Unitstrength function

Back when Axis and Allies first came out, computers were not generally available. In those days, when we wanted to know whether a given attack or defense would succeed, we counted up the combat points as an approximation to which side was stronger. Of course, this is a poor estimate, because it doesn't take into account the critical importance of hit points. We were, however, aware of the FuzzyWuzzy Fallacy; it just wasn't formalized, except perhaps in formal (academic ?) papers.
Now, I think a good estimate of the combat power of a group should start by counting the number of units, modified for combat points. So, does a unitstrength function exist?
I tried using the Battle Calculator to run some simulations, and very quickly found unexpected issues and complications.
For example, if you take 30 1point units against 30 1point units, you would expect the attacker and defender to win 50% of the time, with maybe 1 or 2 units left. Not so! OK, so let's do some simple calculations.
1 v 1: 5/36 A wins, 5/36 D wins, 1/36 draw, 25/36 no effect, thus 45% A wins, 45% D wins, 9% draw.
2 v 2: 8/36 A wins, 8/36 D wins, 4/36 draw, 16/36 no effect, thus 40% A wins, 40% D wins, 20% draw.
3 v 3: 9/36 A wins, 9/36 D wins, 9/36 draw, 9/36 no effect, thus 33% A wins, 33% D wins, 33% draw.
4 v 4: 8/36 A wins, 8/36 D wins, 16/36 draw, 4/36 no effect, thus 25% A wins, 25% D wins, 50% draw.So the combat power has a big effect on the win/draw ratio, but, as the number of units involved goes up, the draw chance goes way down, because the probability of getting exactly the same number of hits on both sides is small. In all cases, the win/lose ratio is in balance.
Back to the 30 v 30 1s. The result after 2k tries is about 50% each win/lose, no draws. It has been established by the smallnumber trials that the defender gets no advantage for holding the territory. But it is also seen that 2k trials is simply not enough to get really good statistics. The surprising result is that the expected number of surviving units is about 2.5 for the attacker and defender. And that means that, whoever wins, they expect to win with 5 units! I guess that, whoever first gets a hit, that advantage propagates through the rest of the combat, so that, after many rounds, it winds up as quite a substantial gain.
So you can see that there are many factors that affect combat results: the number of units, the combat strength, the number of combat rounds, also the distribution of strengths. has any work been done on this issue?

I'm sure the devs will give you more information but these games depend on random/variable computer functions known as checksums ... math is in some ways paradoxial (see Nash Embedding Theorem or the Heisenburg Uncertainty Principle for examples) and so my point is I'm not sure if what you are asking for is even possible ...
that's my best assessment but like I said redrum or some other programming expert might tell you the same thing I just said

@VictorIn_Pacific An army of 10 units with 6 strength beats an army of 2 units with 6 strength and 27 units with 0 strength 100% of the time, which beats an army of 1 unit with 6 strength and 55 units with 0 strength 100% of the time, which in turn beats an army of 10 units with 6 strength 100% of the time. If such a function existed, then the army of 10x 6 would have a higher total value than 2x 6 + 27x 0, which would be higher than the total value of 1x 6 + 55x 0. And so an army of lower total value would beat an army of higher value 100% of the time. No such function exists, not just out of lazyness, but out of sheer logical impossibility. The combat system of TripleA is simple to explain to a human, but mathematically it's very difficult to handle.

Impossible ... You keep using that word. I do not think it means what you think it means.
The difficult, we do immediately. The impossible will take a little longer.

A physics professor was giving a presentation. He put a slide on the projector. "Now, let me explain that", he said, and proceeded to do so. After he was done, he asked "Any questions?" A graduate student in the back of the room put up his hand: "But Professor ... , that slide is upside down!" "Why, so it is!" said the professor, and proceeded to right the slide. He carried on, unfazed: "Now, let me explain that."

I'm not sure what exactly you are hoping to discuss?
There was some considerable discussion about this on the older forum and a few other places on the web, such as Axis and Allies forums.
Hit points are surprisingly valuable, almost everyone overlooks their value as a newer player. Its well established that the 8 PU destroyer (2 attack, 2 defense) was overall a better unit than the 12 PU cruiser (3 attack, 3 defense). Even with cruisers at 11 PUs in some maps, the destroyers remain a good unit to buy.
The next interesting thing is that extra values tend to be stronger than nonextreme values. A 1 attack, 3 defense unit tends to be better than a 2 attack, 2 defense unit. (Also true with a 3 attack, 1 defense unit). This is because the owner chooses what unit to lose, meaning if you are defending you lose the bad defenders, if you are attacking you lose the low attack units.

@VictorIn_Pacific In face to face play, the simplest model is combat strength + hit points. A slightly more complex model is total combat strength * total hit points. However a diverse force is somewhat more effective than a force with all units the same strength.

@alkexr Seven stars, and seven stones, and one white tree.
What you refer to there, surely that is a paradox! And surely it indicates that someone has done a lot of calculations with the combat system, because I cannot believe that that was found on the spur of the moment.
Random thought: If red tank (3/2) beats blue tank, and blue tank beats green tank, does that mean that red tank beats green tank?
Less random thought: The AAA combat system is deterministic, even if it involves probabilities. Thus a nondeterministic outcome is impossible, even if there is no error in the example you showed. And I could not find an error. But there is a solution to this paradox, and it is that the strength of an army is not a fixed value, but is instead a relative value, thus: red army > blue army > green army > red army. Indeed, there is no paradox.

Note: What I am talking about in these posts assumes use of the Low Luck system. Results obtained under the ordinary system have larger variations, and for small sample sets, fluctuations (luck) can dominate the major effects.
When I ran a few simulations, I found that all it takes to bring a combat result from a 50% chance to win to approximately 100% is ONE extra unit, or even ONE extra combat point. The first seems reasonable, the second less so, until you consider that that single extra combat point can add up to multiple extra hits over multiple combat rounds. Accordingly, I propose the following unitstrength function:
P = .5 + .5 U + .5 C ,
where P = probability of winning, U = Unit differential, C = combat strength differential.
Now, I know that this function is wrong, but it is a good start. I expect it to provide a decent result, as long as the the unit count is 10 or more. You can see that it works for the limiting cases. Results for P > 1 are taken to be P = 1.
There are two other factors that should be considered: the Average combat strength, although this may already be accounted for by unit count and combat strength, and Skewness, which is a measure of the deviation of the unit strengths from the average. I would expect one point of Skewness to have almost the same effect as an extra combat point.
Ultimately, I will try to solve the differential equations governing the combat. (I'm sure someone has already done this.) Basically, there are two linked differential equations that describe the situation; they state that the reduction in strength of an army is proportional to the strength of the opposing army. This problem could also be solved using the finitedifference approach, and even a program as simple as Excel can handle that.
Spoiler alert: My secondstage test function, which incorporates Skewness and a fudge factor, can predict the result A > B > C > A. I will post that when I have more time.

@VictorIn_Pacific Impossibility... I do not think it means what you think it means. Mathematical impossibility simply means that the negation of a given statement is true under certain axioms.
You were looking for a function that gives back the strength of a unit. What I have shown is that you were asking the wrong question. You cannot evaluate the value of a unit, not even that of an army.
Linear formulas will also break down. If the value of a unit is scalar1 * hp + scalar2 * power, then the unit will, under its lifetime, kill expected_combat_rounds_survived * power / dicesides hitpoints, so you can write the equation expected_combat_rounds_survived * power / dicesides * scalar1 = scalar2 * power. But expected_combat_rounds_survived depends both on the average power and the deviation of power of ALL units on the board, and so does the ratio of scalar1 and scalar2.
In short, if all units have 1 power, then 1 power is worth X utilons. If all units have 2 power, then 1 power is worth X/2 utilons.
As for the differential equations for which I admittedly do not have a great affinity, my attempts at giving any sort of useful general solution failed miserably, so did those of others I asked about it. The simplified case where every unit has the same power is clearly solvable, but uninteresting, so I decided not to bother.
Anyway... I really shouldn't be spending time writing comments here. Or if I focused my attention on TripleA, I could at least get back to working on my map. So... good luck with those nasty differential equations.

wow Victorin_Pacific has now brought deontic logic into this! Hint, knowing deontic logic will boost your IQ test score 510 points ez believe me!
Oh by the way, I learned from a great tv show years ago back when tv shows were actually educational that the Nash Embedding Theory proved the paradox that "linear equations are modular" ... which won Nash a Nobel and shows OBVIOUS paradox
annyways, I'm waiting for redrum's final word since he programs the battle calculator logic and would know for sure what is and what isn't possible with this game engine

hey how come we can't see who downvotes?

@CaptainCrunch Pokes for a test @ Cap
Interesting you are correct 
@prastle si

@CaptainCrunch just made all votes public interesting @RoiEX ideas?

@prastle I still can't see who is downvoting .

@Victorin_Pacific hey whats next ... the Bell Inequalities??

@CaptainCrunch I downvoted. That comment didn't appear to be very useful, since to my understanding, checksums have little to do with the topic, math is never paradoxical, the Heisenburg Uncertainty Principle is actually called Heisenberg Uncertainty Principle, and that's not even mathemathics, it's quantum physics.

@CaptainCrunch said in Unitstrength function:
annyways, I'm waiting for redrum's final word since he programs the battle calculator logic and would know for sure what is and what isn't possible with this game engine
The battlcalculator just simulates the battle and tells you the % won and lost. Which is an accurate (though sometimes time consuming way) to get a decent prediction. What I (and I think other experienced players) often do is just open the battlecalc to check how many units each side would have, and after seeing the unit numbers the answer is sometimes obvious. Its not 100% accurate with more complex units, but that's okay.
I don't think you can make an equation that produces a single number for the strength of an army (or a unit), because its dependent on your opponent's forces. Its been explained pretty well why. You can't make an equation for individual units either (I tried a while ago), its just not the correct approach.

@CrazyG so his question isn't about the engine but about some chart to help him predict his chances in a battle?
Ok thanks!