Reign of Gaming

Yeah, some of these numbers don't seem right, can you provide more details? (What runes and Masteries did you assume, and did you use an Average value for Base AD and Base AS? Or did you use a specific champion?)

I noticed that you're using the %HP damage as part of the Lifesteal, which is an On-Hit Effect. The lifesteal portion(50% of Damage dealt) was removed with the re-work, and as you should be well aware, you don't get lifesteal from on-hit effects, even if they are physical damage.

It's deceiving in the sense that players could be fooled into thinking that a "more cost efficient" item could provide more damage than a less "cost efficient" item, for example.

However, not every stat scales linearly with Damage, so even though an item may appear to be 'more efficient', even at the same cost, it may not necessarily provide more damage.

In that sense, Gold efficiency is a bad approach to maximize your 'bang' for you buck.

(ADC, for example, scale with 5 different stats: Attack Damage, Attack Speed, Crit Chance, Crit Damage and Armor Penetration. Each stat scales differently depending on the current stats of the champion building it, so it's not always 'efficient' to purchase the most cost efficient item right away, simply because you can obtain more damage by building onto another stat instead. Most common example is the PD vs BT rush.)

I've been using "Damage efficiency" instead for quite a while;

I use it for Ranged AD carries, but It can be applied to the other damage types, too.

It still uses the whole "400 Gold for 10 AD, so 1 AD = 40 Gold, or 1 AD/gold", but instead of it being a linear approach, I use that base value as the "Attack Damage Gold Differential"

And you do the same thing to Crit Chance, Attack Speed, and Armor Penetration to get their respective "X Gold Differential"

Once you get those, you just get the complete differential for DPS on an ADC:

Damage = AD*AS*(1+CritChance*CritDMG)*(Armor*(1-PercentPen) - Flat Pen)

Variables here, you use AD, AS, CritChance, Flat Penetration, and Percent Penetration. (Keep CritDMG and Armor Values as Constant as per individual analysis)

Then complete Partial Derivative, and you get an Expression for d[Damage]/d[gold] , and this value can then be used as a "Damage to Gold Efficiency", which I find as a MUCH better measure.

Of course, there's no simple expression the average user can use, so I guess that's why it's less practical.

Not really. The moment you see someone build it, instead of building a Warmogs, build a Randuins;

You get an Attack Speed Debuff, Physical Damage Reduction, and Health.

Not only that, but Randuins ALSO gives you more Physical Effective Health than a Warmogs.

The statement that "X armor gives X% more Effective HP" is incorrect. The formula, simplified, is as follows:

Effective Health = Health + (Armor/100)*Health, or Effective Health = Health + Bonus Health, where Bonus Health is the "bonus health" that is attributed to the gains in armor. Now, taking Penetration values into account, you can revise this same formula to give you:

Modified Effective Health = Health + (Armor-FlatPenetration)/100)*Health. This can be reduced to :

Modified Effective Health = Health + BonusHealth - (FlatPenetration/100)*Health.

As you can see, 10 Flat Penetration would lower Effective Health by EXACTLY 10% of your Health, not 10% of your Effective Health. 

In the same way as adding 10 Armor, 10 armor would ADD 10% of your Health to your Effective Health, not increase your Effective Health by 10%.