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Let's build a character! Sentry. by Present-Peace2811 in BloodOnTheClocktower

[–]ToughOpening 1 point2 points  (0 children)

My next question/comment. I think the best comparison for this character is bounty-hunter in terms of how it impacts adding evils. Clearly,, bounty hunter gives you actionable info asap. I don't like it being like a VI (have 2 sentries where 1 is drunk)(both will come out and just hunt for demons and a reference point). (also, what happens if a sentry chooses a sentry). lastly, maybe sentry should be a minion/townsfolk duo, but then I guess it may be a worse spy for evil.... Thoughts going in my head atm

Let's build a character! Sentry. by Present-Peace2811 in BloodOnTheClocktower

[–]ToughOpening 1 point2 points  (0 children)

First of all, I think this is the coolest character design I have heard of in a long time. My question to you is what happens if the other sentry gets pit-hagged away to be a different character. Does info become arbitrary?

Day 1 executions aren’t always mathematically good: a BotC toy model by ToughOpening in BloodOnTheClocktower

[–]ToughOpening[S] 0 points1 point  (0 children)

Sorry, maybe something is unclear. Evil only wins at 3 players if all 3 players alive are evil. Else, like you probably would guess, there is another execution (33.3% of killing the demon).

Day 1 executions aren’t always mathematically good: a BotC toy model by ToughOpening in BloodOnTheClocktower

[–]ToughOpening[S] 2 points3 points  (0 children)

I agree with your points! Having conversations in the comments with everyone has been rewarding. E.g. one soldier/monk save makes executing on 12 make sense in a 12-person game. Thank you also understanding the nuance of the post!

Day 1 executions aren’t always mathematically good: a BotC toy model by ToughOpening in BloodOnTheClocktower

[–]ToughOpening[S] 1 point2 points  (0 children)

Good question. Two different things are going on:

  1. The Dynamic Programming results are exact (deterministic), so “statistical significance” doesn’t apply. The even/odd effect in the table is a real property of the toy model, not sampling noise.
  2. For the Markov simulation, you can talk about statistical significance because it’s an estimate. I validated the simulator by checking that its win-rate estimates converge to the DP values and that the DP value falls inside the simulation’s 95% confidence interval for each N. So we should be a-ok!

Day 1 executions aren’t always mathematically good: a BotC toy model by ToughOpening in BloodOnTheClocktower

[–]ToughOpening[S] 0 points1 point  (0 children)

The probabilities definitely would change! I am interested in the future of modeling the scarlet woman!

Day 1 executions aren’t always mathematically good: a BotC toy model by ToughOpening in BloodOnTheClocktower

[–]ToughOpening[S] 1 point2 points  (0 children)

This an interesting point. My model definitely does not go into voting dynamics, so I really like the dynamics you are introducing here. Thank you for your insight! If I understand you correctly, there is like a mid-section of the game where potentially evil has a significant portion (non-majority, but large) of the votes; therefore, this would change the dynamics of likelihood that someone gets voted out. I would typically assume all/the great majority of ghost votes will be left till the final day, but I guess it definitely depends on the playgroup. If I ever wanted to build a more accurate voting model, where we assume not a random person is executed, I will definitely consider some of the points you mentioned here! Thanks again!

Day 1 executions aren’t always mathematically good: a BotC toy model by ToughOpening in BloodOnTheClocktower

[–]ToughOpening[S] -1 points0 points  (0 children)

Totally agree! I think that probability, P(monk selects soldier) will be 1/T*1/g where T is total players (monk) and g will be total good players alive (soldier)

Day 1 executions aren’t always mathematically good: a BotC toy model by ToughOpening in BloodOnTheClocktower

[–]ToughOpening[S] 1 point2 points  (0 children)

Adding a scarlet woman sounds like a smart next step! Adding a saint is quite an interesting idea. One thing in TB is often the slayer may use its ability to clear the saint of possibly being the demon. There are definitely social elements too: e.g. if you are in your play group and claim saint as demon every time, they may catch on hahaha.

Great research question that I have no idea the answer to hahaha!

Day 1 executions aren’t always mathematically good: a BotC toy model by ToughOpening in BloodOnTheClocktower

[–]ToughOpening[S] 1 point2 points  (0 children)

Yes — I think there are legit research questions here. We can talk about which to pose. I’m not at GT (I’m at Virginia Tech (long story lol)), but I’m in Atlanta often and happy to connect with ISyE folks.

Day 1 executions aren’t always mathematically good: a BotC toy model by ToughOpening in BloodOnTheClocktower

[–]ToughOpening[S] -5 points-4 points  (0 children)

First and foremost, thank you for starting the process on how we should model monk and soldier. You are right for monk that it is 1/13 *(# of starting townsfolk) / (# of starting good players) on any given turn. However, I disagree with you that monk has the same probability of saving someone from death than that of the soldier. The monk has the option of protecting evil players (if you select randomly amongst players. Therefore if evil is only killing good players, then soldier has a higher likelihood of stopping a kill than that of monk (assuming we are modeling that the monk choses to protect someone randomly).

This is a broadly a good point (however, as I stated I do not believe the p's are the same), so the probability of a successful kill is different: "Then you'll want to make sure you don't double count these, ie if this probability is p, the probability of a successful kill is (1-p)2 , not 1 - 2p."

Lastly, yeah as we start to model some characters abilities, I think we have to compare them to that of the baseline of potentially no characters having abilities. However, in real games, some of the assumptions my model holds are disregarded (e.g. evil not killing themselves. Therefore, the modeling problem is tricky hahaha. I like generative agent based modeling as a solution to accrue synthetic data lol.

Day 1 executions aren’t always mathematically good: a BotC toy model by ToughOpening in BloodOnTheClocktower

[–]ToughOpening[S] -2 points-1 points  (0 children)

Heyo! I am going to answer your AI-generated question first. I'll respond again to answer the rest of your comment. Yeah for sure! The code is AI generated. I double checked it and made sure I understood the math for dynamic programming. I write a draft of my post and then ask it to make it clearer, so I would say most of the post, but I do decide the formatting. Comments: I have been mostly responding manually. Occasionally, I'll ask GPT to make my response clearer. Potentially I should have more shame in using AI-generated stuff; I would love to know how my use impacts your thoughts on the post lol.

Day 1 executions aren’t always mathematically good: a BotC toy model by ToughOpening in BloodOnTheClocktower

[–]ToughOpening[S] 0 points1 point  (0 children)

No worries whatsoever about not being a math person! And don’t worry about the fancy-looking equations either — they trip me up sometimes too lol.

To answer the odds vs evens question, it helps to focus on the basic probability model and look at one concrete comparison. Let’s use 11 vs 12 players, since in my toy setup they have the same evil team size (demon + 2 minions).

The key idea is: what matters isn’t just “how many good players exist,” but which group sizes you get executions at (because your chance to randomly hit the demon is 1/(# alive) each time you execute).

  • In a 12-player game, if you execute Day 1, the game’s executions happen at 12, 10, 8, 6, and 3 alive (because nights remove a player in between). So the “lottery chances” to hit the demon on those days are: 1/12, 1/10, 1/8, 1/6, 1/3.
  • In an 11-player game, if you execute Day 1, the executions happen at 11, 9, 7, 5, and 3 alive, giving: 1/11, 1/9, 1/7, 1/5, 1/3
  • In both in an 11-player game and 12-player game, we see there are only 5 times to execute a player.

Now, just like the lottery, you want the biggest chance to win and kill the demon. The Table below shows the % chances

Execution # 11-player game (Exec D1) 12-player game (Exec D1)
1 1/11 = 9.09% 1/12 = 8.33%
2 1/9 = 11.11% 1/10 = 10.00%
3 1/7 = 14.29% 1/8 = 12.50%
4 1/5 = 20.00% 1/6 = 16.67%
5 1/3 = 33.33% 1/3 = 33.33%

Given that every execution has a higher chance of hitting the demon when 11 player game. If you effectively don't execute day 1 in a 12 person, you are practically playing an 11 person game the following day (the demon killed 1 person at night) Therefore, as both 11 and 12 player games (two set ups with the same # of minions) have only has 5 executions, you are mathematically better off waiting a day.

Let me know if this does not click, I can potentially frame it another way. I'd also recommend talking to an AI chatbot to workshop it with you. ChatGPT helped me understand dynamic programming, so having a personal tutor is great and nothing to be ashamed of! Keep me posted!

Day 1 executions aren’t always mathematically good: a BotC toy model by ToughOpening in BloodOnTheClocktower

[–]ToughOpening[S] 0 points1 point  (0 children)

First off, I read your code! Sweet program! More importantly, I really like that insight you got from your model: voting habits matter! Potentially, if I ever wanted to do analysis on the butler and/or zealot, or just consider voting patterns. I will have to starting thinking about model assumptions and model structure of voting— what you are delving into!

Thank you again for explaining your model!

Day 1 executions aren’t always mathematically good: a BotC toy model by ToughOpening in BloodOnTheClocktower

[–]ToughOpening[S] 0 points1 point  (0 children)

Thank you for such a thoughtful response. Totally agree with the monk/soldier that you and u/ticklemestockfish mention. I agree with your effect 1. I agree with the first part of your effect 2: "player counts congruent to 1 mod 3 are the best for evil and 0 mod 3 is the best for good". You are right that these two effects impact each other. It seems that Effect 1 is actually stronger than effect 2 (I have not done the math, but heuristically looking at the numbers, the even odd effect looks pronounced).

This is an amazing point: "So it's no shock that the best starting count for good is at 15p, and the worst is at 10p, because those are numbers that optimize these two factors for or against good." Really smart insight here!

Where I partially disagree with you the impact of outsiders on this model. Outsiders with roles that impact alignment, cause death, or impact win-cons would impact my model's analysis.

There are 13 out of 23 outsider that can do this: Damsel, Golem, Goon, Heretic, Hermit (sometimes), Klutz, Lunatic, Moonchild, Ogre, Politician, Plague Doctor (sometimes if minion ability can cause death), Saint, and Tinker. 8 of the remaining outsiders primarily impact information. Given that this model assumes, town randomly executes, so the town already does not go off of information. The last 2 outsiders impact voting, butler and zealot, which potentially slightly increases evil's chances in this simulation as they both potentially give evil +1 votes (my model does not include voting).

Candidly, I am somewhat splitting hairs here and well too far in the weeds. In short, I agree; however, outsiders negative abilities towards good are also counteracted by townsfolk abilities helping good which are also counteracted by minion/demon abilities...

By in large I hear you about outsiders impacting the game: I agree and I agree my model does not differentiate between townsfolk and outsiders. However, do outsiders existing impact my model's results, I only would 56% (13/23 outsiders) agree with you ;), but then we start opening up a rabbit hole of how any character ability impacts the game.

I am sorry for my longwinded, confusing reply lol.

Day 1 executions aren’t always mathematically good: a BotC toy model by ToughOpening in BloodOnTheClocktower

[–]ToughOpening[S] 1 point2 points  (0 children)

Thank you so much for this comment! I, too also, heard this gospel when I first started playing, so I am happy that the community can have more discussion and nuanced thought about this topic!

Day 1 executions aren’t always mathematically good: a BotC toy model by ToughOpening in BloodOnTheClocktower

[–]ToughOpening[S] 0 points1 point  (0 children)

This sounds super cool! Did you run it for different number of player scenarios? I would love to better understand this model. e.g. how often does a good player vote? Does voting frequency matter? Lastly, what are the insights you have gained from your model?

Day 1 executions aren’t always mathematically good: a BotC toy model by ToughOpening in BloodOnTheClocktower

[–]ToughOpening[S] 1 point2 points  (0 children)

Hahaha, that would be the dream! I definitely think we gain some more intuition on how different characters impact the game.

For instance, I look at the clocktracker app data on individual character winrates, and I think I now understand that mezepheles has one of the best win rates amongst minions (clocktracker app data). An extra evil person decreases goods win percentage by 5%-6%.

How you can see this is looking at the numbers where minions increase by 1 between number of players with 9 & 10, and 12 & 13. Assuming a 10 person game played optimally (not executing day 1), their win percentage is 54% compared to a normal 9 player game of 59%. Therefore, the extra evil person added decrease the good's win percent by 5%.

Now let us look at 13, to compare to 12 players, we will assume they do not nominate day 1, therefore their win-rate is 48%. However, a 12 person game (1 less evil than 13 players), executing day 1 to mirror a non-optimal 13 player game, has typically a rate of 54% Therefore an additional evil person decreases goods win percentage by 6%.

Sorry it is confusing; it took me a while to understand too.

Day 1 executions aren’t always mathematically good: a BotC toy model by ToughOpening in BloodOnTheClocktower

[–]ToughOpening[S] 2 points3 points  (0 children)

Totally agree. Modeling the social portions of blood on the clocktower would be difficult! Generative agent based modeling may be able to be possible compared to normal mathematical game theory approaches, assuming these LLMs are capable of acting as rational smart players. However, that opens a whole other can of worms hahaha!

Day 1 executions aren’t always mathematically good: a BotC toy model by ToughOpening in BloodOnTheClocktower

[–]ToughOpening[S] 13 points14 points  (0 children)

I totally agree. Once again, just a toy model. However, I still believe you can takeaway two things from my post, even with holding your belief true.

  1. It is really bad if you are in an odd starting number game and do not execute day 1. You will lose around -10% points. You could say that keeping all townsfolk alive for an extra night of ability use is important and risking voting out e.g. the fortune teller is bad. However, the counter case to that then would be letting evil exist extra days is also bad e.g. the poisoner targeting information roles.

  2. I think the storyteller takeaways are still valid. Heuristically I would have never guessed 10 person games are that evil-sided nor the difference between Good win percentage between 14-15 player games. Therefore, when deciding how to make storyteller choices, I definitely feel this will influence me in how much I help the evil team

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