Hannu Nurmi

Group decision making and consensus under fuzzy preferences and fuzzy majority

Abstract We present how fuzzy logic with linguistic quantifiers, mainly its calculi of linguistically quantified propositions, can be used in group decision making. The fuzzy linguistic quantifiers (exemplified by most, almost all,...) are employed to represent a fuzzy majority which is in many cases closer to a real human perception of the very essence of majority. Fuzzy logic provides here means for a formal handling of such a fuzzy majority which was not possible by using traditional formal apparata. Assuming fuzzy individual and social preference relations, as it is commonly done, and employing in addition a fuzzy majority expressed by a fuzzy linguistic quantifier, we redefine solution concepts in group decision making, and present new ‘soft’ degrees of consensus.

Approaches to collective decision making with fuzzy preference relations

Abstract Recent experimental studies show that the predictive accuracy of many of the solution concepts derived from the collective decision making theory leaves much to be desired. In a previous paper the author attempted to explain some of the inaccuracies in terms of the fuzzy indifference regions of the individuals participating in the voting game. This paper gives straightforward generalizations of the solutions concepts in terms of the fuzzy social or individual preference relations. It turns out that some of these new solution concepts cotain their nonfuzzy counterparts as subsets. Others, in turn, are subsets of their nonfuzzy counterparts. We also discuss a method of aggregating individual nonfuzzy preferences so as to get a fuzzy social preference relation and, furthermore, a nonfuzzy social choice set.

Voting Procedures: A Summary Analysis

Roughly two centuries ago the Marquis de Condorcet and Chevalier Jean-Charles de Borda originated a research tradition – by no means a continuous one – that over the decades has produced results casting doubt on many widely used collective decision-making procedures. The phenomenon known as the Condorcet effect or the Condorcet paradox is the well-known problem of the simple majority rule. The paradox bearing the name of Borda is less commonly known, but it also relates to a procedure that is widely used, namely the plurality principle. Either one of these paradoxes is serious enough to make these procedures suspect unless one is convinced that the situations giving rise to these paradoxical features are extremely rare. In this article we review some voting procedures that have been introduced in the literature. We aim at giving a synthesis of the assessments of procedures with respect to various criteria.

Refereed article: Secret ballot elections in computer networks

The paper presents a secret balloting system for elections carried out in a computer network. The system has some features not possessed by customary secret balloting systems and does not rely on trusted persons and group work to the same extent as customary systems. Our system uses protocols based on public-key cryptography.

On fuzzy tournaments and their solution concepts in group decision making

Abstract Assuming as the point of departure a set of individual or collective fuzzy tournaments (complete and reciprocal binary relations)—representing the preferences of the particular individuals or their group as a whole, respectively—we define some new solution concepts in group decision making as counterparts of the respective solution concepts in the case of nonfuzzy tournaments (complete and asymmetric binary relations). Basically, the new solution concepts are mainly counterparts of some sets of undominated options, uncovered sets, and Banks sets. We consider the cases of both the conventional (nonfuzzy) and fuzzy majorities.

Closeness Counts in Social Choice

A working democratic system of government is based on voting and elections. A glance at the literature reveals an astonishing variety of systems used for electing persons to political offices, for choosing policy alternatives and for enacting legislation. While it is undoubtedly true that voting is just a necessary condition for democracy, it is remarkable how many and how different systems are used for apparently the same purpose, viz. to find out the ‘will of the people’.

Probabilistic, fuzzy and rough concepts in social choice

Abstract We discuss how intrinsic inconsistencies and negative results (concerning opinion aggregation) in social choice may be alleviated by plausible modifications of underlying assumptions and problem formulations, basically by the introduction of some impreciseness of a probabilistic, fuzzy and rough type. First, we discuss briefly probabilistic voting, and the use of fuzzy preference relations and fuzzy majorities. Then, in the main part, we proceed to the use of Pawlaks rough sets theory in the analysis of crucial properties of voting schemes. In this framework we also discuss the concept of a distance between two voting schemes. Finally, we further explore difficult issues of how diverse types of impreciseness can be combined, and we consider in particular the combination of roughness with randomness and fuzziness in the context of spatial voting games.

A priori power measures and the institutions of the European Union

The a priori voting powers of member countries in the council of ministers of the European Union have been discussed in the literature mainly from the view-point of Banzhaf and Shapley-Shubik indices. This paper discusses – in the light of these and other more recent (Colomers, Hollers as well as Deegan and Packels) power indices – the interaction of the council ministers and the European Parliament (EP) under the assumption that the consent of both bodies is needed to carry a motion or piece of union-wide legislation. Moreover, the double-majority principle is discussed from the view-point of voting power distribution. Finally we consider the voting power distribution in a three-chamber system consisting of Commission, Council and EP.

A fuzzy solution to a majority voting game

Abstract In the Fiorina-Plott majority voting game experiment several puzzling phenomena were discovered: (1) In the experiment series where the payoffs were small in general, the predictive performance of the core was markedly worse than in the high payoff series. (2) The core outcome was found to be defeatable by another proposal. (3) Some of the majority winning social outcomes were Pareto-suboptimal. (4) Even when there was no theoretical equilibrium point, there was a clear clustering of outcomes. The paper outlines a fuzzy decision making procedure which seems to be capable of explaining all the puzzling features (1)–(4). In the absence of data on membership functions the explanation provided is of non-predictive nature; it renders the observed outcomes plausible without precisely predicting each one of them.

Group decision making under fuzziness

Group decision making, as meant in this paper, is the following choice problem which proceeds in a multiperson setting. There is a group of individuals (decisionmakers, experts, …) who provide their testimonies concerning an issue in question. These testimonies are assumed here to be individual preference relations over some set of option (alternatives, variants, …). The problem is to find a solution, i.e. an alternative or a set of alternatives, from among the feasible ones, which best reflects the preferences of the group of individuals as a whole. We will survey main developments in group decision making under fuzziness. First, we will briefly outline some basic inconsistencies and negative results of group decision making and social choice, and show how they can be alleviated by some plausible modifications of underlying assumptions, mainly by introducing fuzzy preference relations and, to a lesser extend, a fuzzy majority. Then, we will concentrate on how to derive solutions under individual fuzzy preference relations, and a fuzzy majority equated with a fuzzy linguistic quantifier (e.g., most, almost all, …) and dealt with in terms of a fuzzy logic based calculus of linguistically quantified statements or via the ordered weighted averaging (OWA) operators. Finally, we will discuss a related issue of how to define a “soft ” degree of consensus in the group under individual fuzzy preference relations and a fuzzy majority.