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# Exponential scaling is not the problem

## Introduction

I've seen people complain in many places about exponential scaling making Borderlands 2 weapons obsolete too quickly as you level up. It is true that the growth in weapon damage is faster than in the original Borderlands, which had a shallower damage growth function, especially at higher levels.

## Exponential scaling: the right thing to do

If you want the effect of a difference in levels to have the same gameplay effect, regardless of what level you are, then exponential scaling is exactly the right choice. If you have a gun that does 10 damage and the bandit has 100 hit points, the game will play the same as if your gun did 10,000 damage and the bandit has 100,000 hit points. If your gun does half as much damage (due to e.g. being underleveled), the gameplay effect is the same in both cases. It's the proportion that matters. So for each level to have the same gameplay effect, gun damage at a level should be a multiple of the previous level's gun damage. So if $\ell$ is the level and $x_\ell$ is the damage at that level we have

$x_{\ell + 1} = b x_\ell$

where $b$ is the desired multiple of damage between levels. The equation that fits this inductive definition is

$x_\ell = x_0 b^\ell$

This is why Borderlands 2 has exponential scaling. Okay, so we set b to, say 1.13 for health, shields, and damage for both players and enemies. Everything works, right?

## The monkey wrench: skills

The catch is that things aren't symmetric between players and enemies because of skills. If we gave players and enemies the same scaling, but the players have skills on top of scaling, the game would get easier and easier as the player leveled up. Gearbox's solution was to make the player's stats scale with a smaller base than the enemies. In Borderlands 2, we have

• Player health and shields: 1.08
• Player gun damage[1]: 1.13
• Enemy health[2]: 1.14

So as the players and enemies level up, the players fall further and further behind in terms of base stats. How much further behind? The ratio of two exponentials is itself an exponential. The idea is that the skills need to make up this difference, so your collection of skills needs to contribute an exponentially-increasing multiplier with level to keep up.

## Skills are not exponential

Unfortunately, skills are not exponential. Yes, skills tend to get a little stronger as you move down a skill tree, but eventually you hit the bottom and that stops. Individual stat bonuses stack only linearly, and while you can exploit synergies between stats, that will only take you so far. For example, a +100% bonus to Gun Damage will translate to a x2 multiplier to your DPS. Ignoring reload time, if you instead had +50% Gun Damage and +50% Fire Rate, you would have a x2.25 multiplier to DPS, which is greater than that of the pure Gun Damage bonus, even though the "sum" of the two bonuses is the same. If you cut up your +100% bonus into three pieces that stacked multiplicatively (e.g. Gun Damage, Fire Rate, and an effect like Zer0's Tw0 Fang skill), then you would have a x2.37 multiplier.

Things are more stark when the total amount of bonus increases. If you have a total of +1000% bonus, you would get a x11 multiplier with one stat, but with two stats you could get a x36 multiplier, and with three stats with multiplicative synergy, you could get a x81 multiplier! Respectively, you have a linear, quadratic, and cubic increase. You might ask, what if we could slice things up into an infinite number of slices with multiplicative synergy. Would you get an infinite multiplier? No. But then what kind of multiplier would you get? You would get...

... an exponential multiplier. A fixed, finite number of stats with multiplicative synergy can keep up at first, but the exponential multiplier will always outrun them in the end because it is equivalent to distributing one's bonus among an infinite number of stats with multiplicative synergy.

## Exponential bonus effects?

One possible solution would be to make the effects of stat bonuses exponential as well. At current, if you have a stat bonus of $x\%$, the effective multiplier is

$\text{effective multiplier} = \left\{ \begin{array}{ll}1 + 0.01x & \text{if } x \geq 0 \\ \frac{1}{1 - 0.01x} & \text{if } x \leq 0 \end{array} \right.$

This means that the effect of increasing bonuses on any one stat is linear. If we wanted exponential effects, we could have instead

$\text{effective multiplier} = e^{0.01x}$

or

$\text{effective multiplier} = 1.01^x$

The former case is "continuously compounded", whereas the latter is compounded every percent. They are both exponential and very close to each other, though not exactly identical (the continuously compounded bonus is just slightly steeper).

In this scheme, two "+10%" effects to the same stat would have multiplicative stacking effects, but would still have the same effect as a single "+20%" effect. Pretty neat! There are a few downsides, for example that "+10%" really means a multiplier of about x1.105, and designing effects like Money Shot might be a little more complicated. But I think it could go a long way to smoothing out the difficulty by level.