Collaboration of Untrusting Peers with Changing Interests

1
Collaboration of Untrusting Peers with Changing
Interests

• Baruch Awerbuch, Boaz Patt-Shamir, David Peleg,
  Mark Tuttle
• Review by Pinak Pujari

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Introduction

• Reputation systems are an integral part of
  e-commerce application systems.
• Engines like eBay depend on reputation systems to
  improve customer confidence.
• More importantly, they limit the economic damage
  done by disreputable peers.

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Introduction eBay example

• For Instance in eBay, after every transaction,
  the system invites each party to post its rating
  of the transaction on a public billboard the
  system maintains.
• Consulting the billboard is a key step before
  making a transaction.

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Introduction Possibility of fraud?

• Scene 1 A group of sellers engaging in phony
  transactions, and rating these transactions
  highly to generate an appearance of reliability
  while ripping off other people.
• Scene 2 A single seller behaving in responsible
  manner long enough to entice an unsuspecting
  buyer into a single large transaction, and then
  vanishing.
• Reputation systems are valuable, but not
  infallible.

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Model of the Reputation System

• n players. (Some honest, some dishonest)
• m objects. (Some good, some bad)
• Player probes an object to learn if it good or
  bad.
• The cost of the probe is 1 if the object is bad
  and 0 if the object is good.
• Goal Find a good object incurring minimal cost.

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Model of the Reputation System

• Players collaborate by posting the results of
  their probes on a public billboard.
• And, consulting the board when choosing an object
  to probe.
• Assume that entries are write-once, and that
  billboard is reliable.

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So what is the problem?

• Problem Definition
• Some of the players are dishonest, and can behave
  in an arbitrary fashion, including colluding and
  posting false reports on the billboard to entice
  honest players to probe bad objects.

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Model of the Reputation System (contd.)

• The execution of the system is as follows-
• A player reads the billboard, optionally probes
  an object, and writes to the billboard. (Some
  randomized protocol is used that chooses the
  object to probe based on the contents of the
  billboard)
• Honest players are required to follow the
  protocol.
• But dishonest players are allowed to behave in an
  arbitrary (or Byzantine) fashion, including
  posting incorrect information on the billboard.

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Strategy

• Exploration rule A player should choose an
  object at random (uniformly) and probe it.
• This might be a good idea if there are a lot of
  good objects, or if there are a lot of dishonest
  players posting inaccurate reports to the
  billboard.
• Exploitation rule A player should choose another
  player at random and probe whichever object it
  recommends (if any), thereby exploiting or
  benefiting from the effort the other player.
• This might be a good idea most of the players
  posting recommendations to the billboard are
  honest.

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The Balanced Rule

• In most cases, the player will not know how many
  honest players or good objects are in the system.
  So best option would be to balance between the
  two approaches.
• Flip a coin. If the result is heads, follow
  Exploration rule. If the result is tails,
  follow Exploitation rule.

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Models with Restricted Access to players

• Dynamic object model Objects can enter and
  leave the system over time.
• Partial access model Each player has access to
  a different subset of the objects.

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Model of the Reputation System (contd.)

• The execution of an algorithm is uniquely
  determined by the algorithm, the coins flipped by
  the players while executing the protocol, and by
  three external entities
• Three external entities
• The player schedule that determines the order in
  which players take steps.
• The dishonest players.
• The adversary that determines the behavior of the
  dishonest players.

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Model of the Reputation System (contd.)

• What is the adversary?
• The adversary is a function from a sequence of
  coin flips to a sequence of objects for each
  dishonest player to probe and the results for the
  player to post on the billboard.
• Adversary is quite powerful, and may behave in an
  adaptive, Byzantine fashion.

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Model of the Reputation System (contd.)

• What is an operating environment?
• An operating environment is a triple consisting
  of a player schedule, a set of dishonest players,
  and an adversary.
• The purpose of the operating environment is to
  factor out all of the nondeterministic choices
  made during an execution, leaving only the
  probabilistic choices to consider.

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Models with Restricted Access to players

• Dynamic object model Objects can enter and
  leave the system over time.
• Partial access model Each player has access to
  a different subset of the objects.

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The Dynamic Object Model

• Operating Environment
• The player schedule.
• The dishonest players.
• The adversary.
• The object schedule that determines when objects
  enter and leave the system, and their values.
• m - upper bound on the number of objects
  concurrently present in the system.
• ß - lower bound on the fraction of good objects
  at any time, for some 0 ß 1.

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The Dynamic Object Model Algorithm

• The algorithm is an immediate application of the
  Balanced rule.
• Algorithm DynAlg If the player has found a good
  object, then probe it again. If not, then apply
  the Balanced rule.

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Analysis of Algorithm DynAlg

• Given a probe sequence s, switches(s) denotes the
  number of distinct objects in s.
• Given an operating environment E, let sE(DynAlg)
  be the random variable whose value is the probe
  sequence of the honest players generated by
  DynAlg under E.
• s - the cost of an optimal probe sequence.

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Analysis of Algorithm DynAlg

• Theorem For every operating environment E and
  every probe sequence s for the honest players,
  the expected cost of sE(DynAlg) is at most
• cost(s) switches(s)(2-ß)(m n ln n))

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Proof

• Partition the sequence s into subsequences
• s s1s2 s K such that for all 1iltK,
• -gt all probes in si are to the same object.
• -gt si and si1 probe different objects.
• Similarly, Partition the sequence s into
  subsequences s s1s2 s K such that,
• si si for all 1 i K.

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Proof

• Consider the difference cost(si) - cost(si).
• If the probes in si are to a bad object,
• then trivially cost(si) cost(si).
• To finish the proof, we show that
• If all probes in si are to a good object,
• then cost(si) (2 - ß).(m n ln n).

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Proof

• An object i-persistent if it is good and ifvit is
  present in the system throughout the duration of
  si.
• A probe i-persistent if it probes an i-persistent
  object.
• Partition the sequence si into n subsequences si
  Di0Di1Di2 Din, where Dik consists of all
  probes in si that are preceded by i-persistent
  probes of exactly k distinct honest players.

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Proof

• Obviously, cost (si) Snk0 cost(Dik).
• The expected cost of a single fresh probe in Dik
  is at most 1-ß/2.
• Each fresh probe in Dik finds a persistent object
  with some probability pk.
• The probability that Dik contains exactly l fresh
  probes is (1 - pk)l-1pk.
• Therefore, the expected cost of Dik is at most

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Proof

• For k 0, p0 1/2m.
• For k gt 0, pk k/2n.
• So, expected cost of si is at most

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The Partial Access Model

• Here, each player is able to access only a subset
  of the objects.
• The main problem with this model is that in
  contrast to the full access model (where each
  player can access any object), when we have
  partial access, it is difficult to measure the
  amount of collaboration a player can expect from
  other players in searching for a good object.
• To overcome this difficulty is to concentrate on
  the amount of collective work done by subsets of
  players.

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The Partial Access Model

• Notation
• Model the partial access to the objects with a
  bipartite graph G (P,O,E)
• P is the set of players
• O is the set of objects
• A player j can access an object i only if (j, i)
  belongs to E.
• For each player j, let obj(j) denote the set of
  objects accessible to j, and let deg(j)
  obj(j).
• For each honest player j, let best(j) denote the
  set of good objects accessible to j.
• Let N(j) be the set of all players (honest and
  dishonest) that are at distance 2 from a given
  player j, i.e.,

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The Partial Access Model Algorithm

• Algorithm is same as DynAlg from the dynamic
  model, except that the Balanced rule is adapted
  to the restricted access model.
• In the new rule, a player j flips a coin. If the
  result is heads, it probes an object selected
  uniformly at random from obj(j). Exploration
  Rule
• If the result is tails, it selects a player k
  uniformly at random from N(j) and probes the
  object k recommends, if any and otherwise it
  probes an object selected uniformly at random
  from obj(j). Exploitation Rule

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The Partial Access Model

• Theorem
• Let Y be any set of honest players.
• Denote
• Let
• If X(Y ) in nonempty, then the total work of
  players in Y is at most

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Interpretation

• Consider any set Y of players with common
  interest X(Y) (meaning any object in X(Y) would
  satisfy any player in Y ).
• From the point of view of a player, its load is
  divided among the members of Y the total work
  done by the group working together is roughly the
  same as the work of an individual working alone.
• The first term in the bound is just an upper
  bound on expected amount of work until a player
  finds an object in X(Y).
• The second term is an upper bound on the total
  number of recommendations (times a logarithmic
  factor) a player has to go through.
• This is pleasing, because it indicates that the
  number of probes is nearly the best one can hope
  for.

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Collaboration across groups without common
interest

• Consider sets of players who do not share a
  common interest. Of course, one can partition
  them into subsets SIGs (special interest groups),
  where for each SIG there is at least one object
  that will satisfy all its members.
• The Theorem guarantees that each SIG is nearly
  optimal.
• In the sense that the total work done by a SIG is
  not much more than the total work that must be
  done even if SIG members had perfect coordination
  (thus disregarding dishonest players).
• However, the collection of SIGs may be
  suboptimal, due to overlaps in the neighborhood
  sets (which contribute to the second term of the
  upper bound).

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Collaboration across groups without common
interest

• Does there always exists a good partition of
  players into SIGs, so that the overall work
  (summed over all SIGs) is close to optimal?
• The answer is negative in the general case.
• Even if each good object would satisfy many
  honest players, the total amount of work, over
  all players, is close to the worst case (being
  the sum of work necessary if each player is
  working alone).

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Simulation

• The graph suggests that
• the algorithm works fairly
• well for values of p 0.1
• through p 0.7.
• It suggests that a little
• sampling is necessary, and
• a few recommendations
• can really help a lot

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Conclusion

• This paper shows that, in spite of asynchronous
  behavior, different interests, changes in time,
  and Byzantine behavior of unknown subset of
  peers, the honest peers miraculously succeed in
  collaborating, in the sense that the honest peers
  relatively rarely repeat mistakes of other honest
  peers. One interesting feature of our method is
  that we mostly avoid the issue of discovering who
  the faulty peers are.
• Future Extensions?
• How can it be gained by trying to discover the
  faulty peers.
• Another open question is tightening the bounds
  for the partial access case.