However, even though the attacker rolls an extra dice, the defender has an edge as well. In the event of a tie, the attacker will lose one soldier. Is this enough to offset the attacker's 3dice advantage? What does this mean for rolling strategies?
With 5 dice, there are 7,776 possible number combinations. Clearly, listing every single one down would be quite tedious. Instead, I used Excel to simulate 10,000 dice rolls using a random number generator. I then calculated the "expected net wins for the attacker", which is simply:
Soldiers lost by defender  soldiers lost by attacker
This is expressed as the average for every 100 rolls.
The results are shown below. First look at the topleft corner, which is the typical case where the attacker rolls 3 dice, and the defender rolls 2 dice. While both the attacker and defender will lose some soldiers, the defender will lose 7.5 more soldiers for every 100 rolls. What this means is that the defender loses on average, so it pays to be offensive in Risk. It looks like the 3dice advantage does outweigh the defender's edge in winning ties!
Expected net wins for attacker (per 100 rounds)

Defender rolls 2
dice

Defender rolls 1
dice

Attacker rolls 3 dice

7.5

31.6

Attacker rolls 2 dice

22.3

15.9

However, the defender also has the option of rolling one dice, instead of two. Players sometimes do this, for example when there are only 2 or 3 soldiers left in the territory. Does this raise the defender's odds of winning?
This turns out to be a lousy strategy. Here, the defender loses even more soldiers  32 more than the attacker for every 100 rounds, much more than the 7.5 earlier. Hence, the defender should never use 1 die, unless forced to (i.e. only one soldier left).
Lastly, the attacker might sometimes be down to two dice, when he only has three men left. Should he continue attacking, or wait for reinforcements? The answer to this is "it depends". Firstly, if the defender has two dice, the attacker is at a disadvantage, as he loses 22 men more than the defender (again, per 100 rounds). Secondly, if the defender is down to 1 die, the attacker maintains his edge as the defender loses 16 more men than the attacker. Hence, the attacker should only use 2 dice when the defender uses 1 die.
In a nutshell, attack when you have more dice than the defender, and defend with as many dice as possible. Enjoy La ConquĂȘte du Monde!
Afternote: it turns out that the simulation gets results very close to the theoretical one, shown below.
Expected net wins for attacker (per 100 rounds)

Defender rolls 2 dice

Defender rolls 1 dice

Attacker rolls 3 dice

7.9

31.9

Attacker rolls 2 dice

22.1

15.7

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