Tuesday, September 30, 2014

Linear versus Exponential Progression

The past few days I've been playing Clicker Heroes. Well, "playing". Like Cookie Clicker before it (or even earlier, Candy Box), the game is entirely about finding the most efficient path to getting bigger numbers. It is literally just progression and nothing but. Buy things to automatically do more DPS, which kills enemies faster, to get more money, to buy more things to automatically do even more DPS. It's totally ridiculous, yet it triggers some number-happy portion of my brain.

According to the legend on the right, I currently have 1,162 Quintillion Gold, or 1,162 x 10^18, or 1,162,000,000,000,000,000,000 gold.
To make the game mechanics work, all of these clicker games use exponential progression. It's pretty similar to how WoW does progression these days as well. The pro of exponential progression means that you absolutely must get the new tier's worth of gear to really make a dent in the next tier of content, allowing developers to easily put a stat/gear cap on progression.

In Clicker Heroes, I can't really wail away at level 97 using heroes that only gives me 1 Million DPS, not when enemies have 350 Quintillion health. That would take me over 31,000 years! Or in WoW, you can't use gear from TBC to defeat bosses from Cataclysm. Heck, it's difficult or impossible these days to hit the next tier of content with gear from the previous (i.e.: jumping straight from Heroic ToT to Heroic SoO and beating it completely without any SoO gear at all)

WoW's power curve, from my own calculations nearly a year ago.
It also means you really feel upgrades. Every upgrade feels weighty and like a huge difference to how powerful your character is. Contrast that with a linear power curve, and upgrades become less noticeable over time.

For example, imagine if you had 50 Strength and you find a set of armor that between all the pieces increases your Strength by 10. That's a 20% increase in power, and you'll feel that. But imagine the second set of armor only increases your Strength by 20, 10 more than previous. That's only a 16% increase in power from the previous set of armor (60 Strength + 10 more, versus 50 Strength + 10). Now imagine 5 tiers later, and you have a set of armor that again only increases your Strength by 10 over the previous tier. That'd only be a 9% increase in power that tier (110 + 10). Basically, the more tiers you have, the less each individual tier actually increases your total power.

But there are cases where a linear power curve is handy. If your system uses small numbers, like Paper Mario, an exponential power curve isn't going to be helpful. Or if you purposefully want the differences to be small. Look at D&D 5th Edition, for example:

The difference between a level 1 character and a level 20 character using a skill they're proficient in is only a difference of about +5 to +7 total once you take into account attribute boosts and proficiency bonus boosts. At best, a level 20 character can be twice as good as a level 1 character. Since everything is rolled on a d20, you're looking at values of 5 - 25 or so for a level 1 character, or 12 - 32 or so for a level 20 character. It allows lower level characters the chance (albeit small) to do really heroic things, and gives the level 20 character the chance to still fail at moderately complex tasks (rather than always succeed).

It also all depends on the scale of what you're building. If you expect a really long power curve--like in an MMO--exponential might be a better choice because of that diminishing returns on linear. But if your curve is relatively short, a linear curve is quite easy to balance and maintain.

Both are useful depending on the scenario, just like any other tool in a game designer's toolkit. Either case, you can balance encounters and enemies around them, and they'll feel relatively similar in the moment, but it's all about how things should feel if you're going backwards (or jumping ahead!) in content. Should a high-level character be defeatable by a mid or low-level character? Should even mid-level enemies potentially provide a challenge? Or should you only be able to be effective/balanced within a given tier of content? That's all up to the designer and how you want your game to play/feel.

#DesignExperiment, #DnD, #WoW


  1. I'm a big fan of the linear progression in part because of the "immersion" value it can provide. I find it unsettling when the wolves in one zone are hundreds of thousands of times less powerful than the wolves in another. Same thing with it taking something on the order of 2 million level 1 Warriors to kill one level 90 Warrior...assuming they all get to attack before getting Whirlwind-ed down. But all that is a personal preference, if the designer wants a more DBZ-esque power system that's their call.

    1. Linear doesn't really absolve you of that, though. At least, not unless each individual level is worth very little.

      If your system has 90 levels, and you're getting about +10 strength each level, you still have 950 strength versus a level 1 with 50 strength. It might not take 2 million, but you're still probably going to need hundreds if not thousands.

      I mean, if you made each level +2 instead, then you'd bring them much closer (50 at level 1 versus 230 at 90), but then each level feels nearly immaterial (only 4% increase in power starting, and gets smaller from there). Or move the starting scale down to 10, so you have 190 at 90, but that's actually worse as now the power disparity is larger.

      Basically, linear works quite well on small scales (and arguably, at small scales exponential *looks* linear), but if your power tail is 90 levels, it makes levels difficult to justify. On the other hand Diablo III's Paragon levels is a good example of linear progression over a really long tail, and it seems pretty popular as a post-game progression system. Granted, your gear is still exponential (and is really far more of your power progression than your levels were in the first place).

  2. Well said!

    It's a topic I see come up a lot when people ask how WoW's numbers got inflated to be as large as they are and I have to go "Well, if each tier's gear is 30% stronger than the previous tier and we're at 16 tiers now..."

    Individually, each tier makes sense exponentially and no one objects...but when you look at the net result over a long period of time people are shocked.

    1. WoW is a super interesting experiment in exponential power curves because it's so long-lived. Even after the stat squish, we're probably only going to get 2 - 3 expacs (depending on number of tiers each) before they'll have to number squish again.

      But at the same time, if WoW did a linear power curve, you could swarm a level 100 world boss with twice as many level 50 characters as it would take level 100 characters, and at least to WoW's developers, this is undesirable behaviour.

      But yes, I think many folks underestimate just how quickly those exponential power curves add up, including game designers!