If there are no 100% leads on Day 2 (Sheriff claiming Unseen, Psychic claiming Unseen with public announcement, Observers and Princesses are not 100%) then hanging the King gives BD roughly a 60% (this number is subjective because of Neutral King) chance of removing an enemy while allowing their Nobles, Princess o̶r̶ ̶P̶r̶i̶n̶c̶e̶ to step up to claim the throne. Remember, Neutral Kings are only pro BD until the end of the game.
The chance of the Mastermind converting a Noble or a Princess:
No Nobles, 1 Princess Game: 7.7% chance (Game 1)
2 Nobles, 1 Princess Game: 23% chance (Game 2)
2 Nobles, 2 Princess Game: 30% chance (Game 3)
If we immediately execute the King Day 2 in Game 1, we have a 61% chance of both eliminating an Evil/Neutral King AND electing a BD King. Pretty enormous swing there.
If we immediately execute the King Day 2 in Game 2, we have a 51% chance of both eliminating an Evil/Neutral King AND electing a BD King. The existence of Nobles slightly lowers this chance to a coin flip.
If we immediately execute the King Day 2 in Game 3, we have a 46% chance of both eliminating an Evil/Neutral King AND electing a BD King. At the worst case scenario of this strategy we are faced with a slightly worse coin flip.
Assuming these 3 games all occur just as often as each other, (only accounting for 16 player games for these calculations) we can see that the average chance of killing a Neutral/Evil King Day 2 and electing a BD King is about 53%.
Now let’s look at the numbers involving the King being BD AND Mastermind converting a Royal.
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Game 1: 3% of the time
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Game 2: 8% of the time
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Game 3: 10% of the time
Assuming these games occur about the same time on average, averaging these numbers together gives us 7%. In the worst case scenario of the D2 King Lynch strategy, there is only about 7% of the time it can be catastrophic.
Conclusion:
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53% of the time it is amazing.
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7% of the time it is terrible.
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40% of the time you hang a BD King only to replace them with another BD King, a neutral outcome. This still removes the uncertainty of the first King’s alignment which is valuable and numbers don’t show that.
The law of large numbers says that hanging the King D2 has positive equity 100% of the time.
Does that mean it will backfire sometimes? Yes. Can it lose you the game? Definitely. Just very rarely.
The era of Kings is over.
–all these calculations were done with the assumption the Princess was not killed Night 1–