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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
• 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.

4
Heuristics Race
• PageRank (uses link structure of the web)
• Spammers try to game the system by creating
• 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 ?
• 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
• Looks at a recommendation for a book (Amazon.com)
• Utilize (U)
• User goes to the actual web page (Google/Yahoo)

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
• 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
• 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