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 when it comes to common ized symptoms, traditional counts, commonplace representatives and attribute dependences, one at once or ultimately bargains with aggregation. It comprises revealing the main major features and specific positive factors, quanti tative and qualitative research. hence, the knowledge turns into adaptable for additional processing and handy for human belief. Aggregation is standard in economics, records, administration, making plans, approach research, and plenty of different fields. because of this aggregation is so very important in information professional 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 collection of choices should be ordered with recognize to contradicting standards, or numerous person reviews. despite the fact that, inspite of obvious similarity the issues of multicriteria decision-making and collective selection are a bit varied. certainly, an development in a few standards on the price of aggravate ing others isn't the comparable because the pride of pursuits of a few members to the unfairness 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 self reliant contributors. furthermore, in multicriteria decision-making one usu best friend takes into consideration goal elements, while in collective selection one has to check subjective critiques which can't be measured properly.

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**Sample text**

Since relation P2 is continuous, the sets {Z:ZEX, (z,s)ER2} and {z : z EX, (t, z) E R 2} are closed. 3 39 Continuous Orderings U • Y \t')t • ( • • V • s' s Preference • x ) Fig. 30 and [S n V] c {z : z E X, (t, z) E R2 }. Since x E [S n U] and y E [S n V], we obtain (X,8) E R2 and (t, y) E R 2. 13 we get the required (x, y) E P2 • Similarly, one can prove that (x, y) E P2 implies (x, y) E PI for any x,y E X. The following examples demonstrate that the connectedness of X and the condition that PI and P2 are weak orders are essential for the validity of the above proposition.

Therefore we shall reject alternately axiom A2 (universality), A4 (independence of irrelevant alternatives), and later A5 (prohibition of dictatorship). Giving up the universality principle (A2) in the present chapter, we assume that individual preferences are fixed (a partial restriction of the universality condition we also consider in chapter 7). 4 in a quite general case both individual and collective orderings can be represented by numerical goal functions. If we suppose that collective goal function f (x) is a function of individual utilities, then the task is reduced to the construction of the collective goal function where Ui=U(X), are utility functions of individuals i = 1, ...

6. DEFINITION. A strict order is defined to be an asymmetric, negatively transitive, and weakly-connected binary relation. For these three orders the belonging (x, y) E P to the order P is said to be "x is preferable to y"; we shall also use the notation x >- y, or y -< x. e. (x, y) ¢:. P and (y, x) ¢:. P simultaneously. In this case it is said to be "x is indifferent to y", which is also written as x rv x. The dual relation to P is defined to be R = {(x, y) : (x, y) EX X X, (x, y) ¢:. Pl. If P is a preference, R is interpreted as the preference or indifference.

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

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