Ranking systems

Abstract

This paper initiates research on the foundations of ranking systems, a fundamental ingredient of basic e-commerce and Internet Technologies. In order to understand the essence and the exact rationale of page ranking algorithms we suggest the axiomatic approach taken in the formal theory of social choice. In this paper we deal with PageRank, the most famous page ranking algorithm. We present a set of simple (graph-theoretic, ordinal) axioms that are satisfied by PageRank, and moreover any page ranking algorithm that does satisfy them must coincide with PageRank. This is the first representation theorem of that kind, bridging the gap between page ranking algorithms and the mathematical theory of social choice.

Group Recommendations: Axioms, Impossibilities, and Random Walks

“…A different approach to recommendation systems is the axiomatic one, which seeks to first describe the goals of a system, and then to find the systems that implement such goals. Such an approach has been taken in ranking systems [1,3], including detailed analysis of specific mechanisms [2] including collaborative filtering [18]. More importantly, it has been applied to the individual recommendation system problem, first in Andersen et al [5], and following that, in additional papers complementing it [22,9].…”

“…Once calculated r a (based on other r i s), the recommendation system recommends sgn(r a ) (i.e., sign of the number). 2 Group random-walk, in a sense, calculates a value for each group member based on their individual recommendations, and sums over all of the group members.…”

“…Various properties of such aggregation functions have been considered and have led to various characterizations of particular systems as well as impossibility results showing no system can satisfy certain sets of properties all at once [6]. In the internet setting, the study of ranking and reputation systems, such as page-ranking systems [2], defines a natural extension of classical social choice to the case where the set of voters and the set of alternatives coincide. In such a setting the classical axioms of social choice become less relevant, and are replaced with new axioms which lead to completely new theory.…”

A páros összehasonlításokon alapuló rangsorolás módszertani és alkalmazási kérdései

“…Az aggregált páros összehasonlítási mátrix bizonyos transzformáltjainak domináns sajátvektorát használja az invariáns és a fair bets (Daniels, 1969;Moon és Pullman, 1970), vagy az előbbi egy perturbált változatának megfelelő PageRank módszer (Brin és Page, 1998). Ezeket az eljárásokat, Slutzki és Volij (2005), illetve Slutzki és Volij (2006) karakterizációit részletesen ismertette , bár az invariáns módszer gráfelméleti, ordinális tulajdonságokon alapuló axiomatizációjára (Altman és Tennenholtz, 2005) nem tért ki.…”

“…A megfelelő pontozási eljárás kiválasztásában segítséget jelenthet a társadalmi választások elméletének (social choice theory) axiomatikus szemlélete (Altman és Tennenholtz, 2005):…”