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Algorithms and Incentives for Robust Ranking

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Title: Algorithms and Incentives for Robust Ranking


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.
2
Outline
  • Motivation
  • Model
  • Incentive Structure
  • Ranking Algorithm

3
Content then and now
  • 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)

4
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.

5
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)
High Amp(S) ? S is dishonest Low Amp(S) ? S is
honest Collusion free graph where all sets are
honest
S
6
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, cant put them on top slots
    to gather feedback

7
Ranking reversal
  • Ranking reversal
  • Entity A is better than entity B, but B is
    ranked higher than A

Keyword Search Engine
8
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)

9
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)

10
Why share?
Because they will take it anyway!!!
11
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)?

12
Outline
  • Motivation
  • Model
  • Incentive Structure
  • Ranking Algorithm

13
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)

14
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

15
Sheep and Connoisseurs
  • Sheep can appreciate a high quality entity when
    shown
  • But 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 enough
  • The goal of this scheme is to aggregate the
    information that the connoisseurs have

16
User response
17
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
  • EUi qi (unknown)
  • EI1 EI2 … EIk (known)

18
Outline
  • Motivation
  • Model
  • Incentive Structure
  • Ranking Algorithm

19
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

20
Game theoretic approaches
  • Example Miller et al.
  • Framework to incentivize honest feedback
  • Counter lack of objective outcomes by comparing a
    users 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 itself

21
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

22
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

23
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
  • If a users share goes negative remove it from
    the system for some pre-defined time
  • Let ?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

24
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
  • pi?R/ais
  • Adds up to at most ?R
  • Negative token the revenue share is negative,
    deduct from the users account

25
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 shouldnt 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

26
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?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

27
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

7
4
6
3
5
2
4
1
3
2
1
28
Proof (for separable rev fns)
  • Suppose f(ri, qi) ?i-s
  • ri ?i (?l ?l)-1, rk ?k(?l ?l)-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 is increasing in q
  • f(ri, qk) ?i-s qkri? ?i-s qk ?i?-s (?l ?l)-?
  • Definition of separable function
  • Similarly f(rk, qk) ?k-s qk rk? ?k-s qk ?k?-s
    (?l ?l)-?
  • However qk?i?-s(?l ?l)-???
  • ?i ?k and s ?
  • Hence, f(ri, qi) ?i-s

29
Proof (I-U Model)
  • The rate at which revenue is generated by entity
    i (k) is proportional to (ensured by our ranking
    algorithm) qi?i (qk?k)
  • Rate at which incentives are generated by placing
    a positive token on entity k is qk?k/ ?ks
  • Loss due to placing a negative token on entity i
    is qi?i/ ?is
  • If s1, qk?k1-s qi?i1-s
  • qi
  • ?i ?k (ranking reversal)
  • Thus a profitable arbitrage opportunity exists in
    correcting the system

30
Outline
  • Motivation
  • Model
  • Incentive Structure
  • Ranking Algorithm

31
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

32
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, 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

33
Majorized case
p vector giving the normalized expected
inspections of slots S EI1 EI2 …
EIk p 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 ?
34
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

35
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)

36
Birkhoff von Neumann Theorem
  • Hardy, Littlewood, Pólya theorem on majorization
    doesnt 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

37
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 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 well
  • Resistance to racing

38
Thank You
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