# More Useless Math

I spent some time tonight analyzing the two-cell case of "Dicewars".

(The game is at http://www.gamedesign.jp/flash/dice/dice.html, see my previous entry
http://markgritter.livejournal.com/259531.html for a description of the rules.)

Mainly it was just fun putting the spreadsheet together. The complete solution can only be achieved iteratively, of course, because the value of passing (and gaining one extra die) or attacking and failing (and going down to one die, unless you have reserves) depend upon the value of the new positions.

I capped the number of reserve dice at seven, giving 225 different game positions. 8*8=64 positions with no reserves, plus 8*7=56 where the attacker has at least one reserve die and thus must have 8 dice on his territory, 56 corresponding ones where the defender has at least one reserve die, and 7*7 = 49 situations where both have at least one reserve.

There are two questions with (I thought) nonobvious answers. The defender has a slight advantage when die stacks are equal. Should you always attack if a die or more ahead? When should you attack when stacks are equal? (One answer is 'never', the game is actually a stalemate.)

The first answer is 'yes', you should always attack if a die ahead, pretty much as expected.

The second answer is that you should attack if you have five reserve dice, or if your opponent has more reserve dice than you by at least two. (The two exceptions are that it appears slightly better in the 2 vs. 4 case to wait for 3 vs. 5 before attacking, and it is worth passing with zero reserves when your opponent has fewer than 5.) Some of these are right on the edge and the decision makes less than 1% difference. But, for example, attacking with 3 reserve dice into an opponent with 7 in reserve is almost 20% better than waiting for his attack.

My best understanding as to why 5 is the magic number is that 6 dice are large enough to have a significant chance of standing up to an attack by a stack of 8. (About 16%, compared with 5% for 5 dice.) The extra equity this gives is sufficient to make attacking correct, even though the attack is a slight underdog to succeed.

Later: I'm an idiot. I analyzed a variant in which the attacker doesn't get a die if his attack fails, while in the real game he does.
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