asymptotic risk analysis for trust and reputation systems · asymptotic risk analysis for trust and...

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.


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.


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.


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.



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?



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


Outline

1 Formal Set UpLoss and Decision FunctionsEvaluation of Decision Functions

2 Results

3 Examples

4 Conclusions


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


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


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).


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 .



2 Results

3 Examples

4 Conclusions


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



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))



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



2 Results

3 Examples

4 Conclusions


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



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.



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



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.



2 Results

3 Examples

4 Conclusions


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∗.


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.



2 Results

3 Examples

4 Conclusions


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.



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′)



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.



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).



Goal:

• Analyse a system with respect to the use of different reputationfunctions.



Intuition: map takes advantage of the a priori knowledge represented byπ(·)



2 Results

3 Examples

4 Conclusions


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


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.


Thank you for your attention


asymptotic risk analysis for trust and reputation systems · asymptotic risk analysis for trust and...

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