Collective motion of bird flocks can be explained via the hypothesis of many wrongs and/or a structured leadership mechanism. In pigeons, previous studies have shown that there is a well-defined hierarchical structure and certain specific individuals occupy more dominant positions, suggesting that leadership by the few individuals drives the behavior of the collective. Conversely, by analyzing the same datasets, we uncover a more egalitarian mechanism. We show that both reciprocal relationships and a stratified hierarchical leadership are important and necessary in the collective movements of pigeon flocks. Rather than birds adopting either exclusive averaging or leadership strategies, our experimental results show that it is an integrated combination of both compromise and leadership which drives the group's movement decisions.