affected to some degree, are as follows: Weighted Confidence Sharing and Voting.

Note that, in principle, agents do not know their own slopes, so the order of interactions cannot depend upon the individual (or group) $s_i$. However, as both $s_{WCS}$ and $s_{Vot}$ depend linearly on $s_i$, averaging over every permutation of participants yields a result that is proportional to the arithmetical mean of $s_i$, or $\langle s \rangle$. Consequently, to investigate the influence of the hierarchical information-aggregation, it is sufficient to treat each participant as if his/her performance is equal to $\langle s \rangle$.

For convenience, we consider a more general model with the parameter (the amplification multiplier) depending on $g$ (the group size) as follows: $$\begin{aligned}
s_{a_g}(s_1,\ldots ,s_g) = a_g \frac{s_1+\ldots+ s_g}{g}.\label{eq:sag}\end{aligned}$$ This generalization describes both WCS ($a_2=\sqrt{2}$, $a_3=\sqrt{3}$, $\ldots$) and Voting ($a_3 = 3/2$, $\ldots$), and it allows us to give results in an elegant general form.

Shallow hierarchy {#s:ha}
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Our justification for the Shallow hierarchy is the following: people may locally find their partners and then make a collective decision. Then, iteratively, groups of the same (or similar) size make the collective decision.

The analysis is simple when the number of participants is a power of 