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Most of us know that one of they keys to winning battles is to concentrate your forces on the important objectives. There is an interesting side effect to this that crops up when you have a large advantage over the people you are attacking.
Consider a scenario where a medieval infantry (attack 4) attacks a 2 defense unit (pick another med inf for this example) on a mountain. The defending med inf is unfortified, and so has an effective defense of 4.
First case: 1 MI attacks 1 MI - and surprise surprise, there is a 50% chance of each MI winning the battle.
Second case: 2 MI attack 1 MI - if the defender is not militaristic, the attackers have an 83.3% chance of winning, while if the defender is militaristic, they have an 81.1% chance of winning. No big surprise here either.
The surprise comes in the average number of units lost. In case 1, each side lost 0.5 units on average in the battle. In case 2, the attackers lose 0.67 (0.69) while the defender loses 0.83 (0.81) units on average. Actually, this is pretty obvious - the first fight has a 50% chance of going either way, and if the first attacker loses, the second gets a chance to attack a wounded defender.
Now consider a huge number of attackers (formally, an infinite number). The defender is certain to lose his unit eventually. While the attacker loses on average 0.76 (0.80) units, more or less (if the defender was restored to full health for each battle and was never promoted, the attacker would lose exactly 1 unit on average).
The point is that for one on one battles with a 50% chance of winning, you will on average lose one unit for every enemy unit you kill. If you can attack with more than one unit against the same enemy, you lose less than one unit per enemy unit killed.
Consider two small armies of 4 MI attacking 4 MI, with the defenders always being on mountains, so all fights are 50-50. A lot of one on one fights will lose both sides equal numbers of troops on average. At the other extreme, ganging your four up against each enemy troop individually, and then the survivors (if any) moving on to the next target gives you an average loss on 2.6 units, compared to the enemy's average loss of 3.3 units, giving you a 65% chance of wiping out your opponent (vs 50% in a series of 1 on 1 battles).
The obvious conclusion: concentrate forces where possible. Most of the time it is better two have two units attack a single defender, leaving another one to go free for a while, compared with fighting two 1 on 1 battles. This is probably fairly obvious to most people though.
The other conclusion: the civ that can field the biggest army starts off with an advantage, but that advantage grows as more battles are fought. If you start off with 20 MI, vs your opponents 10, after ten 2 on 1 battles, you end up with 14 MI, and your opponent has 2 (compared to the situation of fighting one on one battles where you have 15 and your opponent 5). The big army's advantage grows since it can take advantage of the stack effect to suffer fewer losses than it's enemy, even in (apparently) equal battles.
This also generalises to some extent to unequal battles. You suffer fewer losses by stacking your troops and attacking small numbers of enemies in one go, than by taking on more equal battles. This hurts the AI in particular in pre-industrial wars, where their units tend to arrive in dribs and drabs to attack you, so you can mass your fast defense units and take them out with small losses compared to what you would suffer if those same units were stacked together.
This also means that the civ with the larger army needs to devote less of its resources to replacing losses than the civ with the smaller army, meaning that it can either do more useful thing while fighting the war, or expand its army faster than the other civ is capable of (assuming equal manufacturing capacities).
Consider a scenario where a medieval infantry (attack 4) attacks a 2 defense unit (pick another med inf for this example) on a mountain. The defending med inf is unfortified, and so has an effective defense of 4.
First case: 1 MI attacks 1 MI - and surprise surprise, there is a 50% chance of each MI winning the battle.
Second case: 2 MI attack 1 MI - if the defender is not militaristic, the attackers have an 83.3% chance of winning, while if the defender is militaristic, they have an 81.1% chance of winning. No big surprise here either.
The surprise comes in the average number of units lost. In case 1, each side lost 0.5 units on average in the battle. In case 2, the attackers lose 0.67 (0.69) while the defender loses 0.83 (0.81) units on average. Actually, this is pretty obvious - the first fight has a 50% chance of going either way, and if the first attacker loses, the second gets a chance to attack a wounded defender.
Now consider a huge number of attackers (formally, an infinite number). The defender is certain to lose his unit eventually. While the attacker loses on average 0.76 (0.80) units, more or less (if the defender was restored to full health for each battle and was never promoted, the attacker would lose exactly 1 unit on average).
The point is that for one on one battles with a 50% chance of winning, you will on average lose one unit for every enemy unit you kill. If you can attack with more than one unit against the same enemy, you lose less than one unit per enemy unit killed.
Consider two small armies of 4 MI attacking 4 MI, with the defenders always being on mountains, so all fights are 50-50. A lot of one on one fights will lose both sides equal numbers of troops on average. At the other extreme, ganging your four up against each enemy troop individually, and then the survivors (if any) moving on to the next target gives you an average loss on 2.6 units, compared to the enemy's average loss of 3.3 units, giving you a 65% chance of wiping out your opponent (vs 50% in a series of 1 on 1 battles).
The obvious conclusion: concentrate forces where possible. Most of the time it is better two have two units attack a single defender, leaving another one to go free for a while, compared with fighting two 1 on 1 battles. This is probably fairly obvious to most people though.
The other conclusion: the civ that can field the biggest army starts off with an advantage, but that advantage grows as more battles are fought. If you start off with 20 MI, vs your opponents 10, after ten 2 on 1 battles, you end up with 14 MI, and your opponent has 2 (compared to the situation of fighting one on one battles where you have 15 and your opponent 5). The big army's advantage grows since it can take advantage of the stack effect to suffer fewer losses than it's enemy, even in (apparently) equal battles.
This also generalises to some extent to unequal battles. You suffer fewer losses by stacking your troops and attacking small numbers of enemies in one go, than by taking on more equal battles. This hurts the AI in particular in pre-industrial wars, where their units tend to arrive in dribs and drabs to attack you, so you can mass your fast defense units and take them out with small losses compared to what you would suffer if those same units were stacked together.
This also means that the civ with the larger army needs to devote less of its resources to replacing losses than the civ with the smaller army, meaning that it can either do more useful thing while fighting the war, or expand its army faster than the other civ is capable of (assuming equal manufacturing capacities).
Or maybe he values the city less after I have pounded it a bit(why do I have 7 pikemen in my capitol, it's only worth 4 now?)
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