: RISK : The game of world domination

: RISK is the popular strategy game of world conquest in which opponents move armies around the world map in an effort to defeat their opponents, capture territory, acquire RISK cards and ultimately rule every territory on the map.
: This task is accomplished primarily through wit, deception, strategy and luck with the dice. While we could say some things about the former topics, here we concentrate on the latter. Specifically, when defending a territory against the invading armies of your opponents, is it better to roll 1 or 2 dice? How much better?

: Let me lay out the problem parameters and describe the rules of engagement:

1. An attacker may roll up to a maximum of 3 dice during an engagement, and must have one more unit in the attacking country than the number of dice thrown (this is because the attack must move at least a number of troops into the conquered country which is equal to the number of dice rolled, yet must leave behind at least one troop in the attacking country).
2. A defender may defend with up to 2 dice and is not restricted in any way.
3. Both participants must choose, before throwing their dice, how many they will use in the engagement.
4. Once the dice are thrown, each participant sorts the dice in descending order and matches them up by rank (i.e. the two high-dice are matched up, the two next highest, etc.). The owner of the lower value then loses one troop unit with the defender winning ties.

: To the answer

: Credit - the image is from iConquer's (a Mac version of RISK) website.