Reconsider Arena Rankings System

[1000esqrtn1]

Hi, you made a lot of valid points, although they are based on a perfect ELO system for skill based games. I agree that it will always be challenging to rate games with luck involved but I think we can find better solutions that the ones we currently have.

Assuming the players are rated properly, there's no reason not to play those games. The reason you get so little vs them is because you're expected to beat them by that much. You should only expect to have your rating go up if you think you are underrated, (or I suppose if you happen to think your opponent is overrated). But the ideal outcome for players is no change on average, not without a change in their play level. Now, as rating gaps get further away, I think the rating's predictions get less accurate, and frankly I find the idea of players regularly playing rated games with such large gaps distasteful as a result. The higher rated players are either getting unfairly nabbed in the greater variance games or they're boosting in the lesser variance games. The main issue here is the lack of depth in the player pool for that game on BGA.  

If there's luck that's not true. ELO system will give you win probability. Since some portion of the games will be decided by luck without skill involved, your win probability is lower on reality.

As an example
Player A ELO Win Prob = 80 %
Games decided by too much luck = 20% (10 % wins, 10 % loses)
Real Winning probability = 10% + 80%*80% = 74%

As your real win probability is lower than the estimated one, on average you will lose points by playing lower players. All your points would hold if we could adapt the ELO formula to consider this luck parameter for every game, however I think this is an almost impossible task to do properly.

As this is almost impossible, that's why I considered taking win % into account, as someone who consistently wins games shows that is winning most of non-luck based games. I agree that this could be exploited by trying to play lower opponents (eventhough you cannot choose opponents on arena) and has to be limited somehow. That's why I always said a formula combining all points, never switching player rating for win %.

A possibility could be

Pts = 5 * win % against 300s ELO + 3 * win % against 200s ELO + 1 * win % against 100s ELO ------- minimum games needed per category

[Ceaseless]

Hi, you made a lot of valid points, although they are based on a perfect ELO system for skill based games. I agree that it will always be challenging to rate games with luck involved but I think we can find better solutions that the ones we currently have.

Assuming the players are rated properly, there's no reason not to play those games. The reason you get so little vs them is because you're expected to beat them by that much. You should only expect to have your rating go up if you think you are underrated, (or I suppose if you happen to think your opponent is overrated). But the ideal outcome for players is no change on average, not without a change in their play level. Now, as rating gaps get further away, I think the rating's predictions get less accurate, and frankly I find the idea of players regularly playing rated games with such large gaps distasteful as a result. The higher rated players are either getting unfairly nabbed in the greater variance games or they're boosting in the lesser variance games. The main issue here is the lack of depth in the player pool for that game on BGA.  

If there's luck that's not true. ELO system will give you win probability. Since some portion of the games will be decided by luck without skill involved, your win probability is lower on reality.

As an example
Player A ELO Win Prob = 80 %
Games decided by too much luck = 20% (10 % wins, 10 % loses)
Real Winning probability = 10% + 80%*80% = 74%

As your real win probability is lower than the estimated one, on average you will lose points by playing lower players. All your points would hold if we could adapt the ELO formula to consider this luck parameter for every game, however I think this is an almost impossible task to do properly.

As this is almost impossible, that's why I considered taking win % into account, as someone who consistently wins games shows that is winning most of non-luck based games. I agree that this could be exploited by trying to play lower opponents (eventhough you cannot choose opponents on arena) and has to be limited somehow. That's why I always said a formula combining all points, never switching player rating for win %.

A possibility could be

Pts = 5 * win % against 300s ELO + 3 * win % against 200s ELO + 1 * win % against 100s ELO ------- minimum games needed per category

Ok, before I comment on my thoughts regarding this suggestion, first I want to make sure I understand your suggestion and where BGA is. So I'll first go over what I think you're saying, showing my work, assumptions, and references along the way. That way if I am off, you can see which of my assumptions, references or math went wrong. Can't say I've messed with elo calculations lately, please let me know if I've completely messed something up. (Edit: I also have a short post with an alternate idea, because these other ones were bugging me for lack of player pairing effect.)

https://boardgamearena.com/faq?anchor=faq_account_elo K x (W - p(D))
We're looking at wins for W = 1, lots of games played for K = 20, D's difference between ratings is calculatable, and p(D) = 100% / [1 + 10^(−D/400)] based on https://en.wikipedia.org/wiki/Elo_ratin ... ite_note-3 if my referencing Wikipedia like the BGA note suggests is accurate.

So if I'm a 400 vs a 200 opponent, my D =200, with p(200) being roughly 76%. 20(1-.76) = 4.8, that's how many points I'd gain for winning. If we have W=0 for defeat, it's 20(0-.76)= -15.2 points for defeat.

With your method, me as a 400 vs a 200 opponent, I get the following.

Your pts = 5*pL(400-300) + 3*pL(400-200) + 1*pL(400-100) = 5*pL(100) + 3*pL(200) + 1*pL(300). I will define pL(D) as p(D) with the luck factor L applied. In this case L = 20, for 20% luck. So pL(D) = L/2 + (1-L)*p(D), an L = 20 being the 10% + 80%*p(D). p(100) is roughly 64%, p(200) is roughly 76%, and p(300) is roughly 85%. pL(100) is therefore roughly 10% + 64%*80% = 61.2%. pl(200) is roughly 10% + 76%*80% = 70.8%. pL(300) is roughly 10% + 85%*80% = 78%. So 5*61.2% + 3*70.8% + 1*78% = 5.964, nearly 6 points of gain.

I don't see how you calculate loss from that direction, but assuming both players have the same k factor (or similar idea for games played in your version), what one gains the other loses, so by calculating what I win if I'm the 200 and my opponent the 400, I will then assume they lose it the other way around. So 5*pL(200-300) + 3*pL(200-200) + pL(200-100) = 5*pL(-100) + 3*pL(0) + pL(100). p(100) is still roughly 64%, so I'd expect p(-100) to be 1 - p(100) or roughly 36%. Naturally p(0) is 50%. pL(-100) is roughly 10% + 36%*80% = 38.8%, pL(0) is 10% + 50%*80% = 50%, and pL(100) is still roughly 10% + 64%*80% = 61.2% So 5*38.8% + 3*50% + 1*61.2% = 4.052 points, which is strangely lower than the 6 points for the 400 beating the 200. So something is off here. Having all of the percentages go down for expected winnings really trolls this method.

Or when you said win% did you mean the expected gain with that win percentage. Meaning applying BGA's numbers with the same W, K, and D, we'd get the gains for pL(100) as roughly 20(1-.612) = 7.76, pL(200) as roughly 20(1-.76) = 4.8, and pL(300) as roughly 20(1-.78) = 4.4, which gives us 5*7.76 + 3*4.8 + 1*4.4 = 57.6 gains.

Applying the same logic for 200 vs 400 we'd get the gains for pL(-100) as roughly 20(1-.36) = 12.8, the gains for pL(0) as roughly 20(1-.5) = 10, and the gains for pL(100) to be roughly 20(1-.64) = 7.2, so 5*12.8 + 3*10 + 1*7.2 = 101.2 gains, which line up a bit better with their respective side, though this method definitely is quite a rewrite for the ratings on BGA. A loss of 101.2 as a 400 vs a 200 with a gain of 57.6 would be a 101.2/(57.6+101.2)% chance of victory (roughly 63.7%).

[Ceaseless]

Pts = 5 * win % against 300s ELO + 3 * win % against 200s ELO + 1 * win % against 100s ELO ------- minimum games needed per category

Oh, it also occurs to me you might mean within the sum of games already played, with a minimum games number, the win% is for the games already finished. I had been reading win % against x Elo as expected win percentage for future games, not a completed win percent over finished ones.

[1000esqrtn1]

Pts = 5 * win % against 300s ELO + 3 * win % against 200s ELO + 1 * win % against 100s ELO ------- minimum games needed per category

Oh, it also occurs to me you might mean within the sum of games already played, with a minimum games number, the win% is for the games already finished. I had been reading win % against x Elo as expected win percentage for future games, not a completed win percent over finished ones.

Hi, thanks for your calculations, but I meant as games already played, so as a formula that gave your final points at the end of the season, not as a constant change like an ELO system.

My thought was to use a method that values win % through a whole season, as already said, I think this gives valuable information. As you mentioned, taking win % and nothing else is a risk as a player could try to farm agains low players, hence the importance to value win quality somehow. Then I would make some considerations.

If we treat Arena as a whole, it shouldn't matter the moment you beat another player. if you beat a 300s player that's everything that should matter for you, regardless if you were 200 or 400 at that point. In fact, chess ratings are not updated match after match, just at the beginning of every month, so precise moment ELO shouldn't factor, just a rough estimation of player quality. This could be done by freezing rating at the beginning of the season (internally only, for calculations purposes) which would neglect the player improvement during a season. Again, seasons may be a bit too long in general.

ELO is based on a 400 points difference, being that the usual difference between professional-amateur or amateur-passionate. You should never win a player 400 points better than you and you would be unlikely to win a player 200-300 points better than you. However, players inside a close range have a close to 50% probability of winning, so in terms to measure win quality, we could just classify players into their respective group.

Considering that I thought that this formula was quite decent and it generates a fair outcome for the season. The general formula would be something like this.
Top Players - (ex 300 + ELO)
Medium Players - (ex 200-299 ELO)
Low Players - (ex 199- ELO)
WT, WM, WL - Win Percentage Against Top/Medium/Low Players respectevily
GT, GM, GL - Value based on games played against Top/Medium/Low Players, This could be something like min(1, games / 20) to ensure a minimum number
KT, KM, KL - Factor to multiply wins against Top/Medium/Low

End of season points = KT*WT*GT + KM*WM*GM + KL*WL*GL

[Lady_Jane]

You won't see threads from a weaker player upset because they just lost 5 games won 1 and gained aes but you constantly see the opposite. Everyone isn't pissed off. Only good players who play in games with a luck factor which is the minority. Good players will play more because we have to compensate for the bad system and weaker players will play more because they can lose a ridiculous amount of games like in the example posted and still bounce back.

I have to admit, a lot of this is way over my head, and hard for me to follow. I also don't play a ton of different games in arena, there's actually one that i play consistently in arena and it's a roll and write game so quite a bit of luck involved.

The thing is, that even with luck based game, skill comes in because you have to have a strategy to mitigate the luck.

What I see (anecdotally) is that yes, the top 5-10 players are often from the same pool of 20 or so players, but they also will frequently drop several ranks and then move back up. I would think this represents them as still playing and being good enough to overcome the losses.

I often play these players, and also often end up playing really low ranked players. Personally, I tend to be most often between rank 15 and 30, but I vary a lot and will move up as high as 6, and drop as low as 50 or so. I assumed this meant I don't win as consistently as the better players. Despite the luck factor, with our strategies there are some players (even one who is consistently in the top 5) where I win almost every time, and then there are others where we go back and forth. There are also a couple of lower ranked players who I cringe when I end up in a game with them, because I know I lose every time. I think I've gotten good with the strategies, but not at adjusting depending on the players strategy. Again, I chalked this up to not being as good as those who are consistently on top.

Of course when I'm at rank 50 a win gives me a lot of points and a loss takes very little, and vice versa if I'm playing while at rank 6. This would be the case for all players playing in arena, so wouldn't that be fair? Wouldn't that still mean the better players end up at top and the lesser end up lower?

When I look at statistics on the current top 5 their win percentage is between 55-65% each. Jump to the 30 somethings, and it's a lot more varied but no one over 50%. And the top 5 are all still playing, seemingly with no lowered frequency. Even if they did stop playing though, wouldn't their points remain static while those who keep playing and winning increase?

Please understand, these are not arguments but legitimately questions... Statistics made me cry in school lol, and I guess I'm not getting it. It just seems that whatever the ranking systems those who consistently win will end up at top and those who don't won't, no?

[ChiefPointThief]

Of course when I'm at rank 50 a win gives me a lot of points and a loss takes very little, and vice versa if I'm playing while at rank 6. This would be the case for all players playing in arena, so wouldn't that be fair? Wouldn't that still mean the better players end up at top and the lesser end up lower?

It depends on how you end your season but a player who had a way more successful season than another can fall below them rather easily. I just seen it happen actually. Someone was chasing #1 and only needed one win and ended up falling hundreds of spots in the blink of an eye.

You won't see threads from a weaker player upset because they just lost 5 games won 1 and gained aes but you constantly see the opposite. Everyone isn't pissed off. Only good players who play in games with a luck factor which is the minority. Good players will play more because we have to compensate for the bad system and weaker players will play more because they can lose a ridiculous amount of games like in the example posted and still bounce back.

Even if they did stop playing though, wouldn't their points remain static while those who keep playing and winning increase?

Not necessarily. They have to win at these rates depending on how much more aes they have over their opponents. to break even +60 aes 3of5 60%,
125 2of3 66%, 200 3of4 75%, 250 4of5 80%, 300 5of6 83%. It is about streaks.

The current number #1 in your game win % over the last 36 games is %83.3 (in arena). But overall they barely have a 50% win rate in arena. I guarantee someone has had a better overall season than them. This is actually a good example for this discussion. They were a 100elo player to start the season. They won 36of104 of their fist games. They don't make elite until game 60 something and they still have a aes over 1520 even though they are more than 30 games below .500 win%. Then 54of96. Then 92of182. During this time they are remaining in the 1600s. They don't play at an elite level until their final run of 36 games. At that point they stopped playing 3 weeks ago because they are smart. They basically have nothing to gain and their next closest contender has to go on a good win streak to catch them. Meanwhile someone can consistently win 70% of their games for the entirety of the season but if they don't manage to put together a good streak to advance them to a higher level they can get stuck in a forever loop of winning 3 losing 1 and remaining in the same spot.

Edit: Some of this players games were ties.

[Ceaseless]

Pts = 5 * win % against 300s ELO + 3 * win % against 200s ELO + 1 * win % against 100s ELO ------- minimum games needed per category

Oh, it also occurs to me you might mean within the sum of games already played, with a minimum games number, the win% is for the games already finished. I had been reading win % against x Elo as expected win percentage for future games, not a completed win percent over finished ones.

Hi, thanks for your calculations, but I meant as games already played, so as a formula that gave your final points at the end of the season, not as a constant change like an ELO system.

My thought was to use a method that values win % through a whole season, as already said, I think this gives valuable information. As you mentioned, taking win % and nothing else is a risk as a player could try to farm agains low players, hence the importance to value win quality somehow. Then I would make some considerations.

If we treat Arena as a whole, it shouldn't matter the moment you beat another player. if you beat a 300s player that's everything that should matter for you, regardless if you were 200 or 400 at that point. In fact, chess ratings are not updated match after match, just at the beginning of every month, so precise moment ELO shouldn't factor, just a rough estimation of player quality. This could be done by freezing rating at the beginning of the season (internally only, for calculations purposes) which would neglect the player improvement during a season. Again, seasons may be a bit too long in general.

ELO is based on a 400 points difference, being that the usual difference between professional-amateur or amateur-passionate. You should never win a player 400 points better than you and you would be unlikely to win a player 200-300 points better than you. However, players inside a close range have a close to 50% probability of winning, so in terms to measure win quality, we could just classify players into their respective group.

Considering that I thought that this formula was quite decent and it generates a fair outcome for the season. The general formula would be something like this.
Top Players - (ex 300 + ELO)
Medium Players - (ex 200-299 ELO)
Low Players - (ex 199- ELO)
WT, WM, WL - Win Percentage Against Top/Medium/Low Players respectevily
GT, GM, GL - Value based on games played against Top/Medium/Low Players, This could be something like min(1, games / 20) to ensure a minimum number
KT, KM, KL - Factor to multiply wins against Top/Medium/Low

End of season points = KT*WT*GT + KM*WM*GM + KL*WL*GL

I'm not a fan of treating the entire season as the judge of a player. If a player's skill changes over the season, then their past skill ends up factoring into their score when it really shouldn't. Improving players would be better off just starting a new account at some point to clean off past poor performance for the season. I do like how elo only cares about past games for how they got to your current rating, but doesn't factor them into your ability to change your rating. Rating changes get harder and harder as game count increases, another reason for players to just get a new account not locking them into a rating set by too many old games.

As for Chess updates, I'm not familiar with how they handle those, but do they really only just update monthly, or is that just convenience to get all the ratings updated at some point, unlike online where they can update immediately. Also, does a player's rating only count as that same number for all of their games, or are the changes factored in for games within that set of time? (If someone is a 1500, and wins 5 games and loses 5 games, are the adjustments all made assuming they're a 1500 for all 10 games, or will any additions/subtractions applied per game factor into the remaining 10?)

[mturton]

I stopped playing Arena because the current system disincentivizes play because losses are punished so heavily.

The simplest thing to do is make each level a fixed number of wins, with the number increasing at each level. Losses dont hurt, but each win helps. The winner would be the one who reached the requisite wins in the fewest games.

Right now it is a zero fun grind that many players avoid.

[Ceaseless]

I stopped playing Arena because the current system disincentivizes play because losses are punished so heavily.

The simplest thing to do is make each level a fixed number of wins, with the number increasing at each level. Losses dont hurt, but each win helps. The winner would be the one who reached the requisite wins in the fewest games.

Right now it is a zero fun grind that many players avoid.

Losses are supposed to be punished heavily, that's how they make it that your rating stops going up, which is one of the goals. If you want to keep getting solid rating increases for wins, you have to beat even higher rated players. Or, in the less ideal case of further spread rating gaps, you have to keep up even higher win rates.

Your method would not be good, the point of your system would be to abuse weaker players so that you don't lose games. High win percentages are a bad thing, it means the pairing was awful.

[1000esqrtn1]

I'm not a fan of treating the entire season as the judge of a player. If a player's skill changes over the season, then their past skill ends up factoring into their score when it really shouldn't. Improving players would be better off just starting a new account at some point to clean off past poor performance for the season. I do like how elo only cares about past games for how they got to your current rating, but doesn't factor them into your ability to change your rating. Rating changes get harder and harder as game count increases, another reason for players to just get a new account not locking them into a rating set by too many old games.

As for Chess updates, I'm not familiar with how they handle those, but do they really only just update monthly, or is that just convenience to get all the ratings updated at some point, unlike online where they can update immediately. Also, does a player's rating only count as that same number for all of their games, or are the changes factored in for games within that set of time? (If someone is a 1500, and wins 5 games and loses 5 games, are the adjustments all made assuming they're a 1500 for all 10 games, or will any additions/subtractions applied per game factor into the remaining 10?)

I agree that on a 3 month season keeping the rankings frozen is not the best idea. Nonetheless the above formula still works with varying ranking, you can defeat the same player twice, but one might count as medium player while the other might count as top player due to player improvement.

Regarding chess, as far as I know, all your matches at a tournament are computed with the ELO you started. On pure skill games, and for tournaments that last at most a couple weeks, is fair to assume that player level won't vary that much.