# Ceasar's Mind

## The Nature of Imbalance

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Before I can explain the combat effectiveness of a heterogeneous group, I need make sure the reader understands a few things first.

First, recall that in homogeneous groups, every additional unit increases the strength of the group exponentially.

Recall also that a rational player will target enemy units according to their Offense to Defense ratio, the highest of which will be targeted first.

With that in mind, I can explain.

Let’s consider the simple case of a group of two units, one unit with 1 Offense (O) and 1 Defense (D), and the other with sqrt(3) O and sqrt(3) D. Since both units in the group have the same Offense to Defense ratio, the order in which our enemy targets our units is unimportant and therefore we can calculate the strength of our group against the Universal Unit to get a K-value for our group. That said, it comes out to be (1 + sqrt(3))*1+sqrt(3)*sqrt(3)=4+sqrt(3) ~= 5.73K.

Notice though, that the unit with sqrt(3) Offense and sqrt(3) Defense has a K-value of 3, which means it worth exactly twice as much the other unit. That is to say, the value of our group is \$1 + \$2 = \$3.

However, notice that \$3 worth of 1O, 1D units produces an effective strength of 1+2+3=6K. This suggests that my previous suggestion of how to price units is not adequate.

Things break down even more when the Offense to Defense ratios are different. If a player rationally targets the unit with the highest Offense to Defense ratio, he will make heterogeneous groups even less effective. Consider for example, another group worth \$3, except this time composed of a unit with 1 Offense and 1 Defense and a unit with 1 Offense a 3 Defense. In this case, the 1 Offense, 1 Defense unit is more threatening, and so with that unit targeted first the group’s strength comes out to a measly (1+1)*1+1*3=4K. Should the player have done otherwise, he would improved the group’s strength to an impressive (1+1)*3+1*1=7K.

This has something interesting implications. What we are basically saying here is that the effective strength for a two unit group is given by,

(O1+O2)D1+O2D2 = D1O1 + D1O2 + D2O2 or (O1+O2)D2+O1D1 = D1O1 + D2O1 + D2O2

What is important here in the middle factor, the D1O2. Typically, if Unit 1 and Unit 2 were the same price we would want to D1=O2, since if D1!=O2 then D2!=O1, and since our opponent gets to choose which unit he will target first, he will always pick the unit that makes the middle term smallest. Thus we try to make D1O2=D2O1, or equivalently D1/O1=D2/O2. Therefore, homogeneous groups are typically the most effective.

However, it is sometimes possible to force our enemy to attack our units in the way that we want him to. In this case, we want very heterogeneous units. In fact, if we could have it, we would have our units as different as possible- give one with infinite Offense and infinitesimal Defense and the other infinite Defense and  infinitesimal Offense, and we get a middle term that equals to infinity^2!

As we can see, prices cannot possibly be determined by looking at stats alone. In fact, prices for units are completely arbitrary- fine-tuned to make the game play as the designer intended. Likewise, the strength of units cannot be measured, since units rarely fight alone or in purely homogeneous groups. Only so long as prices keep the decisions interesting then they are completely left to the designer’s discretion.