1、Algorithms and Incentives for Robust Ranking,Rajat Bhattacharjee Ashish Goel Stanford University,Algorithms and incentives for robust ranking. ACM-SIAM Symposium on Discrete Algorithms (SODA), 2007. Incentive based ranking mechanisms. EC Workshop, Economics of Networked Systems, 2006.,Algorithms and
2、 incentives for robust ranking,Outline,Motivation Model Incentive Structure Ranking Algorithm,Algorithms and incentives for robust ranking,Content : then and now,Traditional Content generation was centralized (book publishers, movie production companies, newspapers) Content distribution was subject
3、to editorial control (paid professionals: reviewers, editors),Internet Content generation is mostly decentralized (individuals create webpages, blogs) No central editorial control on content distribution (instead there are ranking and reco. systems like google, yahoo),Algorithms and incentives for r
4、obust ranking,Heuristics Race,PageRank (uses link structure of the web) Spammers try to game the system by creating fraudulent link structures Heuristics race: search engines and spammers have implemented increasingly sophisticated heuristics to counteract each other New strategies to counter the he
5、uristics Gyongyi, Garcia-Molina Detecting PageRank amplifying structures sparsest cut problem (NP-hard) Zhang et al.,Algorithms and incentives for robust ranking,Amplification Ratio Zhang, Goel, ,Consider a set S, which is a subset of V In(S): total weight of edges from V-S to S Local(S): total weig
6、ht of edges from S to S,10,S,w(S) = Local(S) + In(S) Amp(S) = w(S)/In(S)High Amp(S) S is dishonest Low Amp(S) S is honestCollusion free graph: where all sets are honest,Algorithms and incentives for robust ranking,Heuristics Race,Then why do search engines work so well? Our belief: because heuristic
7、s are not in public domain Is this “the solution”? Feedback/click analysis Anupam et al. Metwally et al. Suffers from click spam Problem of entities with little feedback Too many web pages, cant put them on top slots to gather feedback,Algorithms and incentives for robust ranking,Ranking reversal,Ra
8、nking reversalEntity A is better than entity B, but B is ranked higher than A,Keyword: Search Engine,Algorithms and incentives for robust ranking,Our result,Theorem we would have liked to prove Here is a reputation system and it is robust, i.e., has no ranking reversals even in the presence of malic
9、ious behaviorTheorem we prove Here is a ranking algorithm and incentive structure, which when applied together imply an arbitrage opportunity for the users of the system whenever there is a ranking reversal (even in the presence of malicious behavior),Algorithms and incentives for robust ranking,Whe
10、re is the money?,Examples A: better recommendations more purchases more revenue Netflix: better recommendations increased customer satisfaction increased registration more revenue Google/Yahoo: better ranking more eyeballs more revenue through adsRevenue per entity Simple for A and Netflix For Googl
11、e/Yahoo, we can distribute the revenue from a user on the web pages he looks at (other approaches possible),Algorithms and incentives for robust ranking,Why share?,Because they will take it anyway!,My precious,Algorithms and incentives for robust ranking,Less compelling reasons,Difficulty of eliciti
12、ng honest feedback is well known Resnick et al. Dellarocas Search engine rankings are self-reinforcing Cho, Roy Strong incentive for players to game the system Ballot stuffing and bad mouthing in reputation systems Bhattacharjee, Goel Dellarocas Click spam in web rankings based on clicks Anupam et a
13、l. Web structures have been devised to game PageRank Gyongyi, Garcia-Molina Problem of new entities How should the system discover high quality, new entities in the system? How should the system discover a web page whose relevance has suddenly changed (may be due to some current event)?,Algorithms a
14、nd incentives for robust ranking,Outline,Motivation Model Incentive Structure Ranking Algorithm,Algorithms and incentives for robust ranking,I-U Model,Inspect (I) User reads a snippet attached to a search result (Google/Yahoo) Looks at a recommendation for a book (A)Utilize (U) User goes to the actu
15、al web page (Google/Yahoo) Buys the book (A),Algorithms and incentives for robust ranking,I-U Model,Entities Web pages (Google/Yahoo), Books (A) Each entity i has an inherent quality qi (think of it as the probability that a user would utilize entity i, conditioned on the fact that the entity was in
16、spected by the user) The qualities qi are unknown, but we wish to rank entities according to their qualities Feedback Tokens (positive and negative) placed on an entity by users Ranking is a function of the relative number of tokens received by entities Slots Placeholders for the results of a query,
17、Algorithms and incentives for robust ranking,Sheep and Connoisseurs,Sheep can appreciate a high quality entity when shownBut wouldnt go looking for a high quality entity Most users are sheep,Connoisseurs will dig for a high quality entity which is not ranked high enoughThe goal of this scheme is to
18、aggregate the information that the connoisseurs have,Algorithms and incentives for robust ranking,User response,Algorithms and incentives for robust ranking,I-U Model,User response to a typical query Chooses to inspect the top j positions User chooses j at random from an unknown but fixed distributi
19、on Utility generation event for ei occurs if the user utilizes an entity ei (assuming ei is placed among the top j slots) Formally Utility generation event is captured by random variable Gi = Ir(i) Ui r(i) : rank of entity ei Ir(i),Ui : independent Bernoulli random variables EUi = qi (unknown) EI1 E
20、I2 EIk (known),Algorithms and incentives for robust ranking,Outline,Motivation Model Incentive Structure Ranking Algorithm,Algorithms and incentives for robust ranking,Information Markets,View the problem as an info aggregation problem Float shares of entities and let the market decide their value (
21、ranking) Hanson Pennock Rank according to the price set by the market Work best for predicting outcomes which are objective Elections (Iowa electronic market)Distinguishing features of the ranking problem Fundamental problem: outcome is not objective Revenue: because of more eyeballs or better quali
22、ty? Eyeballs in turn depend on the price set by the market However, an additional lever: the ranking algorithm,Algorithms and incentives for robust ranking,Game theoretic approaches,Example: Miller et al. Framework to incentivize honest feedback Counter lack of objective outcomes by comparing a user
23、s reviews to that of his peers Selfish interests of a user should be in line with the desirable properties of the system Doesnt address malicious users Benefits from the system, may come from outside the system as well Revenue from outcome of these systems might overwhelm the revenue from the system
24、 itself,Algorithms and incentives for robust ranking,Ranking mechanism: overview,Overview: Users place token (positive and negative) on the entities Ranking is computed based on the number of tokens on the entities Whenever a revenue generation event takes place, the revenue is shared among the user
25、s Ranking algorithm Input: feedback scores of entities Output: probabilistic distribution over rankings of the entities Ensures that the number of inspections an entity gets is proportional to the fraction of tokens on it,Algorithms and incentives for robust ranking,Incentive structure,A token is a
26、three tuple: (p, u, e) p : +1 or -1 depending on whether a token is a positive token or a negative token u : user who placed the token e : entity on which the token was placed Net weight of the tokens a user can place is bounded, that is |pi| is bounded User cannot keep placing positive tokens witho
27、ut placing a negative token and vice versa,Algorithms and incentives for robust ranking,User account,Each user has an account Revenue shares are added or deducted from a users account Withdrawal is permitted but deposits are not Users can make profits from the system but not gain control by paying I
28、f a users share goes negative: remove it from the system for some pre-defined timeLet 1 be pre-defined system parameters The fraction of revenue that the system distributes as incentives to the users: Parameter s will be set later,Algorithms and incentives for robust ranking,Revenue share,Suppose a
29、revenue generation event takes place for an entity e at time t R: revenue generated For each token i placed on entity e ai is the net weight (positive - negative) of tokens placed on entity e before token i was placed on e The revenue shared by the system with the user who placed token i is proporti
30、onal to piR/ais Adds up to at most R Negative token: the revenue share is negative, deduct from the users account,Algorithms and incentives for robust ranking,Revenue share,Some features Parameter s controls relative importance of tokens placed earlier Tokens placed after token i have no bearing on
31、the revenue share of the user who placed token i Hence s is strictly greater than 1 Incentive for discovery of high quality entities Hence the choice of diminishing rewards Emphasis is on making the process as implicit as possibleResistance to racing The system shouldnt allow a repeated cycle of act
32、ions which pushes A above B and then B above A and so on We can add more explicit feature by multiplying any negative revenue by (1+) where is an arbitrarily small positive number,Algorithms and incentives for robust ranking,Ranking by quality,Either the entities are ranked by quality, or, there exi
33、sts a profitable arbitrage opportunity for the users in correcting the rankingRanking reversal: A pair of entities (i,k) such that qik qi, qk: quality of entity i and k resp. i, k: number of tokens on entity i and k resp. Revenue/utility generated by the entity: f(r,q) r: relative number of tokens p
34、laced on an entity q: quality of the entity For the I-U Model, our ranking algorithm ensures f(r,q) is proportional to qrObjective: A ranking reversal should present a profitable arbitrage opportunity,Algorithms and incentives for robust ranking,Arbitrage,There exists a pair of entities A and BPlaci
35、ng a positive token on A and placing a negative token on BThe expected profit from A is more than the expected loss from B,1,2,3,4,5,6,7,1,2,3,4,Algorithms and incentives for robust ranking,Proof (for separable rev fns),Suppose f(ri, qi) i-s k and s Hence, f(ri, qi) i-s f(rk, qk) k-s,Algorithms and
36、incentives for robust ranking,Proof (I-U Model),The rate at which revenue is generated by entity i (k) is proportional to (ensured by our ranking algorithm) qii (qkk)Rate at which incentives are generated by placing a positive token on entity k is qkk/ ks Loss due to placing a negative token on enti
37、ty i is qii/ isIf s1, qkk1-s qii1-s qi k (ranking reversal)Thus a profitable arbitrage opportunity exists in correcting the system,Algorithms and incentives for robust ranking,Outline,Motivation Model Incentive Structure Ranking Algorithm,Algorithms and incentives for robust ranking,Naive approach,O
38、rder the entities by the net number of tokens they have Problem? Incentive for manipulationExample: Slot 1: 1,000,000 inspections Slot 2: 500,000 inspections Entity 1: 1000 tokens Entity 2: 999 tokens,Algorithms and incentives for robust ranking,Ranking Algorithm,Proper rankingIf entity e1 has more
39、positive feedback than entity e2, then if the user chooses to inspect the top t (for any t) slots, then the probability that e1 shows up should be higher than the probability that e2 shows up among the top t slotsRandom variable Xe gives the position of entity e Entity e1 dominates e2 if for all t,
40、PrXe1 t PrXe2 t Proper ranking: if the feedback score of e1 is more than the feedback score of e2, then e1 dominates e2 Distribution returned by the algorithm is a proper ranking,Algorithms and incentives for robust ranking,Majorized case,p : vector giving the normalized expected inspections of slot
41、sS = EI1 + EI2 + + EIkp = EI1/S, EI2/S, , EIk/S : vector giving the normalized number of tokens on entities Special case: p majorizes ,For all i, the sum of the i largest components of p is more than the sum of the i largest components of ,Algorithms and incentives for robust ranking,Majorized case,
42、Typically, the importance of top slots in a ranking system is far higher than the lower slots Rapidly decaying tailThe number of entities is order of magnitude more than the number of significant slots Heavy tail Hence for web ranking p majorizes We believe for most applications p majorizes Restrict
43、 to the majorized case here The details of the general case are in the paper,Algorithms and incentives for robust ranking,Hardy, Littlewood, Plya,Theorem Hardy, Littlewood, PlyaThe following two statements are equivalent: (1) The vector x is majorized by the vector y, (2) There exists a doubly stoch
44、astic matrix, D, such that x = DyInterpret Dij as the probability that entity i shows up at position jThis ensures that the number of inspections that an entity gets is directly proportional to its feedback score,Doubly stochastic matrix (Dij 0, j Dij = 1, j Dij = 1),Algorithms and incentives for ro
45、bust ranking,Birkhoff von Neumann Theorem,Hardy, Littlewood, Plya theorem on majorization doesnt guarantee that the ranking we obtain is proper We present a version of the theorem which takes care of thisTheorem Birkhoff, von NeumannAn nxn matrix is doubly stochastic if and only if it is a convex co
46、mbination of permutation matrices Convex combination of permutation matrices Distribution over rankingsAlgorithms for computing Birkhoff von Neumann distribution O(m2) Gonzalez, Sahni O(mn log K) Gabow, Kariv,Algorithms and incentives for robust ranking,Conclusion,Theorem Here is a ranking algorithm
47、 and incentive structure, which when applied together imply an arbitrage opportunity for the users of the system whenever there is a ranking reversalResistance to gaming We dont make any assumptions about the source of the error in ranking - benign or malicious So by the same argument the system is resistant to gaming as wellResistance to racing,Algorithms and incentives for robust ranking,Thank You,
