Aggregation and Representation of Preferences: Introduction by Andranick S. Tanguiane

By Andranick S. Tanguiane

Aggregation is the conjunction of knowledge, geared toward its compact represen­ tation. Any time whilst the totality of information is defined when it comes to basic­ ized signs, traditional counts, usual representatives and attribute dependences, one at once or in a roundabout way offers with aggregation. It comprises revealing the main major features and specific good points, quanti­ tative and qualitative research. therefore, the knowledge turns into adaptable for additional processing and handy for human conception. Aggregation is standard in economics, information, administration, making plans, procedure research, and plenty of different fields. this is why aggregation is so vital in info 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, while a suite of choices needs to be ordered with admire to contradicting standards, or quite a few person critiques. notwithstanding, regardless of obvious similarity the issues of multicriteria decision-making and collective selection are slightly assorted. certainly, an development in a few standards on the rate of aggravate­ ing others isn't the similar because the pride of pursuits of a few participants to the unfairness of the remaining. within the former case the reciprocal compensations are thought of inside a definite entirety; within the latter we infringe upon the rights of self reliant participants. in addition, in multicriteria decision-making one usu­ best friend takes into consideration aim elements, while in collective selection one has to check subjective critiques which can't be measured properly.

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R implies (x, z) E P; 5. (x, y) f/. Rand (y, z) f/. P implies (x, z) E P; 6. (x,y) f/. Rand (y,z) f/. R implies (x,z) E P; 7. (x,y) E P implies either (x,z) E P, or (z,y) E R; 8. (x,y) E P implies either (x,z) E R, or (z,y) E P. 13. PROPOSITION (Properties of Weak Orders). Let P be a weak order on X and R be dual to P. Then for arbitrary x, y, z E X it holds: 1. (x,y) E P implies (x,y) E R; 2. (x, y) E P and (y, z) E R implies (x, z) E P; 3. (x, y) E Rand (y, z) E P implies (x, z) E P; 4. (x,y) E Rand (y,z) E R implies (x,z) E R; 5.

THEOREM (About the Existence of a Goal Function). Let X be a strict, or in dual terms linear ordered set, regarded as a topological space with the topology induced by the given strict order r-. Then the following conditions are equivalent: 1. There exists a goal function on X. 2. There exists a monotone homeomorphism from X into the interval (0; 1). 3. The topology on X induced by the strict order has a countable base. 4. X as a topological space is separable and has no more than a countable set of jumps (empty intervals).

17. THEOREM (About the Extendability of a Partial Order to a Strict Order). Under the axiom of choice each partial order on a given set can be extended to a strict order on the given set. This theorem also implies that partial orders are extendable to weak orders. 14, we can interpret partial orders as preferences, which are indefinite, or revealed worse than that described by weak orders. The indefinitness can be characterized by the number (cardinality) of possible extensions to weak orders, similarly to the dimension of an order understood to be the number of its linear extensions.

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