Where did the values for attack damage and attack speed runes come from?

This week I'm going to continue my analysis of runes in League of Legends with an investigation that started with the question 'which rune page is best for marksmen?'

Basic attacks (aka auto attacks) are critical to ad carry play, especially at the beginning of the game. You want to get favorable trades in with your opponent, and you also need to do as much damage as possible with your basic attacks in order to get cs and farm well. The more damage each basic attack does, the easier it is to get last hits.

Let's recall the official LeagueMath.com basic attack damage per second (dps) equation:

This equation has seven variables: attack damage, attack speed, critical strike chance, critical strike damage, armor, percent armor penetration, and flat armor penetration. All of these except percent armor penetration can be affected by your choice of runes.

I started my investigation by thinking about partial derivatives of this equation. A partial derivative of a function like this expresses the change in the output of the function with respect to a change in any of the inputs. That is, how does your damage output change when the variables in the function change. This kind of analysis makes sense here, because that's precisely what runes do; they change the inputs to this function.

The first thing I noticed is that the derivatives themselves are functions that depend on the other inputs. For example, the partial derivative with respect to attack speed depends on all of the other variables: attack damage, crit chance and damage, armor, etc... If you think about it, that makes sense. The more attack damage you have, the more valuable a given bonus to attack speed will be. In order to evaluate the value of the derivatives I needed to choose some basic values for these variables to plug in.

Here's what I decided on:

These are 100% debatable, and in fact, the whole analytic understanding here depends heavily on these choices. I chose them by looking at all the level 1 statistics of every champion and making some educated averages and guesses. If you have any input about this stage or suggestions for better values to use here, I'd love to discuss it. There's a lot of interesting work to do here; for instance one could look at how the value of each rune differs with each champion and at each champion level.

One note, the average champion base attack damage is closer to 50, but the value 56 takes into account a Doran's Blade start.

These values, together with the derivatives that I calculated, allow me to measure precisely the value of any change in any of the inputs of this function. For instance, increasing your attack damage by a single point (to 57) under this scenario increases your dps by 0.533. And increasing your attack speed by a single point (to 1.63) increases your dps by 29.898. Ha! I guess to keep the game balanced increasing your attack speed by one should cost more than increasing your attack damage by one.

This method of comparison seems a bit lame, but I corrected for this by taking the derivatives with respect to rune choices instead. Given the base values above, the derivative of dps with respect to the number of attack damage marks you use is 0.507 (one half of a point of damage each second per mark). The derivative of dps with respect to the number of attack speed marks you use is 0.508. The difference between these two is probably too small to be a coincidence. Thus, I think I have an answer to the question posed in the title of this article. The values (0.95 attack damage and 1.7% attack speed) of these marks was chosen to make their effect on dps equal under typical circumstances.

As mentioned above, the value of increasing your attack speed is increased by having more attack damage. So, if you choose a champion with especially high base attack damage, you can get more value from attack speed runes, but in general this advantage is traded off with ad scaling ratios on champion skills. That is, even though this analysis is principally about basic attacks, and you can efficiently increase dps from basic attacks by increasing attack speed, that may be sub-optimal for total damage output when you take into account other factors such as champion skills.

Notably, attack speed quintessences give proportionally more bonus than attack damage quints, so they are a lot more efficient in terms of this analysis (about 10%!). One should probably only use ad quints if last hitting is a problem, or if the champion has tremendous ad ratios on their skills.

The derivatives for crit and armor pen runes are low enough that they only provide between a third and two thirds the benefit of ad and attack speed in this kind of analysis. Though this suggests that they are less useful in general, there are tons of gimmicky strategies that can benefit from their use. I'm thinking of Crittlesticks and Critplank when I write that.

The derivative with respect to armor is also kind of interesting. Each point of armor reduces incoming damage per second by about half the value you'd get by increasing your attack damage by one. This seems low, but the values on seal runes (yellows) appear to have been designed to increase the desirability of taking armor seals.

The analytic approach was fun, and has a certain kind of finality to it, but I wanted to corroborate the results with some empirical testing as well. First, some combinatorics.

There are 5 types of runes that make you do more damage. They are attack damage, attack speed, crit chance, crit damage, and flat armor pen. Each of these can be put in any of the nine slots for marks, or the three slots for quintessences. Examining just these variables (that is, only considering these five stats as well as fixing the choice of all seals and glyphs) still allows for 244,140,625 choices for rune pages (244 million!). And this is of course only a tiny fraction of the possible choices when you look at all runes of all types. That could be an interesting number to calculate as well. I bet it would be mind-boggling.

So I tried them all. Well, I calculated them all, anyway.

The tool I used for doing this is ratios. I calculated the base damage per second according to the default statistics in the table above, and then calculated the damage per second of each of the 244,140,625 choices of dps affecting marks and quintessences. The ratio of the dps with the rune bonuses to the base dps is a measure of the value of the runes. I sorted the results by this value, and there were some interesting patterns.

Attack damage and attack speed dominate this measure. The crit and pen runes would need to have their values dramatically increased to have them show up in any meaningful way in this analysis. I assume Riot is aware of this and chooses to leave the values as they are. I suspect that there are some cheesy strategies and tactics that could abuse higher crit and pen rune values. So, the rest of this analysis will be restricted to the choice between attack damage and attack speed.

In all cases, the proportional increase in the value of attack speed quints makes them a clear choice for increasing basic attack dps. If you're going to be using your basic attack on cooldown, either for trading or to push the wave, attack speed quints are a must. The choice of marks is more subtle.

As seen in the derivative analysis above, the value of attack damage and attack speed marks is almost identical. Which to choose depends on how much attack damage you have (again, since bonus attack speed is more valuable the more ad you have). The value hovers around the average ad of 56 mentioned above. If your ad is going to be above that value (and the more above that value it is the more this is true) then you can increase basic attack dps more by taking attack speed marks (reds) than attack damage marks. Though, if your base attack damage is lower than average, it's best to take ad marks. It will be easier to last hit, they will benefit any ad ratios your champion has, and they increase dps more than attack speed marks would until your attack damage reaches that average level.

In summary, to maximize basic attack damage per second, one should choose attack speed quints and a mix of attack speed and attack damage marks that depends heavily on base champion statistics and initial item buys.

An analytic and empirical look at a common decision about runes for ad carry champions. I had fun and I hope you did too. I'm impressed that the value of attack damage and attack speed marks is so close. The team at Riot talks a lot about introducing meaningful strategic choices into the game and I think in this particular instance they've done just that. 8) Take care of yourselves and hang around; I'll be back soon with more league fun and math fun here at LeagueMath.com.

No partial derivatives were harmed in the making of this article. Thanks calculus. Now let's take a break from all this math and get back to practicing last hitting.

Peace.