In risk, who has the advantage? The attacker or the defender?

[Kaz The Original]

What are the odds? To those of you unfamiliar with risk, here is how it goes.

The attacker can roll 3 dice, the defender two. The attackers best two dice are then put up against the defenders roll. You take the highest attacker die and put it against the highest defender die, and same for the lower ones. Tie goes to defender, higher number wins.

What are the odds, who is likely to win?
(I say attacker!)

[SumZero]

The attacker. The possible outcomes for the attacker are +2 (2 defenders die), 0 (1 defender and 1 attacker die), and -2 (2 attackers die). The expected outcome is around +0.158.

[XChamp]

Does anything change when the armies are not large? What if the attacker has 4 armies and the defender has 2 (thus the attacker can only attack with 2 armies after one is defeated because he must leave one on his territory). What are the odds that the defender is destroyed in this situation?

[MickeyHoldem]

65.6%

Risk Odds Calculator

[maurile]

The attacker has the advantage when it's three dice to two. With each roll, an average of 1.08 defending armies will be killed while only 0.92 offensive armies will be killed.

More specifically,

29.26% of the time, the defense will sweep
33.58% of the time, they will split
37.16% of the time, the offense will sweep

Answers to other matchups:


Offense rolls 1 die, Defense rolls 1 die:

41.67% of the time, offense will win 1-0
58.33% of the time, defense will win 1-0

EV: Defense wins, 0.58 to 0.42

Offense rolls 2 dice, Defense rolls 1 die:

57.87% of the time, offense will win 1-0
42.13% of the time, defense will win 1-0

EV: Offense wins, 0.58 to 0.42

Offense rolls 3 dice, Defense rolls 1 die:

65.97% of the time, offense will win 1-0
34.03% of the time, defense will win 1-0

EV: Offense wins, 0.66 to 0.34

Offense rolls 1 die, Defense rolls 2 dice

25.46% of the time, offense will win 1-0
74.54% of the time, defense will win 1-0

EV: Defense wins, 0.75 to 0.25

Offense rolls 2 dice, Defense rolls 2 dice

22.76% of the time, offense will win 2-0
32.41% of the time, offense and defense will tie 1-1
44.83% of the time, defense will win 2-1

EV: Defense wins, 1.22 to 0.78

Offense rolls 3 dice, Defense rolls 2 dice

37.16% of the time, offense will win 2-0
33.58% of the time, offense and defense will tie 1-1
29.26% of the time, defense will win 2-0

EV: Offense wins, 1.08 to 0.92

For a matchup to be even money, one side's percentage disadvantage in EV must be equal to its percentage advantage in armies.

For example, with the offense rolling 3 dice and the defense rolling 2 dice, the defense is at a 1.08 to 0.92 disadvantage. 1.08/0.92 = 1.174. So for the defense to have the overall advantage, it must have more than 1.174 times as many armies as the offense.

Thus an offense with 10 armies would be favored over a defense with 11 armies, but would be an underdog against a defense with 12 armies.