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Who to Investigate?

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8/22/2013 8:44:47 AM
Posted: 4 years ago
One of the eternal mafia questions is: who do you investigate as a cop? Intuitively, people might say that you investigate scummy people. However, there are a number of criticisms of this:

1. If you believe a person is guilty already, then you're wasting an investigation to confirm what you already know. (i.e. if someone has CC'd you it doesn't make sense you investigate them; from your POV you know they are guilty already and if the Town is unsure of who to trust, they aren't going to blindly accept your results anyway).

2. Scummy Town are often targets for framers and scummy Mafia are often targets for Lawyers (or are Godfathers).

The suggestion, then, is to investigate people who you have null reads on. Can we validate this theory?

The first problem I see is that it assumes that people's reads are accurate. Outside of this question I have been working on a way to quantify people's skill. The problem is that a simple right vs. wrong isn't sufficient. If you are 100% wrong, then you are actually accurate since you can just take the opposite of your read. Rather, I think right vs. random is better. That is, the more skilled you are, the more likely someone's alignment is in accordance with your reads as opposed to random chance. We can then rank a person from 0 - 10:

0 - Your reads are no better than random chance.
10 - Your reads are perfect.

As an added element of complexity, your reads themselves can also be ranked from 0 (0% Town/100% Scum) to 10 (100% Town/0% Scum). A caveat of this is that 5 is NOT a null read because the base odds are roughly 33/66. Meaning a read of 3 (30% Town/70% Scum) is closer to a null read.

To relate this to the question at hand, we can use information theory using entropy. One interpretation of entropy within information theory is that it represents that information gained by learning the outcome to some event, based on the amount of information contained within it.

For example, it takes 1 bit to represent the outcome of a coin toss (0 - Tails, 1 - Heads). If the toss is fair (completely random, 50/50) you gain 1 bit of information by learning the outcome. If the toss isn't fair, you gain less information, all the way to 0 if the coin is completely biased toward a single outcome. You gain no information because you already know ahead of time what the outcome will be.

Relating this to the task at hand, let's take an example.
Let's say your skill is 6. This means 60% of the your reads are accurate and 40% of the time, they are no better than chance. Let's say you have a read on a person of 3, that is, there is a 30% chance they are Town and a 70% chance they are Scum.

What are the overall odds they are Town vs. Scum? This is easily calculated:

60% (chance your reads are accurate) * 30% (your read of them as Town) = 18% (Chance they are Town based on your read being accurate)
60% (chance your reads are accurate) * 70% (your read of them as Scum) = 42% (Chance they are Scum based on your read being accurate)
40% (chance your reads are random) * 66% (base chance of them being Town) = 27% (Chance they are Town based on your read being random)
40% (chance your reads are random) * 33% (base chance of them being Scum) = 13% (chance they are Town based on your read being random)

This gives us:
18% + 27% = 45% overall chance they are Town
42% + 13% = 55% overall chance they are Scum

Using this information we can calculate the entropy, in bits, which is 0.8. What does this mean? It means you gain 0.8 bits of information by learning the true alignment of that person (e.g. as a result of a cop investigation), which is pretty good. The entropy for the base chances (66/33) is only around 0.51.

How to apply this information? For any given skill level, we can calculate the entropy based upon the read (0 - 10) for a given individual. We can then find the read associated with the maximum entropy. The conclusion is that you should investigate people whom you read at that level, as you stand to gain the most information from the results of that investigation. Calculating all of the results, I get:

Skill - 0: Irrelevant. A skill of 0 is no better than chance regardless of read, so all of the entropy is the same and it doesn't matter who you investigate.

Skill - 1-5: Investigate your scummiest read (0).

Skill - 6-7: Investigate your next-to-scummiest read (1).

Skill - 8: Investigate slight-scum reads (2).

Skill - 9-10: Investigate null-reads (3).

This seems to validate the initial hypothesis: As your skill in reading people increases, you stand to gain the most information by investigating your null reads (Remember, a read of 3 is 70/30 Town/Scum, which is close to the base probabilities). As your skill in reading people decreases, you should investigate scummier people.

The raw data is here:

The equation for entropy is:

(Overall Chance) * log2 (Overall Chance)

You then sum that for each possibility (Town, Scum) and multiple by -1. Using the example above:

Town: (45%) * log2 (45%) = -0.51935251
Scum: (55%) * log2 (55%) = -0.287375056
-0.51935251 + -0.287375056 = -0.806727566
-0.806727566 * -1 = 0.806727566

Caveat: Yes, I know the immediate objection to this is how do people know what skill level they are? Well, we can perform a statistical analysis based upon a deep dive of the games, but mostly this would have to rely on a self-assessment, but the numbers lend themselves to two general solutions: investigate the scummiest or investigate a null read.