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My wife was a champion table tennis player. This sport uses Elo as well, and I know from watching the sport over time that the rating system has real problems. It doesn't suffer from the weaknesses that you cite, but even so, the problem of "rating inflation" is widely discussed.

It seems that much of the problem comes from rating points brought in by newbie players (and note that, contra TFA, the problem isn't with experienced players losing to newbies, but the opposite).

A newbie is started off with some nominal rating; I forget the number, but let's say it's 800. Most likely that newbie is going to lose his first matches, and some proportion of those newbies will get frustrated and quit. For the ones that stay in the game, things probably work out in the long run. But for those that got discouraged and quit, in the course of their loss they caused a few points (not many, because they're likely way overmatched, but definitely more than 0) to be credited to their opponents. When they quit the sport, they're never going to reclaim any of the rating points that they lost initially. But those points are still in the system, having been added to their winning opponents.

It's hard to quantify because the Elo system is the only objective comparison we have, but over the course of the almost 30 years I've been watching my wife play, the Elo rating enjoyed by a player of a given hypothetical skill level has increased dramatically. Many are saying that for someone of the upper echelons, their rating is maybe 200 points higher than it would have been 30 years ago.

So back in 1991, my wife was in the top 30 women in the USA with a rating in the mid-1700s. Today, someone with that rating isn't even going to be in the top brackets of serious tournament.

Despite all that, the usefulness of the rating system keeps it in use as a valuable tool. It seems that the ability to match players who have never seen each other before, ensuring interesting matches, is part of keeping the game competitive for those in it. And table tennis is also, because of this, one of what I believe is few sports where men and women often play head-to-head (even though men generally have much higher ratings, on account of the sport requiring far more strength than you might suspect).

> It doesn't suffer from the weaknesses that you cite, but even so, the problem of "rating inflation" is widely discussed.

Ah yes! Inflation is also a problem I've seen in competitive online games. Rating inflation was a serious issue with World of Warcraft PvP arenas circa 10 years ago (iirc Blizzard hard capped arena ratings at 3000 during WotLK). I don't follow chess much, and I'm not exactly sure how chess avoids it (or even if it does).

By the point you're playing ranked matches in chess, you're generally invested enough to keep playing. However, chess has a (statistically) significant inflation problem, to the point where you can only compare scores within the same decade or so meaningfully.

It seems there was a lot of rating inflation in chess, but at the top level, at least, it's stopped - the number of players over 2700 has been pretty constant for 5-10 years, a few dozen players. In 1990, only Kasparov and Karpov were rated over 2700.

https://2700chess.com/

https://en.wikipedia.org/wiki/1990_in_chess

I don't think there's an expectation that a skill rating is comparable throughout 20 years, because both individual players and how the game is played (the meta) changes continuously.

But if that's true, then why would rating inflation be a problem?

The game itself has not changed, so it still makes sense to compare players across time. It would be nice if we had a quantitative way of doing this; so we can make statements like 'the average proffessional player today is better than 20 years ago, a typical modern pro would win 60% of the time again one from 20 years ago).

In some sense, it is not surprising that we do not have a system that accomplishes this. Since it is impossible to see the results of a game between players living in different time periods, we cannot get any data to prevent drift. You can still try to normalize the rankings. However, unless you have some independent way of measuring skill, you would need to make an assumption about the relative strength of players. Assuming the average skill of a proffesional is constant across time is probably not accurate, but closer to reality than what you get with unchecked inflation.

the game itself has changed quite a bit, and the number of people playing it, and the dominance that they achieve has also gone up quite a bit.

You can sort of solve the inflation problem by zscoring the elo. Now a person's score will tell you how much better or worse they are than the median player, assuming an underlying normal distribution (reasonable).

Of course, scores will only be comparable if the average skill of all players remain constant. I would imagine this isn't true, but the drift over several decades is probably small.

Unless you start introducing some purely objective criteria for skill, which can never work, this is the best you can do. It's still way way better than a straight elo system though.

Rating distributions are often not normal because some subset of players study the game and take it more seriously resulting in a bimodal distribution. See [0] for an example in Chess.

[0] https://chess.stackexchange.com/questions/2550/whats-the-ave...

That's fascinating, thanks! It looks like you can model it as a mixture distribution made up of two underlying normal distributions.

Even without the bimodality, you wouldn't expect a normal distribution of ratings.

1. Assume that chess ability is normally distributed in the population.

2. Assume that people who are terrible at chess are more likely to stop playing chess than people who are successful.

Then you've sampled the underlying normal distribution mostly from the top end, and that new, highly skewed distribution is what you'll see when you measure everyone's rating.

The idea that chess has not changed in a long time is simply not true. Two huge and relatively recent changes were the addition of chess clocks and premoves.

And aside from the mechanics of how the game is played, there have been massive changes in the popularity of chess (first massively upwards, recently possibly down slightly), as well as how analyses are done.

It would be very difficult to account for these factors in a way that keeps comparisons across 30-year+ time spans meaningful.

Why don't they increase the bar for newbies to get into such a system?

If they know that some people just play a few games and then quit, let's say they only can get Elo when they played a specific amount of time or won at least n games etc.

There is a minimum of 10 games before people start being ranked. People who quit early don't get ranked. People who have played 10 games gain a new long-term goal.

This might not be great for a sporty-sport, but I think that for a video game this would actually be an advantage. This kind of a rating inflation would mean that long-term players would see some numerical progress without really doing much better.

It would also inflate the ratings of people new to the game later in time.

A newbie is started off with some nominal rating; I forget the number, but let's say it's 800. Most likely that newbie is going to lose his first matches, and some proportion of those newbies will get frustrated and quit

That seems like a simple problem to fix. When somebody quits, just subtract 800 points from the remaining ranked players, scaled accordingly such that their relative win probabilities remain the same.

Of course, the other issue is if the number of active players increases over time. In that case, it's not so easy to fix unless you start scaling down the number of starting points given to new players.

Perhaps a better thing to do would be to construct a model of the rating inflation over time and use that to correct for historical comparisons. It's still not particularly meaningful though, because you have no way to measure actual skill inflation.

You don't have to formally quit the game to stop playing. I played one ranked chess tournament in high school, quit for ten years, and then picked it back up. What would you do with my points?

If you choose to delete them, that means that everyone will have constantly eroding ratings unless they keep playing.

Your points could be added back in when you resume playing. There’s no reason to throw the data away.

There's also an inherent deflation effect. Players tend to get better over time. In the simplest case, if we start with a pool of players rated 800 and let them play for a year, at the end they'll be better players but still rated 800 on average.

Most chess Elo systems have an inflationary component where young or new players (who are overall faster improvers than the player pool at large) gain and lose points faster than established players (in detail, either using performance ratings or increased k-factors or both). In a balanced rating system, the sources of inflation and deflation are roughly equal. You can tweak the parameters to keep it this way, though it's not trivial to tell whether there is "real" inflation over the years or whether players are simply playing better - or indeed, what's the difference.

all of this, plus an additional observation that i've had about games w/ tiers/divisions: player skill is assumed to be normally distributed when that is just so demonstrably not the case-- there is a fairly high skill floor to be able to play the game at all, and the right tail (high skill) of the distribution is WAY fatter than the left.

especially with well-established, popular games-- Chess, League of Legends, Overwatch, etc. (where there is even a financial interest in being a top player to boot), the skill levels of the people at the absolute top simply profoundly dwarf players that would even superficially seem "comparable" by the standard of being in adjacent, or even within the same tier.

in League of Legends, for example, it is often claimed that the differences between players in high/low Challenger, high/low Master's, and even high/low "high diamond" (low d2 vs high d1) all constitute distinct "tiers" of player quality that are as substantial as the full-tier jumps closer to the median (e.g. silver -> gold, gold -> platinum), but because of this shoehorned prior about skill distribution it leads to this compression at the very top.

> player skill is assumed to be normally distributed when that is just so demonstrably not the case

This is close but not exactly right, and the small difference matters. Elo does not assume that skill is normally distributed, but rather that "quality of play" in a single game is normally distributed around some average quality level for the player. Obviously this too is an approximation but it's a much smaller one.

hmm, interesting. i did mean to say that this is a problem more in the context of games that add tiers/divisions to their ranked ladders, but i hadn't really thought about elo making assumption about the normal-distributed-ness of player deviation from their "true" skill level. does that not just fall out directly from the Central limit theorem (given the taking of large #s samples (game W/L observations vs. predicted P(win|my elo, their elo)) of means, etc.)?

“player skill is assumed to be normally distributed”

I would think player skill level (at best; there easily can be cases where P typically beats Q, Q beats R and R beats P) is an ordinal (https://en.wikipedia.org/wiki/Ordinal_data), so one can’t say “player P is twice as good as player Q”, “player P is as much better than player Q as player R is better than player S”, and certainly can’t prove or disprove whether skill is being normally distributed. It is customary to assume that, though.

Also, if one assigns numbers to skill levels, those can be normally distributed. It probably is possible to design an ELO-like system that, given enough games, guarantees that the set of skill level numbers of all players approaches being normally distributed.

Another thing to consider with a lot of these games is that they're not static. The game changes and this can boost one player's rating up when their preferred champions/heroes/whatever are strong at that time. Even if the game didn't change, there are so many different characters that's play differently enough that the player's results with them could end up at a rather different rating.

Here's season 13 of rocket league. Free red delicious apple for the first person to correctly identify the shape of the curve:

https://2p1ipt36o1g23g1tt6ba5nou-wpengine.netdna-ssl.com/wp-...

looks gamma, but just like... slightly gamma. i guess my contention would be that it "probably" should be smushed/redistributed with even more mass on the right tail, but i couldn't tell you from personal experience whether that's true, as i'm pretty terrible at rocket league. i will say that the top Rocket League players (to my untrained eye, and jaw on the absolute floor) may have an even higher z-score than top players than any other game(s)... but rocket league is kind of unique in it's being a remake of a game that i guess a lot of (the same) people used to play.

The X axis appears to be an ordinal number, some kind of proprietary rank. How much sense does it make to talk about the shape of a distribution over ordinal numbers? If we converted those to cardinals, the shape of the new, reality-based distribution could be pretty much anything.

To expand on this point, I recently made histograms of gaokao scores for several Chinese provinces. You can see Beijing data here: http://www.gaokao.com/e/20180623/5b2e13c43951c.shtml

This report is nicer than many others I found in that scores are reported all the way down to 0, rather than cutting off at the threshold for university admission. It's much less nice than some in that, below the university admission threshold, scores are reported in brackets of 10 points rather than by individual score. (This, even though the document is called "一分一段表"...)

Anyway, I imported this data into python and plotted it with matplotlib. The histogram you get from this is obviously, wildly flawed -- the ten-points-wide bar from 410 to 419 is also 710 people tall, dwarfing the actual mode of the distribution. To correct this problem, you need to divide the count for bracketed scores by 10 (the width of the bracket) -- the 710 people scoring 410-419 are 71 per score in that range, very comparable to the 70 people scoring 420, but not to the 182 people scoring 548.

Without knowing the width of the rocket league rank brackets, that picture of the population of each rank doesn't tell us anything -- at all -- about the shape of the distribution.

Maybe someone who plays the game could clear it up. I did some searching, and it appears as though the ordinal ranks on the x axis are just buckets of players in ranges of 25 points in each bucket. The underlying rating system for these rating points is apparently something like Elo or Glicko, but I couldn't find a source that explicitly says what.

> Most games aren't chess -- where the only variance is picking who's black and who's white -- in fact, they might include dozens of RNG mechanics (from critical strikes to ability rolls, to spawn points). These mechanics (while fun and well-designed) might pollute your "idealized" model. There's also the problem of RPS (rock-paper-scissors) mechanics or pick-counter-pick mechanics which will also heavily skew win rates. For instance, given a slow combo Magic deck, you will most likely auto-concede to mono red aggro (regardless of skill level). If you're using Elo, this will pollute your model. (Hint: you shouldn't be using Elo.)

None of which matters? All that means is that the results of individual games are a bit higher variance. Elo handles that by design. If you lose a certain proportion of Magic games to less-skilled players then this should be considered a reflection of your skill, because the only reasonable definition of skill at the came is the rate at which you actually win it; anything else can be gamed and so should be ignored.

> Most games also don't have chess' high skill ceiling. Chess has such a high skill ceiling for a number of reasons -- it's one of the oldest games still being actively played, for one. Suppose your "game" is simply the flip of a coin (everyone wins 50% of the time). Zero skill involved. Trying to model win-loss-ratios using a sigmoid curve is silly. Obviously, no game is going to be a coin flip, but there's a world of difference between chess and DOTA.

That's also something that Elo handles just fine? If every game is a coin flip then everyone will end up with the same Elo. If player A has x more Elo points than player B, then they win y% of their games. If your game has a skill ceiling where even a complete beginner always wins, say, 20% of their games, then that just means no-one will ever be able to rise above a corresponding Elo rating.

> That's also something that Elo handles just fine? If every game is a coin flip then everyone will end up with the same Elo. If player A has x more Elo points than player B, then they win y% of their games. If your game has a skill ceiling where even a complete beginner always wins, say, 20% of their games, then that just means no-one will ever be able to rise above a corresponding Elo rating.

That's not how it works. The distribution you end up with will not be uniform, it will look like this (just ran Elo with a coinflip; 11 players, 1000 matches): https://imgur.com/9O82pRj

On the long term, I think this will tend to a geometric distribution with a low p value.

Show your working?

If you're matchmaking players against equal-ranked players, then each match is just +/- 50 points, you'll get a binomial distribution which tends to normal as n gets large (assuming a large player pool so each player's results are independent). If players play players with different ratings then that will tend to push their rating back towards neutral. You certainly don't get a geometric distribution because the rating algorithm is completely symmetric.

> each match is just +/- 50 points

This only happens in the rare cases where you're matching players against (exactly) equally-ranked players. You can mitigate this by always trying to match as "close as possible," but it's only a mitigation. Try simulating random matchmaking with Elo, and you'll get something like this: https://i.imgur.com/1Y08jUB.png (1000 players, 100,000 games). In my simulation, I set k (the Elo constant) = 50.

I think it's going to tend to a geometric distribution for reasons discussed here (which is another interesting and non-intuitive result): http://www.decisionsciencenews.com/2017/06/19/counterintuiti...

Elo can be negative, so this doesn’t apply.

> Try simulating random matchmaking with Elo

I will. I was hoping you'd post the actual simulation details rather than more unlabelled graphs.

https://gist.github.com/dvx/e3311a984e14dca7c5eeb9214e0be3d5

Using: https://github.com/HankSheehan/EloPy (with a custom k-value of 50).

> with a custom k-value of 50

So you've patched this library somehow? Because when I run your code I get a result that's just full of 0 ratings.

But in any case I'm not at all convinced that your charts don't just show the normal distribution that we'd expect, just in some weird way. (Did you test your plotting methodology against some simpler rating system before using it to draw conclusions about Elo?). Plot a normal histogram, or a density plot if you're feeling fancy: https://towardsdatascience.com/histograms-and-density-plots-... . I'm betting the result is just the bell curve that we'd want and expect.

> So you've patched this library somehow?

Yes, as mentioned, I set the k-value to 50 on this line: https://github.com/HankSheehan/EloPy/blob/master/elopy.py#L8...

Author decided to do something fancy which will only work when number of players is less than 1/2 * starting Elo rating.

> But in any case I'm not at all convinced that your charts don't just show the normal distribution that we'd expect, just in some weird way.

As mentioned, you end up with a geometric distribution. I covered a similar phenomenon in a blog post I wrote last year[1]. See Theorem 3.3 in this paper: https://kconrad.math.uconn.edu/blurbs/analysis/entropypost.p... But in short, the geometric distribution has maximal entropy over (0,∞) given a known mean (in our case, the mean will always be 1000).

[1] https://dvt.name/2017/07/10/confusing-math-with-morality/

> As mentioned, you end up with a geometric distribution. I covered a similar phenomenon in a blog post I wrote last year[1]. See Theorem 3.3 in this paper: https://kconrad.math.uconn.edu/blurbs/analysis/entropypost.p.... But in short, the geometric distribution has maximal entropy over (0,∞) given a known mean (in our case, the mean will always be 1000).

Another reply already told you that's irrelevant to Elo, because Elo can go negative (and if it couldn't then the mean wouldn't always be 1000). It's probably going to be normal, and drawing an actual histogram of a simulation like yours comes out looking pretty much like a bell curve: https://imgur.com/YBDp4uI .

As far as I can see none of your claims about Elo stand up. Why do you think you've shown the things that you're claiming?

>Why so many game devs are enamored with Elo when it comes to ranking is a bit bizarre.

Probably just Occam's Razor, they don't know of or care to make something better and can just pull Elo off the shelf.

Another example would be competitive overwatch where the developer's stated goal was an equal distribution of rated players throughout the various ranks (bronze/silver/gold/platinum/diamond/masters/grandmasters). They tweaked the variables until they got their desired distribution, lumping the majority of players in gold and plat. Ranking up became an exercise in either playing hundreds of hours or starting a brand new account with fresh MMR.

Predictably this led to an explosion in boosting and win-trading services.

> the developer's stated goal was an equal distribution of rated players throughout the various ranks

and

> They tweaked the variables until they got their desired distribution, lumping the majority of players in gold and plat

seem to be opposites. What am I missing here?

It also probably doesn't even matter since the main problem is players intentionally losing to rank down and then smash lower rank players.

I also suspect that Chess is exponential because of the "one mistake and you die" nature when playing good players.

Ben Finegold (a Chess Grandmaster) talks about this all the time--"The reason why I'm higher rated than you is that I can play 100 moves without a major mistake and at some point you will hang a piece. The reason why Magnus Carlsson is rated higher than me is that he will play 100 moves that are slightly better than mine and I will lose."

also, what system(s) do you prefer / know of that handle multiplayer matchmaking well? it seems to me that a good system might be necessarily game-specific to some extent, although i'm sure the state of the art is much better than what i've experienced gaming to date xD.

> also, what system(s) do you prefer / know of that handle multiplayer matchmaking well?

None, and actually I don't think it's particularly healthy for the game. For example, I had plenty of fun casually pubbing Counter Strike in the early 2000s. When I wanted to take the game more seriously, I made a team and joined a league which might include group play, single/double elimination, and exhibition games. Actual competitive play (scrims, matches, tournaments) is fundamentally different than what today we call "matchmaking."

Do you believe multiplayer games would be better off without matchmaking at all?

Yeah, 100%.

I'm actually flabbergasted why you can't make a team in games like CSGO or Overwatch. And then play in tournaments or matches (against, you know, other teams). It makes no sense to have individual matchmaking in a team game. Game devs create this individual matchmaking system (which is paradoxically taken seriously by casual players, but totally ignored by actual competitive players), and the community and other organizers (enter FaceIT, ESEA, etc.) have to actually set up leagues, tournaments, and events.

this would probably work better if, in the case of Overwatch, the teams weren't six players (i personally have always felt like the game would be better @ like 4v4 anyway, because of how god damn frustrating dealing with 5 random players on your team every game is in a game that is balanced purely around teamwork and inability to solo carry w/o being much more skilled than everyone else in the game)

I see your standpoint. I don't see it happening from a practical / financial perspective though. Being required to have the right number of same skilled friends ready is quite a high entry bar to playing a game.

In my experience the reason behind devs loving matchmaking is fairly straightforward: being able to solo queue raises engagement. It takes time and effort to make a team in the first place, more time and effort to coordinate games when you're now schedule wrangling n other people, and that extra effort is magnified across all the teams participating. In contrast, hopping into soloqueue is so brainless that the hours spent playing soloqueue end up dwarfing the hours spent playing as teams. Will some people who care enough still play team mode? Sure, but if solo matchmaking is an option it becomes the default simply through being the most-played mode. At the end of the day, devs seem rationally interested in juicing engagement numbers for the vast majority of the playerbase and letting those serious enough to care about not pugging figure it out for themselves.

I think there's a pretty limited space for games that don't compromise on various aspects of design (matchmaking, mtx, etc) with the explicit goal of making a better top-end competitive ecosystem. I'd personally love to see a competitive team-based game without any form of solo queue, but I'm skeptical it would do well in the market. It's almost like Facebook engagement-doomscrolling vs. a mailing list: the format of the latter means there'll probably be better content, but a whole lot more people are going to be hanging out on the former. At least mailing lists don't have to recoup development costs.

> being able to solo queue raises engagement.

This is (unfortunately) probably the answer.

> I'd personally love to see a competitive team-based game without any form of solo queue

I'm okay with solo queue, as long as I can also have a team queue where I could play in traditional seasons or tournaments with a team of friends. It just seems odd that one needs to go outside of the game itself (to ESEA or what-have-you) for this feature.

You can do this in virtually all games. But requiring you to do so, and failing to offer skill based matchmaking is not what most players want.

> You can do this in virtually all games

No you can't. Overwatch, CSGO, etc, etc. don't have a way to make a team and queue as a team (against other teams). You do this by playing on FaceIT, ESEA, CEVO, or in other leagues. Built-in matchmaking is only individual. This is, from a competitive standpoint, a meaningless data point and (from a casual standpoint) only creates toxicity.

I am curious what you mean by matchmaking is only individual, it is common to party up and queue as a 5 stack, both in csgo and valorant. now when you have a bunch of solo qs playing against a 5 stack, the actual team is going to win 9/10 times...

I do miss the old days of CS with "clans" where it wasnt so hard to join up and have a lose group of people you played with regularly and got to know ~20-30 people and whoever was on would join up to play together (maybe this still exists, but I havent found it..)

> I am curious what you mean by matchmaking is only individual, it is common to party up and queue as a 5 stack, both in csgo and valorant. now when you have a bunch of solo qs playing against a 5 stack, the actual team is going to win 9/10 times...

That's exactly the problem. The MMR system isn't based off of team ratings, but off of players. Otherwise, teams (e.g. 5 players) would always play against other teams (another 5 players). Now, even ignoring the model problems this generates (and the gymnastics that something like TruSkill does to mitigate it), it's just a bad experience.

For example, if I go to the beach and join some random volleyball pick-up-game, I'm expecting that the purpose of the game is to "have fun." If I'm joining a team to play in a rec league, the expectation is to try and win. The idea of "matchmaking" mixes these two concepts, so you end up having different people with different expectations. Some are going to say "why are you trying so hard" while others will retort "why aren't you trying harder?" This misalignment of expectation is, imo, the chief cause of toxicity in (competitive) video games these days.

Overwatch, CSGO and virtually all other team based FPS allow you to queue solo, as group, or a full 5 person team. This is outside of a specific league. There are dedicated LFG sites for different games to help find groups ahead of times. Generally you will be matched against a similar team, and different games use some form of skill based matchmaking, but depending on how many players there are, what modes, what region you are in, as a solo player you could be matched against a premade or vice versa.

>No you can't. Overwatch, CSGO, etc, etc. don't have a way to make a team and queue as a team (against other teams).

I don't think this is true. I play Overwatch and I sometimes play with anywhere between 1 and 5 other players as we have arranged to group up before looking for a game. With 5 other players, it's a 6-stack, and I believe that a 6-stack will always be matched against another 6-stack. As far as I know, it takes the average skill rating of your group and finds another group with a similar average skill rating to play against you.

> It makes no sense to have individual matchmaking in a team game.

I kinda feel broadsided by your rather extreme views here. Later on in this thread you say, okay, solo-queue is fine but you need a way to make teams and join tournaments, so it's also not really clear what you think.

Single queue exists because team games are still fun in pick-up groups. Go to any basketball court and you're going to find guys playing pick-up groups of basketball. I don't hold a 5+ basketball team in my pocket, and that's okay. Because playing with strangers in a team-game is still fun. And sometimes even more fun because you're meeting new people and playing with new team dynamics -- solving new human team dynamics on the fly is an underrated fun part of team games. Single queue matching exists because rank gives people a stake in the game and they take it seriously, and it makes the ranking system accessible, and it's fun.

A game that only offers tournaments and requires you to come with a pre-built 5-man team is just a game that excludes most people. The people forming teams for tournaments is the 1% of the gaming population.

I want to come home from work and play a couple CS:GO games with others who will take the game seriously. I don't have time for a tournament. I don't have a team. I don't want to join a no-stakes casual game where people are putting the controller down to answer the front door or just disconnecting. Without ranked-solo queue, what system do you propose for this common use-case?

> Single queue exists because team games are still fun in pick-up groups.

Single queue is fine, I just don't think the "ranked" aspect of it is healthy for the game.

> I don't want to join a no-stakes casual game where people are putting the controller down to answer the front door or just disconnecting.

Maybe not disconnecting, but trolling and just generally being a pain actually ends up being what happens all the time even at high solo queue tiers (last year I had two accounts at Global Elite). ESEA and FaceIT have much more robust pugging systems put in place so that's why people take it more seriously.

But my point is that even though I'm a very competent Global Elite player, my Counter Strike heydays are behind me and if I were to seriously play against even a semi-pro ESEA-Main (or probably even Intermediate) team, I'd get absolutely destroyed. So solo MMR is a pointless metric to have, and just adds toxicity to your game.

yeah, that strikes me as a pretty fair proscription, unfortunately-- the skill gap from coordinated team play in any team game makes it to where teams that play often together are matched against ad-hoc teams of individually more skilled players to make things "balanced", which were it even be possible to do this in the "50% win probability for each team" sense still leads mostly to unfun matches one way or the other. and, of course, queueing with friends you don't play with often, or with high skill variation amongst them just completely screws you from a balance/rank perspective (but hey, at least you get to lose together with all your friends! :).

In your magic example you seem to be arguing that which kind of deck you pick is not part of your skill, which is of course totally incorrect. Picking "fun" decks over "obvious/OP" decks means you're worse at winning games. Or at least that you generally play with a handicap, which is easy to account for in elo.

To your coin-flip example, if you model a league in excel you'll find that elo actually results in a rank distribution very consistent with what your intuition would expect (given enough players and enough matches, of course).

> To your coin-flip example, if you model a league in excel you'll find that elo actually results in a rank distribution very consistent with what your intuition would expect (given enough players and enough matches, of course).

This is incorrect. If you simulate Elo with a coin-flip, you'll get something that looks like this (11 players, 1000 matches): https://imgur.com/9O82pRj -- I think this will tend to a geometric distribution (not sure what the p is though, probably depends on the constants).

>> given enough players and enough matches, of course

Here's 1000 players with 100,000 matches: https://i.imgur.com/1Y08jUB.png

Feel free to try simulating it yourself, but even mathematically it makes no sense to end up with a uniform distribution as we tend to infinity.

Elo, not ELO, after Árpád Élő.

Correct, fixed :)