Re: Lanchester's Laws Applied to the Paladin Upgrade Questio
I tested without patrol (deliberately) with the units mixed in with each other. This actually can happen in extended or messy battles. For patrol, I suggest the intermediate law, because it predicts the same results you have observed.
That's why I developed the intermediate law - the fight initially resembles a phalanx (with some units not being able to fight at first), but over time, the larger numbers begin to matter.
However, I still think the assumption that battles scale with numbers doesn't hold. Although those twenty units don't focus down one unit, they are free to fight again after they've killed a unit from the weaker army. However, they are weaker and hence die quicker. If two spearmen kill one then to say I have one and a half spearmen remaining then means something significant. If I had ten separate 2v1s, your formula would be accurate, but since some units die quicker than others, those units are free to fight again and hammer home their advantage.
I would also draw attention to the fact that your formula
Sigma(n/m)p=p*
is a special case of Lanchester's Square Law (with 0.5 correction) for the case that n/m units fight one unit, with both armies cancelling each other out, as in
a(A^2+A)=b(1^2+1)
where A=n/m, a=p/2 and b=p*/2. You could actually predict how many units survive by setting the surviving army number to be non-zero. That would be the winning margin for one micro-fight. You could then multiply by m to find the total number of surviving units. Granted that implies lots of units on low health, but that's why I don't think the fights scale as you say they do - some units die faster than others, and the surviving units are free to kill more.
Anyway, I just wanted to flag that, because you said you could predict who wins, but not by what margin. Even though you don't think Lanchester's Square Law holds in general, I do think at the very least it's applicable on the level of micro-fights.
That would be interesting, and if you do come up with a space effectiveness factor, let me know.
No, not at all. Look at two cavalry archer armies patrolling into each other in a Hun war. They steadily eat away at each other. Another way of putting it is once you have ballistics, you're going to have large armies you can't realistically micromanage. In that situation, one hit kills don't come into play anymore.
Interestingly, the reason Viper comes out on top is that he picks off units here and there, not allowing his army to engage head on. The point is he gets cost-effective trades and his opponent invests resources into replacing those units, which means Viper is able to move ahead in economy. That's why Viper is able to come back after falling behind in a feudal war.
I have no idea what are the settings you used to get 7 champs remaining for a 19 v 11. But even a proper AI v AI test gives lesser wins for champs (and odd wins for jags) while a real patrol fight which is what matters anyway gives a decently consistent win for champs with IIRC 1-2 injured champs remaining.
I tested without patrol (deliberately) with the units mixed in with each other. This actually can happen in extended or messy battles. For patrol, I suggest the intermediate law, because it predicts the same results you have observed.
I know why my formula deviates from reality for big numbers. Thats because in a 20 v 40 fight the units in back row of 40 army spend a non-insignificant time without fighting. Also why i model 20 v 10 to be approximately 2:1 is because in reality 20 units dont focus attack on a single unit.
That's why I developed the intermediate law - the fight initially resembles a phalanx (with some units not being able to fight at first), but over time, the larger numbers begin to matter.
However, I still think the assumption that battles scale with numbers doesn't hold. Although those twenty units don't focus down one unit, they are free to fight again after they've killed a unit from the weaker army. However, they are weaker and hence die quicker. If two spearmen kill one then to say I have one and a half spearmen remaining then means something significant. If I had ten separate 2v1s, your formula would be accurate, but since some units die quicker than others, those units are free to fight again and hammer home their advantage.
I would also draw attention to the fact that your formula
Sigma(n/m)p=p*
is a special case of Lanchester's Square Law (with 0.5 correction) for the case that n/m units fight one unit, with both armies cancelling each other out, as in
a(A^2+A)=b(1^2+1)
where A=n/m, a=p/2 and b=p*/2. You could actually predict how many units survive by setting the surviving army number to be non-zero. That would be the winning margin for one micro-fight. You could then multiply by m to find the total number of surviving units. Granted that implies lots of units on low health, but that's why I don't think the fights scale as you say they do - some units die faster than others, and the surviving units are free to kill more.
Anyway, I just wanted to flag that, because you said you could predict who wins, but not by what margin. Even though you don't think Lanchester's Square Law holds in general, I do think at the very least it's applicable on the level of micro-fights.
And the formula doesnt even fail by much for 40 sized armies. If i cared i can extend a space effectiveness factor to the formula but as i said i never felt a big need for that as simple allowances can be made in mind.
That would be interesting, and if you do come up with a space effectiveness factor, let me know.
Don't we not use that one because that was like for one hit kill?
No, not at all. Look at two cavalry archer armies patrolling into each other in a Hun war. They steadily eat away at each other. Another way of putting it is once you have ballistics, you're going to have large armies you can't realistically micromanage. In that situation, one hit kills don't come into play anymore.
I also started seeing why micro is so important. The side that has a narrow victory with more units and room to micro starts winning with a lot more left with basic micro. If the losing side micros it's wasted effort in a game situation (unless you're viper).
Interestingly, the reason Viper comes out on top is that he picks off units here and there, not allowing his army to engage head on. The point is he gets cost-effective trades and his opponent invests resources into replacing those units, which means Viper is able to move ahead in economy. That's why Viper is able to come back after falling behind in a feudal war.