# Transparent Task Delegation in Multi-Agent Systems Using the QuAD-V Framework
## 2 Background
### Impressions  
Impressions are evaluative beliefs that a delegator \(\alpha\) produces or receives from third parties about \(\beta\)'s competencies as a delegatee for \(\tau\). These beliefs attest to \(\beta\)'s capability to perform the task \(\tau\) according to its performance estimates [33]. For example, an impression, produced by \(\alpha\), can include scores assigned to \(\beta\) for performing \(\tau\) within the agreed budget and delivery time, based on criteria such as cost and time. These scores reflect how well \(\beta\) met the expectations of \(\alpha\) for \(\tau\). An impression is a five-tuple \(\langle\alpha\), \(\beta\), \(\tau\), \(t\), \(S\rangle\), where  
* \(\alpha\) is the delegator who formed the impression. * \(\beta\) is the assessed delegate. * \(\tau\) is the task performed by \(\beta\). * \(t\) is the time in which the impression was created by \(\alpha\). * \(S=[(c_{1},s_{1}),...,(c_{k},s_{k})]\) is a vector of ratings, where a pair \((c_{i},s_{i})\) represents a score \(s_{i}\in[0,1]\) assigned to \(\beta\) by \(\alpha\) concerning its performance regarding a \(\tau\)'s criterion \(c_{i}\) of \(\tau\). During task execution, a delegator \(\alpha\) evaluates a delegate \(\beta\) by comparing \(\beta\)'s estimated performance for a task \(\tau\) with the actual outcome, for each criterion associated with \(\tau\). The estimation error for a criterion \(c\) is the difference between the estimated and actual performance. If this error is zero, \(\beta\) accurately predicted its performance for \(c\). Otherwise, in the impression produced by \(\alpha\), \(\beta\) is penalized based on its estimation error. Assuming an offer \(Off=\langle\alpha,\beta,\tau,V\rangle\) and an outcome \(Out=\langle\alpha,\beta,\tau,V^{\prime}\rangle\), the delegator \(\alpha\) assigns a score to \(\beta\) based on the criterion \(c\) of task \(\tau\). This score is stored in an impression, represented as a five-tuple, \(Imp=\langle\alpha,\beta,\tau,t,S\rangle\), where \(S\) is a vector of ratings. The score assigned to \(c\) is denoted as \(S[c]\in[0,1]\) and is formally defined as follows:  
\[Imp(S[c])= \begin{cases}\frac{Off(V[c])}{Out(V[c])}&\text{if }isMin(c)\wedge Out(V^{ \prime}[c])\geq Off(V[c])\\ \frac{Out(V^{\prime}[c])}{Off(V[c])}&\text{if }isMax(c)\wedge Out(V^{\prime}[c]) \leq Off(V[c])\\ 1&\text{otherwise}\end{cases}  \tag{1}\]  
where \(isMin(c)\) and \(isMax(c)\) indicate whether \(c\) is a minimization (e.g., time, cost) or maximization criterion (e.g., quality). If \(\beta\)'s estimation is accurate or exceeds \(\alpha\)'s expectations, the score for \(c\) is one [2; 34]. Additionally, impressions can be grouped to build more comprehensive social measures of a partner's competencies, including its social image, reputation, and knowhow [13; 35]. Even though these are distinct concepts, they are complementary in agent interactions. _Social image_ represents a delegator's personal assessment of a delegatee's competencies based on direct interactions. It captures the delegator's immediate impression of how well the delegatee performed in specific tasks. _Reputation_, on the other hand, is a collective measure formed by aggregating impressions from multiple delegators. It reflects a general consensus on a partner's competencies as a delegatee, even when individual delegators lack direct experience with that partner. _Know-how_ is a specialized form of reputation where a delegatee accumulates impressions from delegators about tasks performed in the past, similar to job references. These impressions can be shared with potential new delegators, offering additional information about the partners competencies concerning the execution of a given task. To illustrate these differences, consider a scenario where an agent \(\alpha\) (buyer) is evaluating an agent \(\beta\) (seller) for book and magazine purchases.