The odds in the game risk

adrianopolis

In risk, the attacking party rolls 3 die and the top two numbers of the 3 die rolled get put up against 2 die rolled by the defender. If the die are equal then the defender wins. For example if the offender rolls 5 5 2 and the defender rolls 4 3, then 2 defender men die. If the offender rolls 5 5 2 and the defender rolls 5 4 then they trade kills because when die are equal the defender wins.
If offender rolls 5 5 2 and the defender rolls 6 6 the defender wins. Who has the advantage? Attacking or defending? What is the comparative advantage?

oay

Start with die in the singular and dice in the plural.

haruspex

Here's one way to approach it.
Break it into four cases from defender's perspective:
++ win on both
+- win on high dice, lose on low
-+ etc.
--

Case ++:
For each defender roll, count attacker possibilities:
6+6: 6³
6+5: 5³+3.1.5² (attacker rolls no 6s or one 6)
6+4: 4³+3.2.4² (attacker rolls no 5s nor 6s, or just one such)
:
6+1: 1³+3.5.1² (attacker rolls nothing above 1 or just one such)
(remember to count all above except 6+6 twice)
5+5: 5³
etc.
Summing, we get sum for r = 1 to 6 for each of:
r³, 2r³(6-r), 6r²(6-r) = -2r⁴+7r³+36r²
Sum the series to r and plug in r=6.

Similarly, for case +-:
6+5: 1³+3.1².5
6+4: 2³+3.2².4
etc.

adrianopolis

Thanks man. I'm a little confused by what for example 3.1.5^2 means but thanks for the help

kmwest

Analyzed here:

http://datagenetics.com/blog/november22011/index.html