nodes can completely determine the most important component in a system. In particular, we prove that in k-out-of-n systems, the ranking of component reliabilities determines the ranking of component importance for, at least, three different reliability importance measures.

Circumstantial power: Optimal persuadable voters

Abstract Let us suppose that certain committee is going to decide, using some fixed voting rules, either to accept or to reject a proposal that affects your interests. From your perception about each voter’s position, you can make an a priori estimation of the probability of the proposal being accepted. Wishing to increase this probability of acceptance before the votes are cast, assume further that you are able to convince (at least) one voter to improve his/her perception in favor of the proposal. The question is: which voters should be persuaded in order to get the highest possible increase in the probability of acceptance? In other words, which are the optimal persuadable voters? To answer this question a measure of “circumstantial power” is considered in this paper, which is useful to identify optimal persuadable voters. Three preorderings in the set of voters, based on the voting rules, are defined and they are used for finding optimal persuadable voters, even in the case that only a qualitative ranking of each voter’s inclination for the proposal has been made.

Success and decisiveness on proper symmetric games

This paper provides a complete study for the possible rankings of success and decisiveness for individuals in symmetric voting systems, assuming anonymous and independent probability distributions. It is proved that for any pair of symmetric voting systems it is always possible to rank success and decisiveness in opposite order whenever the common probability of voting for “acceptance” is big enough. On the contrary, for probability values lower than one-half it is not possible to reverse the ranking of these two measures.

An axiomatic characterization of the potential decisiveness index

Let us consider that somebody is extremely interested in increasing the probability of a proposal to be approved by a certain committee and that to achieve this goal he/she is prepared to pay off one member of the committee. In a situation like this one, and assuming that vote-buying is allowed and free of stigma, which voter should be offered a bribe? The potential decisiveness index for simple games, which measures the effect that ensuring one positive vote produces for the probability of passing the issue at hand, is a good tool with which to acquire the answer. An axiomatic characterization of this index is given in this paper, and its relation to other classical power indices is shown.

Circumstantial Power: Some Hints for Finding Optimal Persuadable or Bribable Voters

Two measures of circumstantial or local power are introduced here. In our approach we focus on the viewpoint of an external observer who tries to evaluate the probability of a proposal to be passed by a certain committee. According to this local point of view we analyze the changes in the probability to pass the issue at hand obtained by modifying a voters’ perception about the proposal. In particular, we are interested in finding optimal persuadable voters and optimal bribable voters. To this end three preorderings on the set of voters are considered which are proved to be useful to identify these optimal voters.

A New Tuning of PID Controllers Based on LQR Optimization

Abstract This paper presents a new analytical tuning of Proportional-Integral-Derivative controllers based on the well known Linear Quadratic Regulator optimization problem for a wide class of processes whose behaviour can be modelled with with a constant time and a dead time delay. When the dead time delay is approximated by a first order Pade proposal, an optimal control problem gives an analytical solution for the tuning of the PID parameters, and the performances obtained on the original model, without approximation, are very satisfactory. Also, the positive results obtained by simulation with this new PID control design respect to the conventional empirical formulas to tune PID controllers are very promising to be applied in the future to industrial processes.

Preorders in Simple Games

Any power index defines a total preorder in a simple game and, thus, induces a hierarchy among its players. The desirability relation, which is also a preorder, induces the same hierarchy as the Banzhaf and the Shapley indices on linear games, i.e., games in which the desirability relation is total. The desirability relation is a sub–preorder of another preorder, the weak desirability relation, and the class of weakly linear games, i.e., games for which the weak desirability relation is total, is larger than the class of linear games. The weak desirability relation induces the same hierarchy as the Banzhaf and the Shapley indices on weakly linear games. In this paper, we define a chain of preorders between the desirability and the weak desirability preorders. From them we obtain new classes of totally preordered games between linear and weakly linear games.

Measuring the relevance of factors in the occurrences of events

A new way to compare the relevance of the different factors intervening in the occurrences of an event is presented and developed in this paper. The idea behind the method comes from cooperative game theory but the focus is slightly different because factors are not necessarily rational decision-makers and because the only data available are obtained by repetition of the event. The concept of relevance measure for a factor in a set of data is introduced, some significant examples are given and the main properties of relevance measures are defined and studied. One of these measures, the fair measure, is proved to have interesting properties which characterize it. Two real world situations, one about traffic accidents and the other one about mining accidents, both of them with real data, are used to show the use of relevance measures to compare factors in each one of these events.