SpielByWeb

Knave · Joined: 28 Jun 08 Posts: 258

I'll probably work out the answer for myself later if nobody answers here, but I thought that the question might be of general interest to others, so I figured I would post it.

The situation

You have to cash in either a "bank of nile" or "east/west" bonus card for a round 3 sacrifice. This is a 4 player game. The provinces are distributed 6/6 in terms of nile/not and are distributed 8/4 in terms of east/west (though clearly not in that order).

The question is, which card do you keep? There are of course many extra assumptions, having to do with which provinces have pyramids and which provinces overlap with other potential sets... but if we set aside those considerations, is it better to take the evenly split bonus card or the asymmetrical one? Offhand, a quick back-of-envelope calculation tells me that the asymmetrical is better, but I need to double check.

Dvd Avins · Posted: Wed Aug 12, 2009 5:02 pm Post subject:

Even ignoring the individuality of provinces, I don't think you've adequately defined the problem. You may get one answer if you assume the New Kingdom provinces are randomly distributed and the other answer if you assume that in each round, you get your pick of the available provinces. Reality, of course, is somewhere between those two and depends on the lengths you're willing to go to make your card good.

Nemesis · Joined: 12 May 09 Posts: 32 Location: Ajax

I am intersted in the answer as well. I would have pitched the 8/4 card.

Carmilla · Joined: 13 Jul 08 Posts: 70 Location: Brooklyn, NY

Have to agree. Of course, it's possible to get hosed by the draw either way. (I agree the odds appear to be more in your favor with 8/4, but I'm not completely convinced either.)

But no matter how you slice it, you have trouble. If you go with 8/4, and Round 4 gives you a 1/3 choice, but the 1 has 2+ pyramids, good luck! Particularly as you seem to somewhat cash-shy if you're sacrificing a bonus card in round 3.

Knave · Joined: 28 Jun 08 Posts: 258

Dvd Avins · Posted: Wed Aug 12, 2009 6:10 pm Post subject:

Well, for a + b = k (and a > b > 0), a^2 + b^2 > 2ab. And the greater the difference between a and b, hte more of the pussing ends up in the a^2 + b^2 bowl. Your situation is akin to wanting every draw to be a or every draw to be b, with minimized chance of a mix.

Since that inequality is for continuous variables and this problem is for discrete (and drawing without replacement) it's conceivable that its result would be misleading, but I'd take it as reason for a strong hunch that you're better off holding the 8/4 split.

jmucchiello · Joined: 14 Apr 09 Posts: 80 Location: NJ

I would go with the 8/4 split for the simple fact that the odds of one of the rounds being 0/4 are far slimmer than if the provinces are split 6/6. There's only one combination of provinces that satisfies that condition in the 8/4 setup while there are C(6 4) combinations that satisfy it in the 6/6 setup.

Gaditus · Joined: 05 Feb 07 Posts: 1924 Location: Canterbury, UK

By considering the trvial 12/0 case (I know it's not possible but ...) you can immediately see that 8/4 is better than 6/6.

jglazier · Joined: 22 Nov 05 Posts: 119 Location: Brentwood, CA

After reading this thread, it is very clear to me now why I suck at Amun-Re against the best on this site.