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Asymptotic Risk Analysis for Trust and ReputationSystems
Michele Boreale1 Alessandro Celestini2
1 Universita di Firenze, Italy2 IMT Institute for Advanced Studies Lucca, Italy
39th International Conference on Current Trends in Theory andPractice of Computer Science
Spindleruv Mlyn, 29th January 2013
M. Boreale A. Celestini SOFSEM - January 29, 2013 1 / 26
Context and Motivation
Trust and Reputation Systems: decision support tools used to driveparties’ interactions on the basis of parties’ reputation.
Examples: eBay, TripAdvisor, Amazon, iTunes Store, Android Store, ...
Goal: to assess confidence in the decisions calculated by trust andreputation systems.
M. Boreale A. Celestini SOFSEM - January 29, 2013 2 / 26
Context and Motivation
Trust and Reputation Systems: decision support tools used to driveparties’ interactions on the basis of parties’ reputation.
Examples: eBay, TripAdvisor, Amazon, iTunes Store, Android Store, ...
Goal: to assess confidence in the decisions calculated by trust andreputation systems.
M. Boreale A. Celestini SOFSEM - January 29, 2013 2 / 26
Example: A Generic Trust and Reputation System
Interactions: parties in a trust and reputation system are free to interact.
Rating Values: after each interaction parties rate each other.
Parties
rater
ratee Reputation: aggregate ratings areused to compute reputation scoresfor a given party.
Computational Trust: parties’ trustworthiness is evaluated on the basisof parties’ past behaviours.
M. Boreale A. Celestini SOFSEM - January 29, 2013 3 / 26
Example: A Generic Trust and Reputation System
Interactions: parties in a trust and reputation system are free to interact.
Rating Values: after each interaction parties rate each other.
Parties
rater
ratee Reputation: aggregate ratings areused to compute reputation scoresfor a given party.
Computational Trust: parties’ trustworthiness is evaluated on the basisof parties’ past behaviours.
M. Boreale A. Celestini SOFSEM - January 29, 2013 3 / 26
Example: A Generic Trust and Reputation System
Interactions: parties in a trust and reputation system are free to interact.
Rating Values: after each interaction parties rate each other.
Parties
rater
ratee
Reputation: aggregate ratings areused to compute reputation scoresfor a given party.
Computational Trust: parties’ trustworthiness is evaluated on the basisof parties’ past behaviours.
M. Boreale A. Celestini SOFSEM - January 29, 2013 3 / 26
Example: A Generic Trust and Reputation System
Interactions: parties in a trust and reputation system are free to interact.
Rating Values: after each interaction parties rate each other.
Parties
rater
ratee
Reputation: aggregate ratings areused to compute reputation scoresfor a given party.
Computational Trust: parties’ trustworthiness is evaluated on the basisof parties’ past behaviours.
M. Boreale A. Celestini SOFSEM - January 29, 2013 3 / 26
Example: A Generic Trust and Reputation System
Interactions: parties in a trust and reputation system are free to interact.
Rating Values: after each interaction parties rate each other.
Parties
rater
ratee
Reputation: aggregate ratings areused to compute reputation scoresfor a given party.
Computational Trust: parties’ trustworthiness is evaluated on the basisof parties’ past behaviours.
M. Boreale A. Celestini SOFSEM - January 29, 2013 3 / 26
Probabilistic Trust
Probabilistic Trust: party’s behaviour can be modeled as a probabilitydistribution, drawn from a given family, over a certain set of interactionoutcomes.
Examples:
The simplest case is to assume a set of binary outcomes representingsuccess and failure.
Another possibility is to rate a service’ quality by an integer value in arange of n + 1 values, {0, 1, ..., n}.
Goal: the task of computing reputation scores boils down to inferring thetrue distribution’s parameters for a given party.
M. Boreale A. Celestini SOFSEM - January 29, 2013 4 / 26
Probabilistic Trust
Probabilistic Trust: party’s behaviour can be modeled as a probabilitydistribution, drawn from a given family, over a certain set of interactionoutcomes.
Examples:
The simplest case is to assume a set of binary outcomes representingsuccess and failure.
Another possibility is to rate a service’ quality by an integer value in arange of n + 1 values, {0, 1, ..., n}.
Goal: the task of computing reputation scores boils down to inferring thetrue distribution’s parameters for a given party.
M. Boreale A. Celestini SOFSEM - January 29, 2013 4 / 26
Probabilistic Trust
Probabilistic Trust: party’s behaviour can be modeled as a probabilitydistribution, drawn from a given family, over a certain set of interactionoutcomes.
Examples:
The simplest case is to assume a set of binary outcomes representingsuccess and failure.
Another possibility is to rate a service’ quality by an integer value in arange of n + 1 values, {0, 1, ..., n}.
Goal: the task of computing reputation scores boils down to inferring thetrue distribution’s parameters for a given party.
M. Boreale A. Celestini SOFSEM - January 29, 2013 4 / 26
Context and Motivation
Principal questions
• How do we quantify the confidence in the decisions calculated by thesystem?
• How is this confidence related to such parameters as decision strategyand number of available ratings?
• Is there an optimal strategy that maximizes confidence as more andmore information becomes available?
M. Boreale A. Celestini SOFSEM - January 29, 2013 5 / 26
Context and Motivation
In order to answer these questions, we are interested in:
• a general framework to analyse probabilistic trust systems based onbayesian decision theory
• loss functions for evaluating decisions’ consequences, L(·, ·)• expected and worst-case loss, respectively rn(·, ·) and wn(·), for
quantifying confidence in the systems
• expressions for the limit value as n→∞ and the rate of convergence
M. Boreale A. Celestini SOFSEM - January 29, 2013 6 / 26
Outline
1 Formal Set UpLoss and Decision FunctionsEvaluation of Decision Functions
2 Results
3 Examples
4 Conclusions
M. Boreale A. Celestini SOFSEM - January 29, 2013 7 / 26
Formal Set Up
Observation framework: describes how observations are probabilisticallygenerated.
Observation framework: S = (O,Θ,F , π(·))
• O is a finite non-empty set of observations
• Θ is a set of world states, or parameters
• F = {p(·|θ)}θ∈Θ is a set of probability distributions on O indexed byΘ
• π(·) is an a priori probability measure on Θ
M. Boreale A. Celestini SOFSEM - January 29, 2013 8 / 26
Formal Set Up
Observation framework: S = (O,Θ,F , π(·))
• O is a finite non-empty set of observations
• Θ is a set of world states, or parameters
• F = {p(·|θ)}θ∈Θ is a set of probability distributions on O indexed byΘ
• π(·) is an a priori probability measure on Θ
Assumption: the sequence on = (o1, ..., on) is a realization of a randomvector On = (O1, ...On), where the r.v. Oi ’s are i.i.d. given θ ∈ Θ
M. Boreale A. Celestini SOFSEM - January 29, 2013 8 / 26
Formal Set Up
Example: S = (O,Θ,F , π(·))
A simple possibility is to assume a set of binary outcomes, representingsuccess and failure, O = {o, o}, generated according to a Bernoullidistribution:
p(o|θ) = θ and p(o|θ) = 1− θ, where θ ∈ Θ ⊆ (0, 1).
M. Boreale A. Celestini SOFSEM - January 29, 2013 8 / 26
Formal Set Up
Example: S = (O,Θ,F , π(·))
Another possibility is to rate a service’ quality by an integer value in arange of n + 1 values, O = {0, 1, ..., n}. In this case, we can model parties’behaviour by binomial distribution Bin(n, θ), with θ ∈ Θ ⊆ (0, 1).
The probability of an outcome o ∈ O for an interaction with a party witha behaviour θ is p(o|θ) =
(no
)θo(1− θ)n−o .
M. Boreale A. Celestini SOFSEM - January 29, 2013 8 / 26
1 Formal Set UpLoss and Decision FunctionsEvaluation of Decision Functions
2 Results
3 Examples
4 Conclusions
M. Boreale A. Celestini SOFSEM - January 29, 2013 9 / 26
Loss and Decision Functions
Decision functions: formalise the decision-making process. For any n, adecision function is a function g (n) : On → D .
Two main types of decisions
• Evaluate party’s behaviour (reputation).
• Predict the outcome of the next interaction.
Examples
ML, g (ML)(on) = arg minθ D(ton ||p(·|θ))
MAP, g (MAP)(on) = θ implies p(θ|on) ≥ p(θ′|on) for each θ′ ∈ Θ
M. Boreale A. Celestini SOFSEM - January 29, 2013 10 / 26
Loss and Decision Functions
Decision functions: formalise the decision-making process. For any n, adecision function is a function g (n) : On → D .
Two main types of decisions
• Evaluate party’s behaviour (reputation).
• Predict the outcome of the next interaction.
Examples
ML, g (ML)(on) = arg minθ D(ton ||p(·|θ))
MAP, g (MAP)(on) = θ implies p(θ|on) ≥ p(θ′|on) for each θ′ ∈ Θ
M. Boreale A. Celestini SOFSEM - January 29, 2013 10 / 26
Loss and Decision Functions
Loss functions: evaluate the consequences of possible decisionsassociating a loss to each decision, L(·, ·) = Θ×D → R+.
L(θ, d) quantifies the loss incurred when making a decision d ∈ D , giventhat the real behaviour of the party is θ ∈ Θ.
Assumption: for each θ ∈ Θ, we assume a decision dθ ∈ D exists thatminimizes the loss given θ
Examples
Norm-1 distance, L(θ, θ′) = ||p(·|θ)− p(·|θ′)||1KL-divergence, L(θ, θ′) = D(p(·|θ′)||p(·|θ))
M. Boreale A. Celestini SOFSEM - January 29, 2013 10 / 26
Loss and Decision Functions
Loss functions: evaluate the consequences of possible decisionsassociating a loss to each decision, L(·, ·) = Θ×D → R+.
L(θ, d) quantifies the loss incurred when making a decision d ∈ D , giventhat the real behaviour of the party is θ ∈ Θ.
Assumption: for each θ ∈ Θ, we assume a decision dθ ∈ D exists thatminimizes the loss given θ
Examples
Norm-1 distance, L(θ, θ′) = ||p(·|θ)− p(·|θ′)||1KL-divergence, L(θ, θ′) = D(p(·|θ′)||p(·|θ))
M. Boreale A. Celestini SOFSEM - January 29, 2013 10 / 26
Loss and Decision Functions
Loss functions: evaluate the consequences of possible decisionsassociating a loss to each decision, L(·, ·) = Θ×D → R+.
L(θ, d) quantifies the loss incurred when making a decision d ∈ D , giventhat the real behaviour of the party is θ ∈ Θ.
Assumption: for each θ ∈ Θ, we assume a decision dθ ∈ D exists thatminimizes the loss given θ
Examples
Norm-1 distance, L(θ, θ′) = ||p(·|θ)− p(·|θ′)||1KL-divergence, L(θ, θ′) = D(p(·|θ′)||p(·|θ))
M. Boreale A. Celestini SOFSEM - January 29, 2013 10 / 26
Loss and Decision Functions
Decision framework: describes how decisions are taken.
Decision framework: DF = (S ,D ,L(·, ·), {g (n)}n≥1)
• S = (O,Θ,F , π(·)) is an observation framework
• D is a decision set
• L(·, ·) = Θ×D → R+ is a loss function
• {g (n)}n≥1 is a family of decision functions, one for each n ≥ 1,g (n) : On → D
Reputation framework → D = Θ Prediction framework → D = O
M. Boreale A. Celestini SOFSEM - January 29, 2013 10 / 26
Loss and Decision Functions
Decision framework: describes how decisions are taken.
Decision framework: DF = (S ,D ,L(·, ·), {g (n)}n≥1)
• S = (O,Θ,F , π(·)) is an observation framework
• D is a decision set
• L(·, ·) = Θ×D → R+ is a loss function
• {g (n)}n≥1 is a family of decision functions, one for each n ≥ 1,g (n) : On → D
Reputation framework → D = Θ Prediction framework → D = O
M. Boreale A. Celestini SOFSEM - January 29, 2013 10 / 26
1 Formal Set UpLoss and Decision FunctionsEvaluation of Decision Functions
2 Results
3 Examples
4 Conclusions
M. Boreale A. Celestini SOFSEM - January 29, 2013 11 / 26
Evaluation of Decision Functions
Frequentist risk: for a parameter θ ∈ Θ, the frequentist risk associated toa decision function g after n observation is just the expected losscomputed over On,
Rn(θ, g) =∑
on∈On
p(on|θ)L(θ, g(on)).
Intuition: expected loss for a fixed behaviour θ
M. Boreale A. Celestini SOFSEM - January 29, 2013 12 / 26
Evaluation of Decision Functions
Bayes risk: is the expected value of the risk Rn(θ, g), computed withrespect to the a priori distribution π(·),
rn(π, g) = Eπ[Rn(θ, g)] =∑θ
π(θ)Rn(θ, g).
The minimum bayes risk is defined as r∗ =∑
Θ π(θ)L(θ, dθ).
Intuition: calculating the expected loss of the system considering user’sbelief over possible behaviours.
M. Boreale A. Celestini SOFSEM - January 29, 2013 13 / 26
Evaluation of Decision Functions
Worst risk: is the maximum risk Rn(θ, g) over possible parameters θ ∈ Θ,
wn(g) = maxθ∈Θ
Rn(θ, g).
The minimum worst risk is defined as w∗ = maxθ∈Θ L(θ, dθ)
Intuition: maximum expected loss over all possible behaviours
M. Boreale A. Celestini SOFSEM - January 29, 2013 13 / 26
Evaluation of Decision Functions
Limit values: we study the behaviour of bayes and worst risk when anincreasing number of ratings is available (n→∞).
Exponential convergence: limit values for both risks are achievableexponentially fast (2−nρ).
Rate: the exponent ρ determine how fast the limit is approached.
M. Boreale A. Celestini SOFSEM - January 29, 2013 14 / 26
1 Formal Set UpLoss and Decision FunctionsEvaluation of Decision Functions
2 Results
3 Examples
4 Conclusions
M. Boreale A. Celestini SOFSEM - January 29, 2013 15 / 26
Some results
Best achievable rate: (for any decision function) the upper bound is theleast Chernoff Information
Theorem
if lim rn(π, g) = r∗ then
rate(rn(π, g)) ≤ minθ 6=θ′
C (pθ, pθ′)︸ ︷︷ ︸least Chernoff Information
Similarly for the worst risks wn and w∗.
M. Boreale A. Celestini SOFSEM - January 29, 2013 16 / 26
Some results
Asymptotically optimal: both map and ml are asymptotically optimaldecision functions
Theorem: g either map or ml
limn
rn = r∗ and rate(rn) = minθ 6=θ′
C (pθ, pθ′)
Similarly for wn.
M. Boreale A. Celestini SOFSEM - January 29, 2013 17 / 26
1 Formal Set UpLoss and Decision FunctionsEvaluation of Decision Functions
2 Results
3 Examples
4 Conclusions
M. Boreale A. Celestini SOFSEM - January 29, 2013 18 / 26
Example 1: System assessment
Peers’ behaviour: Bernoulli distribution B(θ) over the set O = {0, 1}.
Parameters set: Θ is a discrete set of N points 0 < γ, 2γ, ...,Nγ < 1, fora positive parameter γ.
Loss function: L(θ, θ′) = ||p(·|θ)− p(·|θ′)||1.
Decision function: g is a ml reputation function.
Priori distribution: uniform distribution π(·) over Θ.
M. Boreale A. Celestini SOFSEM - January 29, 2013 19 / 26
Example 1: System assessment
Peers’ behaviour: Bernoulli distribution B(θ) over the set O = {0, 1}.
Parameters set: Θ is a discrete set of N points 0 < γ, 2γ, ...,Nγ < 1, fora positive parameter γ.
Loss function: L(θ, θ′) = ||p(·|θ)− p(·|θ′)||1.
Decision function: g is a ml reputation function.
Priori distribution: uniform distribution π(·) over Θ.
M. Boreale A. Celestini SOFSEM - January 29, 2013 19 / 26
Example 1: System assessment
Goals:
• Study the rate of convergence of the risk functions depending on γ.
• Compare the analytical approximations of the risk functions with theempirical values.
rn ≈ r∗ + 2−nR and wn ≈ w∗ + 2−nR
where R = minθ 6=θ′ C (pθ, pθ′)
M. Boreale A. Celestini SOFSEM - January 29, 2013 19 / 26
Example 1: System assessment
Intuition: for large values of γ, the incurred loss will be exactly zero. Forsmall values of γ, the incurred loss will be small but nonzero in most cases.
M. Boreale A. Celestini SOFSEM - January 29, 2013 20 / 26
Example 2: System assessment
Peers’ behaviour: Bernoulli distribution B(θ) over the set O = {0, 1}.
Parameters set: Θ = {0 < γ, 2γ, ...,Nγ < 1}, for fixed γ = 0.2.
Loss function: L(θ, θ′) = ||p(·|θ)− p(·|θ′)||1.
Decision functions: g1 ml and g2 map
Priori distribution: binomial distribution centered on the value θ = 0.5,Bin(|Θ|, 0.5).
M. Boreale A. Celestini SOFSEM - January 29, 2013 21 / 26
Example 2: System assessment
Goal:
• Analyse a system with respect to the use of different reputationfunctions.
M. Boreale A. Celestini SOFSEM - January 29, 2013 21 / 26
Example 2: System assessment
Intuition: map takes advantage of the a priori knowledge represented byπ(·)
M. Boreale A. Celestini SOFSEM - January 29, 2013 22 / 26
1 Formal Set UpLoss and Decision FunctionsEvaluation of Decision Functions
2 Results
3 Examples
4 Conclusions
M. Boreale A. Celestini SOFSEM - January 29, 2013 23 / 26
Conclusions
• We proposed a framework based on bayesian decision theory toanalyse trust and reputation systems
• We examinated the behaviour of two risk quantities: bayes and worstrisks to quantify confidency in system’s decisions.
Our results allow to characterize the asymptotic behaviour of probabilistictrust systems :
• showing how to determine limits value of both bayes and worst risks,and their exact exponential rates of convergence
• showing that ml and map decision functions are asymptoticallyoptimal
M. Boreale A. Celestini SOFSEM - January 29, 2013 24 / 26
Further developments
• Extend the present framework to different data models, with ratingvalues released in different ways. (e.g. parties under- or over-evaluateteir interactions)
• How to evaluate the fitness of the model to the data actuallyavailable.
M. Boreale A. Celestini SOFSEM - January 29, 2013 25 / 26
Thank you for your attention
M. Boreale A. Celestini SOFSEM - January 29, 2013 26 / 26