League of Legends: Damage Mitigation

A couple days ago I had what the kids these days would call an 'epic game' of League of Legends. The opposing team picked heroes that did pretty much exclusively magical damage. Someone on my team noticed this and mentioned that we should all just buy some magic resist. 25 minutes into the game 2 people hadn't bought any at all and we were getting obliterated. I think the hero kills at one point were 6-22. So the other guy and I went back to harping over and over about magic resist. 10 minutes later we had a flawless ace against them since 4 of us had enough magic resist that it felt like we were practically unkillable. We stormed into their base while they were dead and won the game despite still being down at least 10 kills. It seemed like stacking on more and more resists was really doing something and I want to take a look at what it actually does and if it really does make sense to stack on as much as we did. (In some games armor just gets better the more you have. In others the marginal gains are worse the more you have.)

The formula (for both armor and magic resistance) is the same and very simple. The amount of damage you take is multiplied by 100/(100+MR). Simple enough. But what impact does that actually have on how much damage it takes to kill you? Assume you have X health and Y MR. Then the amount of damage it takes to kill you (alternatively, your effective health, a stat which WoW players put way too much emphasis on but which works here due to the lack of full resists/dodges.) is:

Z=X*(100+Y)/100

Y' = dZ/dY = X/100

X' = dZ/dX = 1+Y/100

The important things to note here are that Y' doesn't vary with Y and X' doesn't vary with X. How much health you have doesn't matter when you're looking at how good adding more health is. How much magic resistance you have doesn't matter when you're looking at how good adding more magic resistance is. There is no inherent reason who you should stop at any given level of magic resistance, or why you should keep piling it on. They do, however, vary with each other so it is entirely possible that the most efficient stat in terms of gold and inventory space will change as you buy more items. 

The best magic resist item in the game adds 76 MR for 2610 gold. (It also gives 8% movement speed and a hefty amount of health regen in the form of 40hp5+.35% of your health every second.)

The best health item in the game adds 1370 health for 3000 gold, but it takes time to charge up. (It also gives 45hp5.)

Ignore the other bonuses (though the MR item just has strictly better ones) and for the same price you can get 76MR or 1192 health. What's the equilibrium point for those two gains? 

dY*X/100=dX*(1+Y/100)

76*X/100=1192*(1+Y/100)

Y=100/1192*(76*X/100-1192)

Y=.06376X-100

X = 15.68(Y+100)

Lets look at some typical values of Y. With nothing at all you probably have around 30 MR. X = 2039. If you have more than that much health you should get MR. If you have less then you should get health. 

With just the Mercury Treads boots you have 55 MR. X = 2431. 

With a lot of MR I typically get up around 200. X = 4704. 

I must admit I'm a little surprised here. All signs are pointing to just getting more health first. Especially when you consider that most teams will have physical damage dealers too, and MR does nothing against them while more health does plenty. I think I'm going to try getting a basic tanking item and then rushing the super health item and see how it works out in my next game.