The Nature of Imbalance

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 and 1 Defense, and the other with sqrt(3) Offense and sqrt(3) Defense. Since both units in the group have the same Offense to Defense ratio, the order in which our enemy targets our units in unimportant, and thus 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) ~= 5.73.

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 $3.

However, notice that 3 dollars worth of 1 Offense, 1 Defense units produces an effective strength of 6. 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=4. Should the player have done otherwise, he would improved the group’s strength to an impressive (1+1)*3+1*1=7.

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=02, since if D1!=O2 then D2!=01, 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 D102=D201, or equivalently D1/01=D2/02. 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 zero Defense and the other infinite Defense and zero Offense, and we get a middle term that equals to infinity^2!

As we can see, prices then 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.

Research

So a recent post on the TDG forums got me thinking about the value of upgrades and research, and it’s really rather simple, but I think worth saying anyhow since they are so common and to amateurs often a bit bizarre.

Research

Research enables new the construction of new units and technology can often be researched in a variety of ways. New units can often  be unlocked by simply building or upgrading structures, and upgrades or new abilities can often be unlocked via researching the technology directly at a building.

However, the difficult part is really just determining if it’s really worth it to buy many of these upgrades or research the technology. Sometimes the choice can be simple- If for example, a player already owns a hundred soldiers and researching an attack upgrade is cheap, it’s a no brainer. The player will get his money’s worth as soon as the research is completed.

If however, a player is spending resources to simply unlock the ability to construct a new unit (or alternatively use a new ability) the price at first is a little more difficult to figure out. If we assume that the new units is more cost effective than existing units, the player could simply determine if he will reasonably build enough units to get his money’s worth. If for example, the current tier of units get a 100:10 strength to resource ratio, and the new tier would get a 200:10 strength to resource ratio, but initially costs 200 resources, then to make those lost resources back, the player would have to purchase 10 of the new tier units. Beyond that is a profit in strength.

It’s in my opinion though, that having such strict tiers, with each new tier nullifying old ones, is often bad design. (Sometimes, as in the case of Tower Defense War games this is necessary.) The alternative approach, as appears to be most common in RTS games, is to make units cost effective in certain scenarios, say good at defeating groups of units via splash or effective at combating biological units via poison, such that no unit is truly superior to any other, except in certain scenarios. It then becomes the player’s job to guess at how often the player will be able to effectively use unlocked units, and if the benefits outweigh the costs, then to purchase the technology.

To give an example- Consider you are playing StarCraft. You are the Terran (an army of gritty future humans) playing against the Zerg (an alien army that has evolved to kill).  The game begins, and you start building Marines, the basic component of early Terran armies. Very quickly, the option to purchase Medics arrives, which are extremely useful units when the opponent is unable to kill your Marines quickly. Knowing that the Zerg only has access to Zerglings and Hydralisks, both of which are relatively fragile and also unable to overwhelm the healing effects of Medics, it’s a logical decision to buy them. The benefits greatly outweigh the costs, and you will make up the money spent on the research very quickly. You will not however, proceed to buy Firebats, because despite their splash attack, which is effective against clustered enemies, their short range makes them particularly weak against Hydralisk, which your enemy will have many of, and therefore, your Firebats will always be fighting in situations where they are not cost effective, and they will not make up the money spent on research.

As designers, this effectively means that the price of research is effectively arbitrary. Higher costs however, will mean that the research will much less often be purchased, since players will only see it as useful if they can pretty much guarantee that the unlocked units will be able to be deployed in cost effective circumstances.

Research is kind of interesting because players will shape their armies based on the conditions they are fighting in. Based on the map, it may or may not be cost effective to purchase certain units. For example, ground units lose a lot of their effectiveness of island maps. Additionally, the race of the opponent will have to be factored in. For example, Science Vessels are pretty much a must have against Zerg, with their ability to Irradiate enemy units and detect Lurkers. And perhaps even more interestingly, the units that you have already unlocked has to be considered. Certain units may work extremely well together, such as Marines and Medics.

So basically, research is kind of the depth aspect of RTS games. It takes experience to figure out what’s worth researching and what’s not, and these decisions, strategically, are often the most important, and really fuel the interesting interplay behind the RTS.

Prioritizing Targets

Not sure where this belongs exactly yet, but I figured it was worth mentioning. In regards to all of these Hit Points and Damage things, there is a way to prioritize which targets you take out besides what’s most convenient and it’s simple really: Determine which unit is dealing the most damage per hit point, and kill him first.

Take for example, Halo 3. Consider you encounter the two enemies above simultaneously. The one on the left is called a Hunter, armed with a large Assault Cannon, and on the right is a Grunt, armed with a Fuel Rod Cannon. Both weapons are extremely dangerous. However, the Hunter is very difficult to kill, possessing thick armor and a large metallic shield on his left arm which no projectile can penetrate, while the Grunt is very easy to kill, possessing no armor and very little hit points.

The decision before you is this: To target the Hunter first and attempt to dodge both weapons while attempting to find a weak spot in its armor,  or to shoot at the Grunt and kill him quickly and then to fight the Hunter alone. Not much of a choice unless you like a challenge.

Mathematics tell us to take exactly the same course of action in RTS battles. Put simply, to figure out which unit to strike first, determine the DPS of all enemy units and divide it by their hit points and then strike the unit which has the highest result. Usually though, this is intuitively obvious.

Factor in Capital

In strategy games though, there is also capital. That is, while in most cases a damaged unit can be regenerated or repaired, sometimes even for no cost besides time, a dead unit stays dead, and is a permanent loss of resources. With this in mind, the priority can shift to striking the unit with the highest cost to hit point ratio.

Ultimately, it comes down to whether or not you can successfully ensure a kill. If it’s not possible, then it’s best to prioritize by damage to health ratios in order to preserve the hit points of your force. Otherwise, it’s significantly better to take out what you can and then regenerate your units before fighting again on much less level ground.

Why Cheap Units Are Still Useful

The idea of worrying about capital losses usually means trying to invest minimally in weak units since they die very easily. I’ve often wondered if I should make a new formula to adjust for this problem, making expensive units less and less cost effective. I think though that cheap units are underestimated. The one power that they have over their more expensive counterparts is that they can be in multiple places at once. Utilized correctly, this feature can overwork an enemy defense and allow you to penetrate where a more expensive force couldn’t. (The trick is to set it up so that the cheap provides provide an extra kick for your other more expensive units, making each one more powerful individually, and then striking the enemy. He’ll be unable to divert his resources as necessary since they are all centralized in expensive units.) However, this is based off of my experiences with Naval Commander, which accidentally was balanced “incorrectly” since the attack mechanism would choose a random target in range instead of focusing on the weak units, effectively making the strength of units scale linearly. I’ll have to play with the idea a little more in the future.