Battles are resolved by highest total dice roll (ties to the defender), with a winner take all result. Replacement dice are given based on the number of adjacent territories you control, and are assigned randomly. A sound strategy is to leave paths of weakly defended territories open to your biggest piles of dice, inviting other (computer controlled) players to attack you so that you can counter-attack with greater force. You also want to set up paths for your opponents to fight each other, so they do not spend all their efforts on you.
[Updated 11/10]
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