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Algorithms and Incentives for Robust Ranking

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### Algorithms and Incentives for Robust Ranking

Outline

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.

Outline

- Motivation
- Model
- Incentive Structure
- Ranking Algorithm

Algorithms and incentives for robust ranking

Content generation was centralized (book publishers, movie production companies, newspapers)

Content distribution was subject 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)

Content : then and nowAlgorithms and incentives for robust 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 heuristics[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 weight of edges from S to S

10

w(S) = Local(S) + In(S)

Amp(S) = w(S)/In(S)

HighAmp(S) →Sis dishonest

LowAmp(S) →Sis honest

Collusion free graph:where all sets are honest

S

Algorithms and incentives for robust ranking

Heuristics Race

- Then why do search engines work so well?
- Our belief: because heuristics 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, can’t put them on top slots to gather feedback

Algorithms and incentives for robust ranking

Ranking reversal

Ranking reversal

Entity 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 malicious behavior

- Theorem 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

Where is the money?

- Examples
- Amazon.com: 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 ads

- Revenue per entity
- Simple for Amazon.com and Netflix
- For Google/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!!!

Algorithms and incentives for robust ranking

Less compelling reasons

- Difficulty of eliciting 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 al.]
- 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 and 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 (Amazon.com)

- Utilize (U)
- User goes to the actual web page (Google/Yahoo)
- Buys the book (Amazon.com)

Algorithms and incentives for robust ranking

I-U Model

- Entities
- Web pages (Google/Yahoo), Books (Amazon.com)
- 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 inspected 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

Algorithms and incentives for robust ranking

Sheep and Connoisseurs

- Sheep can appreciate a high quality entity when shown
- But wouldn’t go looking for a high quality entity
- Most users are sheep

- Connoisseurs will dig for a high quality entity which is not ranked high enough
- The goal of this scheme is to 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 distribution
- 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
- E[Ui] = qi (unknown)
- E[I1] ≥ E[I2] ≥ … ≥ E[Ik] (known)

- Utility generation event is captured by random variable

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 (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 quality?
- 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’s reviews to that of his peers
- Selfish interests of a user should be in line with the desirable properties of the system

- Doesn’t 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 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 users

- 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 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 without 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 user’s account
- Withdrawal is permitted but deposits are not
- Users can make profits from the system but not gain control by paying
- If a user’s share goes negative: remove it from the system for some pre-defined time

- Let <1 and s>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

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Revenue share- Suppose a 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 proportional to
piR/ais

- Adds up to at most R
- Negative token: the revenue share is negative, deduct from the user’s 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 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 possible

- Resistance to racing
- The system shouldn’t allow a repeated cycle of actions 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 exists a profitable arbitrage opportunity for the users in correcting the ranking
- Ranking reversal: A pair of entities (i,k) such that qi<qk and i>k
- 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 placed on an entity
- q: quality of the entity
- For the I-U Model, our ranking algorithm ensures f(r,q) is proportional to qr

- Objective: A ranking reversal should present a profitable arbitrage opportunity

Algorithms and incentives for robust ranking

5

Arbitrage- There exists a pair of entities A and B
- Placing a positive token on A and placing a negative token on B
- The expected profit from A is more than the expected loss from B

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Algorithms and incentives for robust ranking

Proof (for separable rev fns)

- Suppose f(ri, qi) i-s < f(rk, qk) k-s
- ri = i (ll)-1, rk= k(ll)-1
- It is profitable to put a negative token on entity i and a positive token on entity k

- Assumption: f is separable, that is f(r,q) = qr
- Choose parameter s greater than

- f(ri, qi) i-s < f(ri, qk) i-s
- f is increasing in q

- f(ri, qk) i-s = qkri i-s = qki-s (ll)-
- Definition of separable function

- Similarly f(rk, qk) k-s = qk rk k-s = qkk-s (ll)-
- However qki-s(ll)- < qkk-s (ll)-
- i > k and s >

- Hence, f(ri, qi) i-s < f(rk, qk) k-s

Algorithms and 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 entity i is qii/ is

- If s>1, qkk1-s > qii1-s
- qi < qk (ranking reversal)
- i> k (ranking reversal)

- Thus a profitable arbitrage opportunity exists in correcting the system

Algorithms and incentives for robust ranking

- Motivation
- Model
- Incentive Structure
- Ranking Algorithm

Algorithms and incentives for robust ranking

Naive approach

- Order the entities by the net number of tokens they have
- Problem?
- Incentive for manipulation

- Example:
- 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 ranking
- If entity e1 has more 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 slots
- Random variable Xe gives the position of entity e
- Entity e1 dominates e2 if for all t, Pr[Xe1 ≤t] ≥ Pr[Xe2 ≤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 slots
- S = E[I1] + E[I2] + … + E[Ik]
- p = {E[I1]/S, E[I2]/S, …, E[Ik]/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

- Typically, the importance of top slots in a ranking system is far higher than the lower slots
- Rapidly decaying tail

- The 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 to the majorized case here
- The details of the general case are in the paper

Algorithms and incentives for robust ranking

=1

Hardy, Littlewood, Pólya- Theorem [Hardy, Littlewood, Pólya]
- The following two statements are equivalent: (1) The vector x is majorized by the vector y, (2) There exists a doubly stochastic matrix, D, such that x = Dy

- Interpret Dij as the probability that entity i shows up at position j
- This 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 robust ranking

Birkhoff von Neumann Theorem

- Hardy, Littlewood, Pólya theorem on majorization doesn’t guarantee that the ranking we obtain is proper
- We present a version of the theorem which takes care of this

- Theorem [Birkhoff, von Neumann]
- An nxn matrix is doubly stochastic if and only if it is a convex combination of permutation matrices
- Convex combination of permutation matrices Distribution over rankings

- Algorithms 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 and incentive structure, which when applied together imply an arbitrage opportunity for the users of the system whenever there is a ranking reversal

- Resistance to gaming
- We don’t 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 well

- Resistance to racing

Algorithms and incentives for robust ranking

Thank You

Algorithms and incentives for robust ranking

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