Suppose we have 2 players, there are 2 territories and we have perfect play. Each controls 1 territory. z is income. Let’s say a territory contains x armies. No (1 army must stand guard) What’s the Lowest starting number armies, y, required to take starting x armies? ex. x = 2, y = 4. The map is https://www.warzone.com/SinglePlayer?PreviewMap=465
The list below assumes 60% kill rate and 70% defense rate
INCOME =

*0 y = x+1
Proof for no income: y can keep attacking x with 1 army. 1 army reduces x by 1 and so by y. Eventually, x will have 2 army and y will have 3 armies .
*1 y is almost = 1.0803x

If both kill and defense rate is 0%, y can not take x unless z = 0 and x = 0, which y = 1
If the kill rate is 100%, y = x+z+1 no matter the defense rate
If the kill rate is 0%, y can not take x unless z = 0 and x = 0, which y = 1
The trick is to get y/x as high as possible , modulo come onto play with n/100. We simplify the fraction to 2 coprime number.
We multiply these coprimes to get our r mod q.
Then y attacks with y-r ( So that y can attack most armies while saving most)
The list below assumes 60% kill rate and 70% defense rate

Edited 7/25/2020 02:03:31

Ok

If your not into math and strategy, then this is not for you

If the income is zero, y = x + 1

TL;DR: Duel is a bad map that can't be won on without chance or opponent error

If this is a 2-territory map with even income (Duel?), then if the defensive kill rate is higher than the offensive kill rate, both players will end up waiting forever for the other to attack (or just VTE). To be the player that attacks first in such a scenario is to put yourself at a disadvantage. If the ratio of armies post-deployment (and minus 1 army, if one-army-stands-guard is enabled) is greater than (defending kill rate)/(attacking kill rate), then the player with more armies can just keep amplifying their advantage and eventually win the game. If the ratio of armies is less than the kill rate ratio, then it stays a waiting game- hoping the opponent makes a mistake out of frustration and stupidity.

So your scenario is intractable unless the offensive kill rate is greater than the defensive kill rate, in which case you want to attack (to kill more armies). Here it'd become a lottery based on turn order, since you have to keep attacking so as to not exacerbate your disadvantage but eventually one side will have few-enough armies that the other will be able to just take it. I don't know if this is possible, but if offensive kill rate is over 100%, then the winner depends on specific circumstances. If the offensive kill rate is below 100%, though, then the player who gets the first move on the first turn will win because the very first attack will deal the most damage and their opponent will be at an ever-increasing disadvantage for the rest of the game unless the player who had first move just decides to stop attacking for some reason.

For example, let x = 4, with offensive kill rate of 80% and defensive kill rate of 40%. Let A be the player who has first move, and let's assume one-army-stands-guard is turned off with the luck modifier at 0%, the rounding mode set to straight round, and the move order is cyclic.

A attacks B (4v4) -> A left with 2 armies, B left with 1 army
B attacks A (1v2) -> A left with 1 army, B left with 0 armies
A deploys 1 -> A left with 2 armies
B deploys 1 -> B left with 1 army
B attacks A (1v2) -> A left with 1 army, B left with 0 armies
A attacks B (1v0) -> A left with 1 army, B left with 0 armies, A wins game

But if the offensive kill rate is now 400% and the defensive kill rate 10%:

A attacks B (4v4) -> A:2, B:0
B attacks A (0v2) -> A:2, B:0
A deploys 1 -> A:3
B deploys 1 -> B:1
B attacks A (1v3) -> A:0, B:1, B wins game

Edited 7/23/2020 03:45:48

General Warzone, this is the exact reason why I added the commerce into duel. More cities means more income. However, if you keep building cities without altering to deployments and vice versa, the opponent will overrun you. This is what I already know if both started with same income and armies. I assume, under custom scenario, that one player has more armies, y, than the other player armies, x. The upper bound for y is x = 1.6 y if the income is greater than 0.

Edited 7/24/2020 02:57:47

Edited 7/24/2020 02:56:49

Edited 7/23/2020 13:43:07

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Proof for no income: y can keep attacking x with 1 army. 1 army reduces x by 1 and so by y. Eventually, x will have 2 army and y will have 3 armies .