By Antoine Billot
Fuzzy set conception, which began no longer even more than two decades in the past as a generalization of classical set concept, has meanwhile advanced into a space which scientifically, in addition to from the viewpoint of purposes, is famous as a really worthy contribution to the present wisdom. This e-book presents a amazing contribution to Fuzzy Economics and offers the state-of-the-art in fuzzy concept of price, specifically the aggregated version of microeconomics with fuzzy behaviours. It offers an research of classical issues of new instruments which bring about attention-grabbing results.
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Additional resources for Economic Theory of Fuzzy Equilibria: An Axiomatic Analysis
2 Fuzzy Preferences and Collective Decision row's Axioms Ar- We paradoxically ought to corroborate the general problematic of aggregation before conferring it a different meaning if the agent's preferences are fuzzy. arrow's theorem depends on a group of axioms that we are now going to develop. ), (for any agent i from S). )-reflexive and I-transitive. Under the same assumptions of continuity for the collective preference and connectedness of X, it leads to a continuous social utility function, noted 40 Chapter 2.
2- Second, we can state that if all the agents have the same ordering, society has it too (which is called unanimity). Thus, the relation of social preference is I-transitive and also ordinal. The majority rule studied by Condorcet is based on the whole set of individual preorders. These statements are the basis of the four axioms defined by Arrow. 2 Fuzzy Preferences and Collective Decision row's Axioms Ar- We paradoxically ought to corroborate the general problematic of aggregation before conferring it a different meaning if the agent's preferences are fuzzy.
Let us present the logic denial of the proposition to prove and let us conclude in its mathematical irrelevance. -'x (y). e. -,D(z) = o. -'x (z). To sum up : I'D(z) 1= 0 ~ z E Xi U XJ. e. -'X (z), Vz E DJ. -'X (z). -'f (z) if z E XJ. ) is the membership function of the fuzzy subset D J, function of fij, closure of D J. e. : ;Ji(z) J'X(z), \lz E DJ. Let us now combine these two last equalities: J'D(z) ~ J'x (z) = ;Ji(z). Since X} and are closed then, if DJ ~ X} U it implies ~ X} U XJ, because a union of closed sets is also a closed set.
Economic Theory of Fuzzy Equilibria: An Axiomatic Analysis by Antoine Billot