Does the Condorcification of a Voting Rule Reduce its Manipulability?

We investigate Condorcification of voting rules, where a voting rule is altered to elect the Condorcet winner whenever one exists, and we focus on its impact on coalitional manipulability. We first remind that despite common belief, Condorcet profiles can be prone to manipulation, even in Condorcet-consistent voting rules. We then review a recent result stating that for a large class of voting rules based on strict total orders, their Condorcification is at most as manipulable as the original rule. We show that for most of them, there is a strict improvement. And we prove that under mild assumptions, the Condorcification of a given rule is the least manipulable system among a set of variants called the emph{partial Condorcifications} of the initial rule. Finally, we extend these results to a more general framework that encompasses weak orders and cardinal voting rules.