Searching through all item combinations

Summary: I have tested almost all possible 4-slot item combos with different heroes with an automated offline script. See below for some findings.

Lately, I have been entertaining myself by writing a little program that can calculate hero stats for a given combination of hero, level, and items. It is a bit more convenient than using the hero Demo mode with a dummy (this was explored in another thread here for max DPS). The program outputs relevant things like STR/AGI/INT, HP, effective HP, armor, mana, damage per second etc. It takes into account stacking of crits, evasion and so on, but completely disregards spells and other abilities.

It possible to automate the whole thing and scan over many different combinations of heroes and items. Now, the number of combinations is huge! 6 slots with ca 150 items produces 15,557,597,153,344 combinations, but in practice, it possible to get some interesting results by restricting oneself to a given hero/lvl with 1 slot with only boots, and then say 2 empty slots. Such a search takes around 40 seconds on my machine.

These are my general observations after testing around with Axe, Mirana, and Necrophos. The items combinations are the ones that maximize the respectively property. I believe the results are very general and apply to other STR/AGI/INT heroes too.

Note: this is pure theory crafting. Obviously, these builds are not balanced, or may not even be playable, but they do tell something about the Dota 2 game mechanics and economics.

Level 25 with no restrictions on gold

• Damage per second (DPS): Power Treads(AGI) + Daedalus + Bloodthorne + Manta Style.
• Effective hit points (EHP): Power Treads(STR) + 3 Butterflies
• EHP(magic dmg): Power Treads(STR) + Pipe + Heart + Hood. Replace Heart with extra Hood on Axe.
• Mana: Arcane + Octarine + Soul Booster + Energy Booster

Level 16 with max 13000 gold

• DPS: Boots of Speed + Daedalus + MoM + Manta Style
• EHP: Power Treads(STR) + Butterfly(on Axe) or Halberd(on Mirana/Necro) + 2x Solar crest
• EHP(magic): Power Treads(STR) + Pipe + Heart + Hood
• Mana: Arcane + Octarine + Soul Booster + Energy Booster

Level 11 with max 5000 gold

(This scenario represents a bad start with suboptimal farming.)

• DPS: BoS + Mask of Madness + [Bracer,Wraith,Null] + Crystalys
• EHP: BoS + Vitality Booster + ChainMail + Solar Crest (on Necro/Mirana)
• EHP: BoS + Solar Crest + 2x Buckler (on Axe)
• EHP(magic): BoS + Vitality Booster + 2x Hood
• Mana: Arcane + 3x Energy Booster

For DPS: go Bracer with Axe, Wraith Band for Mirana, and Null Talisman for Necrophos.

Observations

• Many item combinations are simultaneously optimal for Axe, Mirana and Necrophos (STR/AGI/INT heroes). This was surprising to me, there seems to be "universal" build patterns applicable to all heroes.
• Butterfly is primarily a defensive item, which should be bought for EHP.
• Pure attack speed items like Hyperstone seem not economical to maximize DPS late game (but I have seen Gloves of Haste appear in some low gold searches).
• Heavy HP items like Heart is something that you buy on weak AGI/INT heroes, mostly for tanking spell damage, strong heroes do not need them.
• Boosting mana is better done with "boosters" rather than pure INT-items, even on INT-heroes.
• Upgrading the Boots to Power Treads is not economical early game if you are short on gold. It might be beneficial for movement speed, but not for DPS/EHP/Mana.
• Some of these combos might be possible to build in game, e.g. there is clear upgrade path for the max DPS build, and the max mana build. The EHP builds are more difficult recreate in game (you would have to sell a Halberd or a Heart).

The dps is slightly inaccurate because of the animation times and attack points for different heroes, but the rest seems good!

Nice work.

Well, this only applies for heroes with 1.7 BAT and no rightclick steroid...

It is a good point, the heroes I happened to choose were all 1.7 BAT. But different BAT values are taken into account automatically since I read all hero stats from a table. For example, I find that Doom Lvl 25 (BAT 2.0) gets

Max DPS : 1369 using Power Treads(AGI) + Bloodthorne + Daedalus + Manta Style

and Anti-Mage Lvl 25 (BAT 1.65) get

Max DPS : 2391 using Power Treads(AGI) + BloodThorne + Daedalus + Manta Style
(Native mana burn not included here)

I.e. the same proposed item build as the other heroes. I think that since the search doesn't find attack speed items attractive, BAT will also be less influential for item choices for the same reason.

I also agree that the effective DPS is the most difficult ability to estimate, and thus the most inaccurate one. Part of it is the pseudo-random distribution used by Dota and part is some simplifications I have made. I made some tests with the Dota client and estimated DPS seems to be accurate to +/-10% at least. All other properties are spot on, or max 1% off.

(One curious thing I found reverse engineering the stats was that base HP is usually said to be 200, but it actually varies between 180 and 200. I wonder if it is a bug in Dota 2?)

I should also say that my main motivation for making the tool was not for making extreme builds, but rather the opposite: to find the cheapest way to realize a certain amount of DPS, EHP and Mana. In that way, it can be used to tune hero builds for different scenarios.

An example of a way I have used it: let's say you have a given standard hero build, but it is weak in some aspect. Maybe you need more EHP for magic damage when facing a team of nukers. I then input the standard build, make note of the stats, and then run a search asking it to max EHP(magic) given (acceptably) lower EHP and DPS, and some cost.

anti-mage has 1.45 BAT.

My point is that for example AM has far more dps with treads, manta, moonshard and bfury (which is realistic ingame situation, however usually you go for basher first before moonshard) than with your calculated items due to his attack steroid.

I get writing pointless apps due to boredom, it's nice and entertaining (I do it as well) but sorry, you need to consider all factors - such as passive skills procs, item procs, various buffs that different heroes have, enemy armour or other dmg income reductions etc.

Also I think you did some miscalculation about attack speed :-). May I see your math?

(My mistake on BAT, it really was 1.45 in data file for AM)

Here we go...epic theory crafting battle... I know BF is "respectable build" on AM, but there is no way you can max damage (against a single target) with Battle Fury! Please try it out in demo mode. DPS against dummy is 820, and it takes about 3 seconds to kill Level 25 Axe. With the max dps build above, Dummy DPS is 1300 and Axe dies instantly –- I can't even time it.

Again, I emphasize, it is an extreme build, and not something I "recommend" for playing. I included it as a funny example of the tool can do, of "academic interest" so to say. A little bit like the question of what is the max possible DPS in Dota 2.

Regarding the approach as a whole: there is no way to take everything into account exactly without recreating Dota 2 or access to the source code. The question is whether it is "pointless" to try something simpler. Before I tried, I was of your opinion, that it would be impossible to get anything useful due to the enormous search space and too many combinations. It turns out that you can get a lot of insight by studying the search results even in constrained scenarios. I would be surprised if the better Dota teams do not use approaches like this for build planning. They should.

When you say Bloodthorn, do you mean with the bloodthorn active permanently up on your enemy?
Because it's pretty unrealistic in that case. Bloodthorn active can be purged easily in a real game.

For attack speed, I used the info on the Attack Speed page in the Dota 2 wiki (link). The attack speeds I get correspond to what I see in the Dota HUD, so I believe the formulas are correct. It is important to keep in mind that IAS is capped at 500. That is one further reason why the theoretical approach does not favor attack speed items for maximizing DPS. IAS just doesn't stack as well as crits and dmg items do.

Yes, in the approach I have now, all items are considered "active", and sometimes it is difficult to translate their buffed mode into something representative (what's the dmg of Shiva's Guard?). It is unrealistic in some ways, realistic in others. You have to make some assumptions and interpret the results according to those. If you consider an item like Bloodthorn "off", because it can always be dispelled, then probably there would be no case for buying it in the first place. Sometimes I remove specific items from the search because they are not relevant. A typical example would be Divine Rapier.