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Subject: It had to happen. MATH ATTACK! Combinatorics and Bayes' Theorem. rss

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David Goldfarb
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Well, I haven't gotten a chance to actually play this game yet. But just reading the rules and seeing other people play it, I could see some interesting cases for mathematical analysis.

One obvious case to look at: you're a Liberal who has been elected Chancellor, and you have just been handed two Fascist policies by the President. It's possible that the President is a dirty rotten Fascist who has buried a Liberal policy; it's also possible that the President is an innocent Liberal who happened to get screwed by the random draw. How suspicious of the President should you now be?

Rev. Thomas Bayes to the rescue! He gave us a formula that tells us just how far we should adjust our probability estimates, given new data. This is just the sort of situation where we can use it.

Before we can do that, we need to know what the relative chance is of a deck-screw versus a burial. This result turns out to be fun and elegant, something I'm sure the designers intended.

Given the initial composition of the deck, the chance of drawing three Fascist policies is:

11 10 9
---- * ---- * ----
17 16 15


And the chance of two Fascist and one Liberal is:

11 10 6
---- * ---- * ---- * 3
17 16 15


(The times three is because there is only one way to draw three Fascist policies, but three ways to draw two Fascist and one Liberal: you could draw LFF, FLF, or FFL.)

Rather than have to cope with these somewhat hairy fractions, we should notice something: 6 is exactly 2/3 of 9. Since we're multiplying by 2/3, and then multiplying by 3, we can see that FFL must be exactly twice as likely as FFF.

Now, let's get back to our seat as the Chancellor. When the President draws three policies, there are four possible combinations to draw: FFF, FFL, FLL, or (yeah, right) LLL. But we when get FF from her, we can rule out those last two. Now, we paid attention in probability class, and we know that when we are ruling out possibilities like that, the ones that remain keep their original ratios. So then the chance that the President drew FFF was 1/3, and the chance that she drew FFL was 2/3.

Okay, so now we can figure out just how suspicious we should be. We'll start with one specific case: a 7-player game.

In a 7-player game where you are a Liberal, the other players divide 50-50. For any other player you know nothing about there's a 1/2 chance that they're also a Liberal; a 1/3 chance that they're a Fascist, and a 1/6 chance that they're that über-Fascist, Hitler. When the President passes you FF, you know there's a 1/3 chance that they had to pass that, and a 2/3 chance that they buried a Liberal policy.

For simplicity's sake, we'll assume that Fascist players will always pass FF if they can.

We now have to raise that eternal question: WWHD? What would Hitler do?

If you think Hitler would play like a Fascist, and always pass FF when possible, then here's how it works out:

1
--- (the chance that the President is a Fascist/Hitler)
2
P(Fascist) = ----------------
1 1 1
--- + (--- * ---) (the chance that the President is a Liberal who drew FFF)
2 2 3


Add it all up and do the fractions, and it comes out to exactly 3/4 that the President is a Fascist (including Hitler), up from our previous estimate of 1/2. (The chance that the President was Hitler himself comes out to exactly 1/4; which makes sense, since Hitler is 1/3 of the Fascists.)

Now then: what if we think that Hitler would slow-play it, pretending to be a Liberal throughout so as to increase the chances of a late-game Chancellor election? In that case only 1/3 of the other players would always pass FF.

So:
1
--- (the chance that the President is a Fascist)
3
P(Fascist) = ----------------
1 2 1
--- + (--- * ---) (the chance that the President is a Liberal/Hitler who drew FFF)
3 3 3


This works out to exactly 3/5 (60%) chance that the President is a Fascist, up from our previous 33%. So it's the way to bet, but far from certain. The chance that the President is Hitler actually falls a bit: from 1/6 to 1/10. (This will always be true in the Slow-Play Hitler scenario: getting FF means a lower chance of the President being Liberal, but also a lower chance of the President being Hitler.)

Here's a little chart I made up giving the new probabilities for all player counts, given Spiking Hitler and Slow-Play Hitler.


# of players Chance of F/H given Spiking Hitler Chance of F given Slow-Play Hitler
5 .75 .5
6 2/3 (.67) 3/7 (.43)
7 .75 3/5 (.6)
8 ~.69 ~.55
9 .75 ~.64
10 ~.71 3/5 (.6)


I hope you've enjoyed this little adventure into probability.
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