Competitive Prices as a Ranking System over Networks. Ehud Lehrer and Ady Pauzner Tel Aviv University July 2012

Transcription

1 Competitive Prices as a Ranking System over Networks Ehud Lehrer and Ady Pauzner Tel Aviv University July 2012

2 Ranking systems based (only) on network structure Examples: Science Citation Index: Rank of article = number of citations Google s PageRank: Link from a higher ranked item is worth more (Note circular definition)

3 Approaches to ranking Counting citations Citation index (Garfield 1960) Markov chain: PageRank (Wei 1951, Kendall 1955, Brin & Page 1998) Axiomatic: Palacios-Huerta and Volij (2004) Altman and Tennenholtz (2008): Axiomatization of PageRank Demange (2011): Separates quality and refereeing power Dynamics: Demange (2011): Ranking affects citations affect ranking Liebowitz and Palmer (1984): Iteration (impact adjusted) method

4 Our approach Construct economy based on the network of links Derive ranks from the competitive equilibrium prices

5 Pure exchange economy a reminder N consumer Each consumer brings an intial endowment (a basket of L goods) Each has a utility function In equilibrium each consumer sells his initial endowment and buys in exchange the best basket possible (subject to budget constraint) In equilibrium market clears

6 What is a network? N nodes (web page, article, friends) There are (directed) edges connecting between nodes Examples: web-page i gives a link to j; paper i gives a citation to paper j; i and j are friends (two edges)

7 Pure exchange economy in a network

8 Pure exchange economy in a network

9 Example 1

10 Cobb-Douglas utility

11 Example 2 (with Cobb-Douglas utility)

12 Example 2 (with Cobb-Douglas utility)

13 Quasi equilibrium (Debreu 1962) Definition: 1.Markets clear 2.Consumers with positive budget maximize utility subject to budget constraint 3.Consumers with 0 budget only required to satisfy budget constraint Differs from competitive equilibrium only for consumers with 0 budget who derive utility from a 0-priced goods They consume the leftovers, rather than an unbounded amount Quasi equilibrium exists under very mild conditions (utility functions continuous + sets of preferred baskets convex)

14 A little bit of notation

15 Cobb-Douglas general solution

16 Cobb-Douglas general solution

17 PageRank

18 PageRank

19 CES utility

20 Example 1 with CES utility In this example the parameter affects cardinal ranking but not ordinal (except for endpoints) We can easily generate examples where ordinal ranking changes

21 Economic vs. Markov approach The Markov approach can naturally generate only PageRank (equal transition probabilities) In order to replicated the CES equilibrium with the probabilistic approach, one would have to use transition probabilities that depend on the final invariant distribution weights That is, to get something different than PageRank, the model s definition would have to involve the outcome In the economic approach, the model is well defined with no circularity; the dependence of how agents split their budget on the prices of the goods comes from the solution concept (competitive equilibrium)

22 Uniqueness of Quasi Equilibrium

23 Example: multiple equilibria

24 The Citation Index

25 Rank and reviewing power Ranking system gives each article a quality score Each article also has a reviewing power (importance given by ranking system to its links) In PageRank, reviewing power rank In SCI/NCI, reviewing power independent of rank In world of internet, PageRank seems better In world of articles, PageRank is problematic: Case of articles on a timeline, that can only cite older articles. PageRank gives 0 to all of them, but the oldest PageRank works only with sufficient simultaneity

26 Rank and reviewing power

27 Economy with tax

28 Example 2 with tax

29 Tax can also change ordinal ranking

30 Ranking-biased agents

31 Ranking-biased agents

32 Ranking-biased agents

33 Ranking-biased agents

34 Ranking-biased agents

35 Ranking-biased agents

36 Ranking-biased agents

37 Summary Model of competitive economy as a device for ranking Ranking determined by choice of utility function Uniqueness of ranking holds at least for gross substitutes (and of course connected network) Cobb-Douglas economy yields PageRank No exchange economy yields SCI or NCI Minimum economy yields outcome of interaction between PageRank and linearly biased agents By adding a simple taxation scheme we can control how reviewing power depends on assessed quality