Optimal Strategy

1. ## Optimal Strategy

Imagine you are in the following situation:

You are Judge. There are three other players:

• Godfather
• Serial Killer
• Executioner who has already won and is not night immune

You have 1 court call left.

The day just ended. You now have night in order to convince the Serial Killer + Godfather to not kill you. What is the best strategy for achieving that goal?

I found a pretty nice solution to the problem using game theory that almost guarantees a win, although I am curious as to other solutions people can come up with first.

2. ## Re: Optimal Strategy

tell executioner, if judge dies, vote for the person that killed him?

3. ## Re: Optimal Strategy

Well, for the GF the only chance of winning is to not kill. So convince the SK to kill the Exec. If the GF is playing strictly logical, you could even promise the SK to vote on the GF and the GF wouldn't kill you.

Edit: The thing with game theory is, you can't expect anything from the Exec, as he has already won. Also, an SK playing to win will always kill either the Judge or the Exec.

4. ## Re: Optimal Strategy

SK wins by default in a tie, but has the possibility of losing to judge
GF loses by default in a tie, but has the possibility of winning to judge
Exec is AFK

SK has everything to lose by NOT killing the judge. I don't see how you could convince the SK not to kill you.

5. ## Re: Optimal Strategy

Still waiting for solution.

6. ## Re: Optimal Strategy

Specifically, what matters is whether judge lives or dies. Assuming GF and SK don't know each other's identities, then by classical probability:

1. Judge dies = 1 - (4/9) = (5/9) = 56%
2. Judge lives = (2/3) * (2/3) = (4/9) = 44%

But if we optimize for strategy, we assume:

- Exec will side with SK simply because winning as SK is much more respected and harder
- You don't know who anyone is

Then you'd simply night talk that you're the judge and will side with whomever kills the exec. In fact, this is a good opportunity to tell GF and SK to choose first, reveal that you're judge, and then convince them to switch to the other and abuse Monty Hall's problem. Monty Hall's problem works like this:

1. You choose one of three doors (1 door has a car, 2 doors have goats)
2. Host chooses a door with goats and opens it and asks you if you want to switch
3. By probability complement without the information of #2, the probability of you choosing a goat door was initially 2/3 so by switching, you're more likely to hit the exec (2/3) so the chances of exec not dying are (1/3)*(1/3) = (1/9) = 11%

So assuming everyone is knows the Monty Hall problem including SK not realizing it's in his best interests not to kill Judge, then he'll aim for the exec too since both SK and GF have a 2/3 chance each of winning. And you'd automatically win by removing yourself from that equation by exploiting game theory of Monty Hall problem.

Of course, people are emotional, so what will end up happening is classical probability anyway where you only have a 44% chance of living by random choice.

7. ## Re: Optimal Strategy

Originally Posted by SuperJack
Still waiting for solution.
What in the world is this necro...?

Anyway, OP seems young and much less-informed about game theory than they are now - there is no pure game theory solution to guarantee a win.

But this problem is still interesting, even if it doesn't have a clear-cut solution.

As others have noted your biggest threat is the SK.

So at night you tell both killing roles to not attack anyone. The SK has nothing to gain by listening to you, which is why you make a comment at night such as "And Mr. Executioner, let's continue with our plan, we vote up whichever person ignored our plan"

The GF has nothing to gain by attacking, so they very likely won't attack in this instance. And the only way the SK loses is by getting voted up the following day, the presence of two other neutral roles ensures that this is always a risk since the SK can't kill both. So the goal of the Judge should be to maximize the perceived probability that the SK will be voted up if they defect from the Judge's orders. Implying that the executioner is already on-board with the plan is a good method to maximize the doubt in the SK's mind.

And then if you survive the night, who cares what you do, court one, win with the other.

(Oh and you definitely don't reveal who you are if roles haven't been publicly outed - just increases the risk of dying to an SK that is willing to gamble with the executioner)

8. ## Re: Optimal Strategy

This inherently assumes the executioner was in cahoots with the judge beforehand through day PMs. Obviously, if the judge proposes this at the last minute, then it's not going to effective. Also, people are biased towards SK during endgame. If this were a game where executioner needed to survive until the end of the game, I could see it working. But in most cases where they don't, I believe it's a better idea to do the Monty Hall technique assuming SK and GF are familiar with it.

Breakdown under Monty Hall, assuming execution sides with SK by default:

1. SK knows GF
- SK kills Judge to win

2. SK doesn't know GF
- GF kills exec 67-100% of the time, SK kills exec 67% of the time, Judge wins anyway

So if SK doesn't know GF and you've been keeping track, Monty Hall solution is the way to go. If SK knows GF, then I'd use your solution.

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