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A page for general rambling about strategy and tactics, including in the context of game design and balance (To be expanded).

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Lanchester's Square Law

It's no secret that a big army stands a good chance of beating a smaller army. However, the larger force's advantage can be a lot bigger than you think.

Suppose you have a certain amount of artillery, and your enemy has a different amount of identical artillery, and both are steadily shooting at each other and reducing each other's numbers. If everything else is equal, who's stronger? Obviously, whoever has more guns. How much stronger? Obviously, the proportion between the size of the first army and the size of the second, right? Blue has three times as many field guns as red, so blue is three times as strong.

"Wrong!" said Frederick William Lanchester, a British engineer and mathematician who lived from the middle 19th century through the end of World War II.

In 1916, Lanchester devised a series of differential equations to demonstrate the power relationships between opposing forces. Among these were Lanchester's Linear Law (applicable to ancient combat, where engagements on the individual level were purely one-on-one), and Lanchester's Square Law (for modern combat, where multiple individuals can attack a single target). It is the square law that concerns us here in ZK.

According to the square law, the relative strength of two forces is given by the ratio of the squares of their relative numbers. Thus, while two Stumpies are four times as strong as one, four Stumpies are sixteen times as strong as one. Note that the Square Law does not apply to multiple weak units against a single strong unit: for a lone tank A to be able to compete with four opposing B-tanks, tank A needs to be qualitatively four times better (four times damage, four times HP) than an individual tank B.

The Square Law is why ten Reapers take a lot less than 50% losses when squaring off against five Reapers.

Anti-Square Laws

So, if numbers are so important, then why doesn't sheer spam win out every time? There are several reasons for this:

What does all this mean to me?

Due to the squaring effect, you should take note of the following:

Defending vs Attacking

Security against defeat implies defensive tactics; ability to defeat the enemy means taking the offensive.
Sun Tzu

Defense structures in ZK are generally a match for twice their cost in mobiles. This, along with the simplicity of managing them, makes it tempting for players (not all of whom are new) to use them a lot (not just in turtling without expansion, but also during confrontations after boundary lines have been drawn).

It goes without saying that this is a Very Bad Idea™ indeed.

Mobile units being, well, mobile, means that they can gather in a single area (namely, where the action is happening), while statics, being rooted to one spot, must be spread evenly across the defensive perimeter. This means that attackers can easily achieve local net superiority over a static-reliant defense, resulting in a line of useless d-fenz and a breach in the line - and that's before we discuss the effects of artillery against statics. In contrast, mobiles can respond to an attack at any point, relying on the few local statics to give them the small but crucial net advantage over the enemy.

Therefore, a better player will concentrate defense only at key points and sprinkle it over the rest of his/her perimeter, relying primarily on mobiles for defense. But is there an even more effective way?

Numerical weakness comes from having to prepare against possible attacks; numerical strength, from compelling our adversary to make these preparations against us.
Sun Tzu

Yes, the best way to resolve the problem of static vs mobile defense is to leave the question to your opponent. By attacking, you force the enemy into a defensive posture, forcing them to spend more on defense while you have to spend less. With more resources for mobiles, you can contest mex spots and wrecks more effectively, resulting in an even greater resource advantage. Additionally, the psychological pressure will make your opponent more prone to making mistakes, which you can exploit effectively to create a similarly vicious cycle.

So attack. It's the only way to win.


Crossing the T / Enfilade

Most players know that most units (skirmishers in particular) do best in a line or encirclement formation, which allows the maximum amount of forward firepower. The best way to counter this is to run up beside the line, and then down the line (parallel to it). This move, known as crossing the T or enfilade, exploits the weakness of the line formation as units on the far end of the line will often be out of range and/or obstructed by friendly units further up the line. Frontal attacks on a line should be avoided whenever possible.

This maneuver also works on static defenses, although open ends in the line to flank are generally less frequent and it's usually better to ignore the rest of the line and proceed straight to the base once the perimeter is breached.

The Shield and the Rapier

When a head-on assault on a defense line is called for, one trick to increasing your firepower is to mix some raiders with assault units. The assault units close in first to draw fire from enemy defenses, then the raiders move in for the kill with their superior DPS. In addition to increasing the odds of a successful breakthrough, this also makes a raider force immediately available to attack the enemy's logistics once the enemy's defense line has been penetrated.


Looking for a spiritual successor to the Goatly Maxims. - KR

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