# The odds in the game risk

1. Jan 16, 2013

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?

2. Jan 16, 2013

### skiller

3. Jan 17, 2013

### 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.

4. Jan 25, 2013