Collaboration of Untrusting Peers with Changing Interests Baruch Awerbuch, Boaz Patt-Shamir, David Peleg, Mark Tuttle Review by Pinak Pujari

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.

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.

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.

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.

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.

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.

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.

Strategy 1. 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. 2. 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.

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.

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.

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.

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.

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.

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.

The Dynamic Object Model  Operating Environment: 1. The player schedule. 2. The dishonest players. 3. The adversary. 4. 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.  β - lower bound on the fraction of good objects at any time, for some 0 ≤ β ≤ 1.

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.

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

Analysis of Algorithm DynAlg Theorem: For every operating environment E and every probe sequence σ ∗ for the honest players, the expected cost of σ E (DynAlg) is at most cost(σ ∗ ) + switches(σ ∗ )·(2−β)(m + n ln n))

Proof: Partition the sequence σ ∗ into subsequences σ ∗ = σ 1 *σ 2 ∗ · · · σ K ∗ such that for all 1≤i<K, -> all probes in σ i ∗ are to the same object. -> σ i ∗ and σ i+1 ∗ probe different objects. Similarly, Partition the sequence σ into subsequences σ = σ 1 σ 2 · · · σ K such that, |σ i ∗ | = |σ i | for all 1 ≤ i ≤ K.

Proof: Consider the difference cost(σ i ∗ ) − cost(σ i ). If the probes in σ i ∗ are to a bad object, then trivially cost(σ i ) ≤ cost(σ i ∗ ). To finish the proof, we show that If all probes in σ i ∗ are to a good object, then cost(σ i ) ≤ (2 − β).(m + n ln n).

Proof: An object i-persistent if it is good and ifvit is present in the system throughout the duration of σ i ∗. A probe i-persistent if it probes an i- persistent object. Partition the sequence σ i into n subsequences σ i = D i 0 D i 1 D i 2 · · ·D i n, where D i k consists of all probes in σ i that are preceded by i-persistent probes of exactly k distinct honest players.

Proof: Obviously, cost (σi) = Σ n k=0 cost(D i k ). The expected cost of a single fresh probe in D i k is at most 1−β/2. Each fresh probe in D i k finds a persistent object with some probability p k. The probability that D i k contains exactly ℓ fresh probes is (1 − p k )ℓ−1p k. Therefore, the expected cost of D i k is at most

Proof:  For k = 0, p 0 ≥ 1/2m.  For k > 0, p k ≥ k/2n.  So, expected cost of σ i is at most:

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.

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.,

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]

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

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.

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).

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).

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

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? 1. How can it be gained by trying to discover the faulty peers. 2. Another open question is tightening the bounds for the partial access case.