I think if you were to play without dice then you would need to have specialized maps which can give defenders an advantage at certain points and attackers an advantage at others in order to make it so it's not just simply "first to get a bonus is going to win because they will simply be able to overpower you with more troops"
For example, having territories in mountains or whatnot could give the defending army a bonus - you have to spend 2 attackers for every 1 defender to take it (or even 3-1 for very tough attacks). Then, even if your opponent gets a bonus, you still have a chance if you can take up some defensible positions.
Otherwise, I don't see how it's possible to play if EVERY attack produces the same results (1-1 for each side). If you have a spot in the game like this:
A-B-C-D and I have 3 armies at each A and B, and you have 3 at each C and D then this is what happens:
I drop 3 on my side of the border: 3-6-|-3-3
I attack: 6-3 = 3, I have to leave 1 and advance 2, I can't advance further: 3-1-2-|-3
you drop 3 on your side: 3-1-2-|-6
you attack: 6-2 = 4, have to leave 1 behind: 3-1-|-3-1
you attack again: 3-1 = 2, leave 1 behind: 3-|-1-1-1
you can't advance further
I drop 3 on mine: 6-|-1-1-1
I attack: 6-1 = 5, leave 1 behind: 1-4-|-1-1
I attack again: 4-1=3, leave 1 behind: 1-1-2-|-1
I can't advance because I don't have enough to both kill you and take the territory
you drop 3: 1-1-2-|-4
you attack: 4-2=2, leave 1 behind: 1-1-|-1-1
you can't attack any more
I drop 3: 1-4-|-1-1
I attack: 4-1=3, leave 1 behind: 1-1-2-|-1
I still can't attack any further
now we are stuck in an infinite loop and neither of us can win. So the only difference maker would be bonuses. But that is completely dependent on luck too, because, as shown above, the only way to gain territory is to start before the other player (luck) or if you are dropped in a favorable position to get a bonus before the other player can stop you (also luck).
So going 1-1 on every attack is still going to come down to luck, just a different kind of luck rather than rolling dice.