## 4 Adaptive \(\alpha\)-Value Computation
### [Cl/Co,F/Co] and [F/Co,Cl/NC] Cases
#### 4.3.1 [F/Co,Cl/NC] case. Based on the human-operator criterion (set by a threshold value) we want an \(\alpha\)-value that makes, following the scenario depicted in Figure 9, \(T_{3}\) preferred over \(T_{2}\) if and only if \(T_{3}\) belongs to the circular area defined by the \(HO\)-_Threshold_. Next, the existence of such an \(\alpha\) parameter will be demonstrated, and its value will be derived as well. Figure 9: [F/Co,Cl/NC] case. **Proposition 2**. : _When \(T_{j}\) and \(T_{k}\) belong to [F/Co,Cl/NC] is always possible to find an \(\alpha\)-value that satisfies the following inequality:_  
\[\Phi_{i}(\alpha,T_{j},R)=\alpha\cdot\Psi_{i}(T_{j})+\beta\cdot\Omega_{i}(T_{j},R )\geq\alpha\cdot\Psi_{i}(T_{k})=\Phi_{i}(\alpha,T_{k},R) \tag{12}\]  
\[ \begin{split} s.t.&\Omega_{i}(T_{j},R)>0,\Omega_{i}(T_ {k},R)=0\\ &\Psi_{i}(T_{j})\leq\Psi_{i}(T_{k})\end{split} \]  
Proof. : Let \(\Omega_{1}\) the utility assigned to the fact of being connected with only one teammate. Then, from (9), it is possible to state that: if \(T_{j}\) belongs to any \([*/Co]\) class, \(\Omega_{1}\leq\Omega_{i}(T_{j},R)\), \(\forall(i,j)\) over time. Moreover, if the number of robots does not change, it is also possible to state that \(\Omega_{1}\) remains invariant over time. Applying this result into (12) leads to the inequality presented next in (13):  
\[ \begin{split}\Phi_{i}(\alpha,T_{j},R)\geq\alpha\cdot\Psi_{i}(T_{ j})+\beta\cdot\Omega_{1}\geq&\alpha\cdot\Psi_{i}(T_{k})=\Phi_{i}( \alpha,T_{k},R)\\ \alpha\cdot\Psi_{i}(T_{j})+(1-\alpha)\cdot\Omega_{1}\geq& \alpha\cdot\Psi_{i}(T_{k})\\ \alpha\cdot(\Psi_{i}(T_{j})-\Omega_{1})+\Omega_{1}\geq& \alpha\cdot\Psi_{i}(T_{k})\\ \frac{\Omega_{1}}{\Psi_{i}(T_{k})-\Psi_{i}(T_{j})+\Omega_{1}} \geq&\alpha\end{split}  \tag{13}\]  
Besides, substituting \(\Omega_{1}=x\) and \(\Psi_{i}(T_{k})-\Psi_{i}(T_{j})=u\), equation (13) may be rewritten as follows:  
\[\alpha\leq\frac{x}{u+x}\ \implies\alpha\leq inf\left(\frac{x}{u+x}\right) \tag{14}\]  
\[ \begin{split}\text{s.t. }0<c\leq x\leq 1\\ 0\leq u\leq 1\\ \text{given that: }\Omega_{1}=c\\ \Psi_{i}(T_{k})\geq\Psi_{i}(T_{j})\end{split} \]  
From (14) is possible to claim the existence of an \(\alpha\)-value that obey any _HO-Threshold_ if and only if the function \(\frac{x}{u+x}\) presents an absolute minimum on the domain:  
\[D=\{(x,u)\ |\ 0<c\leq x\leq 1,0\leq u\leq 1\}.\]  
This fact can be stated employing _Weierstrass_ theorem (a function \(f\) has an absolute extreme if it is continuous and its domain is compact). Besides, the minimum point might be calculated analysing both: (i) the relative extrema and (ii) the points lying on the border of \(D\). Following this procedure, it is possible to find the absolute extreme of the function \(\frac{x}{u+x}\) in \((x,u)=(c,1)\). Moreover, it is remarkable that this extreme represents a place where the most demanding conditions are reached: task \(T_{j}\) presents the lowest positive connectivity utility, and the distance between both tasks is the largest. Hence, the existence of a positive value \(\alpha\leq\frac{\Omega_{1}}{1+\Omega_{1}}\) (regardless of how demanding can be the distance relation between tasks) that might alter the task selection in favour of connectivity has been demonstrated. Nevertheless, in (13) \(\alpha\) is independent of the _HO-Threshold_. Consequently, its direct application would result in a strictly connectivity-guided exploration, where tasks that offer connectivity are always preferred over the rest no matter how far they are. Therefore, to relate it with an _HO-Threshold_ the value of the term \(\Psi_{i}(T_{j})\) in (13) must be substituted by the utility of being _HO-Threshold_ far from the robot, say \(\Psi_{i}(T_{\text{HO}})\). Next, the value of the term \(\Psi_{i}(T_{k})\) is substituted by \(1\) since \(\Psi_{i}(T_{k})=1\) representsthe necessary condition to reach the extreme coordinate \(u=1\) that arose from (14).