Taking the original question in a different direction (strategy instead of math), I noticed recently that I always try to send just enough force to a region that it would be inefficiently expensive for my opponent to challenge me.
For example, as Japan, when invading eastern Russia, how much force do you need to send? If you are much, much stronger than Russia, and you can sack the capital, fine, do that. Not interesting. If you are much, much weaker than Russia, and you are not going to even be able to keep an active beachhead, fine, stay home. Also not interesting. The interesting case is the middle ground, where you are strong enough to be a real pain in Russia’s behind, but not strong enough yet to be seriously planning an attack on Moscow.
In that case, you want to send enough force that it would cost Russia more to repel your forces than to just let you sit in Kazakh and Novosibirsk and Evenki. If you are conquering, e.g., 5 IPCs’ worth of territory, and you expect to hold it for roughly 3 turns before the global strategic situation changes again, then the stakes are a 30 IPC swing (5 IPCs * 3 turns * 2 players), so you need a force that’s large enough that Russia can’t kill it without losing at least 30 IPCs’ worth of troops. Depending on Russia’s force mix, this could be more or less than 30 IPCs’ worth of your units. If Russia has almost nothing but infantry, then even a small Japanese force of, e.g., 4 infantry and 2 tanks (24 IPCs) might be a fair match on defense against 10 attacking Russian infantry. If Russia has plenty of artillery and air support, then even a large Japanese force of, e.g., 4 infantry and 5 tanks (42 IPCs) might get crushed by a Russian counter-attack of 6 infantry, 3 artillery, and 6 fighters – Russia’s not going to lose more than 30 IPCs on that battle, and the planes are not really going to be taken away from their positions, but Russia can still repel your 42-IPC force.
So you want to think about what it would actually cost your enemy to repel your forces, and then make your invasion force just large enough that your enemy will lose more money from building or diverting the counter-attack than it would lose by letting you have the extra territories. That way you set up a win-win situation: if they let you have the territory without a fight, you win because your income goes up and theirs goes down, but if they fight you for the territory, then they lose so many resources that holding the ground winds up being a net economic loss for them.
The reason why you don’t just go in with everything you can afford to build is that you might need some of those forces to set up a similar no-win scenario for your opponents on another front – sticking with Japan, if you send too many assets into Russia, you usually won’t have enough cash left over to make sure that the Pacific islands are too expensive for America to take, and so on. There are exceptions; sometimes you can just steamroll everyone at once – but in those games, you don’t need a strategy guide!
Bumped. I don’t want this be burried too fast in past page.