I see pockets of mathematical analysis scattered throughout the Elo Hell QQ threads. Typically this will be one person informing the world that your bad teammates are just as likely to be bad opponents next game; that 4 chances for your team to have a leaver and 5 for the other team leads to your net benefit; that probability contradicts Elo Hell. All well and good in the abstract. But the League of Legends solo queue ranked playerbase is a complex system of some 100,000 units being measured individually according to their performance on a 5-man team. One calculation of expected values is not going to illuminate much of anything about such a system; it's far too simplistic. **In this thread I hope to spur an informed investigation into the mathematics of Elo Hell, to find out whether the numbers show a real problem, and whether we can mitigate that problem by tweaking the system and changing the numbers.**

Let's start by noting down some basic things about the Elo system in solo queue.

-Everyone starts at 1200 Elo. This is the original average of the playerbase.

-A match is made by finding a group of 10 people who are close in Elo, and arranging those people so that the match's outcome is 50-50, according to each person's Elo. After the match, winners gain Elo and losers drop in Elo according to how many recent matches they've played, as well as other factors that are presumably Riot-proprietary.

- Duo queueing combines and inflates the Elo of the duos when grouping players up.
- The selectivity of the matchmaker decreases as queue time lengthens.

-Any given match should not change the average Elo. There are exceptions--I'll note the ones I'm aware of:

- Queue-dodging reduces the Elo of the dodger, without adding to anyone else's Elo, hence a drop in the average.
- Inactivity leads to Elo reductions; as far as I know, it's 2 weeks for 50 Elo if you're above 1400. If that is the case, this should be relatively insignificant.
- Leavers may face additional Elo penalties. Whether this is compensated for in the other 9 players' Elo changes is unknown.
- Loss forgiveness leads to Elo inflation. Since this occurs only during server instability, its impact should not be large.
- Accounts with low Elo are probably more likely to be abandoned, which may lead to Elo inflation.

Now, to analyze Elo Hell, we need a definition.

**Elo Hell occurs when a player's Elo remains significantly below his actual level play over a number of games, typically because his teammates are worse than his opponents** (whether it be raging, leaving/AFKing, griefing, or simple poor play that causes this). I hesitate to attach exact numbers, but we can probably use 150 Elo and 50 games as first approximations.

The first obvious conclusion is that

**Elo Hell is a function of probabilistic outliers.** This is why a straight calculation of expected values does not make sense. Were the player's score to follow expected values, he would not be in Elo Hell in the first place. We need analysis of the possible

*deviation* from our expected values. What follows is a slightly better, though still extremely simplistic, calculation of

**how likely it is that someone who has fallen far below their 'true Elo' would remain there after 50 games.**
Assume that any given player may weight their team's chances to win at their Elo by one point--0 to 1. Let us further assume that any given game you play in is 'fair'--no leavers, everyone plays to their weighted score, and the greater number wins. The last assumption is uniform distribution between 0 and 1 for the playerbase at any Elo.

Let's look at a player whose Elo is already far below where it should be, trying to play his way back up the ladder. This player should have a solid influence on any game--say, 0.8. In chess, a 0.75 chance of winning is represented by a 200-Elo difference; not to say that this applies exactly to LoL, but that's to put the number 0.8 in perspective.

So your team's score varies from 0.8 to 4.8, and the other team's score varies from 0 to 5. This gives you a 56% chance of victory in any given match. In the long run, that's to your benefit; but how long does it take for the law of large numbers to come into play?

Typically, after about 50 games,

**about 24% of such players will go no better than 25-25**. That's a pretty significant percentage of players that will feel they're being kept down by their teammates. The consolation is it's hard to fall further: less than 2% of such players will go 20-30 or worse.

I don't pretend that this is a very realistic representation of the playerbase. For example, I by no means expect a uniform distribution of player skill between 0 and 1; more likely one would encounter a normal distribution with mean 0.5 and a standard deviation of 0.3 or so, and a spike at -3 for leavers. I simply want to demonstrate how the expected value is by no means the be-all and end-all of these probability calculations.

This is

*not* because MM is broken and Riot sucks. It's simply the nature of trying to measure individual ability in a 5v5 game with pseudorandom teams. There may be other factors that exacerbate the problems with the Elo system, that can be mitigated; Elo deflation is an example. But the mere existence of outliers is not something Riot can fix by waving a magic wand.

I welcome critique of my math thus far, but what I really hope for is that other people will contribute their own analysis. In the best case, a red post contribution with some insight into any thought Riot has put into this problem would be invaluable.

tl;dr: No, dammit. If you want to skip over the math, you shouldn't be reading a thread about math in the first place.

*(For those who are curious about my numbers: I calculated the probability of winning 25 games exactly (.44^25 * .56^25 * 50!/(25!25!)), then 24 games, then 23 games, and so on down to 15 games, at which point the cumulative probability was only increasing by a fraction of a percentage point over each iteration, so I stopped. The odds of winning less than 15 games out of 50 are infinitesimal. Yes, that was a laborious process. I'm glad Google works as a calculator.)*