The US Senate vote for that destructive bill seems like a game theory problem. Secret ballots might have led to a very different outcome... though arguably bad precedent in a democracy, and may not work given media pressure. Alternative is to just elect people with a least a sliver of courage.
This comes to mind because I've seen the dynamic is SO many professional settings (faculty meetings, etc.), in which folks might have voted differently had their vote been private, for better or for worse. Again, a better solution may be to prioritize professional integrity and courage in hiring!