Improved Risk

The problem is that the ratio change when the battle evolve. And some ratio can be hard to calculate, so you will need a calculator.

Personally I would add a limit in the number of round a battle can take. Which will reflect the time it takes to conquer a territory. For example, 3 battle round max. So a well defended territory will take more than 1 turn to destroy.

Or, if you want to change the rules and strategy deeper, make that all coast territoriers can attack all other coast territories in the same ocean. For example, all territories on the east side of north and south america can attack or be attacked by all terriories on the european coast and west side of affrica.

In this way, you will not be able to defend the antire america with only alaska, groenland and brasil. It will also distribute all the forces around the map. Reducing the number of unit per territory and making the current dice rolling system more convenient. In this case, I would only make 1 battle round for each battle.

(I have not playtested any of this(^_^))

By the way, is it me or there are some modification to the map since the first edition. The map look smaller and I think some territories (like central america) have dissapeared.

I posted the following some time ago. Hope it sparks some ideas:

If you can get your hands on a bunch of different colored dice, you can try this mechanic:
Red=attack, Blue=defense, White=attack type

Presumably, each unit has a maximum number of dice that can be rolled. The players secretly choose the composition (e.g., 2 red, 1 blue, 1 white), with at least one white die always being selected. (So, mininum number of dice =2).
Players simultaneously roll. Only one of each color can be considered in the resolution - and the choice may or may not belong to the player who rolled (depending on resolution of the attack type).

The white dice are resolved first. If the attacker's highest white die is higher than the defender's highest, then the attacker has the initiative (surprise attack). If the defender's highest white die is higher than the attacker's highest white die, then the defender has the initiative (ambush). If both have the same highest value then the attack is normal and no initiative action is taken.

Surprise attack: the attacker removes one of the defender's blue dice. If the defender rolled only one blue die, then the value on the blue die is reduced by one. If the defender did not roll any blue dice, then the attacker gets a plus two on his attack. Each player selects his best die. The damage is the difference between them (adding in the plus two attack bonus, if applicable).

Ambush: the defender removes TWO of the attacker's red dice. If the attacker rolled only one or two red dice, then the defender gets a plus two or a plus one counter-attack bonus, respectively. Since this is an ambush, the defender gets to attack, but only if he rolled at least one red die that is higher than the highest remaining red die held by the attacker. (Otherwise, the ambushed attacker can still make a weakened attack.) Each player selects his best die (keep in mind, if the defender is counter-attacking, then he uses his red die, while the erstwhile attacker must use his blue die, if he has one!) The damage is the difference between them (adding in the plus two or plus one counter-attack bonus, if applicable).

Normal: each player selects his best die. The damage is the difference.

Mitch