# Mathematics of A&A

• Anyone done any work on the math underlying A&A?

Odds simulators based on random number generators are OK but less than elegant solutions, and you certainly couldn’t craft a serious computer opponent from one. Rule of thumb stuff like multiply INF times 1/6 plus same for other units is OK too, but what it fails to reveal are the true overall combat results.

You know what I mean. Never mind the cost, what would you rather go into battle with? 2 subs and a battleship, or 2 battleships? The fractions add to 8/6 either way.

Solid math is what I’m looking for here. I’ve completed most of the math for a precise initial odds calculation, but I’m stuck on a final equation for the coefficients in the battle odds numerator. Anyone already been down this road?

I’m frustrated at being stuck this early, because I’m after bigger game ultimately.

The real overall battle odds are affected by this fact: in a battle that goes multiple rounds, the initial odds are changed by the outcome of the first dice roll - you go into dice roll 2 with different odds. Basically, with each round the strong get stronger and the weak get weaker.

The firepower curve is non-linear to boot. The strong get stronger faster than one might expect. That means there’s more or less a breakover point beyond which you’ve committed more resources than absolutely necessary to assure taking an objective. So what’s the least costly force needed to assure victory?

Kind of an important question when you’re spread thin and trying to take multiple objectives.

A different question along the same lines is “What will I need to take this objective at the lowest total cost?”. We’ve all noticed that more units equals fewer casulaties. The minimum force needed to take a goal is not necessarily the one that suffers the fewest casualties.

Could be an important distinction; after all, you may have to hold the ground afterward. And what will that cost? Ever seen Japan take out the Hawaiian navy, only to lose its expeditionary force to the US response? Who really won?

The mix you throw into a battle is a factor too. Some mixes have a surprisingly higher return on investment than others with exactly the same initial attack vs. defense points.

Chess pieces have rough numerical values of relative worth, but no grandmaster-level computer has ever been based on those alone. The same is true of A&A.

Anyway, there’s more, but this should be enough to kick off the topic.

• I don’t know about all that math stuff but there are some interesting delimnas[sp] involved. Buy a fighter or 4inf. If thinking about attacking both addup to 4. But how do you factor the fact that the fighter can land somewhere else after battle? Correction the fighter attacks at 3. Even so, the flexibility of the fighter and the value of it, goes beyond simple math. I undestand your interest, but considering the flow of a particular game, I don’t see much use.

• What has a plane landing elsewhere after the battle got to do with figuring odds during the battle?

• Maybe guest ment a ftr’s movement won’t effect the attack, but it could effect the casualty choices since planes can’t land in or capture a new territory, their retreatability or the prospect of being able to use them rd after rd after rd, where a tank may get taken out on the counter strike, but this has more to do with AI stuff then straight up odds calculating.

Anyway this brings me to my question, are you looking to just create an odds calculator or do you have a greater purpose in mind, meaning trying to create something that could then be used to make an AI?
Or do you want something to be able to just figure out the least amount of each unit I need to bring into a certain battle?

I’m just a little confused on what your looking for, cause it seems like you have the odd’s calculator stuff down pretty good.

• DM has it about right. As it might be fun or interesring to develop an odds calculator, the decision to use a particular mix of weapons depends much on what position you want to be in after the attack. For instance, attack with fighter or armor? Attack with fighter , and the armor could be used to reinforce elsewhere. Attack with armor, the conquered territory will have more defense. My small brain can’t imagine an odds calculator takung these considerations into account.

• I only play the board game, so my little system is very crude, but, I’ve found, reasonably effective for the purpose. When I’m considering an attack or a defense I simply add up all the unit values for the appropriate circumstances and compare that to the enemy’s. What I mean is like, for instance, 2 INF & 2 ARM attack 4 INF. The attackers attack at 2-at-1 (INF) and 2-at-3 (ARM). The defenders defend at 4-at-2. If a unit’s attack/defense value is considered as “points”, then the attacker has 2 “attack points” for the INF, 6 “attack points” for the ARM, for a total value of 8 “attack points”. The defender has 8 “defense points” (4 units defending at “2” each) so in this battle the odds would theoretically be even. Works pretty well with very large battles, less well with smaller ones–but it’s the big battles I need an “odds calculator” for anyway so there you go. In smaller battles, experience and intuition serve just as well…

I know this is gonna really bug the odds wizards out there but I’m only posting it because its a system any dope (such as myself) can use on the fly in a live boardgame situation…

Ozone27

• I do a similar thing, Ozone. However, I add up all the points and divide by 6 to determine how many hits i score, and do the same for my opponant, mentally subtract units on each side for my kills, rinse and repeat

• Hey that’s a really good system ozone! It’s simple and it works! I’ve done calculations by writing out odds of winning like 2/3 for each bomber attacking, 1/2 for each tank attacking, and added up each sides’ chances of winning by counting every total more than one (say, 4/3) as a hit. If there are any fraction odds left you carry them over to the next round. Carrying on the battle counting hits for both sides, you seem to get the same results as ozone’s method, for example, 4 armour and 4 inf attacking 8 inf is an even battle.

• wehrmacht you made a very good point when you said that the strong get stronger and the weak get weaker. This is something many people don’t see in A & A. Because of this a probability calculator does not actually help very much in deciding what to do in a battle. I downloaded a small program from somewhere once that would give the attacker the odds of victory.

Several years ago my brother, math prof. at U. of Chicago, wrote a simple program that was by far the most useful I have ever seen for A & A. You entered in the units on both sides and told the computer how many times to run the battle. It then created a histogram showing the number of times every result occured. From this information you could easily figure out that after the attack you there would be a 40% probability that you would have 5 units left and a 80% probability that you would have between 6 and 4 units left and so on. This type of information is much more valuable to the player than just there is a 75% chance you will win. Unfortunately, the pc crashed and burned before I copied the program and have never put forth the effort to write one myself. It was not hard to do and it only took him a couple of hours to write. We also decided that we could only use the program during the game after the battle had been fought, because the information was too valuable.

• Indeed, if it can be made such a program will be very valuable in a computer-game context, because it’s much more useful to know how many units you’d have left than just whether you would win or not. Personally, though, I would need a system one could use on the fly in a real-time situation since I only play the board game. If someone could come up w/ something like that it’d be extremely worthwhile…

Ozone27

• DarthMaximus has it right - I’m looking to create an accurate odds calculator as part of a broader A&A AI project. For now, I’m able to manually predict the coefficients in a term of the numerator (the coefficients are the sum of a series), but I’m unable to reduce that to a straightforward calculation. Frustrating, because it’s the last piece I need. In the interim, I’ve substituted a truth table for the required coefficients, so the AI project forges onward.

The AI has two goals. First, to create a computer opponent that will give you a real fight. Then, a tool for developing new strategies. Really the same tool, just depends on how you want to run it.

Got pretty low specs for this thing. Nothing you can play over the internet, just a freebie you could download and run standalone mano AI mano.

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