# Transparent Task Delegation in Multi-Agent Systems Using the QuAD-V Framework
## 2 Background
### QuAD-V Framework  
The QuAD-V framework, proposed by [31], extends the quantitative argumentation debate framework (QuAD) [36]. Its main advantage is the ability to resolve debates using a voting system, where users vote for or against arguments. As defined in [31], QuAD-V is a six-tuple \((A,C,P,R,U,V)\), where  
* A is a finite set of answer arguments (possible responses to a question).
* C is a finite set of con arguments (arguments opposing another argument).
* P is a finite set of pro arguments (arguments supporting another argument).
* R is an acyclic binary relation over \((C\cup P)\times(A\cup C\cup P)\).
* U is a finite set of users.
* V is a total function assigning votes \(V(u,a)\in-,?,+\), where \(u\in U\) votes on argument \(a\in(A\cup C\cup P)\).  
In the QuAD-V framework, arguments can attack or support each other. _Attackers_ and _supporters_ of an argument are defined based on con and pro arguments, respectively. For any argument \(a\in(A\cup C\cup P)\), the set of attackers is \(R^{-}(a)=b\in C|(b,a)\in R\), and the set of supporters is \(R^{+}(a)=b\in P|(b,a)\in R\). Moreover, each argument \(a\) has a vote-based score \(bS(a):A\cup C\cup P\rightarrow\mathbb{I}\), with \(\mathbb{I}=[0,1]\), computed from user votes:  
\[bS(a)=\begin{cases}0.5&\text{if }|U|=0\\ 0.5+(0.5*\frac{N^{+}(a)-N^{-}(a)}{|U|})&\text{if }|U|\
eq 0\end{cases} \tag{2}\]  
where \(N^{+}(a)=|V^{+}(a)|\) is the count of positive votes, with \(V^{+}(a)=u\in U:V(u,a)=+\), and \(N^{-}(a)=|V^{-}(a)|\) is the count of negative votes, with \(V^{-}(a)=u\in U:V(u,a)=-\). Note that in this modeling, an argument with no votes (i.e., \(|U|=0\)) is assigned a base score of 0.5, representing an initial state where it has not yet been evaluated positively or negatively by users [37]. Conversely, if all users vote in favor of an argument \(a\) (i.e., \(|V^{+}(a)|=|U|\)), its base score is 1 (fully positive), whereas if all users vote against it (i.e., \(|V^{-}(a)|=|U|\)), its base score is 0 (fully negative).  
On the other hand, the _acceptance score_ of an argument \(a\) (\(aS\in[0,1]\)) determines its strength. For example, 1 for accepted, 0.5 for neutral, and 0 for rejected [37]. This measure is computed by considering the influence of the attackers and supporters of the argument \(a\). In particular, for each argument (parent), the attackers and supporters are extracted, forming two sequences of child arguments. These sequences can be normalized by the number of arguments, considering the strength of each argument within the sequence. The aggregated strength of a sequence of arguments is computed as follows:  
\[\mathbf{s}(a)=\begin{cases}0,&\text{if }\mathbf{s}=\emptyset\\ \frac{1}{|\mathbf{s}|}\sum\limits_{i=1}^{|\mathbf{s}|}\mathbf{s}_{i},&\text{ otherwise}\end{cases} \tag{3}\]where \(s\) represents the strength of a sequence of arguments, either from the attackers or the supporters. Taking into account the aggregate strength of the attackers (\(s_{att}\)) and supporters (\(s_{supp}\)), the acceptance score of the argument is computed as follows:  
\[aS(a)=\begin{cases}bS(a)+(s_{supp}-s_{att})\times(1-bS(a)),&\text{if }s_{supp}\geq s _{att}\\ bS(a)+(s_{supp}-s_{att})\times bS(a),&\text{otherwise}\end{cases} \tag{4}\]  
Note that the process used to compute the acceptance score of an argument \(a\) ensures that stronger supporters amplify \(a\)'s acceptability without exceeding 1, while stronger attackers reduce it but never below 0.