Sid Meier's Colonization/War against the King

The King is weak.

At the toughest level, his Expeditionary Force consists of a set number of units (with lower levels shown in parentheses): He will increase that force by a few units later in the game.
 * 47 Regulars (15, 23, 31, 39)
 * 25 Cavalry (5, 10, 15, 20)
 * 26 Artillery (2, 8, 14, 20)
 * 14 Man o'Wars (2, 5, 8, 11)

Once you have the industrial base, the war is trivial. An army of 30 Veteran Dragoons + 20 Artillery, 100% support and at least a Fort in all your coastal cities are enough to suppress the King's forces. However, the fun would be lost. Colonization is an economic strategy, and the fun is in building a thriving economy, not in defeating those pathetic royalists.

You can easily get 100% support and a fortress everywhere at production levels of 2-5 units per turn. If you're interested in the economy more than the fight, start the revolution when your army is big enough to invade the home country, not the other way around.

Remember to have all your veteran dragoons/soldiers in the cities with 100% rebel sentiment when the revolution starts. Only then will they get a free upgrade to Continental Army and a +1 strength bonus (those in cities with 50%-99% support also have a chance of upgrade).

Attacking
Unit strengths are:
 * 2 Soldiers
 * 3 Veteran Soldiers
 * 3 Dragoons
 * 3 Damaged Artillery (defending)
 * 4 Veteran Dragoons
 * 4 Continental Army
 * 5 Continental Cavalry
 * 5 Regulars
 * 5 Artillery (defending)
 * 5 Damaged Artillery (attacking)
 * 6 Cavalry
 * 7 Artillery (attacking)

Artillery defending in the open gets a -75% penalty in addition to having 2 fewer points of strength.

Some quick simulation. You need:
 * 25 wins against Cavalry (6)
 * 72 wins against Regulars (5)
 * 26 wins against Artillery (7 or 5)
 * 26 wins against Damaged Artillery (5 or 3)

If they're attacking, and you defend with Artillery or Continental Cavalry, they have a +50% attack bonus against your +200% Fortress bonus:
 * 26 times: 10.5 vs 25
 * 25 times: 9 vs 25
 * 98 times: 7.5 vs 25

If you're attacking with Continental Cavalry and they're on a forest square, you have +50% attack and +50% ambush bonus (Artillery gets -75% if attacked in the open):
 * 25 times: 10 vs 6
 * 72 times: 10 vs 5
 * 26 times: 10 vs 1.25
 * 26 times: 10 vs 0.75

I'm not sure how the battle system works exactly (in particular I'm not sure if two +50% bonuses mean +100% or +125%). A quick experiment with only 30 samples seems to indicate that the probability of winning is roughly proportional to a unit's strength. If there is any deviation it's quite small, but as it's unlikely that a system with just a small deviation from something as simple would be introduced, I will assume the probability of winning is exactly proportional to unit strength.

The probability of your win in various attacks:
 * 10.5 vs 25: 70%
 * 9 vs 25: 74%
 * 7.5 vs 25: 77%
 * 10 vs 6: 62%
 * 10 vs 5: 66%
 * 10 vs 1.25: 89%
 * 10 vs 0.75: 93%

Because every unit that can attack with p% probability will cause no loss with probability p, and 1 + as much as itself loss otherwise, expected number of losses is: ELoss = p * 0 + (1-p) (1 + ELoss) ELoss = (1-p) + (1-p) ELoss p ELoss = (1-p) ELoss = (1-p) / p

Expected number of losses from a unit fighting its way (at least in "chance of winning is proportional to strength" model, the expected number of losses is equal to exactly proportion of strength, so a strength 5 unit will on average defeat 0.5 strength 10 units before losing):
 * 10.5 vs 25: 0.42
 * 9 vs 25: 0.36
 * 7.5 vs 25: 0.3
 * 10 vs 6: 0.6
 * 10 vs 5: 0.5
 * 10 vs 1.25: 0.125
 * 10 vs 0.75: 0.075

So in an (almost) pure defensive strategy the expected number of losses is: Total: 49.32 (each means 1 Dragoon loses his horses or 1 Artillery is damaged)
 * 26 times 0.42: 10.92
 * 25 times 0.36: 9.0
 * 98 times 0.3: 29.4

So if you have 30 Dragoons and 20 Artillery you will destroy all the king's forces before he gets to attack your damaged Artillery or Soldiers, without using any extra horses, without foreign armies etc.

Some things increase and some things decrease this estimate, but I don't think they change it by more than ±10.

That of course means you can probably win against the king with half as much (15 Dragoons + 10 Artillery), but there's not much point unless you like war much more than economy.

Note: the random number generator is horrible. Some people who investigated it claim the game reseeds the random number generator too often and there's even a binary patch available somewhere online to make it work better. So most of the voodoo relationships like "probability of loss increases when you have more ships" are probably a result of people trying to see patterns where there's nothing except for the bad random number generator's artifacts.

Dealing with the Royal Navy
Don't bother making your own navy to take on the king's. Even frigates can't stand up to Men o'War.

A Fortress with a decent number of artillery units is going to sink or at least heavily damage every Man o'War that comes close. You may even want to get the artillery out of the fortresses, because you can't win the war if you sink all the king's ships before they transport enough of the Expeditionary Force (it's a bug in some versions of the game).

The main problem with Men o'War is the bombardment bonus they give to attacking land units. Just fortress-sink them (your artillery automatically fires on them every turn) or attack the land units.

Dealing with land units
You're like the Indians now: if you attack in heavy terrain you get the ambush bonus instead of the defender getting a defense bonus. But - it only applies if you attack from outside of a city.

If the king's army lands in a Forest or on Hills (Mountains would be simply too good), get your Continental Cavalry and crush them. With the ambush bonus (+150% mountains, +100% hills, +50% forest) and +50% attack bonus they stand no chance, especially as you have a lot of free horses and more numerous armies.

Don't fight with units that lost horses; they're already weaker and on the next loss they'll lose their Continental status. Better get them those horses. Also never take their muskets away for any reason or, again, they permanently revert to plain Veteran Soldiers.

If you can destroy all Cavalry and Regulars from the stack, attack the Artillery with everything you have, even the unmounted soldiers. These are king's most annoying units, and you should destroy them when they're so defenseless.

It's a bit of a problem if the king lands on 0%-ambush-bonus terrain. Remember that you're still going to have an advantage whether attacking (your +50% + ambush attack bonus) or defending (the king's +50% attack bonus versus your +200% fortress bonus), and you can win by outnumbering the king (just keep resupplying the horses faster than they're being lost and the losses will be minimal).

Dealing with Indians
The king may give Indians guns and horses to attack you. Better get rid of all nearby Indians in advance.

Dealing with the royalists
If one of the colonies is dominated by the royalists, they may start a counter-revolution. Just get a majority everywhere. Decrease the size of the royalist colonies if necessary.

Even if a colony is 99% rebel, the remaining 1% traitors are going to help the king, giving a bonus to his attacking units. Simply get 100% support in every coastal colony.