Using of Rectangular Stochastic Matrices for the Problem of Evaluating and Ranking Alternatives

Abstract

The paper investigates the issue related to a possible generalization of the “state-probability of choice” model so that the generalized model could be applied to the problem of ranking alternatives, either individual or by a group of agents. It is shown that the results obtained before for the problem of multi-agent choice and decision making by majority of votes can be easily transferred to the problem of multi-agent alternatives ranking. On the basis of distributions of importance values for the problem of ranking alternatives, we can move on to similar models for the choice and voting with the help of well-known exponential normalization of rows.So we regard two types of matrices, both of which belonging to the sort of matrices named balanced rectangular stochastic matrices. For such matrices, sums of elements in each row equal 1, and all columns have equal sums of elements. Both types are involved in a two-level procedure regarded in this paper. Firstly a matrix representing all possible distributions of importance among alternatives should be formed, and secondly a “state-probability of choice” matrix should be obtained on its base. For forming a matrix of states, which belongs and the rows of which correspond to possible distributions of importance, applying pairwise comparisons and the Analytic Hierarchy Method is suggested. Parameterized transitive scales with the parameter affecting the spread of importance between the best and the worst alternatives are regarded. For further getting the matrices of choice probabilities, another parameter which reflects the degree of the agent’s decisiveness is also introduced. The role of both parameters is discussed and illustrated with examples in the paper.The results are reported regarding some numerical experiments which illustrate getting distributions of importance on the basis of the Analytic Hierarchy Process and which are connected to gaining the situation of dynamic equilibrium of alternatives, i.e. the situation when alternatives are considered as those of equal value.