Subjective logic operators in trust assessment: an empirical study

Abstract

Computational trust mechanisms aim to produce trust ratings from both direct and indirect information about agents’ behaviour. Subjective Logic (SL) has been widely adopted as the core of such systems via its fusion and discount operators. In recent research we revisited the semantics of these operators to explore an alternative, geometric interpretation. In this paper we present principled desiderata for discounting and fusion operators in SL. Building upon this we present operators that satisfy these desirable properties, including a family of discount operators. We then show, through a rigorous empirical study, that specific, geometrically interpreted, operators significantly outperform standard SL operators in estimating ground truth. These novel operators offer real advantages for computational models of trust and reputation, in which they may be employed without modifying other aspects of an existing system.

Appendices

Appendix A: The geometry of subjective logic

A SL opinion $O \triangleq {{\langle {b_{O}}, {d_{O}}, {u_{O}} \rangle }}$ is a point in the ${\mathbb {R}}^{3}$ space, identified by the coordinate b _O for the first axis, d _O for the second axis, and u _O for the third axis. However, due to the requirement that b _O +d _O +u _O =1, an opinion is a point inside (or at least on the edges of) the triangle ${\mathop {BDU}\limits ^{\triangle }}$ shown in Fig. 5, where B=〈1,0,0〉,D=〈0,1,0〉,U=〈0,0,1〉.

Fig. 5

The subjective logic plane region

Definition 14

The Subjective Logic plane region ${\overset {\triangle }{BDU}}$ is the triangle whose vertices are the points $B \triangleq {\langle 1,0,0 \rangle }$, $D\triangleq {\langle 0,1,0 \rangle }$, and $U\triangleq {\langle 0,0,1 \rangle }$ on a ${\mathbb {R}}^{3}$ space where the axes are respectively the one of belief, disbelief, and uncertainty predicted by SL.

Since an opinion is a point inside triangle ${\mathop {BDU}\limits ^{\triangle }}$, it can be mapped to a point in Fig. 6. This representation is similar to the one used in Jøsang (2001) for representing opinions in SL, but here the belief and disbelief axes are swapped.

Fig. 6

An opinion $O \triangleq {{\langle {b_{O}}, {d_{O}}, {u_{O}} \rangle }}$ in SL after the $1:\frac {\sqrt {3}}{\sqrt {2}}$ scale. The belief axis is the line from B ₀ (its origin) toward the B vertex, the disbelief axis is the line from D ₀ toward the D vertex, and the uncertainty axis is the line from U ₀ toward the U vertex

In order to keep the discussion consistent with Jøsang’s work (Jøsang 2001), in what follows we will scale triangle ${\mathop {BDU}\limits ^{\triangle }}$ by a factor $1:\frac {\sqrt {3}}{\sqrt {2}}$ thus obtaining that $|\overrightarrow {B_{0} B}| = |\overrightarrow {D_{0} D}| = |\overrightarrow {U_{0} U}| = 1$.

These geometric relations lie at the heart of the Cartesian transformation operator, which is the subject of the next subsection.

A.1 The cartesian representation of opinions

As shown in Appendix A, an opinion in SL can be represented as a point in a planar figure (Fig. 6) laying on a Cartesian plane. In this section we will introduce the Cartesian transformation operator, which returns the Cartesian coordinate of an opinion.

First of all, let us define the axes of the Cartesian system we will adopt.

Definition 15

Given the SL plane region $\mathop {BDU}\limits ^{\triangle }$ , the associated Cartesian system is composed by two axes, named respectively x,y, where the unit vector of the x axis ${\overrightarrow {e_{x}}} = \frac {1}{|\overrightarrow {BD}|} \overrightarrow {BD}$, the unit vector of the y axis the y axis ${\overrightarrow {e_{y}}} = {\overrightarrow {e_{{u_{}}}}}$, and B is the origin.

Figure 7 depicts this Cartesian system.

Fig. 7

An opinion and its representation in the Cartesian system

The correspondence between the three values of an opinion and the corresponding coordinate in the Cartesian system we defined is shown in the following proposition.

Proposition 4

(Cerutti et al. 2013b, Proposition 1) Given a SL plane region $\mathop {BDU}\limits ^{\triangle }$ and its associated Cartesian system 〈x,y〉, an opinion $O \triangleq {{\langle {b_{O}}, {d_{O}}, {u_{O}} \rangle }}$ is identified by the coordinate 〈x _O ,y _O 〉 s.t.:

• ${x_{O} \triangleq \frac {{d_{O}} + {u_{O}}~\cos (\frac {\pi }{3})}{\sin (\frac {\pi }{3})}};$

• $y_{O} \triangleq {u_{O}}.$

Proof

Proving that $y_{O} \triangleq {u_{O}}$ is trivial.

Let us focus on the first part of the proposition. Consider Fig. 7. Given O, we note that the for the point P, $\frac {1}{|\overrightarrow {P O}|} \overrightarrow {P O} = {\overrightarrow {e_{b}}}$ (i.e. $\overrightarrow {PO}$ is parallel to the disbelief axis) and $\frac {1}{|\overrightarrow {BP}|} \overrightarrow {BP} = \frac {1}{|\overrightarrow {BU}|} \overrightarrow {BU}$ (i.e. P is on the line $\overrightarrow {BP}$), and therefore ${\angle _{BPO}} = \frac {\pi }{2}$. Then we must determine Q and R s.t. $\overrightarrow {Q R} = \overrightarrow {P O}$ and y _R =0. By construction $|\overrightarrow {P O}| = |\overrightarrow {Q R}| = {d_{O}}$, y _R =0. By construction $|\overrightarrow {P O}| = |\overrightarrow {Q R}| = {d_{O}}$, ${\angle _{Q R B}} = \frac {\pi }{6}$, ${\angle _{ORD}} = \frac {\pi }{3}$, and $x_{O} \triangleq |\overrightarrow {BS}| = |\overrightarrow {BR}| + |\overrightarrow {RS}|$, where $|\overrightarrow {BR}| = \frac {{d_{O}}}{\sin (\frac {\pi }{3})}$, and $|\overrightarrow {RS}| = \frac {{u_{O}}}{\sin (\frac {\pi }{3})} \cos (\frac {\pi }{3})$. □

There are some notable elements of Fig. 7 that we will repeatedly use below, and we therefore define them as follows:

• the angle α _O determined by the x axis and the vector $\overrightarrow {BO}$;

• the three angles (γ _O ,δ _O , and 𝜖 _O ) of the triangle ${\overset {\triangle }{ODU}}$, namely the triangle determined by linking the point O with the vertex D and U through straight lines.

Definition 16

Given the SL plane region $\mathop {BDU}\limits ^{\triangle }$, given O=〈b _O ,d _O ,u _O 〉 whose coordinates are 〈x _O ,y _O 〉 where ${x_{O} \triangleq \frac {{d_{O}} + {u_{O}}~\cos (\frac {\pi }{3})}{\sin (\frac {\pi }{3})}}$ and $y_{O} \triangleq {u_{O}}$, let us define and (via trivial trigonometric relations) compute the following.

• ${\alpha _{O} \triangleq {\angle _{OBD}}} =$

  $\left \{ \begin {array}{l l} 0 & \text {if } {b_{O}} = 1\\ {\arctan \left (\frac {{u_{O}}~\sin (\frac {\pi }{3})}{{d_{O}} + {u_{O}}~\cos (\frac {\pi }{3})}\right )} & \text {otherwise} \end {array} \right .$;

• ${\beta _{O} \triangleq {\angle _{ODB}} } =$

  $\left \{ \begin {array}{l l} {\frac {\pi }{3}} & \text {if } {d_{O}} = 1\\ {\arctan \left (\frac {{u_{O}}~\sin (\frac {\pi }{3})}{1-({d_{O}}+{u_{O}}~\cos (\frac {\pi }{3}))}\right )} & \text {otherwise} \end {array}\right .$;

• ${\gamma _{O} \triangleq {\angle _{ODU}} = \frac {\pi }{3}} - \beta _{O}$;

• ${\delta _{O} \triangleq {\angle _{OUD}} = }$

  $\left \{ \begin {array}{l l} 0 & \text {if } {u_{O}} = 1\\ {\arcsin \left (\frac {{b_{O}}}{|\overrightarrow {OU}|}\right )} & \text {otherwise} \end {array}\right .$;

• $\epsilon _{O} \triangleq {\angle _{DOU}} = \pi - \gamma _{O} - \delta _{O}$;

where ${|\overrightarrow {OU}| = \sqrt {\frac {1}{3} (1 + {d_{O}} - {u_{O}})^{2} + {b_{O}}^{2}}}$.

The angle α _O is called the direction of O.

Equivalently, we can write $\overrightarrow {BO}$ or ${\langle B, \alpha _{O}, |\overrightarrow {BO}| \rangle }$.

Finally, as an element of SL is bounded to have its three components between 0 and 1, we are also interested in determining the point M _O such that the vector $\overrightarrow {BM_{O}}$ has the maximum magnitude given (a) the direction α _O of an opinion O, and (b) M _O is a SL opinion. In other words, determining the magnitude of $\overrightarrow {BM_{O}}$ will allow us to re-define the vector $\overrightarrow {BO}$ as a fraction of $\overrightarrow {BM_{O}}$.

Definition 17

Given the SL plane region ${\mathop {BDU}\limits ^{\triangle }}$, and $O \triangleq {{\langle {b_{O}}, {d_{O}}, {u_{O}} \rangle }}$ whose coordinates are 〈x _O ,y _O 〉 where ${x_{O} \triangleq \frac {{d_{O}} + {u_{O}}~\cos (\frac {\pi }{3})}{\sin (\frac {\pi }{3})}}$ and $y_{O} \triangleq {u_{O}}$, and ${\alpha _{O} \triangleq {\angle _{OBD}} = \arctan \left (\frac {{u_{O}}~\sin (\frac {\pi }{3})}{{d_{O}} + {u_{O}}~\cos (\frac {\pi }{3})}\right )}$, let us define $M_{O} \triangleq {\langle x_{M_{O}}, y_{M_{O}} \rangle }$ as the intersection of the straight line passing for O and B, and the straight line passing for U and D, and thus define the following.

• $x_{M_{O}} \triangleq \frac {2 - y_{O} + \tan (\alpha _{O})~ x_{O}}{\tan (\alpha _{O}) + \sqrt {3}}$;

• ${y_{M_{O}} \triangleq -\sqrt {3}~ x_{M_{O}} + 2}$.

Appendix B: Proof

Proof

• i. Let us prove that 0≤b _W ≤1, 0≤d _W ≤1, 0≤u _W ≤1.

  To prove that ${b_{{W_{}}}} \geq 0$, ${d_{{W_{}}}} \geq 0$, and ${u_{{W_{}}}} \geq 0$ is trivial since Cand Tare opinions.

  ${b_{{W_{}}}} = {b_{{C_{}}}} \cdot {b_{{T_{}}}} \leq 1$ is immediate since Cand Tare opinions.

  ${d_{{W_{}}}} = {b_{}} \cdot {d_{{T_{}}}} + {d_{{C_{}}}} \leq 1$ can be rewritten as ${d_{{T_{}}}} \leq 1 + \frac {{u_{{C_{}}}}}{{b_{{C_{}}}}}$ if ${b_{{C_{}}}} \neq 0$, or ${d_{{C_{}}}} \leq 1$ otherwise. Both in-equations are verified since Cand Tare opinions.

  ${u_{{W_{}}}} = {b_{}} \cdot {u_{{T_{}}}} + {u_{{C_{}}}} \leq 1$ can be rewritten as ${d_{{T_{}}}} \leq 1 + \frac {{d_{{C_{}}}}}{{b_{{C_{}}}}}$ if ${b_{{C_{}}}} \neq 0$, or ${u_{{C_{}}}} \leq 1$ otherwise. Both in-equations are verified since Cand Tare opinions.

  Finally, ${b_{{W_{}}}} + {d_{{W_{}}}} + {u_{{W_{}}}} = {b_{{C_{}}}} ({b_{{T_{}}}} + {d_{{T_{}}}} + {u_{{T_{}}}}) + {d_{{C_{}}}} + {u_{{C_{}}}} = 1$

• ii.Given C=〈1,0,0〉, W=T∘ _n Cis such that:

  • ${b_{{W_{}}}} = {b_{{C_{}}}} \cdot {b_{{T_{}}}} = {b_{{T_{}}}}$;

  • ${d_{{W_{}}}} = {b_{{C_{}}}} \cdot {d_{{T_{}}}} + {d_{{C_{}}}} = {d_{{T_{}}}}$;

  • ${u_{{W_{}}}} = {b_{{C_{}}}} \cdot {u_{{T_{}}}} + {u_{{C_{}}}} = {u_{{T_{}}}}$.

• iii.Given C=〈0,0,1〉, W=T∘ _n Cis such that:

  • ${b_{{W_{}}}} = {b_{{C_{}}}} \cdot {b_{{T_{}}}} = 0 = {b_{{C_{}}}}$;

  • ${d_{{W_{}}}} = {b_{{C_{}}}} \cdot {d_{{T_{}}}} + {d_{{C_{}}}} = {d_{{C_{}}}}$;

  • ${u_{{W_{}}}} = {b_{{C_{}}}} \cdot {u_{{T_{}}}} + {u_{{C_{}}}} = {u_{{C_{}}}}$.

• iv.By contradiction, ${b_{{W_{}}}} = {b_{{C_{}}}} \cdot {b_{{T_{}}}} > {b_{{T_{}}}}$ leads to ${b_{{C_{}}}} > 1$, which is impossible.

□

Proof

By Definition 8, ${\mathbb {O}_{{T_{}}{}}} = {\{X \in {\mathbb {O}}{} | {b_{X}} \leq {b_{{T_{}}{}}}\}}$. From Proposition 4,

$$\begin{array}{ll} {b_{X}} & \leq {b_{{T_{}}}}\\ y_{X} & \geq \sqrt{3}x_{X} + 2(1 - {b_{{T_{}}}})\\ {u_{X}} & \geq 1 - {d_{X}} - {b_{{T_{}}}} \end{array} $$

Therefore:

• if d _X =0, u _X ≥1−b _T (limit case ${\langle {b_{{T_{}}}}, 0, 1 - {b_{{T_{}}}} \rangle }$;

• if u _X =0, d _X ≥1−b _T (limit case ${\langle {b_{{T_{}}}}, 1-{b_{{T_{}}}}, 0 \rangle } = Q$).

□

Proof

Proving the thesis in the limit case is trivial. In the following we will assume, without loss of generality, that $\alpha _{C^{\prime }} \neq \frac {\pi }{2}$, $\alpha _{C^{\prime }} \neq -\frac {\pi }{3}$, $\alpha _{C^{\prime }} \neq \frac {2}{3} \pi$.

• (i.)W=〈b _W ,d _W ,u _W 〉 must respect

$$ {u_{W}} + {d_{W}} \leq 1 $$

(1)

From Definition 9 it is clear that Eq. 1 can be rewritten as follows.

$$ \begin{array}{ll} {u_{T}}{}+ {d_{T}}{}+{}\frac{r_{C}}{2} \frac{2 \sqrt{\tan^{2}(\alpha_{C^{\prime}}) + 1}}{|\tan(\alpha_{C^{\prime}}) + \sqrt{3}|} ~{b_{T}}{} ~(\sin(\alpha_{C^{\prime}}) {}+{}\sqrt{3}\cos(\alpha_{C^{\prime}})) \leq 1 \end{array} $$

(2)

In turn, using the relation $\tan (\alpha _{C^{\prime }}) = \frac {\sin (\alpha _{C^{\prime }})}{\cos (\alpha _{C^{\prime }})}$, this can be rewritten as

$${u_{T}} + {d_{T}} + r_{C} {b_{T}} \leq 1 $$

which entails the requirement that $r_{C} \leq \frac {1 - {u_{T}} - {d_{T}}}{{b_{T}}} = \frac {{b_{T}}}{{b_{T}}} = 1$. However, from Definition 9, we know that r _C ≤1, fulfilling this requirement.

• (ii.) C=〈1,0,0〉 implies that $|\overrightarrow {BC}| = 0$ and thus r _C =0. Therefore, from Definition 9, ${u_{W}} = {u_{T}} + \sin (\alpha _{C^{\prime }}) r_{C} |\overrightarrow {TM_{C^{\prime }}}| = {u_{T}}$ and this results also implies that d _W =d _T . Since Point 1 shows that W is an opinion in SL, we conclude that W=T.

• (iii.) C=〈0,0,1〉 implies r _C =1, $\frac {\frac {\pi }{3} \epsilon _{T}}{\frac {\pi }{3}} - \beta _{T} \leq \alpha _{C^{\prime }} \leq \epsilon _{T} - \beta _{T}$, and thus $\alpha _{C^{\prime }} = \epsilon _{T} - \beta _{T} = \epsilon _{T} - \beta _{T} = \frac {2}{3} \pi - \delta _{t}$. Therefore, we obtain that ${u_{W}} = {u_{T}} + \frac {{b_{T}}}{2} (1 + \frac {\sqrt {3}}{\tan (\delta _{T})})$. From Definition 16 and the trigonometric property that $\tan (\arcsin (v)) = \frac {v}{\sqrt {1 - v^{2}}}$ we obtain that ${u_{W}} = {u_{T}} + \frac {{b_{T}}}{2} + \frac {\sqrt {3}}{2} \sqrt {|\overrightarrow {TU}|^{2} - {b_{T}}^{2}}$. From Definition 16 we can write:

  $$ \begin{array}{ll} {u_{W}} & = {u_{T}} + \frac{{b_{T}}}{2} + \frac{\sqrt{3}}{2} \frac{1 + {d_{T}} - {u_{T}}}{\sqrt{3}}\\ & = \frac{1}{2} (1 + {b_{T}} + {d_{T}} + {u_{T}} ) = 1 \end{array} $$

  (3)

  Similarly, ${d_{W}} = {d_{T}} + \frac {{u_{T}}}{2} - \frac {1}{2} + \frac {\sqrt {3}}{2} \frac {1}{\sin (\delta _{T})} {b_{T}} = {d_{T}} + \frac {{u_{T}}}{2} - \frac {1}{2} + \frac {\sqrt {3}}{2} |\overrightarrow {TU}|$. From Definition 16 we have

  $$ \begin{array}{ll} {d_{W}} & = {d_{T}} + \frac{{u_{T}}-1}{2} + \frac{3}{4}{b_{T}} - \frac{\sqrt{3}}{4} {b_{T}} \frac{1 + {d_{T}} - {u_{T}}}{\sqrt{3} {b_{T}}} \\ & = \frac{1}{4} (4{d_{T}} + 2 {u_{T}} - 2 + 3 {b_{T}} - 1 + {u_{T}} - {d_{T}}) = 0 \end{array} $$

  (4)

  From Eqs. 3 and 4, together with Point 1, it follows that W=〈0,0,1〉=C.

• Suppose instead b _W >b _T .

$$\begin{array}{@{}rcl@{}}1 - {d_{W}} - {u_{W}} > 1 - {d_{T}} - {u_{T}} \end{array} $$

$$\begin{array}{@{}rcl@{}} {d_{W}} + {u_{W}} < {d_{T}} + {u_{T}} \end{array} $$

$$\begin{array}{@{}rcl@{}} {d_{T}} + \sin(\alpha_{C^{\prime}} + \frac{\pi}{3}) \frac{r_{C}}{\sin(\alpha_{C^{\prime}} + \frac{\pi}{3})} + {u_{T}} < {d_{T}} + {u_{T}} \end{array} $$

$$\begin{array}{@{}rcl@{}} r_{C} < 0 \end{array} $$

but 0≤r _C ≤1. Quod est absurdum. □

Proof

• (i.) To prove that 〈b ₁(W ₁,…,W _n ),d ₁(W ₁,…,W _n ),u ₁(W ₁,…,W _n )〉 is an opinion, we have to show that u ₁(W ₁,…,W _n ) + d ₁(W ₁,…,W _n ) Subscript>≤1 holds.

  $$\begin{array}{rl} {u_{{{\Gamma}_{1}}(W_{1}, \ldots, W_{n})}} + {d_{{{\Gamma}_{1}}(W_{1}, \ldots, W_{n})}} = & {\frac{1}{{\sum}_{i=1}^{n} K_{i}} \left( \sum\limits_{i=1}^{n} K_{i}({u_{W_{i}}} + {d_{W_{i}}})\right)}\\ = & {\frac{1}{{\sum}_{i=1}^{n} K_{i}} \left( \sum\limits_{i=1}^{n} K_{i}(1 - {b_{W_{i}}})\right)}\\ = & 1 - {\frac{1}{{\sum}_{i=1}^{n} K_{i}} \left( \sum\limits_{i=1}^{n} K_{i}~{b_{W_{i}}} \right)} \end{array}$$

• (ii.) From Proposition 4,

  $$\left\{\begin{array}{l} {x_{{{\Gamma}_{1}}(W_{1}, \ldots, W_{n})} = \frac{{d_{{{\Gamma}_{1}}(W_{1}, \ldots, W_{n})}}}{\sin(\frac{\pi}{3})} + \frac{1}{2~\sin(\frac{\pi}{3})~{\sum}_{i=1}^{n} K_{i}} \left( \sum\limits_{i=1}^{n} K_{i}~ {u_{W_{i}}}\right)}\\ {x_{{{\Gamma}_{1}}(W_{1}, \ldots, W_{n})} = \frac{1}{\sin(\frac{\pi}{3}) {\sum}_{i=1}^{n} K_{i}} \left( \sum\limits_{i=1}^{n} K_{i}~ \left({d_{W_{i}}} + \frac{{u_{W_{i}}}}{2}\right) \right)} \end{array}\right.$$

Thus we obtain:

$$\begin{array}{rl} {d_{{{\Gamma}_{1}}(W_{1}, \ldots, W_{n})}} = & {\sin(\frac{\pi}{3})\left(\frac{1}{\sin(\frac{\pi}{3}) {\sum}_{i=1}^{n} K_{i}} \left( \sum\limits_{i=1}^{n} K_{i}~ \left({d_{W_{i}}} + \frac{{u_{W_{i}}}}{2}\right) \right) + \right. }\\ & ~~~~ {\left.- \frac{1}{2~\sin(\frac{\pi}{3})~{\sum}_{i=1}^{n} K_{i}} \left( \sum\limits_{i=1}^{n} K_{i}~ {u_{W_{i}}}\right) \right)}\\ = & {\frac{1}{{\sum}_{i=1}^{n} K_{i}} \left( \left(\sum\limits_{i=1}^{n} K_{i}~ \left({d_{W_{i}}} + \frac{{u_{W_{i}}}}{2}\right)\right) - \left( \sum\limits_{i=1}^{n} \frac{{u_{W_{i}}}}{2} \right) \right)}\\ = & {\frac{1}{{\sum}_{i=1}^{n} K_{i}} \left( \sum\limits_{i=1}^{n} K_{i}~ {d_{W_{i}}}\right)} \end{array}$$

Since ${\frac {1}{{\sum }_{i=1}^{n} K_{i}} \left (\sum \limits _{i=1}^{n} K_{i}~{b_{W_{i}}} \right )} \geq 0$, then u ₁(W ₁,…,W _n ) + dΓ₁(W ₁,…,W _n )1 holds.

• (iii.) Immediate from Proposition 4.

□