By Andranick S. Tanguiane

ISBN-10: 3642765165

ISBN-13: 9783642765162

ISBN-10: 3642765181

ISBN-13: 9783642765186

Aggregation is the conjunction of data, aimed toward its compact represen­ tation. Any time while the totality of knowledge is defined by way of common­ ized signs, traditional counts, standard representatives and attribute dependences, one at once or in a roundabout way bargains with aggregation. It comprises revealing the main major features and specific positive aspects, quanti­ tative and qualitative research. for that reason, the data turns into adaptable for additional processing and handy for human conception. Aggregation is widespread in economics, data, administration, making plans, method research, and plenty of different fields. for this reason aggregation is so vital in information seasoned­ cessing. Aggregation of personal tastes is a selected case of the overall challenge of ag­ gregation. It arises in multicriteria decision-making and collective selection, whilst a suite of choices needs to be ordered with appreciate to contradicting standards, or a number of person critiques. even if, even with obvious similarity the issues of multicriteria decision-making and collective selection are a bit diverse. certainly, an development in a few standards on the price of irritate­ ing others isn't the similar because the delight of pursuits of a few contributors to the unfairness of the remaining. within the former case the reciprocal compensations are thought of inside a undeniable entirety; within the latter we infringe upon the rights of self sufficient members. additionally, in multicriteria decision-making one usu­ best friend takes under consideration target components, while in collective selection one has to check subjective critiques which can't be measured properly.

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27). PROOF. Define the sets Il=In{x:xEX, (x,y)ER} and I2 = I n {x: X E X, (y, x) E R}. Since P is continuous, II and 12 are closed subsets of I; they are not empty, because Xl E II and X2 E 12 ; finally (otherwise there exists X E I such that (x, y) ~ Rand (y, x) ~ R, consequently, (y,x) E P and (x,y) E P in contradiction to asymmetry of P). 3 Continuous Orderings 35 {x: x E X,(x,y) E R} I ......................... Y X3 {x: x E X,(y,x) E R} Fig. 27 which is obviously indifferent to y. The following propositions, proving the analogy between continuous orders and functions, are valid for weak orders only.

37). 3. On each axis Ui , i = 2, ... , n, determine the sequence such that k ( Si' SiK-k) E M i for all k = 0, ... ,K (fig. 38). 4. For each i = 2, ... 2 Construction of Approximation of Additive Goal Function K-2 81 K-2 81 Fig. 37 57 58 3 AGGREGATION OF FIXED INDEPENDENT PREFERENCES '" .. '" '" '" . '" .. '" '" '" '" . '" '" . '" .. u·, , K-l 8· 1 o K-l 8· • Fig.

Y X3 {x: x E X,(y,x) E R} Fig. 27 which is obviously indifferent to y. The following propositions, proving the analogy between continuous orders and functions, are valid for weak orders only. 5. PROPOSITION (On Attaining Maxima and Minima on a Compactum). Let P be a continuous weak order on a compactum X and let R be dual to P. e. m,M E X such that (x,m) E Rand (M,x) E R for any x EX. e. PROOF. M(x) = {x': x' E X, (x', x) E R}, which is closed by continuity of P. We shall show that the family of sets M(x) has the finite intersection property.

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Aggregation and Representation of Preferences: Introduction to Mathematical Theory of Democracy by Andranick S. Tanguiane


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