Level 3: Transitive Mechanics and Cost Curves

Readings/Playings

None for this week, other than this post. This is a pretty long post, though, so it should be enough. As with last week, you’ll be doing a bit of outside research to compensate.

This Week’s Topic

This week is one of the most exciting for me, because we really get to dive deep into nuts-and-bolts game balance in a very tangible way. We’ll be talking about something that I’ve been doing for the past ten years, although until now I’ve never really written down this process or tried to communicate it to anyone. I’m going to talk about how to balance transitive mechanics within games.

As a reminder, intransitive is like Rock-Paper-Scissors, where everything is better than something else and there is no single “best” move. In transitive games, some things are just flat out better than others in terms of their in-game effects, and we balance that by giving them different costs, so that the better things cost more and the weaker things cost less in the game. How do we know how much to cost things? That is a big problem, and that is what we’ll be discussing this week.

Examples of Transitive Mechanics

Just to contextualize this, what kinds of games do we see that have transitive mechanics? The answer is, most of them. Here are some examples:

  • RPGs often have currency costs to upgrade your equipment and buy consumable items. Leveling is also transitive: a higher-level character is better than a lower-level character in nearly every RPG I can think of.
  • Shooters with progression mechanics like BioShock and Borderlands include similar mechanics. In BioShock, for example, there are costs to using vending machines to get consumable items, and you also spend ADAM to buy new special abilities; higher-level abilities are just better (e.g. doing more damage) than their lower-level counterparts, but they cost more to buy.
  • Professional sports in the real world do this with monetary costs: a player who is better at the game commands a higher salary.
  • Sim games (like The Sims and Sim City) have costs for the various objects you can buy, and often these are transitive. A really good bed in The Sims costs more than a cheap bed, but it also performs its function of restoring your needs much more effectively.
  • Retro arcade games generally have a transitive scoring mechanism. The more dangerous or difficult an enemy, the more points you get for defeating it.
  • Turn-based and real-time strategy games may have a combination of transitive and intransitive mechanics. Some unit types might be strong or weak against others inherently (in an intransitive fashion), like the typical “footmen beat archers, archers beat fliers, fliers beat footmen” comparison. However, you also often see a class of units that all behave similarly, but with stronger and more expensive versions of weaker ones… such as light infantry vs. heavy infantry.
  • Tower Defense games are often intransitive in that certain tower types are strong against certain kinds of attacks, like splash damage is strong against enemies that come clustered together (intransitive), but in most of these games the individual towers are upgradeable to stronger versions of themselves and the stronger versions cost more (transitive).
  • Collectible-card games are another example where there may be intransitive mechanics (and there are almost always intransitive elements to the metagame, thank goodness – that is, a better deck isn’t just “more expensive”), but the individual cards themselves generally have some kind of cost and they are all balanced according to that cost, so that more expensive cards are more useful or powerful.

You might notice something in common with most of these examples: in nearly all cases, there is some kind of resource that is used to buy stuff: Gil in Final Fantasy, Mana in Magic: the Gathering, ADAM in BioShock. Last week we talked about relating everything to a single resource in order to balance different game objects against each other, and as you might expect, this is an extension of that concept.

However, another thing we said last week is that a central resource should be the win or loss condition for the game, and we see that is no longer the case here (the loss condition for an RPG is usually running out of Hit Points, not running out of Gold Pieces). In games that deal with costs, it is common to make the central resource something artificially created for that purpose (some kind of “currency” in the game) rather than a win or loss condition, because everything has a monetary cost.

Costs and Benefits

With all that said, let’s assume we have a game with some kind of currency-like resource, and we want to balance two things where one might be better than the other but it costs more. I’ll start with a simple statement: in a transitive mechanic, everything has a set of costs and a set of benefits, and all in-game effects can be put in terms of one or the other.

When we think of costs we’re usually thinking in terms of resource costs, like a sword that costs 250 Gold. But when I use this term, I’m defining it more loosely to be any kind of drawback or limitation. So it does include resource costs, because that is a setback in the game. But for example, if the sword is only half as effective against demons, that is part of a cost as well because it’s less powerful in some situations. If the sword can only be equipped by certain character classes, that’s a limitation (you can’t just buy one for everyone in your party). If the sword disintegrates after 50 encounters, or if it does 10% of damage dealt back to the person wielding it, or if it prevents the wielder from using magic spells… I would call all of these things “costs” because they are drawbacks or limitations to using the object that we’re trying to balance.

If costs are everything bad, then benefits are everything good. Maybe it does a lot of damage. Maybe it lets you use a neat special ability. Maybe it offers some combination of increases to your various stats.

Some things are a combination of the two. What if the sword does 2x damage against dragons? This is clearly a benefit (it’s better than normal damage sometimes), but it’s also a limitation on that benefit (it doesn’t do double damage all the time, only in specific situations). Or maybe a sword prevents you from casting Level 1 spells (obviously a cost), but if most swords in the game prevent you from casting all spells, this is a less limiting limitation that provides a kind of net benefit. How do you know whether to call something a “cost” or a “benefit”? For our purposes, it doesn’t matter: a negative benefit is the same as a cost, and vice versa, and our goal is to equalize everything. We want the costs and benefits to be equal, numerically. Whether you add to one side or subtract from the other, the end result is the same.

Personally, I find it easiest to keep all numbers positive and not negative, so if something would be a “negative cost” I’ll call it a benefit. That way I only have to add numbers and never subtract them. Adding is easier. But if you want to classify things differently, go ahead; the math works out the same anyway.

So, this is the theory. Add up the costs for an object. Add up the benefits. The goal is to get those two numbers to be equal. If the costs are less than the benefits, it’s too good: add more costs or remove some benefits. If the costs are greater than the benefits, it’s too weak; remove costs or add benefits. You might be wondering how we would relate two totally different things (like a Gold cost and the number of Attack Points you get from equipping a sword). We will get to that in a moment. But first, there’s one additional concept I want to introduce.

Overpowered vs. Underpowered vs. Overcosted vs. Undercosted

Let’s assume for now that we can somehow relate everything back to a single resource cost so they can be directly compared. And let’s say that we have something that provides too many benefits for its costs. How do we know whether to reduce the benefits, increase the costs, or both?

In most cases, we can do either one. It is up to the designer what is more important: having the object stay at its current cost, or having it retain its current benefits. Sometimes it’s more important that you have an object within a specific cost range because you know that’s what the player can afford when they arrive in the town that sells it. Sometimes you just have this really cool effect that you want to introduce to the game, and you don’t want to mess with it. Figure out what you want to stay the same… and then change the other thing.

Sometimes, usually when you’re operating at the extreme edges of a system, you don’t get a choice. For example, if you have an object that’s already free, you just can’t reduce the cost anymore, so it is possible you’ve found an effect that is just too weak at any cost. Reducing the cost further is impossible, so you have no choice: you must increase the benefits. We have a special term for this: we say the object is underpowered, meaning that it is specifically the level of benefits (not the cost) that must be adjusted.

Likewise, some objects are just too powerful to exist in the game at any cost. If an object has an automatic “I win / you lose” effect, it would have to have such a high cost that it would be essentially unobtainable. In such cases we say it is overpowered, that is, that the level of benefits must be reduced (and that a simple cost increase is not enough to solve the problem).

Occasionally you may also run into some really unique effects that can’t easily be added to, removed, or modified; the benefits are a package deal, and the only thing you can really do is adjust the cost. In this case, we might call the object undercosted if it is too cheap, or overcosted if it is too expensive.

I define these terms because it is sometimes important to make the distinction between something that is undercosted and something that’s overpowered. In both cases the object is too good, but the remedy is different.

There is a more general term for an object that is simply too good (although the cost or benefits could be adjusted): we say it is above the curve. Likewise, an object that is too weak is below the curve. What do curves have to do with anything? We’ll see as we talk about our next topic.

Cost Curves

Let’s return to the earlier question of how to relate things as different as Gold, Attack Points, Magic Points, or any other kinds of stats or abilities we attach to an object. How do we compare them directly? The answer is to put everything in terms of the resource cost. For example, if we know that each point of extra Attack provides a linear benefit and that +1 Attack is worth 25 Gold, then it’s not hard to say that a sword that gives +10 Attack should cost 250 Gold. For more complicated objects, add up all the costs (after putting them in terms of Gold), add up all the benefits (again, converting them to their equivalent in Gold), and compare. How do you know how much each resource is worth? That is what we call a cost curve.

Yes, this means you have to take every possible effect in the game, whether it be a cost or a benefit, and find the relative values of all of these things. Yes, it is a lot of work up front. On the bright side, once you have this information about your game, creating new content that is balanced is pretty easy: just put everything into your formula and you can pretty much guarantee that if the numbers add up, it’s balanced.

Creating a Cost Curve

The first step, and the reason it’s called a “cost curve” and not a “cost table” or “cost chart” or “cost double-entry accounting ledger” is that you need to figure out a relationship between increasing resource costs and increasing benefits. After that, you need to figure out how all game effects (positive and negative) relate to your central resource cost. Neither of these is usually obvious.

Defining the relationship between costs and benefits

The two might scale linearly: +1 cost means +1 benefit. This relationship is pretty rare.

Costs might be on an increasing curve, where each additional benefit costs more than the last, so incremental gains get more and more expensive as you get more powerful. You see this a lot in RPGs, for example. The amount of currency you receive from exploration or combat encounters is increasing over time. As a result, if you’re getting more than twice as much Gold per encounter as you used to earlier, even if a new set of armor costs you twice as much as your old one, it would actually take you less time to earn the gold to upgrade. Additionally, the designer might want incremental games for other design reasons, such as to create more interesting choices. For example, if all stat gains cost the same amount, it’s usually an obvious decision to dump all of your gold into increasing your one or two most important stats while ignoring the rest; but if each additional point in a stat costs progressively more, players might consider exploring other options. Either way, you might see an increasing curve (such as a triangular or exponential curve), where something twice as good actually costs considerably more than twice as much.

Some games have costs on a decreasing curve instead. For example, in some turn-based strategy games, hoarding resources has an opportunity cost. In the short term, everyone else is buying stuff and advancing their positions, and if you don’t make purchases to keep up with them, you could fall hopelessly behind. This can be particularly true in games where purchases are limited: wait too long to buy your favorite building in Puerto Rico and someone else might buy it first; or, wait too long to build new settlements in Settlers of Catan and you may find that other people have built in the best locations. In cases like this, if the designer wants resource-hoarding to be a viable strategy, they must account for this opportunity cost by making something that costs twice as much be more than twice as good.

Some games have custom curves that don’t follow a simple, single formula or relationship. For example, in Magic: the Gathering, your primary resource is Mana and you generally are limited to playing one Mana-generating card per turn. If a third of your deck is cards that generate Mana, you’ll get (on average) one Mana-genrating card every three card draws. Since your opening hand is 7 cards and you typically draw one card per turn, this means a player would typically gain one Mana per turn for the first four turns, and then one mana every three turns thereafter. Thus, we might expect to see a shift in the cost curve at or around five mana, where suddenly each additional point of Mana is worth a lot more, which would explain why some of the more expensive cards have crazy-huge gameplay effects.

In some games, any kind of cost curve will be potentially balanced, but different kinds of curves have different effects. For example, in a typical Collectible Card Game, players are gaining new resources at a constant rate throughout the game. If a game has an increasing cost curve where higher costs give progressively smaller gains, it puts a lot of focus on the early game: cheap cards are almost as good as the more expensive ones, so bringing out a lot of forces early on provides an advantage over waiting until later to bring out only slightly better stuff. If instead you feature a decreasing cost curve where the cheap stuff is really weak and the expensive stuff is really powerful, this instead puts emphasis on the late game, where the really huge things dominate. You might have a custom curve that has sudden jumps or changes at certain thresholds, to guide the play of the game into definite early-game, mid-game and late-game phases. None of these are necessarily “right” or “wrong” in a universal sense. It all depends on your design goals, in particular your desired game length, number of turns, and overall flow of the gameplay.

At any rate, this is one of your most important tasks when balancing transitive systems: figuring out the exact nature of the cost curve, as a numeric relationship between costs and benefits.

Defining basic costs and benefits

The next step in creating a cost curve is to make a complete list of all costs and benefits in your game. Then, starting with common ones that are used a lot, identify those objects that only do one thing and nothing else. From there, try to figure out how much that one thing costs. (If you were unsure about the exact mathematical nature of your cost curve, something like this will probably help you figure that out.)

Once you’ve figured out how much some of the basic costs and benefits are worth, start combining them. Maybe you know how much it costs to have a spell that grants a damage bonus, and also how much it costs to have a spell that grants a defense bonus. What about a spell that gives both bonuses at the same time? In some games, the cost for a combined effect is more than their separate costs, since you get multiple bonuses for a single action. In other games, the combined cost is less than the separate costs, since both bonuses are not always useful in combination or might be situational. In other games, the combined cost is exactly the sum of the separate costs. For your game, get a feel for how different effects combine and how that influences their relative costs. Once you know how to cost most of the basic effects in your game and how to combine them, this gives you a lot of power. From there, continue identifying how much new things cost, one at a time.

At some point you will start also identifying non-resource costs (drawbacks and limitations) to determine how much they cost. Approach these the same way: isolate one or more objects where you know the numeric costs and benefits of everything except one thing, and then use basic arithmetic (or algebra, if you prefer) to figure out the missing number.

Another thing you’ll eventually need to examine are benefits or costs that have limitations stacked on them. If a benefit only works half of the time because of a coin-flip whenever you try to use it, is that really half of the cost compared to if it worked all the time, or is it more or less than half? If a benefit requires you to meet conditions that have additional opportunity costs (“you can only use this ability if you have no Rogues in your party”), what is that tradeoff worth in terms of how much it offsets the benefit?

An Example: Cost Curves in Action

To see how this works in practice, I’m going to use some analysis to derive part of the cost curve for Magic 2011, the recent set that was just promoted recently for Magic: the Gathering. The reason I’m choosing this game is that CCGs are among the most complicated games to balance in these terms – a typical base or expansion set may have hundreds of cards that need to be individually balanced – so if we can analyze Magic then we can use this for just about anything else. Note that by necessity, we’re going into spoiler territory here, so if you haven’t seen the set and are waiting for the official release, consider this your spoiler warning.

For convenience, we’ll examine Creature cards specifically, because they are the type of card that is the most easily standardized and directly compared: all Creatures have a Mana cost (this is the game’s primary resource), Power and Toughness, and usually some kind of special ability. Other card types tend to only have special, unique effects that are not easily compared.

For those of you who have never played Magic before, that is fine for our purposes. As you’ll see, you won’t need to understand much of the rules in order to go through this analysis. For example, if I tell you that the Flying ability gives a benefit equivalent to 1 mana, you don’t need to know (or care) what Flying is or what it does; all you need to know is that if you add Flying to a creature, the mana cost should increase by 1. If you see any jargon that you don’t recognize, assume you don’t need to know it. For those few parts of the game you do need to know, I’ll explain as we go.

Let us start by figuring out the basic cost curve. To do this, we first examine the most basic creatures: those with no special abilities at all, just a Mana cost, Power and Toughness. Of the 116 creatures in the set, 11 of them fall into this category (I’ll ignore artifact creatures for now, since those have extra metagame considerations).

Before I go on, one thing you should understand about Mana costs is that there are five colors of Mana: White (W), Green (G), Red (R), Black (B), and Blue (U). There’s actually a sixth “type” called colorless which means any color you want. Thus, something with a cost of “G4” means five mana, one of which must be Green, and the other four can be anything (Green or otherwise). We would expect that colored Mana has a higher cost than colorless, since it is more restrictive.

Here are the creatures with no special abilities:

  • W, 2/1 (that is, a cost of one White mana, power of 2, toughness of 1)
  • W4, 3/5
  • W1, 2/2
  • U4, 2/5
  • U1, 1/3
  • B2, 3/2
  • B3, 4/2
  • R3, 3/3
  • R1, 2/1
  • G1, 2/2
  • G4, 5/4

Looking at the smallest creatures, we immediately run into a problem with three creatures (I’m leaving the names off, since names aren’t relevant when it comes to balance):

  • W, 2/1
  • R1, 2/1
  • G1, 2/2

Apparently, all colors are not created equal: you can get a 2/1 creature for either W (one mana) or R1 (two mana), so an equivalent creature is cheaper in White than Red. Likewise, R1 gets you a 2/1 creature, but the equivalent-cost G1 gets you a 2/2, so you get more creature for Green than Red. This complicates our analysis, since we can’t use different colors interchangeably. Or rather, we could, but only if we assume that the game designers made some balance mistakes. (Such is the difficulty of deriving the cost curve of an existing game: if the balance isn’t perfect, and it’s never perfect, your math may be slightly off unless you make some allowances.) Either way, it means we can’t assume every creature is balanced on the same curve.

In reality, I would guess the designers did this on purpose to give some colors an advantage with creatures, to compensate for them having fewer capabilities in other areas. Green, for example, is a color that’s notorious for having really big creatures and not much else, so it’s only fair to give it a price break since it’s so single-minded. Red and Blue have lots of cool spell toys, so their creatures might be reasonably made weaker as a result.

Still, we can see some patterns here just by staying within colors:

  • W, 2/1
  • W1, 2/2
  • B2, 3/2
  • B3, 4/2

Comparing the White creatures, adding 1 colorless is equivalent to adding +1 Toughness. Comparing the Black creatures, adding 1 colorless mana is equivalent to adding +1 Power. We might guess, then, that 1 colorless (cost) = 1 Power (benefit) = 1 Toughness (benefit).

We can also examine similar creatures across colors to take a guess:

  • W, 2/1
  • R1, 2/1
  • W4, 3/5
  • U4, 2/5

From these comparisons, we might guess that Red and Blue seem to have an inherent -1 Power or -1 Toughness “cost” compared to White, Black and Green.

Is the cost curve linear, +1 benefit for each additional colored mana? It seems to be up to a point, but there appears to be a jump around 4 or 5 mana:

  • W, 2/1 (3 power/toughness for W)
  • W4, 3/5 (5 additional power/toughness for 4 additional colorless mana)
  • G1, 2/2 (4 power/toughness for G1)
  • G4, 5/4 (5 additional power/toughness for 3 additional colorless mana)

As predicted earlier, there may be an additional cost bump at 5 mana, since getting your fifth mana on the table is harder than the first four. Green seems to get a larger bonus than White.

From all of this work, we can take our first guess at a cost curve. Since we have a definite linear relationship between colorless mana and increased power/toughness, we will choose colorless mana to be our primary resource, with each point of colorless representing a numeric cost of 1. We know that each point of power and toughness provides a corresponding benefit of 1.

Our most basic card, W for 2/1, shows a total of 3 benefits (2 power, 1 toughness). We might infer that W must have a cost of 3. Or, using some knowledge of the game, we might instead guess that W has a cost of 2, and that all cards have an automatic cost of 1 just for existing – the card takes up a slot in your hand and your deck, so it should at least do something useful, even if its mana cost is zero, to justify its existence.

Our cost curve, so far, looks like this:

  • Cost of 0 provides a benefit of 1.
  • Increased total mana cost provides a linear benefit, up to 4 mana.
  • The fifth point of mana provides a double benefit (triple for Green), presumably to compensate for the difficulty in getting that fifth mana on the table.

Our costs are:

  • Baseline cost = 1 (start with this, just for existing)
  • Each colorless mana = 1
  • Each colored mana = 2
  • Total mana cost of 5 or more = +1 (or +2 for Green creatures)

Our benefits are:

  • +1 Power or +1 Toughness = 1
  • Being a Red or Blue creature = 1 (apparently this is some kind of metagame privilege).

We don’t have quite enough data to know if this is accurate. There may be other valid sets of cost and benefit numbers that would also fit our observations. But if these are accurate, we could already design some new cards.

How much would a 4/3 Blue creature cost? The benefit is 1 (Blue) + 4 (Power) + 3 (Toughness) = 8. Our baseline cost is 1, our first colored mana (U) is 2, and if we add four colorless mana that costs an extra 4… but that also makes for a total mana cost of 5, which would give an extra +1 to the cost for a total of 8. So we would expect the cost to be U4.

What would a 4/1 Green creature cost? The benefit is 5 (4 Power + 1 Toughness). A mana cost of G2 provides a cost of 5 (1 as a baseline, 2 for the colored G mana, and 2 for the colorless mana).

What if I proposed this card: W3 for a 1/4 creature. Is that balanced? We can add it up: the cost is 1 (baseline) + 2 (W) + 3 (colorless) = 6. The benefit is 1 (power) + 4 (toughness) = 5. So this creature is exactly 1 below the curve, and could be balanced by either dropping the cost to W2 or increasing it to 2/4 or 1/5.

So you can see how a small amount of information lets us do a lot, but also how we are limited: we don’t know what happens when we have several colored mana, we don’t know what happens when we go above 5 (or below 1) total mana, and we don’t know how to cost any special abilities. We could take a random guess based on our intuition of the game, but first let’s take a look at some more creatures. In particular, there are 18 creature cards in this set that only have standard special abilities on them:

  • W3, 3/2, Flying
  • WW3, 5/5, Flying, First Strike, Lifelink, Protection from Demons and Dragons
  • WW2, 2/3, Flying, First Strike
  • WW3, 4/4, Flying, Vigilance
  • W1, 2/1, Flying
  • WW, 2/2, First Strike, Protection from Black
  • W2, 2/2, Flying
  • U3, 2/4, Flying
  • BB, 2/2, First Strike, Protection from White
  • B2, 2/2, Swampwalk
  • B1, 2/1, Lifelink
  • R3, 3/2, Haste
  • GG5, 7/7, Trample
  • GG, 3/2, Trample
  • G3, 2/4, Reach
  • GG3, 3/5, Deathtouch
  • G, 0/3, Defender, Reach
  • GG4, 6/4, Trample

How do we proceed here? The easiest targets are those with only a single ability, like all the White cards with just Flying. It’s pretty clear from looking at all of those that Flying has the same benefit of +1 power or +1 toughness, which in our math has a benefit of 1.

We can also make some direct comparisons to the earlier list of creatures without abilities to derive benefits of several special abilities:

  • B2, 3/2
  • B2, 2/2, Swampwalk
  • R3, 3/3
  • R3, 3/2, Haste

Swampwalk and Haste (whatever those are) also have a benefit of 1. And we can guess from the B1, 2/1, Lifelink card and our existing math that Lifelink is also a benefit of 1.

We run into something curious when we examine some red and blue creatures at 4 mana. Compare the following:

  • W3, 3/2, Flying
  • W4, 3/5 (an extra +1 cost but +2 benefit, due to crossing the 5-mana threshold)
  • U3, 2/4 Flying (identical total cost to the W3 but +1 benefit… in Blue?)
  • R3, 3/3 (identical total cost and benefit to the W3, but Red?)

It appears that perhaps Red and Blue get their high-cost-mana bonus at a threshold of 4 mana rather than 5. Additionally, Flying may be cheaper for Blue than it is for White… but given that it would seem to have a cost of zero here, we might instead guess that the U3 creature is slightly above the curve.

We find another strange comparison in Green:

  • G3, 2/4, Reach (cost of 6, benefit of 6+Reach?)
  • G4, 5/4 (cost of 8, benefit of 9?)

At first glance, both of these would appear to be above the curve by 1. Alternatively, since the extra bonus seems to be consistent, this may have been intentional. We might guess that Green gets a high-cost bonus not just at 5 total mana, but also at 4 total mana, assuming that Reach (like the other abilities we’ve seen) has a benefit of 1. (In reality, if you know the game, Reach gives part of the bonus of Flying but not the other part, so it should probably give about half the benefit of Flying. Unfortunately, Magic does not offer half-mana costs in standard play, so the poor G3 is probably destined to be either slightly above or below the curve.)

Let’s assume, for the sake of argument, that the benefit of Reach is 1 (or that the original designers intended this to be the benefit and balanced the cards accordingly, at least). Then we can examine this card to learn about the Defender special ability:

  • G, 0/3, Defender, Reach

The cost is 1 (baseline) + 2 (G mana) = 3. The benefit is 3 (toughness) + 1 (Reach) + ? (Defender). From this, it would appear Defender would have to have a benefit of negative 1 for the card to be balanced. What’s going on?

If you’ve played Magic, this makes sense. Defender may sound like a special ability, but it’s actually a limitation: it means the card is not allowed to attack. We could therefore consider it as an additional cost of 1 (rather than a benefit of -1) and the math works out.

We’ve learned a lot, but there are still some things out of our immediate grasp right now. We’d love to know what happens when you have a second colored mana (does it also have a +2 cost like the first one?), and we’d also like to know what happens when you get up to 6 or 7 total mana (are there additional “high cost” bonus adjustments?). While we have plenty of cards with two colored mana in their cost, and a couple of high-cost Green creatures, all of these also have at least one other special ability that we haven’t costed yet. We can’t derive the costs and benefits for something when there are multiple unknown values; even if we figured out the right total level of benefits for our GG4 creature, for example, we wouldn’t know how much of that benefit was due to the second Green mana cost, how much came from being 6 mana total, and how much came from its Trample ability. Does this mean we’re stuck? Thankfully, we have a few ways to proceed.

One trick is to find two cards that are the same, except for one thing. Those cards may have several things we don’t know, but if we can isolate just a single difference then we can learn something. For example, look at these two cards:

  • GG4, 6/4, Trample
  • GG5, 7/7, Trample

We don’t know the cost of GG4 or GG5, and we don’t know the benefit of Trample, but we can see that adding one colorless mana that takes us from 6 to 7 gives us a power+toughness benefit of 4. A total cost of 7 must be pretty hard to get to!

We can also examine these two cards that have the same mana cost:

  • WW3, 5/5, Flying, First Strike, Lifelink, Protection from Demons and Dragons
  • WW3, 4/4, Flying, Vigilance

From here we might guess that Vigilance is worth +1 power, +1 toughness, First Strike, Lifelink, and the Protection ability, making Vigilance a really freaking awesome special ability that has a benefit of at least 4. Or, if we know the game and realize Vigilance just isn’t that great, we can see that the 5/5 creature is significantly above the curve relative to the 4/4.

We still don’t know how much two colored mana costs, so let’s use another trick: making an educated guess, then trying it out through trial and error. As an example, let’s take this creature:

  • GG, 3/2, Trample

We know the power and toughness benefits are 5, and since most other single-word abilities (Flying, Haste, Swampwalk, Lifelink) have a benefit of 1, we might guess that Trample also has a benefit of 1, giving a total benefit of 6. If that’s true, we know that the cost is 1 (baseline) + 2 (first G), so the second G must cost 3. Intuitively, this might make sense: having two colored mana places more restrictions on your deck than just having one.

We can look at this another way, comparing two similar creatures:

  • G1, 2/2
  • GG, 3/2, Trample

The cost difference between G1 and GG is the difference between a cost of 1 (colorless) and the cost of the second G. The benefit difference is 1 (for the extra power) + 1 (for Trample, we guess). This means the second G has a cost of 2 more than a colorless mana, which is a cost of 3.

We’re still not sure, though. Maybe the GG creature is above the curve, or maybe Green has yet another creature bonus we haven’t encountered yet. Let’s look at the double-colored-mana White creatures to see if the pattern holds:

  • WW, 2/2, First Strike, Protection from Black
  • WW2, 2/3, Flying, First Strike
  • WW3, 4/4, Flying, Vigilance

Assuming that Protection from Black, First Strike, and Vigilance each have a +1 benefit (similar to other special abilities), most of these seem on the curve. WW is an expected cost of 6; 2/2, First Strike, Protection from Black seems like a benefit of 6. WW3 is a cost of 10 (remember the +1 for being a total of five mana); 4/4, Flying, Vigilance is also probably 10.

The math doesn’t work as well with WW2 (cost of 8); the benefits of 2/3, Flying and First Strike only add up to 7. So, this card might be under the curve by 1.

Having confirmed that the second colored mana is probably a cost of +3, we can head back to Green to figure out this Trample ability. GG, 3/2, Trample indeed gives us a benefit of 1 for Trample, as we guessed earlier.

Now that we know Trample and the second colored mana, we can examine our GG4 and GG5 creatures again to figure out exactly what’s going on at the level of six or seven mana, total. Let’s first look at GG4, 6/4, Trample. This has a total benefit of 11. The parts we know of the cost are: 1 (baseline) + 2 (first G) + 3 (second G) + 4 (colorless) + 1 (above 4 mana) + 1 (above 5 mana) = 12, so not only does the sixth mana apparently have no extra benefit but we’re already below the curve. (Either that, or Trample is worth more when you have a really high power/toughness, as we haven’t considered combinations of abilities yet.)

Let’s compare to GG5, 7/7, Trample. This has a benefit of 15. Known costs are 1 (baseline) + 2 (first G) + 3 (second G) + 5 (colorless) + 1 (above 4 mana) + 1 (above 5 mana) = 13, so going from five to seven mana total has an apparent additional benefit of +2. We might then guess that the benefit is +1 for 6 mana and another +1 for 7 mana, and that the GG4 is just a little below the curve.

Lastly, we have this Deathtouch ability that we can figure out how, from the creature that is GG3, 3/5, Deathtouch. The cost is 1 (baseline) + 2 (first G) + 3 (second G) + 3 (colorless) + 1 (above 4 mana) + 1 (above 5 mana) = 11. Benefit is 8 (power and toughness) + Deathtouch, which implies Deathtouch has a benefit of 3. This seems high, when all of the other abilities are only costed at 1, but if you’ve played Magic you know that Deathtouch really is a powerful ability, so perhaps the high number makes sense in this case.

From here, there are an awful lot of things we can do to make new creatures. Just by going through this analysis, we’ve already identified several creatures that seem above or below the curve. (Granted, this is an oversimplification. Some cards are legacy from earlier sets and may not be balanced along the current curve. And every card has keywords which don’t do anything on their own, but some other cards affect them, so there is a metagame benefit to having certain keywords. For example, if a card is a Goblin, and there’s a card that gives all Goblins a combat bonus, that’s something that makes the Goblin keyword useful… so in some decks that card might be worth using even if it is otherwise below the curve. But keep in mind that this means some cards may be underpowered normally but overpowered in the right deck, which is where metagame balance comes into play. We’re concerning ourselves here only with transitive balance, not metagame balance, although we must understand that the two do affect each other.)

From this point, we can examine the vast majority of other cards in the set, because nearly all of them are just a combination of cost, power, toughness, maybe some basic special abilities we’ve identified already, and maybe one other custom special ability. Since we know all of these things except the custom abilities, we can look at almost any card to evaluate the benefit of its ability (or at least, the benefit assigned to it by the original designer). While we may not know which cards with these custom abilities are above or below the curve, we can at least get a feel for what kinds of abilities are marginally useful versus those that are really useful. We can also put numbers to them, and compare the values of each ability to see if they feel right.

Name That Cost!

Let’s take an example: W1, 2/2, and it gains +1 power and +1 toughness whenever you gain life. How much is that ability worth? Well, the cost is 4, the power/toughness benefit is 4, so that means this ability is free – either it’s nearly worthless, or the card is above the curve. Since there’s no intrinsic way to gain life in the game without using cards that specifically allow it, and since gaining life tends to be a weak effect on its own (since it doesn’t bring you closer to winning), we might guess this is a pretty minor effect, and perhaps the card was specifically designed to be slightly above the curve in order to give a metagame advantage to the otherwise underpowered mechanic of life-gaining.

Here’s another: W4, 2/3, when it enters play you gain 3 life. Cost is 8; power/toughness benefit is 5. That means the life-gain benefit is apparently worth 3 (+1 cost per point of life).

Another: UU1, 2/2, when it enters play return target creature to its owner’s hand. The cost here is 7; known benefits are 5 (4 for power/toughness, 1 for being Blue), so the return effect has a benefit of 2.

And another: U1, 1/1, tap to force target enemy creature to attack this turn if able. Cost is 4, known benefit is 3 (again, 2 for power/toughness, 1 for Blue), so the special ability is costed as a relatively minor benefit of 1.

Here’s one with a drawback: U2, 2/3, Flying, can only block creatures with Flying. Benefit is 5 (power/toughness) + 1 (blue) + 1 (Flying) = 7. Mana cost is 1 (baseline) + 2 (U) + 2 (colorless) = 5, suggesting that the blocking limitation is a +2 cost. Intuitively, that seems wrong, when Defender (complete inability to block) is only +1 cost, suggesting that this card is probably a little above the curve.

Another drawback: B4, 4/5, enters play tapped. Benefit is 9. Mana cost is 1 (baseline) + 2 (B) + 4 (colorless) + 1 (above 5 mana) = 8, so the additional drawback must have a cost of 1.

Here’s a powerful ability: BB1, 1/1, tap to destroy target tapped creature. Mana cost is 7. Power/toughness benefit is 2, so the special ability appears to cost 5. That seems extremely high; on the other hand, it is a very powerful ability, it combos well with a lot of other cards, so it might be justified. Or we might argue it’s strong (maybe a benefit of 3 or 4) but not quite that good, or maybe that it’s even stronger (benefit of 6 or 7) based on seeing it in play and comparing to other strong abilities we identify in the set, but this at least gives us a number for comparison.

So, you can see here that the vast majority of cards can be analyzed this way, and we could use this technique to get a pretty good feel for the cost curve of what is otherwise a pretty complicated game. Not all of the cards fit on the curve, but if you play the game for awhile you’ll have an intuitive sense of which cards are balanced and which feel too good or too weak. By using those “feels balanced” creatures as your baseline, you could then propose a cost curve and set of numeric costs and benefits, and then verify that those creatures are in fact on the curve (and that anything you’ve identified as intuitively too strong or too weak are correctly shown by your math as above or below the curve). Using what you do know, you can then take pretty good guesses at what you don’t know, to identify other cards (those you don’t have an opinion on yet) as being potentially too good or too weak.

In fact, even if you’re a player and not a game designer, you can use this technique to  help you identify which cards you’re likely to see at the tournament/competitive level.

Rules of Thumb

How do you know if your numbers are right? A lot of it comes down to figuring out what works for your particular game, through a combination of your designer intuition and playtesting. Still, I can offer a couple of basic pieces of advice.

First, a limited or restricted benefit is never a cost, and its benefit is always at least a little bit greater than zero. If you have a sword that does extra damage to Snakes, and there are only a few Snakes in the game in isolated locations, that is a very small benefit but it is certainly not a drawback.

Second, if you give the player a choice between two benefits, the cost of the choice must be at least the cost of the more expensive of the two benefits. Worst case, the player takes the better (more expensive) benefit every time, so it should be costed at least as much as what the player will choose. In general, if you give players a choice, try to make those choices give about the same benefit; if it is a choice between two equally good things, that choice is a lot more interesting than choosing between an obviously strong and an obviously weak effect.

Lastly, sometimes you have to take a guess, and you’re not in a position to playtest thoroughly. Maybe you don’t have a big playtest budget. Maybe your publisher is holding a gun to your head, telling you to ship now. Whatever the case, you’ve got something that might be a little above or a little below the curve, and you might have to err on one side or the other. If you’re in this situation, it’s better to make an object too weak than to make it too strong. If it’s too weak, the worst thing that happens is no one uses it, but all of the other objects in the game can still be viable – this isn’t optimal, but it’s not game-breaking. However, if one object is way too strong, it will always get used, effectively preventing everything else that’s actually on the curve from being used since the “balanced” objects are too weak by comparison. A sufficiently underpowered object is ruined on its own; a sufficiently overpowered object ruins the balance of the entire game.

Cost curves for new games

So far, we’ve looked at how to derive a cost curve for an existing game, a sort of design “reverse engineering” to figure out how the game is balanced. This is not necessarily an easy task, as it can be quite tedious at times, but it is at least relatively straightforward.

If you’re making a new game, creating a cost curve is much harder. Since the game doesn’t exist yet, you haven’t played it in its final form yet, which means you don’t have as much intuition for what the curve is or what kinds of effects are really powerful or really weak. This means you have to plan on doing a lot of heavy playtesting for balance purposes, after the core mechanics are fairly solidified, and you need to make sure the project is scheduled accordingly.

Another thing that makes it harder to create a cost curve for a new game is that you have the absolute freedom to balance the numbers however you want. With an existing game you have to keep all the numbers in line with everything that you’ve already released, so you don’t have many degrees of freedom; you might have a few options on how to structure your cost curve, but only a handful of options will actually make any sense in the context of everything you’ve already done. With a new game, however, there are no constraints; you may have thousands of valid ways to design your cost curve – far more than you’ll have time to playtest. When making a new game, you’ll need to grit your teeth, do the math where you can, take your best initial guess… and then get something into your playtesters’ hands as early as you can, so you have as much time as possible to learn about how to balance the systems in your game.

There’s another nasty problem when designing cost curves for new games: changes to the cost curve are expensive in terms of design time. As an example, let’s say you’re making a 200-card set for a CCG, and one of the new mechanics you’re introducing is the ability to draw extra cards, and 20 cards in the set use this mechanic in some way or other. Suppose you decide that drawing an extra card is a benefit of 2 at the beginning, but after some playtesting it becomes clear that it should actually be a benefit of 3. You now have to change all twenty cards that use that mechanic. Keep in mind that you will get the math wrong, because no one ever gets game balance right on the first try, and you can see where multiple continuing changes to the cost curve mean redoing the entire set several times over. If you have infinite time to playtest, you can just make these changes meticulously and one at a time until your balance is perfect. In the real world, however, this is an unsolved problem. The most balanced CCG that I’ve ever worked on, was a game where the cost curve was generated after three sets had already been released; it was the newer sets released after we derived the cost curve that were really good in terms of balance (and they were also efficient in terms of development time because the basic “using the math and nothing else” cards didn’t even need playtesting). Since then, I’ve tried to develop new games with a cost curve in mind, and I still don’t have a good answer for how to do this in any kind of reasonable way.

There’s one other unsolved problem, which I call the “escalation of power” problem, that is specific to persistent games that build on themselves over time – CCGs, MMOs, sequels where you can import previous characters, Facebook games, expansion sets for strategy games, and so on. Anything where your game has new stuff added to it over time, rather than just being a single standalone product. The problem is, in any given set, you are simply not going to be perfect. Every single object in your game will not be perfectly balanced along the curve. Some will be a little above, others will be a little below. While your goal is to get everything as close to the cost curve as possible, you have to accept right now that a few things will be a little better than they’re supposed to… even if the difference is just a microscopic rounding error.

Over time, with a sufficiently large and skilled player base, the things that give an edge (no matter how slight that edge) will rise to the top and become more common in use. And players will adapt to an environment where the best-of-the-best is what is seen in competitive play, and players become accustomed to that as the “standard” cost curve.

Knowing this, the game designer faces a problem. If you use the “old” cost curve and produce a new set of objects that is (miraculously) perfectly balanced, no one will use it, because none of it is as good as the best (above-the-curve) stuff from previous sets. In order to make your new set viable, you have to create a new cost curve that’s balanced with respect to the best objects and strategies in previous sets. This means, over time, the power level of the cost curve increases. It might increase quickly or slowly depending on how good a balancing job you do, but you will see some non-zero level of “power inflation” over time.

Now, this isn’t necessarily a bad thing, in the sense that it basically forces players to keep buying new stuff from you to stay current: eventually their old strategies, the ones that used to be dominant, will fall behind the power curve and they’ll need to get the new stuff just to remain competitive. And if players keep buying from us on a regular basis, that’s a good thing. However, there’s a thin line here, because when players perceive that we are purposefully increasing the power level of the game just to force them to buy new stuff, that gives them an opportunity to exit our game and find something else to do. We’re essentially giving an ultimatum, “buy or leave,” and doing that is dangerous because a lot of players will choose the “leave” option. So, the escalation-of-power problem is not an excuse for lazy design; while we know the cost curve will increase over time, we want that to be a slow and gradual process so that older players don’t feel overwhelmed, and of course we want the new stuff we offer them to be compelling in its own right (because it’s fun to play with, not just because it’s uber-powerful).

If You’re Working On a Game Now…

If you are designing a game right now, and that game has any transitive mechanics that involve a single resource cost, see if you can derive the cost curve. You probably didn’t need me to tell you that, but I’m saying it anyway, so nyeeeah.

Keep in mind that your game already has a cost curve, whether you are aware of it or not. Think of this as an opportunity to learn more about the balance of your game.

Homework

I’ll give you three choices for your “homework” this week. In each case, there are two purposes here. First, you will get to practice the skill of deriving a cost curve for an existing game. Second, you’ll get practice applying that curve to identify objects (whether those be cards, weapons, or whatever) that are too strong or too weak compared to the others.

Option 1: More Magic 2011

If you were intrigued by the analysis presented here on this blog, continue it. Find a spoiler list for Magic 2011 online (you shouldn’t have to look that hard), and starting with the math we’ve identified here, build as much of the rest of the cost curve as you can. As you do this, identify the cards that you think are above or below the curve. For your reference, here’s the math we have currently (note that you may decide to change some of this as you evaluate other cards):

  • Mana cost: 1 (baseline); 1 for each colorless mana; 2 for the first colored mana, and 3 for the second colored mana.
  • High cost bonus: +1 cost if the card requires 4 or more mana (Red, Blue and Green creatures only); +1 cost if the card requires 5 or more mana (White, Black, and Green creatures only – yes, Green gets both bonuses); and an additional +1 cost for each total mana required above 5.
  • Special costs: +1 cost for the Defender special ability.
  • Benefits: 1 per point of power and toughness. 1 for Red and Blue creatures.
  • Special benefits: +1 benefit for Flying, First Strike, Trample, Lifelink, Haste, Swampwalk, Reach, Vigilance, Protection from White, Protection from Black. +2 benefit for Deathtouch.

You may also find some interesting reading in Mark Rosewater’s archive of design articles for Magic, although finding the relevant general design stuff in the sea of  articles on the minutiae of specific cards and sets can be a challenge (and it’s a big archive!).

Option 2: D&D

If CCGs aren’t your thing, maybe you like RPGs. Take a look at whatever Dungeons & Dragons Players Handbook edition you’ve got lying around, and flip to the section that gives a list of equipment, particularly all the basic weapons in the game, along with their Gold costs. Here you’ll have to do some light probability that we haven’t talked about yet, to figure out the average damage of each weapon (hint: if you roll an n-sided die, the average value of that die is (n+1)/2, and yes that means the “average” may be a fraction; if you’re rolling multiple dice, compute the average for each individual die and then add them all together). Then, relate the average weapon damage to the Gold cost, and try to figure out the cost curve for weapons.

Note that depending on the edition, some weapons may have “special abilities” like longer range, or doing extra damage against certain enemy types. Remember to only try to figure out the math for something when you know all but one of the costs or benefits, so start with the simple melee weapons and once you’ve got a basic cost curve, then try to derive the more complicated ones.

If you find that this doesn’t take you very long and you want an additional challenge, do the cost curve for armors in the game as well, and see if you can find a relation between damage and AC.

Option 3: Halo 3

If neither of the other options appeals to you, take a look at the FPS genre, in particular Halo 3. This is a little different because there isn’t really an economic system in the game, so there’s no single resource used to purchase anything. However, there is a variety of weapons, and each weapon has a lot of stats: effective range, damage, fire rate, and occasionally a special ability such as area-effect damage or dual-wield capability.

For this exercise, use damage per second (“dps”) as your primary resource. You’ll have to find a FAQ (or experiment by playing the game or carefully analyzing gameplay videos on YouTube) to determine the dps for each weapon; to compute dps, take the amount of damage and multiply by fire rate (in shots-per-second) and that is dps.

Relate everything else to dps to try to figure out the tradeoffs between dps and accuracy, range, and each special ability. (For some things like “accuracy” that can’t be easily quantified, you may have to fudge things a bit by just making up some numbers).

Then, analyze. Which weapons feel above or below the curve based on your cost curve? How much dps would you add or remove from each weapon to balance it? And of course, is this consistent with your intuition (either from playing the game, or reading comments in player forums)?

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3 Responses to “Level 3: Transitive Mechanics and Cost Curves”

  1. hamiltonianurst Says:

    Nitpick:
    “Here’s one with a drawback: U2, 2/3, Flying, can only block creatures with Flying. [...] suggesting that the blocking limitation is a +2 cost. Intuitively, that seems wrong, when Defender (complete inability to block) is only +1 cost, suggesting that this card is probably a little above the curve.”
    Defender is the complete inability to attack, as defined earlier in the article.

    Overall, however, this lesson was fantastic; just the right length and amount of detail.

    • ai864 Says:

      Fair point, and one that I actually did mention in the “live” lecture and forgot to transcribe here. I left out a logical step, that you’d think “complete inability to attack” and “complete inability to block” would be in the same cost ballpark, and if anything inability to attack (Defender) might be more expensive since attacking brings you closer to your win condition while blocking does not. So you would still expect “partial inability to block” to be a low cost, certainly not +2.

  2. Using Cost Curves to Balance Card Costs | Ruby Cow Games Says:

    [...] of the best discussions I could find on the topic of balance is a lecture from Ian Schreiber on Transitive Mechanics and Cost Curves. In the lecture, Ian talks about breaking down all the individual effects of your game into two [...]

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