While all nations did use them differently or had different doctrines, their result was basically the same anyways so what nations intended to use them for doesn't matter much for our game.
Subs achieved 3 results which only 1-2 work to various degrees in HoI3:
# Sink enemy convoy tonnage (and cargo) weakening their overseas trades and operations.
# Sink or Damage enemy capital ships that happen to pass by with "lucky" shots (warship attrition).
# Work as scouts giving you a good general Idea off enemy ship and convoy movements.
Excellent summary of the main uses for subs.
A. For Germany, their main sub mission was convoy attack, with some scouting. For Japan mainly capital ship attack, and scouting. For the US, a pretty good balance between all 3 missions. Players, and the AI, should be able to give mission priorities to sub fleets, so they can order the priority of these 3 missions for each sub fleet.
B. From what I heard in other threads, the convoy raiding part of this is reasonably modeled, except for 2 problems:
1. Subs are not surrently cost effective enough at sinking convoys. If you have subs attacking enemy convoys, they should be extremely cost effective, with the cost of the convoys they sink being much more than their own cost. This could probably be fixed by giving subs a larger sinking probability in convoy attack.
2. To preserve balance though, escorts should be more effective at sinking subs, provided you have enough of them per convoy, and the escorts are not technologically behind the subs. Sub effectiveness should also be reduced if their operating area is being patrolled by either land aircraft or CAGs.
3. If these changes are made, then a typical result would be that a nation would build subs, and start sinking a huge amount of convoys, then the convoy nation, will have to build more and better escorts, and provide air patrols over the sub operating areas, thus paring back the sub threat, until the subs are no longer cost effective, at least until sub research provides better subs. Pretty much what happened historically in the atlantic.
C. On scouting, I agree that subs should have a very good search value in finding enemy ships and task forces, and with a low chance of being harmed, provided their primary mission is selected as scouting. You could fix this, by allowing a scout mission for subs. They would have good search value, but lower convoy and ship attack, and a very low chance at being detected. On mission selection, if the mission is scouting, then they will only scout. If the mission is convoy attack, they will scout, until they find a convoy, then attack it. If fleet attack, they will scout, until they find a fleet, then attack. If balanced, they will scout until they find a fleet or a convoy, and then attack it.
D. On capital ship attack, the problem is you can't have subs fight like normal surface ships, like happens now, because they will either be far too effective, devestating whole fleets in mega stacks, or not effective at all and just getting sunk. You need a system where subs are very good at hit and run capital ship attrition, but ineffective in a running battle. You could do this by special sub rules doing the following:
1. When a surface fleet encounters a group of subs, they dont fight as a group. Instead would attack individually, at widely spaced intervals, simulating each sub laying in wait, while the fleet comes to it. Once the attack is resolved for a sub, it then breaks off combat, and will not get another chance for a specified period, while they slowly try to reach an attack position again. While there would be a small chance to detect and sink a sub between attacks, it should be relatively small. Subs were mainly only vulnerable when they were trying to attack something.
2. In resolving the attack, one sub at a time, you would first determine whether the sub manages to get an attack on a capital ship or troop transport without being detected. This would depend on the effectiveness of the sub, vs the effectiveness of the escorts, and the ratio of escorts to capital ships. The capital ship would also have some chance at detecting the sub, but much less than an escort.
3. If the sub manages to approach undetected, then it gets an attack against a capital ship or troop transport or possibly also an escort, with a good chance at damage, and a chance to sink, depending on the strength of the sub vs the size/armor of the target. After this attack, you again determine whether the sub is found by the escorts, with the chance being much higher now. If not detected, the sub gets to attack another target.
4. Once the sub is detected, either before they get to attack, or after an attack, they will then immediately break off. The escorts will pursue, with a chance to sink the sub, determined by the effectiveness of the sub, vs the numbers and effectiveness of the escorts. Once this escort pursuit ends, the sub will have to wait a specified period, before it can try to approach and attack again.
5. The probabilities of the escorts detecting and sinking the sub will have to be carefully balanced, so they have a decent chance to evade detection and sink targets, but if those targets have enough escorts, the subs have a fair chance to be detected before they can attack, and a good chance to be detected and sunk after they attack. Fleets with a decent ratio of escorts to targets, should be much less vulnerable to sub attack than a fleet with insufficient escorts. And very good subs should have a better chance to attack without detection, and evade and retreat after attack. Also, any fleet with air cover should be less vulnerable to sub attack, because the planes have a chance to detect the subs before they approach.