Aggregation in multiagent and multicriteria decision models: interaction, dynamics, and maximum entropy weights in the framework of choquet integration

Abstract

In the context of MCDM, the Choquet integral constitutes an interesting aggregation model which generalizes both the classical and the ordered weighted means. In the Choquet integration framework, an additive capacity generates a classical weighted mean, whereas a symmetric (non-additive) capacity generates an ordered weighted mean. Moreover, a general (non-additive) Choquet capacity induces a natural weighted mean, the Shapley mean, whose weights correspond to the so-called Shapley power indices. In the first part of the thesis, we examine a negotiation model which combines the Choquet integration framework with the classical Morris H. DeGroot 1974 model of consensus linear dynamics, in interactive multicriteria and multiagents networks. We consider a set $N={1,ldots,n }$ of interacting criteria (or agents) whose single evaluations (or individual opinions) are expressed in some domain $mathbb{D}subseteq mathbb{R}$. The interaction among the criteria (or agents) is expressed by a symmetric interaction matrix with null diagonal and off-diagonal coefficients in the open unit interval. The interaction network structure is thus that of a complete graph with edge values in $(0,1)$.

In the Choquet integration framework, the interacting network structure is the basis for the construction of a capacity $mu$, whose Shapley indices are proportional to the average degree of interaction between criterion (or agent) $i$ and the remaining criteria (or agents). In relation with this interactive multicriteria (or multiagent) network model, we discuss three types of linear consensus dynamics, each of which represents a progressive aggregation process towards a consensual evaluation (or opinion) of the single criteria (or agents), corresponding to some form of mean of the original evaluations (or opinions). In the first type, the progressive aggregation converges simply to the plain mean of the original evaluations (or opinions) of the single criteria (or agents), while the second type converges to the Shapley mean of the original evaluations (or opinions).

The third type, instead, converges to an emphasized form of Shapley mean, which we call superShapley mean. The interesting relation between Shapley and superShapley aggregation is investigated. In the second part of the thesis, we focus on entropy constrained optimization in the context of ordered weighted means, both in the classical Shannon entropy case, and in the more general Tsallis entropy case. The maximum entropy method is based on the solution of a nonlinear constrained optimization problem in which the OWA weights are obtained by maximizing the entropy, given a specified degree of orness. In the Shannon entropy case, we begin by reviewing the analytic solution of the maximum entropy method proposed by Filev and Yager in 1995, and later by Fuller and Majlender in 2001, and we consider the maximum entropy method in the binomial decomposition framework. Then, we present the optimization of the parametric Tsallis entropy function associated with Ordered Weighted Averaging.

We examine the meaning of the entropic parameter $gamma$ in the context of OWA functions and how it affects the behavior of the associated entropy function. We introduce the nonlinear constrained optimization problem of Tsallis entropy for parameter values $gamma in (0,1)$ and we obtain the solution for the optimal weights in terms of the two Lagrange multipliers. Both in Shannon and Tsallis entropy cases for parameter $gamma in (0,1)$, the optimal weights for orness values in the open unit interval are positive (except for the extreme orness values $0,1$) and monotonic (increasing or decreasing) over the whole orness range $Omega in[0,1]$.