Economic Theory of Fuzzy Equilibria

With regards to dictionaries, five possible definitions may be alternatively used to define the adjective precise, each one referring to exactitude. For the word imprecision, there is generally a brief definition: ‘which lacks precision’. However, the english language is rich with many expressions that can help us to understand imprecision. ‘vague’, ‘ambiguous’, ‘general’, or ‘fuzzy’ are indistinctively used adjectives expressing the imprecise character of a phenomena or an idea.

The main purpose of the fuzzy approach of individual preferences is to introduce some new behaviors inside the standard theory. The existing literature is generally divided in two schools, according to the position authors adopt relatively to the axiom of independence of Irrelevant Alternatives (IIA).¹ Actually, most of the works using fuzzy preferences are based on additive measures of satisfaction (Orlovsky 1978, 1980, Kaufmann 1973, Zadeh 1971b, Nahmias 1980, Dubois & Prade 1979, Bazu 1984 and Butnariu 1980, 1985, 1987).

The first part of our work was devoted to the new agent that we desired to consider in an economical and social situation. We then tried to define his behavior, his rationality and the insertion of his decision in a larger social structure where he would be considered no more like a lone Robinson but just like one of the agents of the social decision. The intersection of both first chapters was the preference theory. Henceforth, to define equilibria inside a theory of fuzzy preferences, we introduce a new concept ; the conflict. By trying to model it, we enter the domain of game theory.

Here, we are going to assume preferences to be fuzzy in the case of an exchange economy with production. Hence, we introduce factors or inputs and the associated markets. The former influence individuals’ choices (especially in terms of labor) and modify the structure of models by generating some new markets (of factors) and new agents (the producers).

Since L.A. Zadeh’s initial article ‘Fuzzy sets’ in 1965 in Information and Control, the scope of the theory of fuzzy sets (linguistics, artificial intelligence, biology, etc…) has been wide and varied. As far as economic science is concerned, the influence of this new mathematical theory — which it should be said is essentially a generalization in the sense of a ‘general theory’ which accommodates the standard approach as a borderline case — has been evenly spread beyond the traditional areas of micro-economic analysis:¹ preference theory (Chakraborty, Das, Orlovsky, Ovchinnikov, Roubens, Vincke), aggregation of preferences (Barret, Montero, Nurmi, Pattanaïk, Salles, Tejada), game theory (Aubin, Butnariu, Fedrizzi, Heilpern, Hüsseinov), theory of value (Ponsard), operational research (Farreny, Okuda & Asai, Tanaka, Zimmermann), multi-criteria decision making (Roy), data analysis (Ponsard), investment choice (Buckley) and decision making theory (Bazu, Bellman, Dubois, Fung & Fu, Prade, Sugeno). Even though a presentation of this kind of dispels any differences there may be between the direct application of fuzzy set theory (preference relations, fixed point theorems, topological structure of an economy, utility functions and axioms) and the use of non-additive uncertainty measure which are derived therefrom (possibility, necessity), Zadeh’s 1978 article ‘fuzzy sets as a basis for a theory of possibility’ means the two currents can be linked together unequivocally — and so paradoxically told apart.

Our purpose is to introduce fuzziness in microeconomics thanks to fuzzy preferences.