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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

Traditional

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

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)

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

7

6

5

4

3

2

1

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

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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