By Andranick S. Tanguiane

ISBN-10: 3642765165

ISBN-13: 9783642765162

ISBN-10: 3642765181

ISBN-13: 9783642765186

Aggregation is the conjunction of knowledge, aimed toward its compact represen tation. Any time whilst the totality of information is defined when it comes to normal ized symptoms, traditional counts, normal representatives and attribute dependences, one without delay or not directly offers with aggregation. It contains revealing the main major features and specific beneficial properties, quanti tative and qualitative research. for that reason, the knowledge turns into adaptable for extra processing and handy for human notion. Aggregation is customary in economics, records, administration, making plans, procedure 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 final challenge of ag gregation. It arises in multicriteria decision-making and collective selection, whilst a suite of possible choices needs to be ordered with appreciate to contradicting standards, or quite a few person evaluations. in spite of the fact that, regardless of obvious similarity the issues of multicriteria decision-making and collective selection are a little bit various. certainly, an development in a few standards on the expense of irritate ing others isn't the comparable because the pride of pursuits of a few members to the bias of the remaining. within the former case the reciprocal compensations are thought of inside of a undeniable entirety; within the latter we infringe upon the rights of autonomous contributors. additionally, in multicriteria decision-making one usu best friend takes under consideration target elements, while in collective selection one has to match subjective reviews which can't be measured properly.

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**Example text**

Since pI is continuous, there exists an open neigborhood U of the point 0 such that U={x:xEX, (s,x) E PI}. Since X is a circumference, U contains all points sufficiently close to 27r. If x is sufficiently small, then (s,27r - x) E p'. 3 Continuous Orderings 41 Preference o A Y2 Yl X 1/27r B Fig. 32 On the other hand, if x is sufficiently small, then 27r - x > s, whence by definition of P and pI we have (27r - x,s) E pI, in contradiction to the asymmetry of P'. The second possibility (0, s) E pI is brought to contradiction in the same way.

Let R' be dual to P'. Define two sequences and Put and Note that (xn, A) E P' C R' and (B,Yn) E P' c R' for all n = 1,2 .... Since by assumption P' is continuous on X, the sets {x: x E X, (x,A) E R'} and {x: x E X, (B,x) E R'} are closed, whence we obtain that (0, A) E R' and (B, 0) E R'. 13) we have (B, A) E R', and therefore (A, B) ~ P' in contradicition to (A, B) E P. To end the section we conclude that the proved properties of continuous weak orders make them similar to continuous functions, which is convinient for their analysis and applications in decision-making models.

If P is a preference, R is interpreted as the preference or indifference. The notation (x, y) E R is read as "x is preferred or indifferent to y". Sometimes it is written as x t y, or y ~ x. 3, the relation R can be defined not by means of relation P but directly. Note that the concept of strict order is dual to the concept of linear order. Let us give simple examples of preferences, which will be useful in the sequel. 2 26 PREFERENCES AND GOAL FUNCTIONS y y x o x o 1 2 Fig. 23 Fig. 7. EXAMPLE.

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

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