Who has the advantage in Risk: Attacker or Defender?

[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?

 

[skiller]

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]

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.

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