[proposition] town:scum winrate calculator

[zoraster]

http://games-net.de/hosted/tggc/trash/m ... 0&do=Go%21

[zoraster]

Yeah, that calculator is wrong and here's why:

Let's assume it's 5p with 2 Mafia, 3 Town, Day Start Mountainous

On D1, there are 5 different possible lynches. Out of those,

3 of those lead to an immediate loss.

- That's 3 Outcomes
The other two lynches leads to a 3p LyLo situation (which we know is 33%, but the more important number to know is that each instance has 3 different possibilities with 1 win in each)

Do the addition and multiplication for the total number of outcomes: 3 + (2 * 3) = 9 Total Outcomes
Do the addition and multiplication for the number of outcomes where towm wins: 2 * 1 = 2 Town Win Outcomes
If you divide, you get 2/9 or 22.2%.

This is the way the calculator gets its answer of 13.3% though:

3/5 chance Town Loses Immediately.
2/5 chance Town goes to D2 3p LyLo (1/3 win chance)
---------
Town Win Chance = 2/5 * 1/3 = 2/15 = 13.3% which is NOT the correct probability if town is lynching randomly. (Cuz there are NOT 15 different outcomes for the game; there are ONLY 9)

But why the difference?

The difference is because the 3/5 chances town loses immediately DO NOT have three more possible putcomes; they only have 1 outcome each there. You are trying to count an extra 6 outcomes which simply do not exist.

I think you're confusing the day start for night start. Day start is 0 in the system. I get 22.86% with 2s/3t daystart

[zoraster]

oh wait. I entered a number wrong.

[zoraster]

I don't see how you're getting what you are. There's a 40% chance of lynching scum on Day 1. Then scum night kill bringing it to 2-1. Then there's a 33.3% chance of lynching scum on Day 2.

.3333333*.4 = .133333333

Using pure outcomes to determine things tries to act as if every outcome is equally as likely when certain outcomes are dependent on other outcomes that are more or less likely.

[zoraster]

Day 1 there's a 2-5 chance you lynch scum for 40%. So that branch of the tree has a 40% chance of happening.

ASSUMING the above (meaning we're in the 40% world) there's now a 1-3 chance of lynching scum for 33.3%. So the 33.3% is multiplied by the 40% because that's how probabilities work. Because no other branch of the tree leads to a town win, the chance of a town win is 13.3%

[zoraster]

I've read what you're saying, but you're wrong. It's not that there are 9 or 15 outcomes. There's actually only 3 outcomes (Day 1 Scum win, Day 2 scum win, Day 2 town win). But it doesn't matter how many outcomes there are because they're not equally possible.

[zoraster]

Right. Because P(T|M) as you put it is 33.3%. But a mafia goon being lynched day 1 is only a 40% chance of happening. So given a 40% chance of something happening, you have a 33.3% chance of winning. Which is, once again, 13.3%

[zoraster]

so what? if I tell you that you win 100,000 if you flip heads and if you don't then you get a second chance of 100,000 dollars if you roll a 5 on a 100000000000000000000-sided die, your expected value is about $50,000 regardless of the number of technical outcomes possible.

Or I guess once again I dispute your 9, 21 or any number that isn't 3. There are 3 different outcomes for the game. Either you you lynch BOTH Day 1 and 2 or you lose, either on day 1 or day 2. And the probability of that isn't hard to calculate. And it's 13.333%

[zoraster]

No problem! I forget who wrote that, but it's been a go to for a while! Though certainly keep in minds its limitations.

[zoraster]

what does that mean?

[zoraster]

You keep treating possible outcomes as having the same probability when they do not. Consider that you're treating X being lynched Day 1 as EQUALLY as probable as the following scenario: A is lynched Day 1, X is lynched Day 2. When in reality the first has a 20% chance of occurring and the latter a 6.6% chance of happening.

[zoraster]

How so? The Monty Hall problem's insight is that you should switch because suddenly you've consolidated a 1/3 or 1/3 choice into a 2/3 choice. Here, there's no such mechanic is there?

[zoraster]

Well, I've theorized that it might be different:

In 3 person lylo, a town member has a 50/50 shot of placing his vote on mafia. Then he only has a 50/50 shot of the other town person placing his vote on mafia. So as town, voting first leads to a 25% EV win rate, and not a 33% which normal EV analysis would lead you to believe.

If Mafia DOES vote first, then town has a 50/50 shot of winning. The vote target can reflexively vote the mafia player, and the other town decides. The fact that mafia can quick lynch isn't really relevant from a theoretical standpoint, but this is:

mafia theoretically never have to be the first mover

. They can always force town to vote first (unless you're using a non-standard deadline mechanic) because the deadline reached leads to a no lynch.

In reality, not voting as mafia can come at a cost and seem scummy and if the above is acknowledged, voting first can be seen as town.

[zoraster]

I don't see why that'd be the case. I guess there's a theoretical thing where you could say Town A votes Mafia, Mafia votes Town B... and then Town B votes Town A, then Mafia switches to Town A, but that's a pretty silly scenario.