Competitive Prices as aRanking System over Networks

Ehud Lehrer and Ady Pauzner

Tel Aviv University

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)

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

Our approach

• Construct economy based on the network of links• Derive ranks from the competitive equilibrium prices

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

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)

Pure exchange economy in a network

Pure exchange economy in a network

Example 1

Cobb-Douglas utility

Example 2 (with Cobb-Douglas utility)

Example 2 (with Cobb-Douglas utility)

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)

A little bit of notation

Cobb-Douglas – general solution

Cobb-Douglas – general solution

PageRank

PageRank

CES utility

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

CES utility improve

• In the Markov approach one can naturally generate only PageRank. The degree of freedom in choosing the CES parameter can only be replicated by having transition probabilities that depend on the final invariant distribution weights.

Uniqueness of Quasi Equilibrium

Example: multiple equilibria

The Citation Index

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

Rank and reviewing power

Economy with tax

Example 2 with tax

Tax can also change ordinal ranking

Ranking-biased agents

Ranking-biased agents

Ranking-biased agents

Ranking-biased agents

Ranking-biased agents

Ranking-biased agents

Ranking-biased agents

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