# An Adaptive Dempster-Shafer Theory of Evidence Based Trust Model in Multiagent Systems
## 3 Dempster-Shafer Theory of Evidence  
The Dempster-Shafer theory of evidence, proposed by [PERSON] and [PERSON] to handle uncertainty [39; 40], has a wide range of applications in risk assessments, medical diagnosis, and target recognition [41; 42; 43]. It is a typical method of uncertainty information fusion because of its adjustability in uncertainty modeling [44]. Some preliminaries are introduced below. **Definition 1**. : _The frame of discernment and BPAs._  
Evidence theory is defined in the frame of discernment denoted by \(\Omega\), which consists of \(n\) mutually exclusive and collectively exhaustive elements; \(2^{\Omega}\) is the set of all subsets of \(\Omega\), which is represented by \(2^{\Omega}=\{\emptyset,\{\theta_{1}\},\{\theta_{2}\},\ldots,\{\theta_{n}\},\{ \theta_{1},\theta_{2}\},\ldots,\{\theta_{1},\theta_{2},\ldots,\theta_{n}\}\}\). Mathematically, a basic probability assignment (BPA, also known as mass function) is \(2^{\Omega}\rightarrow[0,1]\), with the following conditions satisfied:  
\[\left\{ \begin{array}{l}\sum_{A\in\Omega}m(A)=1;\\ m(\emptyset)=0,\end{array} \right. \tag{1}\] \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Definition** & **Why the Factor Is Influential** \\ \hline Uncertainty & Uncertainty caused by fading, & To ensure trust accuracy in \\ (Direct trust) & randomness, incompleteness, etc. & dynamic MASs \\ \hline Consistency & The degree of similarity with & To tell high-quality evidence; \\ (Confidence) & third-party testimony & Avoid sudden change \\ \hline Credibility & The credibility degree of & To distinguish trusted witnesses \\ (Confidence) & third-party witnesses & and avoid group deception \\ \hline Certainty & The certainty degree of & To capture the certainty \\ (Confidence) & third-party testimony & of testimony \\ \hline Motivation & Inspire witnesses to be honest & Ensure the system runs correctly \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Influential factors for trust estimation in MASs. where \(\emptyset\) is the empty set, the subsets \(A\in 2^{\Omega}\) such that \(m(A)>0\) is named as focus element, and the set of all focus elements is called core. The other two functions (belief function and plausibility function) are defined as follows. **Definition 2**. : _Belief function._  
The belief function Bel of BPA \(m\) is defined as follows:  
\[Bel_{m}(A)=\sum_{B\in 2^{\Omega}:B\subseteq A}m(B) \tag{2}\]  
**Definition 3**. : _Plausibility function._  
The plausibility function Pl of BPA \(m\) is defined as follows:  
\[Pl_{m}(A)=\sum_{B\in 2^{\Omega}:B\bigcap A\
eq\emptyset}m(B) \tag{3}\]  
In the Dempster-Shafer theory of evidence, [PERSON]'s combination rule can be used to combine multiple pieces of evidence to reach the final evaluation. Given two BPAs \(m_{1}\) and \(m_{2}\), the [PERSON]'s combination rule denoted by \(m=m_{1}\oplus m_{2}\) can fuse them as follows. **Definition 4**. : _[PERSON]'s combination rule._  
\[m(A)=\left\{ \begin{array}{ll}\frac{1}{1-K}\sum_{B\cap C=A}m_{1}(B)m_{2}(C),& A\
eq\emptyset;\\ 0,&A=\emptyset;\end{array} \right. \tag{4}\]  
with \(K=\sum_{B\cap C=\emptyset}m_{1}(B)m_{2}(C)\), \(A\), \(B\), and \(C\) are elements in \(2^{\Omega}\). The normalization constant \(K\) shows the conflicting degree between defined BPAs. If \(K=1\), the two pieces of evidence totally conflict, and [PERSON]'s combination rule is not applicable. If \(K=0\), the two pieces of evidence are non-conflicting. **Definition 5**. : _Evidence distance._  
With the full application of evidence theory, the study of the distance of evidence has attracted increasing interest. The distance measure of evidence can represent the dissimilarity between two BPAs, and several definitions of distance in evidence theory have also been proposed, such as [PERSON]'s distance [45], [PERSON]'s cosine similarity [46], and [PERSON] transferable belief model (TBM) global distance measure [47].