Uncertainty in Probabilistic Trust Models - Semantic Scholar
Uncertainty in Probabilistic Trust Models Sadegh Dorri Nogoorani, Rasool Jalili Department of Computer Engineering Sharif University of Technology Tehran, Iran e-mails: [email protected], [email protected]
Abstract—Computational models of trust try to transfer the concept of trust from the real to the virtual world. While such models have been widely investigated in the past decade, the uncertainty involved in trust computation has been overlooked in the literature. In this paper, uncertainty of probabilistic trust models is quantified using confidence intervals and its factors are determined through simulation. The results confirm the importance and highlight the amount of uncertainty in the Beta and HMM (Hidden Markov Model) trust models. In addition, an uncertainty-driven method is proposed which reduces the risk involved in the trust-based utility maximization according to uncertainty. Keywords-probabilistic trust model; uncertainty; risk reduction;
I. I NTRODUCTION Trust plays an important role in coordinating interpersonal interactions when there is a possibility of risk. It enables a more accurate assessment of the risk, and relaxes the strict traditional control mechanisms. Various computational models of trust have been proposed to transfer such a concept to the virtual world and the idea has been successfully applied to security, routing, and collaborative solutions. In spite of such promising developments, many of the proposals suffer from the fundamental problem of ignoring uncertainty or improperly handling its factors [1], [2]. Without a proper consideration and evaluation of uncertainty, the result of trust assessment cannot be relied on. Our simulations in Section IV confirm that the well-known Beta trust model [3] has a great amount of uncertainty. Even a recent extension of this model [4], requires a relatively long history of past interactions in order to reach an acceptable degree of certainty. In this paper, the uncertainty of probabilistic trust models is analyzed and quantified in the form of confidence interc 2012 IEEE. Personal use of this material is permitted. Permission from
IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. Citation: S. Dorri Nogoorani and R. Jalili, “Uncertainty in Probabilistic Trust Models,” 26th IEEE Int’l Conf. Adv. Info. Networking and Applications, Japan, Mar. 2012, pp. 511–517. DOI 10.1109/AINA.2012.73
vals. As two concrete examples, the Beta and HMM (Hidden Markov Model) trust models are studied and their uncertainty factors are identified using simulation experiments. In order to calculate the associated confidence intervals, the bootstrapping method [5] is applied. The method is almost general and can be used in other scenarios with similar trust models. We also propose an uncertainty-driven risk reduction method which takes into account the risk involved in utility maximization. The applicability of the method is also investigated on the simulation results. The remainder of this paper is organized as follows. In Section II, the uncertainty in probabilistic trust models is discussed in more detail. In Section III, two trust models are described and their uncertainty is analyzed in Section IV. In Section V the uncertainty-driven risk reduction method is proposed. Related work is discussed in Section VI and the paper is concluded through Section VII. II. U NCERTAINTY AND T RUST Various kinds of uncertainty are associated with different stages of trust evolution and application (observation, modeling, and prediction). Observations are inherently uncertain as their possible error cannot be simply ignored. Trust models are prone to uncertainty due to the existence of some degree of abstraction or simplification in their construction. Moreover, the trustee is not fully observable and model inputs are barely samples of reality. The predictions of trust models, even the most flexible ones, have some output errors and are uncertain. A. Basic Notation This paper is concentrated on probabilistic uncertainty, and other forms of uncertainty are left for further research. Therefore, the trustee behavior is characterized by a probability distribution into which all the information about the trustee is fed. In this case, trust is based on the probability of the outcome of future interactions with trustee. Assume that the outcome of an interaction can be represented by a binary variable taking one of the values of x or x ¯. Also assume that trust is based on the history of past interactions, only. Accordingly, the probability of success in
a future interaction of a trustor tr with a trustee te at any time instance t can be calculated from
In particular, the four arithmetic operations on two intervals I1 = [a, b] and I2 = [c, d] are defined as
ptr,te = Pro(Ottr,te = x | Httr,te ), t
(1)
I1 + I2 = [a + c, b + d],
(4)
where Ottr,te represents the future outcome, and Httr,te = {Ot1 , . . . , Otn } (t1 < t2 < . . . < tn < t) the history of interactions until t. Usually a Bayesian approach is taken and ptr,te itself is t assumed to be a random variable following a probability distribution. Therefore, the problem of trust assessment is translated into estimating the expected probability of success. Hence, the trust of tr on te at time t is evaluated using
I1 − I2 = [a − d, b − c],
(5)
τttr,te = E [ptr,te ]. t
(2)
I1 × I2 = [min {ac, ad, bc, bd}, max {ac, ad, bc, bd}], (6) I1 /I2 = I1 × [1/d, 1/c]
(0 6∈ I2 ).
(7)
Note that these operators do not take dependencies between the values represented by the intervals into account. Consequently, dependencies should be resolved before using them. III. C ASE S TUDIES
The super/subscripts in the above formula emphasize on the subjectivity (the point of view of a specific trustor tr), and time-dependency of trust. From now on, a specific trustor tr, trustee te, and time instance t are implicitly assumed and the super/subscripts are only used for clarification where different parties or another time instance are involved in a formula.
In this section, the well known probabilistic Beta trust model (proposed in the Beta reputation system) [3], and one of its extensions, HMM (Hidden Markov Model) trust model [4] are used to evaluate and exemplify the uncertainty of trust assessment.
B. Quantifying Uncertainty
In the Beta trust model, behavior of the trustee is assumed to follow a Bernoulli distribution with a different success probability for each trustee. Hence, the probability of success (p) follows the Beta distribution [3]:
In this paper, confidence intervals are used to represent uncertainty of the result of trust assessment. For example, the 0.95 confidence interval of [0.4, 0.6] means that the real trust value is possibly between 0.4 and 0.6 with the probability of 0.95. Confidence intervals are widely used to represent uncertainty of measurement information, and have a clear probabilistic interpretation. Additionally, there is an almost general method (the bootstrapping [5]) to calculate them. This is in contrast with some ad hoc uncertainty measures which are bound to specific factors, or do not have a clear interpretation. If H = {O1 , O2 , . . . , On } is the history of interactions of tr with te, (2) is used to build a point estimator of τ . On the other hand, an interval estimator of τ is the interval ∆τ = [τ1 , τ2 ] where τ1 and τ2 are functions of H. We refer to [τ1 , τ2 ] as the δ confidence interval of τ if Pro(τ1 ≤ τ ≤ τ2 ) = δ,
(3)
where the constant δ is the confidence coefficient of the estimated trust (usually 0.95 or 0.99). The smaller the confidence interval is, the more certain is the estimation of trust τ . In this regard, the length of the confidence interval is a good measure of uncertainty. C. Propagation of Uncertainty If the trust value is to be used in a calculation, interval arithmetic is used to propagate its uncertainty to the result.
A. The Beta Trust Model with Forgetting Factor
Γ(α + β) α−1 p (1 − p)β−1 Γ(α)Γ(β) where α = r + 1, β = s + 1
f (p | α, β) =
(8) (9)
where r and s are respectively the number of past successful and unsuccessful interactions between the trustee and trustor. Hence, the trust can be estimated using [3] τ=
α r+1 = . α+β r+s+2
(10)
In order to account for changes in the trustee behavior, a forgetting factor λ ∈ [0, 1] is introduced in the model which controls the effect of past history on τ according to (11) and (12) [3]: rtn = rt(n−1) .λ + I{x} (Otn ),
(11)
stn = st(n−1) .λ + I{¯x} (Otn ),
(12)
where IA (.) is an indicator function checking if the outcome is (un)successful, and rti (or sti ) is the value of r (or s) after feeding the ith interaction outcome into the model. With λ = 1, all interactions have the same effect (no forgetting) while with λ = 0, trustor is very forgetful and no history is taken into account. 512
B. HMM Trust In the HMM trust model, a 2-state Hidden Markov Model Ω = (Q, π, A, R, B) is used to keep pace with dynamics of the trustee behavior. In this model, the probability of a successful outcome is again assumed to follow a Bernoulli distribution but in this case dependent on the current internal state of the trustee. The parameters of the model are: • Q = {s0 , s1 }: the set of (hidden) states (e.g. being malicious or benevolent), • π = {π0 , π1 }: the initial probability of being in each state, • A = {Ai,j | i, j ∈ {0, 1}}: the probability of a transition between two states (from i to j), • R = {x, x ¯}: the possible outcomes, • B = {Bi | i ∈ {0, 1}}: the probability distribution of the possible outcomes in each state (Bernoulli). In the HMM model, the trustor uses the history of past interactions with the trustee to calculate (estimate) the parameters π, A, and B of the HMM by the Baum-Welch algorithm (see [6] for a tutorial on Hidden Markov Models and related algorithms). Having the estimated model of trustee, the probability of success in the future interaction is calculated using [4] p = Pro (O = x | H, Ω) =
Pro (O = x, H | Ω) , Pro (H | Ω)
(13)
where Ω is the estimated trustee HMM. Accordingly, the expected probability of success (trust) can be calculated using the Forward-Backward algorithm [6]. C. Theoretical Analysis In order to analyze the uncertainty of the Beta and HMM models, the confidence intervals associated with their trust estimates must be calculated. Bearing in mind that τ in the Beta model follows the Beta distribution (8), the bounds of the δ confidence interval ∆τ = [τ1 , τ2 ] of the Beta model can be calculated from τ1 = F−1 α,β (c),
(14)
F−1 α,β (c
(15)
τ2 =
+ δ),
max{F(τ ) − δ, 0} ≤ c ≤ min{F(τ ), 1 − δ},
(16)
where Fα,β and F−1 α,β are the Beta cumulative distribution function (CDF) and its inverse respectively, and τ is the estimated trust value. c is an arbitrary constant real number (e.g. c = 1−δ 2 ) which controls the relative position of τ in the interval. The constant is constrained to (16) in order to ensure that the interval is a probability interval (subset of [0, 1]) around τ . The parameters α and β are calculated from H, and there are efficient computational methods which can be used to calculate Fα,β and F−1 α,β . The equations can be straightforwardly derived from (3) using the general properties of CDFs.
In order to take advantage of (14) and (15), the real behavior of the trustee must be assumed to follow a fixed known probability distribution (the Beta distribution in this case). Additionally, this method do not produce a closedform formula which can be mathematically analyzed. Another disadvantage of these equations is that they are not practical for the HMM model because the CDF of p (and its inverse) in (13) cannot be determined efficiently. There are approximations to the confidence interval under normality (of the estimator) and independence (of observations) assumptions which cannot be used too because normality is already violated by (8), and a fundamental assumption of the HMM model is the dependence of observation outcomes. Hence, we use the bootstrapping method to determine the confidence intervals. D. The Bootstrapping Method Bootstrapping is a resampling-based method to measure accuracy of almost any statistic using a simple general procedure [5]. In order to compute the confidence interval of τ based on the history of interaction outcomes H of length n, the following steps must be followed in the bootstrapping method: 1) The trust value τ is estimated according to H and the trust model in use. 2) A new bootstrap sample H ∗ of length n is randomly chosen with replacement from the empirical distribution of outcomes in H. 3) A new trust value τ ∗ is calculated according to H ∗ and the trust model. 4) The steps 2 and 3 are repeated B times (B is a large number at least 1000) and the respective bootstrap estimates τ ∗ (1), . . . , τ ∗ (B) are calculated. 5) The confidence interval of τ is determined according to the (empirical) distribution of τ ∗ (.) (e.g. by calculating its percentiles). Before that, the distribution must be shifted so that its mean becomes equal to τ (in order to ensure the interval forms around τ ). Note that the bootstrapping method is supposed to analyze the (possible) deviation of the estimator around a specific estimation. Hence, the relative distribution of τ ∗ (.) around its mean is of concern in the aforementioned procedure, not the absolute distribution. This method approximates the real unknown or complex distribution of τ with the resampling technique. Our coverage analysis shows that the resulting confidence intervals are nearly tight. Using a larger B reduces the resampling error; however, it does not eliminate all the errors and does not produce an exact confidence interval. The bootstrapping method is based on few assumptions and is very general, therefore can be applied to other similar trust models. The bootstrapping method in contrast with the analytic method of (14) and (15) is applicable to estimators with a complex or unknown probability distribution (such as the 513
B0 (x) = 1.0 B0 (¯ x) = 0.0
b,w
1−s 3
b,f
B1 (x) = 0.7
1−s 3
π0 = 0.25 1−s 3 B2 (x) = 0.3 B2 (¯ x) = 0.7
m,f
π2 = 0.25
π1 = 0.25 1−s 3
1−s 3 1−s 3
m,w
λ = 0.1 λ = 0.3 λ = 0.5 λ = 0.7 λ = 0.9 HMM
0.1
B1 (¯ x) = 0.3 8 · 10−2 6 · 10−2
B3 (x) = 0.0 B3 (¯ x) = 1.0 π3 = 0.25
4 · 10−2 2 · 10−2
0.2
Figure 1. The HMM which simulates the trustee behavior (Ω0 ). State labels: b/m: benevolent/malicious, w/f: working, faulty.
result of the forward-backward algorithm in the HMM trust model). However, it does not produce a closed-form formula again. We have also used a variant of the basic bootstrapping (described earlier) which is called parametric bootstrapping. In this variant, the bootstrap samples (step 2) are chosen from the parametric distribution of the estimated model (the trust model instantiated with τ ). Although the bootstrapping method imposes an overhead proportional to B on the computation of trust, the assessment of uncertainty is very valuable to decision processes. We believe that the overhead is tolerable with regard to the nowadays computationally powerful computers and devices. Besides that, uncertainty information can be used to reduce costs and overhead in other aspects of trust computation. IV. S IMULATION R ESULTS AND D ISCUSSION The Beta trust model with various forgetting factors has already been compared to the HMM model with respect to prediction error [4]. However, we studied the effect of different settings and various history sizes on the uncertainty of the models, as well as their error.
0.4
0.6
0.8
Stability (s) Figure 2. Average relative entropy error of the Beta and HMM models with 300 observations. ·10−2
λ = 0.1 λ = 0.3 λ = 0.5 λ = 0.7 λ = 0.9 HMM
3
2
1
0 0
200
400
600
Observation Count (n) Figure 3. Average relative entropy error of the Beta and HMM models with s = 0.4.
in 100 independent rounds and average of the results is reported. Without loss of generality, the 0.95 confidence interval is of interest in all simulations.
A. Simulation Setting The behavior of the trustee is simulated using a 4-state HMM Ω0 (similar to the one used in [4]) depicted in Figure 1. The transition probabilities of Ω0 are dependent on the stability of trustee (s ∈ [0, 1]). The more stable the trustee, the less probable is a transition to another state. In each round of a simulation, a sample outcome sequence (history) of specified length is generated from Ω0 and performances of the trust models are compared with respect to the sample. Their accuracies are compared using the relative entropy [4] (with reference to the real distribution calculated from Ω0 and the distribution suggested by each trust model). To compare uncertainties of the models, the length of their confidence intervals are considered. The basic bootstrap with B = 1000 is used to compute the confidence interval of the Beta model and a parametric bootstrap [7] for the HMM model. Each simulation is run
B. Relative Error The effects of changing stability on relative entropy error is depicted in Figure 2. Not surprisingly, the HMM model has relatively less error than the Beta model. However, as stated in [4], the performance of the HMM model diminishes by increasing stability, and in some settings the Beta model performs better. Nevertheless, the trustor cannot use this information to tune his/her model because stability and relative entropy error are hidden from the trustor. In Figure 3, the effect of the observation count (n) is studied on the error of the models (not studied in [4]). The Beta model does not have a consistent behavior in response to changes in this respect. The HMM model has greater error for small ns. However, in contrast with the Beta model, the error rapidly decreases as n increases. With n < 100, the two models have relatively the same amount of error whereas 514
λ = 0.1 λ = 0.3 λ = 0.5 λ = 0.7 λ = 0.9 HMM
0.5
0.4
0.3
0.2
factor (in most settings, the length is greater than 0.35). In contrast, the HMM model gives more certain results with more observations. However, the performance of the HMM model is very unsatisfiable with a small number of observations. For example, with n ≤ 30, the interval length exceeds 0.45. More specifically, with n < 75, the Beta model is more certain while the HMM model is preferred with more observations. D. Discussions
0.2
0.4
0.6
0.8
Stability (s) Figure 4. Average confidence interval length of the Beta and HMM models with 300 observations. 0.8
λ = 0.1 λ = 0.3 λ = 0.5 λ = 0.7 λ = 0.9 HMM
0.6
0.4
0.2
0
200
400
600
Observation Count (n) Figure 5. Average confidence interval length of the Beta and HMM models with s = 0.4.
with more observations, the HMM model is superior to the Beta model. C. Uncertainty of the Models Figure 4 shows the effect of stability as well as the model parameters on confidence interval length (which has inverse relationship with certainty). According to this figure, the HMM model calculates trust with higher certainty because the interval length of the HMM model is very smaller than that of the Beta model. The Beta model has very disappointing certainty specially with λ = 0.5, 0.7, and the best results belong to λ = 0.1. Even in this case (λ = 0.1), interval length is still greater than 0.35, compared to 0.11 ∼ 0.13 for the HMM model (less than half). Moreover, the interval length in all models shows a very small decrease with increasing stability, and is almost independent of stability (in the Beta model) or weakly dependent on it (in the HMM model). Figure 5 shows the effect of observation count on the uncertainty (confidence interval length). It confirms that the certainty of the Beta model is not dependent on this
According to the performance results of the Beta and HMM models, uncertainty of the Beta model is independent of observation count and the stability of the trustee, yet nonmonotonically dependent on the forgetting factor. By comparing these results with relative entropy error of the model, we conclude that a trustor with a Beta trust model, has no trivial option to decrease uncertainty of its trust estimations. On the other hand, uncertainty of the HMM model weakly depends on the stability of the trustee, and is directly related to observation count. Therefore, a trustor with an HMM trust model can decrease its uncertainty by testing trustee in unimportant situations, or consulting other agents who may have more experience with the trustee. In this way, the trustor will reach a more certain as well as a more accurate estimation of trust. These results are very valuable to the trustor because confidence intervals are based solely on the information available to the trustor and can be computed in real time with a moderate overhead. In the following section an uncertainty-driven risk reduction method is proposed which can be used to take advantage of uncertainty information. V. U NCERTAINTY-D RIVEN R ISK R EDUCTION In a multi-agent environment, a rational agent acts as though it is maximizing a utility function [8]. The utility in a trust-based agent, or trustor, is dependent on trust relationships between agents as well as the application specific factors. Particularly, trust supports the agent in a better estimation of the other agents’ behavior, while the application determines the utility function of the trustor. An straightforward strategy is to select the trustee with the maximum expected utility. In this strategy, the expected utility of an interaction is calculated according to U = E [u] = E [O = x].u(x) + E [O = x ¯].u(¯ x) (17) = τ.u(x) + (1 − τ ).u(¯ x)
(18)
= τ.(u(x) − u(¯ x)) + u(¯ x),
(19)
where u(.) is the application-specific utility function that maps the possible outcomes to their corresponding supposed utility. Practically, utility on its own is insufficient to make a rational decision, and the more risky a decision is, the 515
Table I M AX . ACCEPTABLE U NCERTAINTY FOR C RITICALITY C LASSES . Criticality
Low
Medium
High
Max. Confidence Interval Length
15
10
5
more cautious the trustor must be to make it. Risk has direct relationship with criticality and inverse relationship with certainty. Criticality of a decision is dependent on the situation and cannot be reduced. In contrast, uncertainty may be decreased by considering the uncertainty factors (highlighted in Section IV). However, the cost incurred by decreasing uncertainty (time, bandwidth, . . . ) can be adjusted by considering the required minimum certainty during evaluation. This way, more investment is made on highly critical decisions (to lower risk), whereas a modest investment will suffice for noncritical ones. The trustee can use (20) to propagate uncertainty of the trust to the expected utility using interval arithmetic: ∆U = ∆τ .(u(x) − u(¯ x)) + u(¯ x),
(20)
where a scalar value a is assumed to be a zero-length interval [a, a] in interval arithmetic. Equation (20) is based on (19) in order to resolve the dependency between different parts of (18). Having ∆U , confidence interval length (uncertainty) is compared to the prespecified maximum acceptable interval length threshold corresponding to the criticality of the decision. If the requirement is not met, trustor may increase the certainty by using the uncertainty factors, or leave out the trustee from the possible interaction alternatives. For example it may consult other agents, or examine the trustee in experimental scenarios. This method is exemplified with the results of the Beta and HMM trust models in the following sections. A. Uncertain Utility Estimation In this section, the estimation of the trustor’s expected utility and its uncertainty is examined in a randomly selected run of the simulations to illustrate its application. The simulation settings were nearly optimal for both models: n = 300, s = 0.9, and λ = 0.4. Suppose the utility function of the trustor is specified by +50 units O = x u(O) = . (21) −20 units O = x ¯ Moreover, the maximum acceptable uncertainty for three criticality classes of low, medium, and high are specified according to Table I. The estimated trust values and their corresponding 0.95 confidence intervals in the selected run are stated in τ b = 0.478, ∆b = [0.285, 0.673], (22) τ h = 0.519,
∆h = [0.431, 0.601],
(23)
where the superscripts of b and h distinguish between the Beta and HMM models respectively. In this case, the resulting expected utilities and their confidence intervals are as in (24) and (25), according to the interval arithmetic. U b = 13.450, ∆U,b = [−0.037, 27.087] (24) U h = 16.353,
∆U,h = [10.151, 22.058]
(25)
According to the results, the Beta model has a great amount of uncertainty in its trust estimation, and the length of the expected utility confidence interval is about two times greater than the expected utility. B. Risk Reduction Both models suggest a positive expected utility. However, according to the confidence intervals, the true value may be far away from the expectations. With the Beta model, this value can vary in an interval of length 27.124 units while with the HMM model, the interval length is 11.907 units. According to the uncertainty-driven risk reduction method applied to the Beta model results, the trustee should not be selected even in low-critical situations. Moreover, according to the analyses of uncertainty factors in the past sections, there is no way to improve the certainty. However, the result of the HMM model can be used for low or medium-critical situations. Moreover, in case of high-critical decisions, collecting further observations will result in a more certain estimation with the HMM model. VI. R ELATED W ORK Traditionally, security researchers were concerned about trust in their systems (e.g. in Public Key Infrastructure, and Trusted Computing). Yet, initial formation and evolution of trust were implicit or external to the proposed systems [9], [10]. The current trend in the literature, however, is to explicitly consider these aspects and many trust models are proposed to compute the trust between parties based on past history, social networks, and other social factors (for a survey on computational trust models see [9], [11], [12]). Uncertainty is disregarded in many of the proposals, or combined into trust [1]. The probabilistic and fuzzy trust models are the most susceptible ones to uncertainty analysis. A fuzzy trust model for peer-to-peer systems has been proposed in [13] with special focus on uncertainty. Particularly, a threshold on observation count has been put in order to decide whether to include history information in the trust evaluation or not. Referential trust as an uncertainty factor is put forward in SUNNY which is a trust derivation algorithm in social networks [1], [14]. Referential trust in SUNNY is represented by a binary value and is used to calculate confidence. Confidence in this model is the probability of positive trustor belief in the correctness of trust information received from another agent. Unlike the confidence intervals in this paper, 516
the confidence in SUNNY is an internal uncertainty measure that cannot be used in trust-based decision making. TRAVOS [15] has followed an approach which is more comparable to ours. In TRAVOS, evaluation of trust follows the same base formula of the Beta model, and the confidence coefficient of a specific interval around the trust value is used as the measure of uncertainty. The coefficient is calculated using the Beta distribution and does not take the effect of forgetting factor into account. In the case where past interaction history does not estimate a confident trust value, TRAVOS uses other agents’ recommendations to improve the result. Nevertheless, we have shown in Section IV that confidence of the Beta model is (at least sometimes) independent of observation count. Moreover, according to our analyses which are not reported in this paper, the Beta distribution produces relatively wide confidence intervals when the trustee behavior does not exactly follow a Bernoulli distribution. Confidence intervals are superior to other proposals because as our simulations indicate, uncertainty factors are not the same in all models, and these intervals are not bound to a specific uncertainty factor. Additionally, the bootstrapping method helps in studying the effect of various parameters of a trust model on uncertainty, and all the effects are unified in the confidence interval. The risk involved in the decision of a trustor with respect to uncertainty is not considered in the existing trust models even in the most general ones such as [12]. To our knowledge, our uncertainty-driven risk reduction is the first proposal which propagates the uncertainty to the decisionmaking process and helps to maintain a balance between the risk induced by uncertainty, and cost.
relying on a bigger history that may not be affordable in many applications. The disappointing amount of uncertainty in the models cannot be avoided because it is inherent to the models. We believe that new models of trust must be proposed with special attention to uncertainty, and all forms of uncertainty should be considered in the models. In order to achieve this goal, a mathematical framework other than the probability theory should be used that is capable to handle various forms of uncertainty.
VII. C ONCLUSIONS AND F UTURE W ORK
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Computational models of trust are useful tools in the assessment of the risk involved in future interactions and help agents in a multi-agent environment maximize their utility. Confidence intervals calculated via the bootstrapping method were proposed in this paper to quantify the uncertainty of probabilistic trust models. The uncertainty of two trust models was studied and the related factors were determined. Both models suffered from a great amount of uncertainty when a small number of past interaction outcomes were used to calculate trust. However, the performance of the HMM model was improved by increasing the size of history. In contrast, the Beta model was indifferent to the size of history, and nonmonotonically dependent on its forgetting factor (fixed over time). We also proposed a risk reduction method which takes advantage of trust uncertainty to include risk in decision making. Studying the effect of referential trust and other factors on uncertainty of the models are planned for further research. These factors can help in reducing uncertainty without solely
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Abstract—Computational models of trust try to transfer the concept of trust from the real to the virtual world. While such models have been widely investigated in the past decade, the uncertainty involved in trust computation has been overlooked in the literature. In this paper, uncertainty of probabilistic trust models is quantified using confidence intervals and its factors are determined through simulation. The results confirm the importance and highlight the amount of uncertainty in the Beta and HMM (Hidden Markov Model) trust models. In addition, an uncertainty-driven method is proposed which reduces the risk involved in the trust-based utility maximization according to uncertainty. Keywords-probabilistic trust model; uncertainty; risk reduction;
I. I NTRODUCTION Trust plays an important role in coordinating interpersonal interactions when there is a possibility of risk. It enables a more accurate assessment of the risk, and relaxes the strict traditional control mechanisms. Various computational models of trust have been proposed to transfer such a concept to the virtual world and the idea has been successfully applied to security, routing, and collaborative solutions. In spite of such promising developments, many of the proposals suffer from the fundamental problem of ignoring uncertainty or improperly handling its factors [1], [2]. Without a proper consideration and evaluation of uncertainty, the result of trust assessment cannot be relied on. Our simulations in Section IV confirm that the well-known Beta trust model [3] has a great amount of uncertainty. Even a recent extension of this model [4], requires a relatively long history of past interactions in order to reach an acceptable degree of certainty. In this paper, the uncertainty of probabilistic trust models is analyzed and quantified in the form of confidence interc 2012 IEEE. Personal use of this material is permitted. Permission from
IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. Citation: S. Dorri Nogoorani and R. Jalili, “Uncertainty in Probabilistic Trust Models,” 26th IEEE Int’l Conf. Adv. Info. Networking and Applications, Japan, Mar. 2012, pp. 511–517. DOI 10.1109/AINA.2012.73
vals. As two concrete examples, the Beta and HMM (Hidden Markov Model) trust models are studied and their uncertainty factors are identified using simulation experiments. In order to calculate the associated confidence intervals, the bootstrapping method [5] is applied. The method is almost general and can be used in other scenarios with similar trust models. We also propose an uncertainty-driven risk reduction method which takes into account the risk involved in utility maximization. The applicability of the method is also investigated on the simulation results. The remainder of this paper is organized as follows. In Section II, the uncertainty in probabilistic trust models is discussed in more detail. In Section III, two trust models are described and their uncertainty is analyzed in Section IV. In Section V the uncertainty-driven risk reduction method is proposed. Related work is discussed in Section VI and the paper is concluded through Section VII. II. U NCERTAINTY AND T RUST Various kinds of uncertainty are associated with different stages of trust evolution and application (observation, modeling, and prediction). Observations are inherently uncertain as their possible error cannot be simply ignored. Trust models are prone to uncertainty due to the existence of some degree of abstraction or simplification in their construction. Moreover, the trustee is not fully observable and model inputs are barely samples of reality. The predictions of trust models, even the most flexible ones, have some output errors and are uncertain. A. Basic Notation This paper is concentrated on probabilistic uncertainty, and other forms of uncertainty are left for further research. Therefore, the trustee behavior is characterized by a probability distribution into which all the information about the trustee is fed. In this case, trust is based on the probability of the outcome of future interactions with trustee. Assume that the outcome of an interaction can be represented by a binary variable taking one of the values of x or x ¯. Also assume that trust is based on the history of past interactions, only. Accordingly, the probability of success in
a future interaction of a trustor tr with a trustee te at any time instance t can be calculated from
In particular, the four arithmetic operations on two intervals I1 = [a, b] and I2 = [c, d] are defined as
ptr,te = Pro(Ottr,te = x | Httr,te ), t
(1)
I1 + I2 = [a + c, b + d],
(4)
where Ottr,te represents the future outcome, and Httr,te = {Ot1 , . . . , Otn } (t1 < t2 < . . . < tn < t) the history of interactions until t. Usually a Bayesian approach is taken and ptr,te itself is t assumed to be a random variable following a probability distribution. Therefore, the problem of trust assessment is translated into estimating the expected probability of success. Hence, the trust of tr on te at time t is evaluated using
I1 − I2 = [a − d, b − c],
(5)
τttr,te = E [ptr,te ]. t
(2)
I1 × I2 = [min {ac, ad, bc, bd}, max {ac, ad, bc, bd}], (6) I1 /I2 = I1 × [1/d, 1/c]
(0 6∈ I2 ).
(7)
Note that these operators do not take dependencies between the values represented by the intervals into account. Consequently, dependencies should be resolved before using them. III. C ASE S TUDIES
The super/subscripts in the above formula emphasize on the subjectivity (the point of view of a specific trustor tr), and time-dependency of trust. From now on, a specific trustor tr, trustee te, and time instance t are implicitly assumed and the super/subscripts are only used for clarification where different parties or another time instance are involved in a formula.
In this section, the well known probabilistic Beta trust model (proposed in the Beta reputation system) [3], and one of its extensions, HMM (Hidden Markov Model) trust model [4] are used to evaluate and exemplify the uncertainty of trust assessment.
B. Quantifying Uncertainty
In the Beta trust model, behavior of the trustee is assumed to follow a Bernoulli distribution with a different success probability for each trustee. Hence, the probability of success (p) follows the Beta distribution [3]:
In this paper, confidence intervals are used to represent uncertainty of the result of trust assessment. For example, the 0.95 confidence interval of [0.4, 0.6] means that the real trust value is possibly between 0.4 and 0.6 with the probability of 0.95. Confidence intervals are widely used to represent uncertainty of measurement information, and have a clear probabilistic interpretation. Additionally, there is an almost general method (the bootstrapping [5]) to calculate them. This is in contrast with some ad hoc uncertainty measures which are bound to specific factors, or do not have a clear interpretation. If H = {O1 , O2 , . . . , On } is the history of interactions of tr with te, (2) is used to build a point estimator of τ . On the other hand, an interval estimator of τ is the interval ∆τ = [τ1 , τ2 ] where τ1 and τ2 are functions of H. We refer to [τ1 , τ2 ] as the δ confidence interval of τ if Pro(τ1 ≤ τ ≤ τ2 ) = δ,
(3)
where the constant δ is the confidence coefficient of the estimated trust (usually 0.95 or 0.99). The smaller the confidence interval is, the more certain is the estimation of trust τ . In this regard, the length of the confidence interval is a good measure of uncertainty. C. Propagation of Uncertainty If the trust value is to be used in a calculation, interval arithmetic is used to propagate its uncertainty to the result.
A. The Beta Trust Model with Forgetting Factor
Γ(α + β) α−1 p (1 − p)β−1 Γ(α)Γ(β) where α = r + 1, β = s + 1
f (p | α, β) =
(8) (9)
where r and s are respectively the number of past successful and unsuccessful interactions between the trustee and trustor. Hence, the trust can be estimated using [3] τ=
α r+1 = . α+β r+s+2
(10)
In order to account for changes in the trustee behavior, a forgetting factor λ ∈ [0, 1] is introduced in the model which controls the effect of past history on τ according to (11) and (12) [3]: rtn = rt(n−1) .λ + I{x} (Otn ),
(11)
stn = st(n−1) .λ + I{¯x} (Otn ),
(12)
where IA (.) is an indicator function checking if the outcome is (un)successful, and rti (or sti ) is the value of r (or s) after feeding the ith interaction outcome into the model. With λ = 1, all interactions have the same effect (no forgetting) while with λ = 0, trustor is very forgetful and no history is taken into account. 512
B. HMM Trust In the HMM trust model, a 2-state Hidden Markov Model Ω = (Q, π, A, R, B) is used to keep pace with dynamics of the trustee behavior. In this model, the probability of a successful outcome is again assumed to follow a Bernoulli distribution but in this case dependent on the current internal state of the trustee. The parameters of the model are: • Q = {s0 , s1 }: the set of (hidden) states (e.g. being malicious or benevolent), • π = {π0 , π1 }: the initial probability of being in each state, • A = {Ai,j | i, j ∈ {0, 1}}: the probability of a transition between two states (from i to j), • R = {x, x ¯}: the possible outcomes, • B = {Bi | i ∈ {0, 1}}: the probability distribution of the possible outcomes in each state (Bernoulli). In the HMM model, the trustor uses the history of past interactions with the trustee to calculate (estimate) the parameters π, A, and B of the HMM by the Baum-Welch algorithm (see [6] for a tutorial on Hidden Markov Models and related algorithms). Having the estimated model of trustee, the probability of success in the future interaction is calculated using [4] p = Pro (O = x | H, Ω) =
Pro (O = x, H | Ω) , Pro (H | Ω)
(13)
where Ω is the estimated trustee HMM. Accordingly, the expected probability of success (trust) can be calculated using the Forward-Backward algorithm [6]. C. Theoretical Analysis In order to analyze the uncertainty of the Beta and HMM models, the confidence intervals associated with their trust estimates must be calculated. Bearing in mind that τ in the Beta model follows the Beta distribution (8), the bounds of the δ confidence interval ∆τ = [τ1 , τ2 ] of the Beta model can be calculated from τ1 = F−1 α,β (c),
(14)
F−1 α,β (c
(15)
τ2 =
+ δ),
max{F(τ ) − δ, 0} ≤ c ≤ min{F(τ ), 1 − δ},
(16)
where Fα,β and F−1 α,β are the Beta cumulative distribution function (CDF) and its inverse respectively, and τ is the estimated trust value. c is an arbitrary constant real number (e.g. c = 1−δ 2 ) which controls the relative position of τ in the interval. The constant is constrained to (16) in order to ensure that the interval is a probability interval (subset of [0, 1]) around τ . The parameters α and β are calculated from H, and there are efficient computational methods which can be used to calculate Fα,β and F−1 α,β . The equations can be straightforwardly derived from (3) using the general properties of CDFs.
In order to take advantage of (14) and (15), the real behavior of the trustee must be assumed to follow a fixed known probability distribution (the Beta distribution in this case). Additionally, this method do not produce a closedform formula which can be mathematically analyzed. Another disadvantage of these equations is that they are not practical for the HMM model because the CDF of p (and its inverse) in (13) cannot be determined efficiently. There are approximations to the confidence interval under normality (of the estimator) and independence (of observations) assumptions which cannot be used too because normality is already violated by (8), and a fundamental assumption of the HMM model is the dependence of observation outcomes. Hence, we use the bootstrapping method to determine the confidence intervals. D. The Bootstrapping Method Bootstrapping is a resampling-based method to measure accuracy of almost any statistic using a simple general procedure [5]. In order to compute the confidence interval of τ based on the history of interaction outcomes H of length n, the following steps must be followed in the bootstrapping method: 1) The trust value τ is estimated according to H and the trust model in use. 2) A new bootstrap sample H ∗ of length n is randomly chosen with replacement from the empirical distribution of outcomes in H. 3) A new trust value τ ∗ is calculated according to H ∗ and the trust model. 4) The steps 2 and 3 are repeated B times (B is a large number at least 1000) and the respective bootstrap estimates τ ∗ (1), . . . , τ ∗ (B) are calculated. 5) The confidence interval of τ is determined according to the (empirical) distribution of τ ∗ (.) (e.g. by calculating its percentiles). Before that, the distribution must be shifted so that its mean becomes equal to τ (in order to ensure the interval forms around τ ). Note that the bootstrapping method is supposed to analyze the (possible) deviation of the estimator around a specific estimation. Hence, the relative distribution of τ ∗ (.) around its mean is of concern in the aforementioned procedure, not the absolute distribution. This method approximates the real unknown or complex distribution of τ with the resampling technique. Our coverage analysis shows that the resulting confidence intervals are nearly tight. Using a larger B reduces the resampling error; however, it does not eliminate all the errors and does not produce an exact confidence interval. The bootstrapping method is based on few assumptions and is very general, therefore can be applied to other similar trust models. The bootstrapping method in contrast with the analytic method of (14) and (15) is applicable to estimators with a complex or unknown probability distribution (such as the 513
B0 (x) = 1.0 B0 (¯ x) = 0.0
b,w
1−s 3
b,f
B1 (x) = 0.7
1−s 3
π0 = 0.25 1−s 3 B2 (x) = 0.3 B2 (¯ x) = 0.7
m,f
π2 = 0.25
π1 = 0.25 1−s 3
1−s 3 1−s 3
m,w
λ = 0.1 λ = 0.3 λ = 0.5 λ = 0.7 λ = 0.9 HMM
0.1
B1 (¯ x) = 0.3 8 · 10−2 6 · 10−2
B3 (x) = 0.0 B3 (¯ x) = 1.0 π3 = 0.25
4 · 10−2 2 · 10−2
0.2
Figure 1. The HMM which simulates the trustee behavior (Ω0 ). State labels: b/m: benevolent/malicious, w/f: working, faulty.
result of the forward-backward algorithm in the HMM trust model). However, it does not produce a closed-form formula again. We have also used a variant of the basic bootstrapping (described earlier) which is called parametric bootstrapping. In this variant, the bootstrap samples (step 2) are chosen from the parametric distribution of the estimated model (the trust model instantiated with τ ). Although the bootstrapping method imposes an overhead proportional to B on the computation of trust, the assessment of uncertainty is very valuable to decision processes. We believe that the overhead is tolerable with regard to the nowadays computationally powerful computers and devices. Besides that, uncertainty information can be used to reduce costs and overhead in other aspects of trust computation. IV. S IMULATION R ESULTS AND D ISCUSSION The Beta trust model with various forgetting factors has already been compared to the HMM model with respect to prediction error [4]. However, we studied the effect of different settings and various history sizes on the uncertainty of the models, as well as their error.
0.4
0.6
0.8
Stability (s) Figure 2. Average relative entropy error of the Beta and HMM models with 300 observations. ·10−2
λ = 0.1 λ = 0.3 λ = 0.5 λ = 0.7 λ = 0.9 HMM
3
2
1
0 0
200
400
600
Observation Count (n) Figure 3. Average relative entropy error of the Beta and HMM models with s = 0.4.
in 100 independent rounds and average of the results is reported. Without loss of generality, the 0.95 confidence interval is of interest in all simulations.
A. Simulation Setting The behavior of the trustee is simulated using a 4-state HMM Ω0 (similar to the one used in [4]) depicted in Figure 1. The transition probabilities of Ω0 are dependent on the stability of trustee (s ∈ [0, 1]). The more stable the trustee, the less probable is a transition to another state. In each round of a simulation, a sample outcome sequence (history) of specified length is generated from Ω0 and performances of the trust models are compared with respect to the sample. Their accuracies are compared using the relative entropy [4] (with reference to the real distribution calculated from Ω0 and the distribution suggested by each trust model). To compare uncertainties of the models, the length of their confidence intervals are considered. The basic bootstrap with B = 1000 is used to compute the confidence interval of the Beta model and a parametric bootstrap [7] for the HMM model. Each simulation is run
B. Relative Error The effects of changing stability on relative entropy error is depicted in Figure 2. Not surprisingly, the HMM model has relatively less error than the Beta model. However, as stated in [4], the performance of the HMM model diminishes by increasing stability, and in some settings the Beta model performs better. Nevertheless, the trustor cannot use this information to tune his/her model because stability and relative entropy error are hidden from the trustor. In Figure 3, the effect of the observation count (n) is studied on the error of the models (not studied in [4]). The Beta model does not have a consistent behavior in response to changes in this respect. The HMM model has greater error for small ns. However, in contrast with the Beta model, the error rapidly decreases as n increases. With n < 100, the two models have relatively the same amount of error whereas 514
λ = 0.1 λ = 0.3 λ = 0.5 λ = 0.7 λ = 0.9 HMM
0.5
0.4
0.3
0.2
factor (in most settings, the length is greater than 0.35). In contrast, the HMM model gives more certain results with more observations. However, the performance of the HMM model is very unsatisfiable with a small number of observations. For example, with n ≤ 30, the interval length exceeds 0.45. More specifically, with n < 75, the Beta model is more certain while the HMM model is preferred with more observations. D. Discussions
0.2
0.4
0.6
0.8
Stability (s) Figure 4. Average confidence interval length of the Beta and HMM models with 300 observations. 0.8
λ = 0.1 λ = 0.3 λ = 0.5 λ = 0.7 λ = 0.9 HMM
0.6
0.4
0.2
0
200
400
600
Observation Count (n) Figure 5. Average confidence interval length of the Beta and HMM models with s = 0.4.
with more observations, the HMM model is superior to the Beta model. C. Uncertainty of the Models Figure 4 shows the effect of stability as well as the model parameters on confidence interval length (which has inverse relationship with certainty). According to this figure, the HMM model calculates trust with higher certainty because the interval length of the HMM model is very smaller than that of the Beta model. The Beta model has very disappointing certainty specially with λ = 0.5, 0.7, and the best results belong to λ = 0.1. Even in this case (λ = 0.1), interval length is still greater than 0.35, compared to 0.11 ∼ 0.13 for the HMM model (less than half). Moreover, the interval length in all models shows a very small decrease with increasing stability, and is almost independent of stability (in the Beta model) or weakly dependent on it (in the HMM model). Figure 5 shows the effect of observation count on the uncertainty (confidence interval length). It confirms that the certainty of the Beta model is not dependent on this
According to the performance results of the Beta and HMM models, uncertainty of the Beta model is independent of observation count and the stability of the trustee, yet nonmonotonically dependent on the forgetting factor. By comparing these results with relative entropy error of the model, we conclude that a trustor with a Beta trust model, has no trivial option to decrease uncertainty of its trust estimations. On the other hand, uncertainty of the HMM model weakly depends on the stability of the trustee, and is directly related to observation count. Therefore, a trustor with an HMM trust model can decrease its uncertainty by testing trustee in unimportant situations, or consulting other agents who may have more experience with the trustee. In this way, the trustor will reach a more certain as well as a more accurate estimation of trust. These results are very valuable to the trustor because confidence intervals are based solely on the information available to the trustor and can be computed in real time with a moderate overhead. In the following section an uncertainty-driven risk reduction method is proposed which can be used to take advantage of uncertainty information. V. U NCERTAINTY-D RIVEN R ISK R EDUCTION In a multi-agent environment, a rational agent acts as though it is maximizing a utility function [8]. The utility in a trust-based agent, or trustor, is dependent on trust relationships between agents as well as the application specific factors. Particularly, trust supports the agent in a better estimation of the other agents’ behavior, while the application determines the utility function of the trustor. An straightforward strategy is to select the trustee with the maximum expected utility. In this strategy, the expected utility of an interaction is calculated according to U = E [u] = E [O = x].u(x) + E [O = x ¯].u(¯ x) (17) = τ.u(x) + (1 − τ ).u(¯ x)
(18)
= τ.(u(x) − u(¯ x)) + u(¯ x),
(19)
where u(.) is the application-specific utility function that maps the possible outcomes to their corresponding supposed utility. Practically, utility on its own is insufficient to make a rational decision, and the more risky a decision is, the 515
Table I M AX . ACCEPTABLE U NCERTAINTY FOR C RITICALITY C LASSES . Criticality
Low
Medium
High
Max. Confidence Interval Length
15
10
5
more cautious the trustor must be to make it. Risk has direct relationship with criticality and inverse relationship with certainty. Criticality of a decision is dependent on the situation and cannot be reduced. In contrast, uncertainty may be decreased by considering the uncertainty factors (highlighted in Section IV). However, the cost incurred by decreasing uncertainty (time, bandwidth, . . . ) can be adjusted by considering the required minimum certainty during evaluation. This way, more investment is made on highly critical decisions (to lower risk), whereas a modest investment will suffice for noncritical ones. The trustee can use (20) to propagate uncertainty of the trust to the expected utility using interval arithmetic: ∆U = ∆τ .(u(x) − u(¯ x)) + u(¯ x),
(20)
where a scalar value a is assumed to be a zero-length interval [a, a] in interval arithmetic. Equation (20) is based on (19) in order to resolve the dependency between different parts of (18). Having ∆U , confidence interval length (uncertainty) is compared to the prespecified maximum acceptable interval length threshold corresponding to the criticality of the decision. If the requirement is not met, trustor may increase the certainty by using the uncertainty factors, or leave out the trustee from the possible interaction alternatives. For example it may consult other agents, or examine the trustee in experimental scenarios. This method is exemplified with the results of the Beta and HMM trust models in the following sections. A. Uncertain Utility Estimation In this section, the estimation of the trustor’s expected utility and its uncertainty is examined in a randomly selected run of the simulations to illustrate its application. The simulation settings were nearly optimal for both models: n = 300, s = 0.9, and λ = 0.4. Suppose the utility function of the trustor is specified by +50 units O = x u(O) = . (21) −20 units O = x ¯ Moreover, the maximum acceptable uncertainty for three criticality classes of low, medium, and high are specified according to Table I. The estimated trust values and their corresponding 0.95 confidence intervals in the selected run are stated in τ b = 0.478, ∆b = [0.285, 0.673], (22) τ h = 0.519,
∆h = [0.431, 0.601],
(23)
where the superscripts of b and h distinguish between the Beta and HMM models respectively. In this case, the resulting expected utilities and their confidence intervals are as in (24) and (25), according to the interval arithmetic. U b = 13.450, ∆U,b = [−0.037, 27.087] (24) U h = 16.353,
∆U,h = [10.151, 22.058]
(25)
According to the results, the Beta model has a great amount of uncertainty in its trust estimation, and the length of the expected utility confidence interval is about two times greater than the expected utility. B. Risk Reduction Both models suggest a positive expected utility. However, according to the confidence intervals, the true value may be far away from the expectations. With the Beta model, this value can vary in an interval of length 27.124 units while with the HMM model, the interval length is 11.907 units. According to the uncertainty-driven risk reduction method applied to the Beta model results, the trustee should not be selected even in low-critical situations. Moreover, according to the analyses of uncertainty factors in the past sections, there is no way to improve the certainty. However, the result of the HMM model can be used for low or medium-critical situations. Moreover, in case of high-critical decisions, collecting further observations will result in a more certain estimation with the HMM model. VI. R ELATED W ORK Traditionally, security researchers were concerned about trust in their systems (e.g. in Public Key Infrastructure, and Trusted Computing). Yet, initial formation and evolution of trust were implicit or external to the proposed systems [9], [10]. The current trend in the literature, however, is to explicitly consider these aspects and many trust models are proposed to compute the trust between parties based on past history, social networks, and other social factors (for a survey on computational trust models see [9], [11], [12]). Uncertainty is disregarded in many of the proposals, or combined into trust [1]. The probabilistic and fuzzy trust models are the most susceptible ones to uncertainty analysis. A fuzzy trust model for peer-to-peer systems has been proposed in [13] with special focus on uncertainty. Particularly, a threshold on observation count has been put in order to decide whether to include history information in the trust evaluation or not. Referential trust as an uncertainty factor is put forward in SUNNY which is a trust derivation algorithm in social networks [1], [14]. Referential trust in SUNNY is represented by a binary value and is used to calculate confidence. Confidence in this model is the probability of positive trustor belief in the correctness of trust information received from another agent. Unlike the confidence intervals in this paper, 516
the confidence in SUNNY is an internal uncertainty measure that cannot be used in trust-based decision making. TRAVOS [15] has followed an approach which is more comparable to ours. In TRAVOS, evaluation of trust follows the same base formula of the Beta model, and the confidence coefficient of a specific interval around the trust value is used as the measure of uncertainty. The coefficient is calculated using the Beta distribution and does not take the effect of forgetting factor into account. In the case where past interaction history does not estimate a confident trust value, TRAVOS uses other agents’ recommendations to improve the result. Nevertheless, we have shown in Section IV that confidence of the Beta model is (at least sometimes) independent of observation count. Moreover, according to our analyses which are not reported in this paper, the Beta distribution produces relatively wide confidence intervals when the trustee behavior does not exactly follow a Bernoulli distribution. Confidence intervals are superior to other proposals because as our simulations indicate, uncertainty factors are not the same in all models, and these intervals are not bound to a specific uncertainty factor. Additionally, the bootstrapping method helps in studying the effect of various parameters of a trust model on uncertainty, and all the effects are unified in the confidence interval. The risk involved in the decision of a trustor with respect to uncertainty is not considered in the existing trust models even in the most general ones such as [12]. To our knowledge, our uncertainty-driven risk reduction is the first proposal which propagates the uncertainty to the decisionmaking process and helps to maintain a balance between the risk induced by uncertainty, and cost.
relying on a bigger history that may not be affordable in many applications. The disappointing amount of uncertainty in the models cannot be avoided because it is inherent to the models. We believe that new models of trust must be proposed with special attention to uncertainty, and all forms of uncertainty should be considered in the models. In order to achieve this goal, a mathematical framework other than the probability theory should be used that is capable to handle various forms of uncertainty.
VII. C ONCLUSIONS AND F UTURE W ORK
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Computational models of trust are useful tools in the assessment of the risk involved in future interactions and help agents in a multi-agent environment maximize their utility. Confidence intervals calculated via the bootstrapping method were proposed in this paper to quantify the uncertainty of probabilistic trust models. The uncertainty of two trust models was studied and the related factors were determined. Both models suffered from a great amount of uncertainty when a small number of past interaction outcomes were used to calculate trust. However, the performance of the HMM model was improved by increasing the size of history. In contrast, the Beta model was indifferent to the size of history, and nonmonotonically dependent on its forgetting factor (fixed over time). We also proposed a risk reduction method which takes advantage of trust uncertainty to include risk in decision making. Studying the effect of referential trust and other factors on uncertainty of the models are planned for further research. These factors can help in reducing uncertainty without solely
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