this end, we provide a simple but optimal method to compute which agents to rewire or not in $O(n^2)$ time (further explained later), which is inspired from Pandora’s Rule[^2] [@weitzman1979optimal].

Supposing that agent $i$ has current baseline $y_i$ and a potential partner $j$ in $\bar O_i$, to determine whether to rewire partner $j$ through our $K$-sight optimal rewiring approach, we have:

-   In the following $K$ rounds, the expected interaction value we could obtain from neighborhood $O_i$ is at least: $$K y_i.$$

-   If choose to rewire agent $j$, we could get a net benefit of the following $K$-rounds interactions in the worst cases as follows: $$-c_i^j + K (\int_{-\infty}^{y_i}x \ \;\text{d}F_j^i(x) + {y_i}' \int_{y_i}^{\infty}\;\text{d}F_j^i(x) ).$$ where ${y_i}'$ is the new baseline if $x \ge y_i$ after breaking the connection with old baseline one. We define ${y_i}'$ as ${y_i}' = \min\{x, y_i^{sec}\}$ where $y_i^{sec}$ is the second minimum expected payoff in $O_i$ given that agent $i$ has at least two neighbors before rewiring.[^3] Formally we have: $${y_i}' = \int_{-\infty}^{y_i^{sec}}x \ \;\text{d}F_j^i(x) + y_i^{sec} \int_{y_i^{sec}}^{\infty}\;\text{d}F_j^i(x).$$

-   Further, we use $\Lambda$ for the single-round payoff difference between not rewiring (Eq.6) and rewiring agent $j$ (Eq.7) which can be formalized as follows: $$\Lambda 