This is part two of a four-part series on Elo and Elo Hell in League of Legends. View the Elo Hell category for the whole series.



True Elo

We know how the League of Legends Elo rating system works. The result of this system, ideally, is that matchmaking can collect two groups of five people whose average skill level is nearly equal – a 50-50 shot. However, this is based on the players’ current ratings, may or may not be accurate. A player’s True Elo is the rating he or she would reach if they played an infinite amount of games at that point in their career. If a player’s True Elo is higher than their current Elo rating, they will shift their team’s expected result higher than the 50-50 matchmaker expects. They will, on average, get favorable game matchups and be able to win, until they reach their True Elo and become a 50-50 contributor to the team (and vice versa).

Of course, your True Elo is fluid, changing with anything that affects your play: learning a new champion, falling out of practice, pulling an all-nighter or having a bad day at work. The bigger the gap between your current Elo and True Elo, the larger of an effect you will have, on average, on each game. If you deserve to be 2000 Elo but your account says 1200, you’re very likely to carry hard and run the table. However, if your True Elo is only 1300, your theoretical win percentage might shift from 50% to 52%, meaning you’ll have to play a lot of games to see a real change.

Basic Probability and Expected Value

Expected value is pretty straightforward – it’s the most likely average outcome of a probabilistic event. It’s calculated by multiplying all the possible outcomes by their likelihood, and adding the results together. For an example, let’s look at the expected number of heads flipped on a coin. There’s a 50-50 chance that the coin will land on heads versus tails, so our calculation looks like this:

eV(heads) = (0.5 * 1) + (0.5 * 0)

eV(heads) = 0.5

So the expected value is half a heads landing per flip. What this actually means is that if we flip a coin an infinite amount of times, we can expect that the coin will have landed heads half the time.

Variance



Luck is probability taken personally.

-Penn Jillette

We don’t have infinity to work with. While we know that the expected value of heads is 0.5, that does not mean that’s what always happens. You can flip a coin ten times right now, and it’s actually more likely that you won’t get five heads. Why? Variance. Variance is a mathematical term for the “gray area” in probability – the chance that the actual result of a limited set will differ from the expected value. While the expected value is what is most likely to happen, that doesn’t automatically mean it happens a majority of the time. The rest of the time, variance takes over.

This is why you hear the term “sample size” tossed around a lot – the more times you repeat something, the closer you’ll get to the expected value and the less variance will affect the result. That’s why playing a lot of games will level everything out. Of course, there’s no way to tell what the “right” number of games are – variance will mess with you just as much in every individual game – but the more games you play, the more sure you can be that you’re getting closer to where you should be.

Variance is the thing that keeps gamblers, athletes, and LoL players alike playing through the best and the worst – good luck can come at any time, and bad luck can go away. While there’s no such thing as probability karma, if you can’t tell if you’re losing because of variance or not, you can always cross your fingers and wait for the leaver to be on THEIR team.