Exponential Math and Games.

Note: This post is a spiritual follow-up to an earlier post, which can be found here.

Exponentials are hard to understand.

Most of us understand instinctively how big 100 is compared to 10. If I gave you 10 dollars a week for a year, you'd have a fairly decent grasp of how much money you'd have at the end of the time period. 10 x 52 = 520.

What if I gave you a choice? I could give you 10 dollars a week for a year or instead, I would start by giving you a dollar each week, and I would give you a 10% raise every week for a year, how would you respond? On the one hand, 520 dollars is relatively easy to understand. But starting with one dollar and going 10% bigger every week.. that seems rather small compared to 520 dollars, no? I mean, how big could that dollar get?

Fast forward to 51 weeks from now, and realize that you would get 142 dollars on the 52nd week. In total you'd rake up about... 2000 dollars for the last 4 months alone. Raise your hands if you guessed anywhere near that range.

Some of us have some grasp that exponentials tend to grow out of hand really fast, but the human mind tends to be really bad at grasping exactly how much those exponentials will be. We tend to dramatically overshoot or undershoot our guesses. Part of it is due to the complex and compound math involved but there's some evidence to suggest that it is hard-wired into our systems. Human beings tend to live for the immediate moment and look towards the future in a linear fashion.

So, what does this have to do with games? It has to do with the difficulties in grasping what is truly effective at first glance. It has to do with balance and the interplays between choices in games.

Qualitative Example: Contra

If you have ever played a little gem called Contra, perhaps you've noticed a slight discrepancy in the weapon power-up selection. It has to do with a weapon dearly beloved and known as 'Spread' or 'Shotgun.'

For those of you unfamiliar with the Contra series, imagine you have a tiny little pistol that can shoot one bullet a second.

One bullet a second seems reasonable.

Now, we could improve your gun in several ways. We could make the pistol fire more powerful shots, like say, a flaming shot. Or we could make your pistol fire faster and fire many bullets at a time.

One powerful fireball a second seems reasonable.
Several bullets a second also seems reasonable.

In practice, the machine gun is somewhat more effective than the fireball weapon simply because increasing the Rate of Fire on a gun is far more effective as the shots can hit multiple targets and sometimes a fireball is simply overkill on a small enemy (like a chicken.) The discrepancy isn't very high, however, as the fireball's sheer stopping power and 'fire and forget' mentality makes up for the versatility and ease of use of the machine-gun.

Now, let us examine the 'Spread' or 'Shot' gun. This is meant to be a slightly rarer power-up than the machine gun or fireball power ups. So, maybe we can make it a little more powerful. But not too much.

The 'Spread' gives you the ability to shoot out three balls every half a second. These balls are pitched at an angle to give it a wide arc of attack. The Spread gun can't really attack a single enemy since the balls spread out in an arc, and so it is less powerful than a machine-gun at long range and somewhat more powerful than the flame-thrower at close range. These balls are slightly bigger and stronger than the machine gun shots, but nowhere near as powerful as the flame shot.

This seems like a reasonable power up, in comparison with the machine gun and the flame shot weapon, no? It fires slightly more shots, slightly faster, for slightly more damage and is slightly safer. This seems on far with the ridiculous firing rate of the machine gun or the superb damage of the flame thrower.

However, in practice, any Contra player would tell you that the 'Spread' gun is the weapon of the game. This is in part due to the fact that although the 'Spread' guns *damage* is about equivalent to the flame thrower or the machine gun (In reality, the spread gun is slightly less damaging than either of the two) ; the fact that the spread guns *arc* gives the player far more survivability. It also shoots faster than the flame shot with about the same power as the flame-shot up close if all 3 shots connect.

What gives? Each individual ball of the 'Spread' is far weaker than a single fire shot. The 'Spread' gun fires slower than the machine gun which literally is an unending hailstorm of bullets. The 'Spread' guns marginally safer attack angle doesn't seem to make it *so* much more powerful than the other two. What gives?

It is again, the power of exponential math.

The Spread Gun is faster than your regular gun.
The Spread Gun has slightly more powerful bullets.
The Spread Gun shoots more bullets at once.
The Spread Gun lets the player survive longer by not having to be in the direct path of fire.

The combination of all four gives the final weapon a dramatically increased value in relation to everything else. Imagine each bonus as a numerical value.

The machine gun would increase your rate of fire 5 to 7 times. This would mean that roughly speaking, it is 5 to 7 times better than your old gun.

The flame shot increases your damage 6 or 7 times over. This would mean that roughly speaking, it is 6 or 7 times better than your old gun.

Let us perform a very very rough analysis of the spread gun's benefits, however.

The Spread gun only doubles your rate of fire. (2x) However, it lets you shoot 3 balls at a time (3x, Running Total: 2x * 3x = 6x). It can't hit a single target with all three balls which costs it some points. Usually, you'll only be able to land 2 out of 3 balls on a target. However, the third ball might actually hit something else as a nice bonus. (2/3 x, Running Total: 2x * 3x * 2/3x = 4x) However, these pellets are also stronger than your regular gun's pellets, they are about twice as strong (2x, Running Total: 2x * 3x * 2/3x *2x = 8x). And the 'Spread' gun lets you survive a lot longer against bosses by not putting you in the direct line of fire, maybe even twice as long. (2x? , Running Total: 2x, * 3x, * 2/3x * 2x * 2x? = 16x?)

And so we have a gun that, roughly speaking, is about a hojillion times better than your pellet gun and 3 to 4 times better than the machine gun. The cumulative effect of small incremental bonuses (slightly faster speed, slightly more damaging shots, slightly more shots) gives the final effect a huge boost in the end.

Linear vs. Exponential: MMORPG's

What is more effective in an online MMORPG to be the best player? Raising your attack speed? Raising your chance to hit? Raising your chance to dodge attacks? Raising your chance to block? Raising your raw damage potential? Raising your chance to critically hit?

Let's say I could choose to choose to improve 3 attributes by 100% -or- two attributes by 150% -or- one attribute by 300%.

So I could double 3 things, or 2.5 x two things, or quadruple one thing.

On the surface, this seems fairly intuitive and straightforward.

However, in practice, this is what happens.

If someone swings twice as fast, and hits twice as often and hits twice as hard, he will be doing eight times the damage. He will not be doing (100% + 100% + 100% + 100% = 400%) damage, he will instead be doing 800% damage.

If instead he simply chose to quadruple his damage, he would only be doing 4 times as much damage.

What if we could distribute things farther? What if we could improve 6 attributes by 50%.

So I could swing 1.5 times as fast, hit 1.5 times as often, hit 1.5 times harder, hit 1.5 times critically, dodge 1.5 times as much, take 1.5 as many hits. I would be twelve times as effective as someone else on average.

This is the insane power of exponentials, something that's not quite easily grasped. The difference between offering the player a choice of doing a few more points of damage, and attacking slightly faster, realize the difference between a small linear boost and an exponential percent based boost. If balance is your goal, then you need to carefully consider how different exponential gains will produce dramatic increases in power that need to be accounted for.

So the next time someone offers you to double your damage or double your speed, instead ask for 50% more damage and 50% more speed. This will give you a net increase of 25% over merely doubling one or the other.