WIFOM - A mathematical analysis

So I was going through mafiascum wiki, reading about all the weird roles and came across their WIFOM page.
https://wiki.mafiascum.net/index.php?title=WIFOM

Sadly, although it gives some extremely nice examples, the answers to the questions at the end are not provided.

So let's take a look at the first example given: What are the asnwers to the questions? Which way is the best way to go as the criminal? Which way is the best way to go as the policeman?

(Note: in the calculations below, I'm going to use 33% to mean 1/3 and 66% to mean 2/3. Unfortunately fractions do not appear nicely when you don't have access to proper mathematical notation).

Let's assume that if both go into the well-lit boulevard, the criminal definitely gets caught, while if both go into the dark alleyway, the policeman has 50% chance of catching the criminal. If they go into different directions, the criminal escapes.

First of all, you can use make use of your knowledge of the other person's knowledge/ignorace/stupidity. If the criminal knows that the policeman is ignorant of which path lies which way and makes his choice randomly, then the criminal's strategy should be to always go to the dark alley. There is a 50% chance that the policeman goes the other way, and if the policeman goes same way, there is a 50% chance of escaping anyway, giving him an overall 75% chance of escaping. Sehr gut!
Similarly, if the policeman knows that the criminal is ignorant and makes his choice randomly, then the policeman should always choose the well-lit boulevard, giving him a 50% chance of catching the criminal. Wunderbar!
If the criminal knows that the policeman follows the above strategy (i.e. always goes to the well-lit boulevard), he should always go to the dark alleyway. If the policeman knows that the criminal knows that the policeman... etc... etc... etc...

So what, should be the optimal strategy assuming the layout of both paths is common knowledge?

First of all, let's define what "common knowledge" is. It doesn't simply mean "everyone knows it". It can be recursively defined as:
X is common knowledge if:
_ Everyone knows X. AND
_ Everyone knows that X is common knowledge.

This means that: Everyone knows X AND Everyone knows that everyone knows X AND Everyone knows that everyone knows that everyone knows X... etc.

So what's the best strategy for the criminal is if the layout of both paths is common knowledge? It's not to go to the dark alley. If he chooses that strategy then the policeman can choose a counter-strategy of always going that way too, giving the criminal only a 50% chance of winning. Furchtbar! He can do better.

The perfect strategy for the criminal is to randomly choose the dark alley 66% of the time, and the well-lit boulevard 33% of the time.

This gives him a solid 66% chance of winning to matter what counter-strategy the policeman adopts. Richtig!

If the policeman always goes to the well-lit boulevard, he catches the criminal only if he goes there too, which is 33% of the time. If the policeman always goes to the dark alley, he catches the criminal half the time he goes there too, meaning half of 66% which again gives 33% chance of catching him. If the policeman does what the criminal does, and goes 66% of the time into the dark alley, then the chances of catching the criminal are:
_ If both go to the well-lit boulevard, which is 1/3 * 1/3 = 1/9 of the time.
_ Half the time both go to the dark alley, which is 1/2 * 2/3 * 2/3 = 2/9 of the time.
Giving a total of 3/9 of the time = 33%.
In fact the above is the optimal strategy for the policeman.
If any of the two deviates from their strategy, then the other one gets a counter-strategy that increases his chances of winning (i.e. the criminal can have a better chance of winning than 66%, and the policeman a better chance of winning than 33% if either deviates from the strategy above).

The situation above (one where each player has an optimal strategy regardless of what the other player does, but causes the other player to improve their odds if he deviates from the optimal strategy), is called a Nash equilibrium. In this case, it's called a mixed strategy equilibrium (since the optimal play is a mixed strategy, as opposed to a pure strategy, where the optimal play is to always choose one way). John Nash proved in 1950 that every finite game has at least one such equilibrium point.

The above is of course a simple example. If you enjoy this sort of thing, you can calculate the optimal strategy when there are more paths and more different payoff values (e.g. there can be 3 outcomes: the criminal escapes, the criminal gets caught and the criminal escapes but breaks his leg which for the criminal is a worse payoff than escaping but better than getting caught).

Till next time. Auf Wiedersehen.

PS. One more thing about that WIFOM page which I didn't like is that the first sentence describes WIFOM as circular reasoning. The phrase "circular reasoning" is often used to describe something else: a phallacy where one tries to prove a claim with a premise that presupposes the claim e.g. A is bigger than B because B is smaller than A.