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All you really need to know is this:
Following text covers most of battle calculation details. While this details are quite important, you can still skip this boring treatise and return here after you have played D&C at least a week.
All battles can have two stages: space battle, if it is requested and there is an enemy, and surface, if invasion was requested. This means some units (like Shuttle) will fight twice first in the space and then on surface. All battles are calculated identically, no matter in space or at a surface.
All units may have two kind of weapons rockets or lasers. They also have two independent defense systems from rockets and from lasers. Level 5 laser will hit unit with level 5 antilaser defense with about 60% chance. Level 20 laser will hit level 5 defense with about 100% chance, as well as 5-to-20 shot will have close to zero chance to hit.
In addition to attack rating all weapons have damage level. Rockets have higher damage, but it is really hard to hit some units (like infantry or a fighter) with them because of a good antirocket defense. If your technology allows you, you can destroy enemy fighters with single shot, but if your enemy has close to yours tech level, some units will be invulnerable for your rockets.
Every unit also has those "hitpoints", and when unit's hitpoints are equal to zero this unit is equated to zero as well. If you have a carrier with 1% hitpoints left, it is time to move it to one of your worlds, where damaged unit will be automatically repaired.
Depending on the race you have chosen, your units can be stronger or weaker from the start. Next important thing is "level". If you have high technology level, all the units you create will be stronger then the base prototype.
From level to level unit's stats grow dramatically and two-level gap between you and your opponent could mean he has a double force with the same unit count and weight.
All fleet moves end BEFORE the battles are calculated. All planet's production and support costs are calculated AFTER the battles are calculated. Battle starts when one fleet attacks another. Enemy's fleets can remain in one system forever and never fight if they do not get an order to attack.
Once you are involved in the battle, ALL your fleets in the system will fight. You will fight if you are attacking, if you were attacked or if one of your fleets has an order to help an ally attacked or attacking. Ally relations are not inherited, so you will not fight, if your ally will fight to help his ally.
If you are not attacking and you are not attacked and you have no intent to help an ally, you are not involved in battle. If you allies fight with each other, you are not fighting at all too.
After all fleets were moved, every fleet checks, if there are a targets it must attack nearby. If there are it does attack and it's fighting in attack order. Suppose you have assembled a "Mad" fleet, order it to attack all vassals and moved it to some vassal's planet. Fleet will arrive and attack all the vassal fleets it will find. Your "Mad" fleet will go in attack order, while your vassals fleets will be in defense order, even if they will expect you and order to attack a lord arriving fleet has a preference in attack.
Now when base order is defined for both players, units start firing at each other one-by-one (or one-by-two, depending on the number of units you have). First go light units, then heavy. Your bombers will crack vassals carriers, while vassal's fighters will start to hunt for your bombers, and at the end of the battle-turn your heavy ships will fire too.
Every unit has his own list of targets, like bombers will first attack heavy jumpships while fighters will attack bombers. You can manually define a "Mission" for one of your fleets, like "Destroy carriers", and that fleet will do as you ask, all units will target selected unit class.
Inside of the target class, the target is chosen randomly.
This whole order thing will happen several times (currently, 5) and then stop. Keep in mind, that your bombers will launch all rockets and can became a low-end fighter after two battle-turns. Then all your transports and carriers are checked to find units without a transport. All homeless units are killed, others will be resorted among free space in your fleets, if needed. If you are fighting on your own planet, all "lost units" are simply moved to garrison.