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Other Everything You Ever Wanted to Know About Ratings

Discussion in 'Technical Projects' started by Antar, Aug 20, 2013.

  1. Antar

    Antar
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    This post has been revised to reflect the current state of Pokemon Showdown rating systems

    Recently, PS' ladder has come under a lot of fire, mostly from people who have no understanding of how a rating system works or its fundamental limitations. This article, which could either go in the forums alongside the Weighted Stats FAQ or on-site somewhere, was written to educate members of the community about our rating system, to answer fundamental questions about how it works and, ideally, to lead to a more informed discussion of our ladder practices. I welcome any and all constructive feedback, especially questions that will help populate the FAQ section.

    -------------------------

    Introduction
    Rating systems and ladders are a fundamental component of many competitive games, from college football to tennis to chess to competitive Pokemon. Both Pokemon Online and Pokemon Showdown implement rating systems to attempt to rank players based on skill and to pair players with similar skill levels for battles. This is not an easy job, and there is not a system on the planet that can do it perfectly. In what follows, I will be attempting to explain the concepts, theory and execution behind Showdown's rating system (and will in the process explain Pokemon Online's as well). For those who are mathematically inclined, I will include technical details. For those of you who aren't, this article was written so as to allow you to skip those sections without negatively impacting your ability to understand the rest of the article.

    Background and History
    The goal of most rating system is, most fundamentally, to determine a player's skill level. This can be useful for ranking players, for tournament seeding, or for pairing players of similar skill levels for optimally "interesting" battles, but at the end of the day, the goal is simply to try to determine how good a given player is, to the greatest degree of accuracy.

    This is not a new problem, and competitive leagues have been at this for hundreds of years. Easily the simplest rating system is the win-loss record, but that has the problem of rewarding players who only play poor players and punishing those who seek out challenging opponents. Throughout the centuries, more complex points-based systems have been developed to try to correct for this issue but were often criticized as being arbitrary and unfair. Then in 1960, the United States Chess Federation adopted a new rating system designed by chess master and physicist Arpad Elo. His statistics-based rating system became widely popular due to its simplicity and fairness, and half a century later, almost all modern rating systems are built upon his concepts. Pokemon Online and Pokemon Showdown both use variants on Elo as their primary rating systems, although Showdown also implements an extension of Elo called Glicko (more on that in a bit). The old Shoddy Battle used the Glicko-2 rating system, which is an extension of Glicko.

    Ratings, Parameters and Estimates
    I would say that 90% of people's confusion regarding the various ladders comes down to a lack of conceptual understanding concerning the difference between parameters and estimates. The issue is that a player's true skill level, the piece of information that all rating systems are fundamentally trying to determine, is fundamentally unknowable--there is no realistic way to have every player battle every other player on the ladder, and if there were, there would be no way to guarantee that the same matchup, if repeated, would have the same results. Instead, we have to approximate a player's skill level based on observation of the battles that actually took place.

    In the language of statistics, a player's skill level is a "parameter"--if you know that, you know everything about how the player behaves. In contrast, the rating we generate based on the results of his or her battles is an "estimate" of this parameter. If you take nothing else away from this article, that is it, so I will repeat:

    The rating shown on a ladder is simply an estimate of a player's skill level, nothing more.

    When most people think about climbing the ladder, they think of it in terms of earning "points" for wins and losing them for defeats. This leads to frustration when a highly ranked player sees his or her rating change not at all when they defeat a low-rated opponent (since they didn't gain any points by winning), but this is a mistake in thinking--the "reward" for the highly-rated player is that he or she has had his or her skill estimate "validated"--he or she has successfully shown that the ladder was justified in rating him or her so highly. This is something that bears emphasizing:

    Your rating is not something that you "earn." It is something you discover about yourself.

    I realize that the above statement may be shocking--the idea that a player should be "rewarded" for battling more is ingrained into most players' psyches. And certainly, we want to encourage players to battle more frequently. What we don't want is players who reach the top of the ladder and "park" there, refusing to battle anyone further, for fear of losing his or her #1 spot when he or she suffer an unlucky defeat, and this is something I will discuss further in a later section. In the meantime, remember that the fundamental purpose of a ladder is not to reward or to punish, but to determine the skill levels of the players on it.

    Both Glicko and Elo ratings are estimates of a player's skill, and in the sections that follow, I will explain some of the mechanics underlying these systems and explain how you should interpret your ratings.

    Pokemon as a Different Sort of Card Game
    Before I get into describing the rating systems themselves, I'm going to describe the mathematical theory at the heart of both systems. If you're not interested in the technical details, feel free to simply skip this section.

    An important feature of most rating systems is that they do not require that a better player will always defeat an inferior one. Whether due to chance ("hax") or one player being "in the zone" while another is having an off-day, there will always be some uncertainty regarding the outcome of a match between two players, even if their skill levels are well defined.

    To account for this, the Elo and Glicko systems use at their heart a model of "pairwise comparison" called Bradley-Terry, which simplifies all Pokemon battles (or chess games or tennis matches) down to a simple game of chance:

    Picture two players, sitting across from each other. Each has his or her own deck of cards, and each card is marked with a number. The players shuffle their decks, and on the count of three, each pulls a single card and places it on the table. The winner is the player whose card has the higher number. The key aspect of this model--and what makes it different from, say, a simple coin toss--is that the two players need not have identical decks, and if one player's deck is stacked with numbers that are generally higher, then that player will be more likely to win.

    For the mathematically inclined... (open)

    While in principle, the distributions of cards in the deck could follow any form, in the Bradley-Terry model, the distribution of cards follows the Extreme Value (or Fisher-Tippett) distribution, which is asymmetric about its mean, having a longer positive "tail." This says that it is more likely that a player will occasionally play far above their typical skill level than it will be for them to play far below it. Furthermore, Elo (and consequently Glicko) assumes that these Extreme Value distributions have the same width and vary only by their center.


    While this model obscures the thrill of a well-made prediction and the frustration of an unfortunately-timed bit of hax, the point is that it's built into the model that a player will not always have the same "performance" every time he or she plays the game, even if his or her skill level hasn't changed, and the up-shot is that there's always a chance that an inferior player will defeat a superior one.

    The Unknowable "True Rating"
    The underlying principle behind both Elo and Glicko is that a player has a single "parameter" representing his or her skill level, and that parameter governs how the player will perform: as per the previous section, a player at a given skill level will sometimes end up performing under his or her skill level and sometimes over (due to luck, "hax" or simply being "in the zone"), but given enough battles, his or her "average" performance should converge to a well defined value. This average, or expected, performance I will refer to as the player's "true rating." This "true rating" is the end-all, be-all, the holy grail of ratings. Given two player's rating, it is, in principle, trivial to calculate the odds that one will defeat the other, and it is the goal of both the Elo and Glicko rating systems to estimate a player's "true rating" as closely and as accurately as possible. An important point here is that ratings are absolute--there's no "rock-paper-scissors" element here where Player A usually defeats Player B, Player B usually defeats Player C, but Player C usually defeats Player A--if Player A usually defeats Player B, and Player B usually defeats Player C, then it's assumed that Player A will usually defeat Player C as well.

    The Elo Rating System
    I'll start with Elo, which is the system behind the Pokemon Online and Pokemon Showdown ladders. Given a series of wins and losses against a collection of opponents, a player's Elo rating is calculated so as to be the "true rating" that has the greatest likelihood of producing those results. Keep in mind, once again, that sometimes a less skilled player will defeat a more-skilled one, so if you defeat three players with 1200 ratings and then proceed to lose to a player with an 800 rating, that's fine, Elo can account for that (what exactly your rating would be in that case depends on a choice of parameter).

    Without going to go into details about how Elo ratings are actually calculated or updated, one of its main advantages is its simplicity (you can readily calculate how two players' ratings will change as a result of a single match) and transparency. For those used to points-based rating systems where ones rating goes up or down as a direct consequence of winning or losing, Elo feels comforting, aside from the extremes where a player with a much higher rating than his or her opponent will not gain any points from a victory. It was this simplicity and it's "fairness" and accuracy when compared to previously-used points-based rating systems that led to its wide-scale adoption, and after more than fifty years, its ubiquity is one of its big selling points.

    Elo does have a number of downsides, though, the most prominent being that players with high ratings often have little incentive to continue playing and GREAT incentive NOT to play: if you have an rating of 1800 on PO and you end up facing a player with a rating of 1000, a loss will knock 50 points off your rating, whereas a win will net you zero. Consequently, it is not uncommon for top players on Elo-based ladders to "park" themselves at the top and refuse to continue battling altogether. To combat this, Pokemon Online introduced rating decay, which automatically lowers a player's displayed rating for every so-many hours the player stays inactive. Note that this doesn't actually change the player's rating, only what is displayed, and after a few battles, this decay is erased. This system effectively forces players to keep battling to retain their ranking, but there are other related issues that must be guarded against.

    Pokemon Showdown's Elo implementation also includes a few tweaks, such as its own decay and a sliding "K-factor" which makes a player gain or lose fewer points the higher his or her Elo score.

    The Glicko Rating System
    In 1995, Mark Glickman, a statistician who is currently the chair of the United States Chess Federation's Ratings Committee, introduced an extension of the Elo rating system which he called Glicko (which was later extended to Glicko-2). Whereas Elo estimates a player's "true rating" by using only a single value, the player's rating, Glicko adds a second value, called RD (short for "rating deviation"), which quantifies the level of uncertainty in the estimate of a player's "true rating" (Glicko-2 adds on one more variable, called volatility, which measures the erraticness of a player's performance).

    Thus, where Elo attempts to directly estimate a player's "true rating," Glicko instead estimates that a player's "true rating" falls within a probability distribution, in this case, a normal distribution of width RD and center R.

    Below are two sample distributions: one for a player of rating 1600±100 and one whose rating is 1575±50.
    [​IMG]
    For those unfamiliar with probability distributions, the idea is that the probability of a player's "true rating" being in the infinitesimal interval r<rating<r+dr is f(r)dr.

    So looking at these two graphs the question is, who's the better player? Using the Glicko system, it's impossible to know for certain, but what we can determine is the likelihood that each player's "true rating" is above some specific value. For instance, in the case pictured above, there's an 84% chance that the blue player's "true rating" is above 1500 and a 93% chance that the same is true for the red player. On the other hand, there's less than a 1% chance that the red player's true rating is at least 1700, whereas the odds that the blue player is at least at that skill level is nearly 16%.

    This showcases the primary problem with Glicko rating systems--whereas under Elo, comparing players was a simple matter of comparing their ratings, under Glicko it's much harder. In addition, Glicko ratings are updated only once per "rating period" (Showdown uses two days or 15 battles). Since players for the most part wouldn't stand for only seeing their ratings change every fifteen battles, players are assigned "provisional" ratings at the end of each battle which are designed to estimate how the new ratings will change. This can lead to confusion when a rating period ends, and the player's rating actually updates.

    As with Elo, I won't be going into the technical details concerning implementation, although I will point out a key feature of Glicko: where Elo ladders have to manually implement rating decay to discourage periods of inactivity, Glicko builds inactivity into the rating system, with RD increasing the longer a player remains inactive.

    Rating Estimates
    So I said that since a player's Glicko rating is not one number but two, it's not nearly as straightforward to directly compare two players' skill levels. Ideally, a player's skill level would always be represented by his or her R and RD, and applications requiring an assessment of a player's skill level would be implemented probabilistically (see: Weighted Stats). Back in the real world, however, things like ladder rankings require us to have a set method of comparing two players and determining which one is "better." The way we do this is through "Conservative Rating Estimates" or CREs. CREs basically say, "I don't know for certain how good a player is, but I'm pretty sure he or she is better than this.

    The most famous of these around here is ACRE, which is what Showdown's ladder used to use to rank players. ACRE (the "A" originally stood for aeo, aka Zarel, who designed it) corresponds to about the 8th percentile of a player's rating distribution, meaning there's about a 92% chance that a player's "true rating" is at least as high as his or her ACRE. The primary problem with ACRE is that it is extremely inaccurate when RD is high. In addition, top-ranked players see the same problem as in Elo--at that level it takes a lot of battles to change one's ACRE.

    One proposed alternative was designed for Smogon's Shoddy ladder by X-Act, a mathematician who was very active in the community in those days. His Glicko-X-Act Estimate (or GXE) measures the odds that a player would win a battle against a randomly selected opponent from the ladder. Players' GXEs are displayed along with their Elo and Glicko ratings on the Showdown ladder. While GXE is designed to have concrete meaning and should, in principle, outperform ACRE, it shares many of ACRE's problems when RD is high, and the 0-100 rating scale is not as intuitive to players used to Elo, so it hasn't seen wide-scale adoption. That being said, GXE is the primary component of COIL, an achievement-centric ladder score we currently use to determine suspect voting requirements.

    There's an important point that bears emphasis about GXE and ACRE: people often erroneously refer to these rating estimates as if they were alternatives to Glicko, the same way Elo and Glicko are different rating systems, but in truth,

    ACRE and GXE are measures for interpreting Glicko ratings and are not independent systems.

    Quick note: Trueskill
    Microsoft's Trueskill is another rating system that's been gaining traction lately, and it is occasionally suggested that PS implement Trueskill for its ladder. The problem is that Trueskill is proprietary (read: might require a license) and the differences between Trueskill and Glicko are mostly trivial.

    Summary
    At the end of the day, if you come away with nothing else after reading this article, I hope it's the following:
    1. Ratings are not rewards: while one's rating may go up after winning a battle and down after losing one, at the end of the day one's rating, as implemented on Pokemon Online and Pokemon Showdown is designed to be an accurate measure of one's skill level.
    2. It is important to differentiate between a player's "true rating," which is not knowable, his or her Glicko rating, which is not a single number but rather a rating an an uncertainty, RD, that together define the likelihood of his or her "true rating" falling within a given range, and a player's Rating Estimate, either expressed as ACRE or GXE, which whitewashes most of the nuances of your Glicko rating.

    Further Reading
    Frequently Asked Questions
    Ask away!
    Last edited: Jun 25, 2014
  2. Aldaron

    Aldaron
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    By this you mean the rating systems you described here seek to do that right; what if the purpose of the desired rating system is not to find a person's skill level (which, from what we've observed in the past 6 years of popular online Pokemon laddering, is probably not practically viable, as pretty much no thinks the ladder rankings are "true measures" of people's skill) but actually to measure performance in that ladder?

    By this I mean, what if we don't care about finding a player's "true skill" but just want to find out how he has performed in that ladder whenever he has participated in it?

    In that scenario, isn't the rating system actually a reward system that we hope to measure?

    I know you started this article off with "inform about our ladder practices", which means you're describing the perspective with which we've approached ladder rating systems historically and presently, but I'm asking why we assume that perspective is even what we desire? What if we have no desire to attempt to find "true skill" (considering how few people actually consider any of the shoddy / PO / PS rating systems to ever represent true skill) and simply want to measure how a person has performed in a specific ladder during a specific time?

    This is a very important question, particularly because what most people use as a measure of true skill is actually tournament performance, so I would further ask why endeavoring to have a system to find "true skill" on the ladder is even worth our time.
  3. Antar

    Antar
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    @Aldaron, I have updated that bullet point to clarify, but the issue you raise is a valid one. If people wanted to, we *could* implement a point-based rating system. I approached this article, as you correctly inferred, from the perspective of "this is what we've done up until now," but you are correct that point-based rating systems exist to this day, and the article needs to reflect that.

    Edit: done. Let me know if you see anywhere else where the language should be changed to reflect this alternate perspective.
  4. Zarel

    Zarel Not a Yuyuko fan
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    Oh, @Antar, if you were wondering why ACRE stands for "Advanced CRE"; it originally stood for "aeo's CRE" but then I changed my username and around the same time decided that naming a CRE after myself was really narcissistic.

    Also, at the time I renamed it, the ACRE formula was much more complicated, it was something like:

    R
    - (RD - 50) if RD > 50
    - (RD - 100) if RD > 100
    - (RD - 150) if RD > 150
    - (RD - 200) if RD > 200

    I simplified it after too many complaints about people gaining points for losing battles.
  5. Erico9001

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    While you may wish for people to not view their ranks as rewards, unfortunately it is impossible. The biggest reason people compete in showdown is to prove their skill and smartness and in effect make themselves feel competent. This means that a number or a series of numbers used to estimate the player's skill level will be seen as a reward no matter what because it is the player's only evidence of their skill that they can show other people. At least people can find comfort in knowing PS's rating system is not absolute and convince themselves their true skill level is higher than what is shown, whether they're right or wrong. You know, perhaps it is a good thing that the ranks system isn't perfect for this reason.

    As a side note – I just noticed Zarel's message count is 1337. Very nice.
  6. josehand1

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    Hi there,

    Could anyone explain how it's now? I don't need ACRE anymore and I see Elo, but Glicko 2 and GXE are still there. If I understood correctly, ELo and Glicko are two different rating systems. Are both being used together?
  7. Antar

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    Hi, josehand1! PS is currently using Elo for its primary rating system and is calculating and displaying Glicko2 rating as well (though that's going to be replaced with Glicko1 as soon as possible). Moving forward, we'll be implementing a new kind of rating system that's entirely different from what's described in this post, in that the goal will be to reward achievement on the ladder rather than to accurately assess player skill. This system will exist in tandem with one of our skill-based rating systems--we're ironing out the details now. Stay tuned!
  8. asbdsp

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    Wow, was just thinking about how they ought to do something like this on the toilet yesterday. An exciting new dawn for PS, one feels.
  9. Qoseph

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    Thanks, this changed my view. One part in particular especially: Rating is not something you earn, it's something you discover about yourself. I never thought of it like that.
    Last edited: Jan 31, 2014
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  10. Mulan15262

    Mulan15262

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    For Elo and Glicko-1's median ranking value, does the amount of battles you played in the past affect your ranking?

    If that's not clear, suppose I have an Elo of 1300. I win several battles and raise it up to a 1500, and then lose several and lower it to that same 1300. Would it be any easier or harder to bring it up to 1500 again than last time, or would it be about the same?
  11. Zarel

    Zarel Not a Yuyuko fan
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    For Elo: the same

    For Glicko-1: harder
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  12. Arcanuke

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    Sorry if this question was already answered in the article but with the Glicko-1 rating system does having a more erratic performance result in a lower rating overall than if you won and lost a consistent amount and how is consistency measured
  13. Antar

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    Arcanuke IIRC this is the chief improvement of Glicko-2, that it takes into account performance consistency. Off the top of my head I don't know how differently two players with the same average performance but different volatilities would be scored, but I suspect Glickman may talk about it in one of his papers.
  14. MacChaeger

    MacChaeger formerly OU banned my masquerain

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    From what I can tell, in a pure ELO system, the winner gains as many points as the loser loses. The PS! ladder definitely doesn't adhere to that rule, so the formula has clearly been modified in some way. Would it be possible for the actual formula that PS! uses to be released? I fear that the ladders may be subject to extreme deflation if a very highly skilled player floods the top of the ladder with many, many alts instead of sticking to just one or two. This would be easy to prove in a pure ELO system, as each alt is gaining all but a few points (the ones awarded for beating players rated at 1000) by taking them from the rest of the players, but PS! doesn't use one, so for all I know it might have proper safeguards in place. Still, it would be nice to have the real formula, so I can be sure.
  15. Zarel

    Zarel Not a Yuyuko fan
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    I'll write up a detailed description of how it works in a bit...

    The short answer is that PS increases gains and decreases losses when your Elo is between 1000-1200 (and doesn't allow it to fall below 1000), and is regular Elo with K-scaling above 1200, with the addition of rating decay.
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  16. Zarel

    Zarel Not a Yuyuko fan
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    MacChaeger, Antar

    Our ladder displays four ratings: Elo, GXE, Glicko-1, and COIL.

    Elo is the main ladder rating. It's a pretty normal ladder rating: goes up when you win and down when you lose.

    GXE (Glicko X-Act Estimate) is an estimate of your win chance against an average ladder player.

    Glicko-1 is a different rating system. It has rating and deviation values.

    COIL (Converging Order Invariant Ladder) is mainly used for suspect tests. It goes up as you play games, but not too many games.

    Note that win/loss should not be used to estimate skill, since who you play against is much more important than how many times you win or lose. Our other stats like Elo and GXE are much better for estimating skill.

    PS Elo

    Your rating starts at 1000.

    Our Elo implementation uses K-scaling. The K factor is:
    • K = 50 if Elo is 1100 – 1299
    • K = 40 if Elo is 1300 – 1599
    • K = 32 if Elo is 1600 or higher
    We have a rating floor of 1000 (If your rating would fall below 1000, it is set to 1000). This makes it unnecessary to create new accounts to "fix" your rating.

    Between 1000 and 1100, we have some special behavior:

    If Elo is 1000, K = 80 for the winner and K = 20 for the loser. Between 1001 to 1099, K scales linearly from 80 to 50 for the winner and from 20 to 50 for the loser. This helps spread out low ladder people between 1000 and 1100 instead of causing the rating floor to cluster them all at 1000.

    In OU and randbats, we have rating decay above 1400. Every day:
    • If you played 6 or more games, there is no decay
    • If you played 1-5 games, you lose 1 point for every 100 points above 1500 you are
    • If you played 0 games, you lose 1 point for every 50 points above 1400 you are
    In all other tiers, we have rating decay above 1500. Every day:
    • If you played 6 or more games, there is no decay
    • If you played 1-5 games, you lose 1 point for every 100 points above 1700 you are
    • If you played 0 games, you lose 1 point for every 50 points above 1500 you are
    This helps combat rating inflation.

    Note that there's no "official" Elo standard. K-scaling and rating floors are common, rating decay somewhat common, and our dynamic K-scaling seems to be unique.

    PS Glicko-1

    Your rating starts at R = 1500, RD = 130.

    We use a rating period of 24 hours and an RD range of 25 to 130, with a system constant of 6.6775026092.
    Last edited: Aug 28, 2016
  17. OrdA

    OrdA That which can be destroyed by the truth, should be.
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    Based on the source code found here, I think there's a few mistakes in this. Some possible corrections:
    • If you played over 5 games, there is no decay
    • If you played 1-5 games, you lose 1 point for every 100 points above 1400 you are
      Code:
      $decay = 0 + intval(($elo-1400)/100)
    • If you played 0 games, you lose 1 point, plus one for every 50 points above 1400 you are
      Code:
      $decay = 1 + intval(($elo-1400)/50)
    Ladders besides OU and randbats decay by 2 points less, meaning that the decay will be 1 point for every 100 points above 1600, and 1 point plus one for every 50 points above 1500, respectively.
    Code:
    default:
         $decay -= 2;
  18. Zarel

    Zarel Not a Yuyuko fan
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    "above 1500" and omitting "plus one" are intentional. It's mathematically equivalent.

    1-5 games: lose 1 point at 1500, 2 points at 1600, 3 points at 1700, etc...
    0 games: lose 1 point at 1400, 2 points at 1450, 3 points at 1500, etc...
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  19. OrdA

    OrdA That which can be destroyed by the truth, should be.
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    Oh. Maybe it's just me reading this wrong. To me, 1 point for every 100 points above 1500 would mean that I lose 1 at 1600 (because that's the first time I have 100 points over 1500). Then 2 at 1700, and so on.

    To you, it seems to be "1 point for every started 100" (similar to how you pay for phone calls for every minute you started).
  20. ABR

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    Can you clarify what "every day" means? I asked this somewhere else but it was never answered and this post explains my question well:
    http://www.smogon.com/forums/thread...posting-a-thread.3520646/page-53#post-6933426
  21. Zarel

    Zarel Not a Yuyuko fan
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    Apparently SQSA isn't getting answers reliably? That kinda sucks.

    Anyway, the answer is "once per calendar day". Rating periods roll over (Elo calculates decay and Glicko-1 rolls over a rating period) at midnight GMT (4pm-8pm in the US).
  22. ABR

    ABR
    is a Tiering Contributor

    Joined:
    Mar 4, 2014
    Messages:
    336
    Thank you!

    e: I lost points when I was sleeping (at least 4 hours after you said decay happens)?
    Last edited: Sep 2, 2016

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