Tactical Theory: Focused Fire

In this series, I’m going to look at some squad based tactics and see how they stack up. I’ll start by looking at the popular focused fire tactic.

Scattered Fire

Before we can examine focused fire, we need something to compare it to. We’ll start with a “scattered fire” tactic that involves spreading damage evenly among all targets. Since this is close to firing more or less at random, this makes a good base to check others against.

When scattered fire is used against enemies with similar defense, all enemies should drop at roughly the same time. We can estimate this as: total enemy defense / total allied offense.

This means the damage taken by the enemy at any given point equals: time * total allied offense / total enemy defense.

Note that if both teams use this tactic and start with equal numbers, this works out the same as a 1 on 1 battle in terms of time taken. We’ll refer to the time for a 1 on 1 battle against basic enemies (offense 1, defense 1, health 100%) as 1 tic.

Focused Fire

This tactic involves picking a “mark” and having everyone on the team direct damage on them until they drop. At that point, a new mark is picked and the cycle begins again.

The time to drop each enemy works out as: enemy defense / total allied offense. By the same token, damage taken equals: time * total allied offense / enemy defense.

For the basic combatants mentioned in the last section, this works out to: time to drop = 1 / allied team size, damage = time * allied team size.

Focus Fire Match Up

If two evenly matched teams both use focused fire, the time to each mark on each side drops equals 1 / enemy team size.

In a 2 on 2 battle, that means 1 mark drops in 0.5 tics and the last mark drops in 1 tic (since both sides are down to one member each).

By the same token, a 3 on 3 has the first mark down in 1/3rd a tic, the second down in half a tic, and the last drops in 1 tic.

This pattern continues with as team size grows and can be summed up as follows: battle duration = 1 + 1/2 + 1/3 … + 1 / initial team size.

Scattered vs Focused

We’ve already seem that scattered match up are actually faster than focused matches due to keeping each team at full strength longer. Now lets see how squads of basic units using these tactics do when matched against each other.

2 on 2

The scattered is set to drop the focused team at 1 tic. However, the focused team drops their strength by one within half a tic. That gives us:
Scattered Team: 1 member at 100% health
Focused Team: 2 members at 50% health

It takes the scattered team 1 tic to wipe the rest of the focused team’s health. However, the focused team can take the last scatter member down in half a tic, leaving both focus members at 25% health.

3 on 3

The scattered is set to drop the focused team at 1 tic. However, the focused team drops their strength by one within 1/3rd a tic. That gives us:
Scattered Team: 2 members at full health
Focused Team: 3 members at 2/3 health (1/3 time * 3 attackers / 3 defenders = 1/3 damage)

It takes the scattered team 1 tic to wipe the rest of the focused team’s health ((3 defender * 2/3 health) / 2 attackers). However, the focused team can take another scatter member down in a 1/3rd a tic, leaving both focus members at:
Scattered Team: 1 member at full health
Focused Team: 3 members at 4/9 health (1/3 time * 2 attackers / 3 defenders = 2/9 damage)

It takes the scattered team 1 1/3 tic to wipe the rest of the focused team’s health ((3 defender * 4/9 health) / 1 attacker). However, the focused team can take the last scatter member down in a 1/3rd a tic, leaving both focus members at 1/3rd health (1/3 time * 1 attackers / 3 defenders = 1/9 damage).

Progression

We’ve got a progression of 1 – 1/2 – 1/4 for the first run and 1 – 1/3 – 2/9 – 1/9 for the second. This looks like it can be generalized as follows: 1 – initial team size/(initial team size ^ 2) – (initial team size – 1) / (initial team size ^ 2) .. – 1/ (initial team size ^ 2). This in turn becomes: 1 – (initial team size + (initial team size – 1) .. + 1) / (initial team size ^ 2). That addition series is a standard arithmetic progression, so it can be shortened to: initial team size * (initial team size + 1) / 2. Plugging that into the formula gives us:
health remaining (hr) = 1 – sum / initial team size (its) ^ 2
hr = 1 – (its * (its +1)/2) / its^2
hr = 1 – (its + 1) / (2 * its)

That suggests 3/8 health for 4 on 4, 40% health for 5 on 5, and so on. As the team size increase, the remaining health approaches 50%.

We can also use these numbers to estimate how extra offense the scattered team would need to match the focused fire tactic from these numbers. For example, we can get health loss as follows:
health loss (hl) = 1 – hr
hl = 1 – (1 – (its + 1) / (2 * its))
hl = (its + 1) / (2 * its)

To match the focused team, the scattered team’s offense has to raise to the point that health loss equals 100% per member. This gives us:
allied offense (ao) * hl = 1
ao * (its + 1) / (2 * its) = 1
ao = 2 * its / (its + 1)

So for teams of 2, that means the focused fire is roughly equivalent to increasing offense by a third. This comes out to a 50% increase for a team of 3, a 60% increase for a team of 4, a 2/3rds increase for a team of 5, and so on. At larger team sizes, the offense multiplier approaches 2. That means for very large teams, focused fire can be almost as effective as doubling the team’s offense.

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Published in: on May 19, 2011 at 10:26 pm  Leave a Comment  
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