Attack Damage VS. Attack Speed

[quote=Fireup6]I've long tried to determine what's the optimal balance between attack speed and attack damage. So today i set off on trying to find my answer. At first i needed to know what the price was for each. I tried to get the average amount of money spent on either a % of AS or 1 point AD. This is what i came up with: [quote=Averages]*AVERAGES UPDATED ONLY TO INCLUDE HIGH-TIER ITEMS* Average Gold per %AS = 50 Average Gold per AD = 55 [/quote] Seemed pretty reasonable. Now for the functions. First let's start with attack speed. Every character starts with a BASE attack speed. This is what the percentages for the AS items are based on. If you buy a 15% AS Dagger, that will increase your AS by 15% of your base. This can be written as: AS = BASE*(1+(x/100)) x being the bonus AS you're getting from the items you bought. From a Damage per Second perspective, buying attack speed items simply increases your Damage Per Second by that % amount. A dagger will increase your damage by 15% Damage per second, a Recursive bow will increase your damage by 40$ Damage Per Second...and so on. They are directly proportional, and not exponential making them very easy to calculate. To figure out how much damage per second you'll be doing, simply multiply your current AD by the Attack speed. Thus the formula becomes: Damage Per Second = AD * (BASE*(1+(x/100))) Now, x is simply by how much % you have increased your base. What determines x? Well since we've already previously established that 1% AS costs us on average 29gp, we know that x is a function of how much gold you have spent on AS items: %AS = GOLD/29 So let's stick that into our formula: Damage Per Second = AD * (BASE*(1+((GOLD/29)/100))) Replace Damage Per Second and GOLD with x and y respectivly so it's usable in a graph y = AD * (BASE*(1+((x/29)/100))) Now let's go just a bit further. This function will tell us how our DPS increases the more gold we spend on AS. But since we're going to be spending some money on AD and some money on AS in combination, we need to determine how AD increases when we spend gold on it. That is just as simple as AS. GOLD/39 = Bonus AD from items. AD = BaseAD + Bonus AD from items Into our function: y = (BASEAD+(x/39)) * (BASE*(1+((x/29)/100))) We're almost done. The only problem now is that we need some constants. We will assume that the base AD of a champion is 60, and that the BASE AS is .75. Lets stick that in. y = (60+(x/39)) * (.75*(1+((x/29)/100))) Great, however, there is still a problem. We only have a certain amount of money to spend (x) yet in this function we're spending ALL of x on AD and ALL of x on AS. We're spending twice as much money as we have, this isn't possble. Thus x needs a coefficient. How much of X we are spending on AD, and how much we are spending on AS. This can easily be done: y = (60+((.5x)/39)) * (.75*(1+(((.5x)/29)/100))) In this case, we are spending .5 of x (50%) on AD and .5 of x (50%). Together, these coefficients must always equal 1 (100%) since it's impossible to spend more than 100% of all the money you have (unless you live in America). So that's our final function. If we need to check 90% on AD and 10% on AS, this would be done simply: y = (60+((.9x)/39)) * (.75*(1+(((.1x)/29)/100))) And so on. I hope this all makes sense, and i didn't make it too confusing for you all. Mostly i wrote this out so that someone could find something doing incorrectly and correct me. If you see something please do. Anyway, now lets get on TO THE GOOD STUFF! :) This graph shows how much DPS we will be dishing out as a function of how much gold we have spent divided in various proportions between AD and AS: [img]http://i.imgur.com/mL9Xj.png[/img] In case you can't see it the Cyan line very near the top is 60% AD and 40% AS. Just on top of it, is the optimal DPS black line: 50% AD and 50% AS. A perfect balance. However, this isn't the nicest way to do it since you can't clearly make out what the true optimal is. We were simply incrementing by 10% in the graph above. The nice way to do it is to make x represent the proportion that we're splitting a static amount of gold in to spend on either AD or AS. The static gold can be 12,000, and if x = 70 then we spent (70/100)12000 on AD, and 1-(70/100)12000 (the remainder) on AS. We do that here: y = (60+(((x/100)12000)/39)) * (.75*(1+((((1-(x/100))12000)/29)/100))) [img]http://i.imgur.com/s2EGM.png[/img] The extremum (highest point in this case) of the function is 492.7347, when x = 52.33... What this means is that you will achieve the highest DPS possible with 12k gold by spending 52% of your gold on AD, and 48% on AS. Ideally, this means that you'll be doing the most damage per second (with auto attacks) by simply spending an equal amount of money on both AD and AS. What this doesn't take into account of course are the other effects that items have to offer such as Madreds. However, you must balance this against the fact that most champions who buy AD have skills which benefit from AD, whereas there are very few skills that benefit from AS. EDIT: If you want to simplify this function to cut out the gold, and simply calculate DPS based on specific combinations of AD and %AS then just use this: y= (60+(BONUSAD))*(.75*(1+(%BONUSAS/100))) Be sure to realize that in this case there is no ratio between the AD and AS, thus you'll always receive the greatest benefit from whenver AD and AS are the greatest. Making this function rather useless. EDIT2: The same graph as above, only with base values of various champions: [img]http://img217.imageshack.us/img217/3103/ui8yj.png[/img] Larger: http://i.imgur.com/UI8yJ.png[/quote]

[quote=NicknameMy]That means, [[Master Yi]] should buy mainly AS because of Wuju Style and he has no spell which syngernises with AD. But his Ult gives him also AS, so... Hmm, a bit tricky^^. y= (60+(BONUSAD))*(.75*(1+([color=red]([/color]%BONUSAS/100))) One ( is too much, marked red btw: [URL=http://www.livephysics.com/ptools/online-3d-function-grapher.php?xmin=-1&xmax=1&ymin=-1&ymax=1&zmin=Auto&zmax=Auto&f=%2860%2B%28x%29%29%2A%280.75%2A%281%2B%28y%2F100%29%29%29]Graph: (60+(x))*(0.75*(1+(y/100)))[/URL] (PS: The edit-funktion is not nice, i know^^.)[/quote]

[quote=Fireup6][quote=NicknameMy] That means, [[Master Yi]] should buy mainly AS because of Wuju Style and he has no spell which syngernises with AD. But his Ult gives him also AS, so... Hmm, a bit tricky^^. y= (60+(BONUSAD))*(.75*(1+([color=red]([/color]%BONUSAS/100))) Where is x? And one ( is too much, marked red btw: [URL=http://www.livephysics.com/ptools/online-3d-function-grapher.php?xmin=-1&xmax=1&ymin=-1&ymax=1&zmin=Auto&zmax=Auto&f=%2860%2B%28x%29%29%2A%280.75%2A%281%2B%28y%2F100%29%29%29]Graph: (60+(x))*(0.75*(1+(y/100)))[/URL] [/quote] Ah thank you. I didn't know there was a 3d graphing source online. I'll mess around with that.[/quote]

[quote=deathalo44]I just think that if you have skills that gets boosted by AD (Caitlyn's Ace in the Hole etc) AD should be the way to build (For the time being). However, attack speed builds usually do magic damage (Madred's Razors) with a hint of physical damage (Infinity Edge etc).[/quote]

[quote=Jebus McAzn]You have to realize that mathcrafting is incredibly impractical in practice. You almost always get AD early game because you are NOT autoattack spamming. You have abilities and you're just farming up. Infinity Edge is a prime example. In addition, there are stats that you can't calculate such as I-Edge's crit damage bonus, which is not included in the graph. Late game you're obviously right-clicking more and ability spamming less so AS becomes more prominent there. In short you can't really go wrong with a decent mix - if anything, this graph verifies that. There's really no point in establishing an optimization point, though, because that's not at all practical.[/quote]

[quote=Fireup6][quote=Jebus McAzn] You have to realize that mathcrafting is incredibly impractical in practice. You almost always get AD early game because you are NOT autoattack spamming. You have abilities and you're just farming up. Infinity Edge is a prime example. In addition, there are stats that you can't calculate such as I-Edge's crit damage bonus, which is not included in the graph. Late game you're obviously right-clicking more and ability spamming less so AS becomes more prominent there. In short you can't really go wrong with a decent mix - if anything, this graph verifies that. There's really no point in establishing an optimization point, though, because that's not at all practical. [/quote] Absolutely true of course. But nonetheless it's useful to do these calculations in case there are clear balance issues that you can exploit, at least that's what I'm looking for whenever i do some number crunching. ;)[/quote]

[quote=Temzilla][quote=Fireup6] [quote=Jebus McAzn] You have to realize that mathcrafting is incredibly impractical in practice. You almost always get AD early game because you are NOT autoattack spamming. You have abilities and you're just farming up. Infinity Edge is a prime example. In addition, there are stats that you can't calculate such as I-Edge's crit damage bonus, which is not included in the graph. Late game you're obviously right-clicking more and ability spamming less so AS becomes more prominent there. In short you can't really go wrong with a decent mix - if anything, this graph verifies that. There's really no point in establishing an optimization point, though, because that's not at all practical. [/quote] Absolutely true of course. But nonetheless it's useful to do these calculations in case there are clear balance issues that you can exploit, at least that's what I'm looking for whenever i do some number crunching. ;) [/quote] IF you're looking for balance issues, you should look at champions instead of AtS vs AD on random base damages and base attack speeds. For example, let's look at Cho'Gath, Cho has a base AtS of .625, and a level 18 base AD of 130+80(Vorpal Spikes). This means that flat AtS gives Cho'Gath .00625 attacks per second every 1 point of Attack Speed. .00625 attacks is 1.3125 damage, or 1.3125 dps BASE. Let's look at another hero who is commonly built Attack Speed (commonly, not optimally.). Caitlyn has a base attack speed of .668 and a base AD @ 18 of 101. 1% AtS on Caitlyn is worth 0.67468dps. Basically, you are forced to build tons of AD to make AtS worth it on Caitlyn, but you see people all the time rush Phantom Dancer haha. It's less about the stats themselves, and more about their synergy with the character itself. I find, that unless the character has a specific steroid, that using the most cost effective items, and a reasonable item synergy is the most effective way to build.[/quote]

[quote=TheJohn]Theory and Experience are archenemies. While theory has the strategy and math, experience got action and real time strategy. And getting x% of AS to make the AD worthy or vice-versa depends a lot on the champion. Getting 3 Phantom Dancers, 1 Starks, 1 Black Cleaver and 1 Bloodthirster for Yi makes him impossible to kill (CC only) and your Thornmail will be useless (I tryed it). But if you do that on Ashe, the result will not be the expected.[/quote]

[quote=NicknameMy]What happens if you get youmuu's and trinity instead of two phantom dancers?[/quote]

[quote=Fireup6][quote=Temzilla]IF you're looking for balance issues, you should look at champions instead of AtS vs AD on random base damages and base attack speeds.[/quote] http://i.imgur.com/UI8yJ.png Proportion (x) is the percentage of 12,000 gold to spend on AD items, and the rest (100-x) will be spent on AD items. Cho wins by a landslide with his vorpal spikes. And Kassadin wins by simply having great base AD/AS and simply amazing bonuses every level.[/quote]