Game Theory and Impeachment:
https://www.theatlantic.com/ideas/archive/2019/10/mitt-romney-versus-trump/600508/
This article concerns how members of the GOP have been markedly more critical of President Trump in recent weeks, with a specific focus on how Mitt Romney has seemingly built up his vocal opposition to President Trump over the course of the Trump Presidency. The most interesting part of the article references Jeff Flake, a former Republican Senator, who made a statement that “35 or more Republican Senators might vote in a trial to remove Trump from Office, but only if the vote was held privately”. The article then enumerates how it is in the best interest of the potentially anti-Trump senators, if they exist, to move as one unit. Trump can make whatever statements he would like to make about one dissenting Senator, which would likely be extremely politically damaging to the senators, but if multiple senators were in a defined bloc against President Trump, Trump couldn’t credibly make dismissing claims against them.
This is a problem that is a true game theory scenario. If all of the anti-Trump Republican Senators decide to identify themselves, then they effectively are able to shield themselves from Trump’s criticism due to their strength in numbers, and they can achieve the political victory they desire. However, if some of them end up being unaligned, and some decide to be vocal in their dislike of President Trump, those that have identified themselves as against the President will suffer a significant political defeat. Finally, if none of the Senators come out against Trump, there is no result, apart from them not achieving their political goal of impeaching the President. If each Senator has two options: come out against Trump, or continue to support Trump, they stand to benefit the most if they act unanimously for either option. However, if there is discrepancy in their decisions, those that go agains the President open themselves to political annihilation.
The fact that a vote for impeachment will likely be public gives us the most insight into what will likely happen. Unless the Senators identify themselves as anti-Trump before the proceedings, it is unlikely that any will actually vote against him if the impeachment goes to the Senate: to do so would simply be too risky without some sort of confirmation that others would do the same. If the vote is private, it would be in the best interest of the Senators to vote their true value, so to speak, because the repercussions of their decision being public would not occur.