By Andranick S. Tanguiane

ISBN-10: 3642765165

ISBN-13: 9783642765162

ISBN-10: 3642765181

ISBN-13: 9783642765186

Aggregation is the conjunction of data, geared toward its compact represen tation. Any time whilst the totality of knowledge is defined when it comes to common ized signs, traditional counts, average representatives and attribute dependences, one at once or in some way bargains with aggregation. It contains revealing the main major features and certain good points, quanti tative and qualitative research. for that reason, the knowledge turns into adaptable for extra processing and handy for human conception. Aggregation is general in economics, information, administration, making plans, method research, and plenty of different fields. this is the reason aggregation is so vital 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 suite of choices should be ordered with recognize to contradicting standards, or quite a few person reviews. in spite of the fact that, regardless of obvious similarity the issues of multicriteria decision-making and collective selection are a little diverse. certainly, an development in a few standards on the fee of aggravate ing others isn't the comparable because the delight of pursuits of a few contributors to the unfairness of the remainder. within the former case the reciprocal compensations are thought of inside a definite entirety; within the latter we infringe upon the rights of self sufficient participants. furthermore, in multicriteria decision-making one usu best friend takes into consideration aim components, while in collective selection one has to check subjective critiques which can't be measured properly.

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Sand (y, z) tI. S implies (x, z) tI. S for every connected if X,y E X implies (x,y) E S, or (y,x) E S, or both; weakly-connected if X,y E X and x t= y implies either (x,y) E S, or (y,x) E S. Binary relations, satisfying some of the above properties, can be defined by means of their complements. It is stated by the following proposition. 3. PROPOSITION (A Duality Principle for Binary Relations). Let two binary relations S1 and S2 on a set X be bound up by the condition S2 = {(x,y): (x,y) E X x X, (y,x) tI.

That there exists such a weak order P'. 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.

Rand (y, z) ~ P implies (x, z) R implies (x, z) f/. f/. R; R; 8. (x, y) E P impies either (x, z) E P, or (z, y) E P. Now we formulate a test for weak orders. 14. PROPOSITION (A Test for Weak Orders). Every weak order is at the same time a partial order. e. a reflexive, symmetric, and transitive binary relation. 2 Binary Relations and Orderings 29 PROOF. Let P be a weak order on X. 13 the relation P is transitive, whence P is a partial order. Let us show that the indifference", defined with respect to P is an equivalence.

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

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