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Nathan Gao

USATT#: 202320

Introduction
This page explains how Nathan Gao (USATT# 202320)'s rating went from 1728 to 1795 at the US Open held on 17 Dec 2019 - 21 Dec 2019. These ratings are calculated by the ratings processor which goes through 4 passes over the match results data for a tournament. The following values are produced at the end of each of the 4 passes of the ratings processor for Nathan Gao for this tournament.

Initial Rating Pass 1 Pass 2 Pass 3 Final Rating (Pass 4)
1728 1765 1728 1728 1795


You can click here to view a table of all the resultant values from each of the 4 passes (and the initial rating) of the ratings processor for all of the 684 players in this tournament. Sections below for further details on the initial rating and the 4 passes of the ratings processor.

Note: We use mathematical notation to express the exact operations carried out in each pass of the ratings processor below. Whenever you see a variable/symbol such as for example Xi3, we are following the convention that the superscript part of the variable (in this case "3") indicates an index (such as in a series), and it should not be misconstrued to be an exponent (which is how it is used by default).

Initial Rating
The initial rating of a player for a tournament is the rating the player received at the end of the most recent tournament prior to the current tournament. If this is the first tournament the player has ever participated in (based on our records), then the player has no initial rating.

The initial rating for US Open held on 17 Dec 2019 - 21 Dec 2019 for Nathan Gao, and its source tournament are as follows:
Initial Rating From Tournament Start Day End Day
1728 2019 JOOLA North American Teams 29 Nov 2019 1 Dec 2019

Click here to view the details of the initial ratings for all the players in this tournament.

Pass 1 Rating
In Pass 1, we only consider all the players that come into this tournament with an initial rating while ignoring all the unrated players. If a rated player has a match against an unrated player, then that match result is ignored from the pass 1 calculations as well. We apply the point exchange table shown below to all the matches participated in by the rated players:

Point Spread Expected Result Upset Result
0 - 12 8 8
13 - 37 7 10
38 - 62 6 13
63 - 87 5 16
88 - 112 4 20
113 - 137 3 25
138 - 162 2 30
163 - 187 2 35
188 - 212 1 40
213 - 237 1 45
238 and up 0 50

Suppose player A has an initial rating of 2000 and player B has an initial rating of 2064, and they played a match against each other. When computing the impact of this match on their rating, the "Point Spread" (as it is referred to in the table above) between these two players is the absolute value of the difference their initial ratings. When the player with the higher rating wins, presumably the better player won, which is the expected outcome of a match, and therefore the "Expected Result" column applies. If the player with the lower rating wins the match, then presumably this is not expected, and therfore it is deemed as an "Upset Result" and the value from that column in the table above is used. So, in our example of player A vs player B, if player B wins the watch, then the expected outcome happens, and 5 points are added to player B's rating and 5 points are deducted from player A's rating. Looking at Nathan Gao's match results and applying the point exchange table, gives us the following result:

Nathan Gao's Wins
Winner Loser
Point Spread Outcome Gain Player USATT # Rating Player USATT # Rating
25 EXPECTED 7 Nathan Gao 202320 1728 Tarun Sankuratri 96756 1703
379 EXPECTED 0 Nathan Gao 202320 1728 Andrea Tang 202987 1349
420 EXPECTED 0 Nathan Gao 202320 1728 Jason M Liu 263284 1308
103 UPSET 20 Nathan Gao 202320 1728 Alex Luo 202029 1831
241 EXPECTED 0 Nathan Gao 202320 1728 Diyan Shah 219347 1487
7 EXPECTED 8 Nathan Gao 202320 1728 Chelsea Elston 34201 1721
34 UPSET 10 Nathan Gao 202320 1728 Blake Osborn 229351 1762
675 EXPECTED 0 Nathan Gao 202320 1728 Joshua Dinu Joseph 1157745 1053
1138 EXPECTED 0 Nathan Gao 202320 1728 William Sun 230594 590
602 EXPECTED 0 Nathan Gao 202320 1728 Vincent Liu 1157718 1126

Nathan Gao's Losses
Winner Loser
Point Spread Outcome Loss Player USATT # Rating Player USATT # Rating
249 EXPECTED -0 Husen Saidov 215307 1977 Nathan Gao 202320 1728
254 EXPECTED -0 Andy Hou 218172 1982 Nathan Gao 202320 1728
24 EXPECTED -7 Rignesh Achar Padamanur 215565 1752 Nathan Gao 202320 1728
326 EXPECTED -0 Muskan Gupta 213603 2054 Nathan Gao 202320 1728
313 EXPECTED -0 Thanh Lee 11817 2041 Nathan Gao 202320 1728
363 EXPECTED -0 Matheo Boutin 237723 2091 Nathan Gao 202320 1728
221 EXPECTED -1 Kevin Nguyen 221139 1949 Nathan Gao 202320 1728
0 -0 Theo Galais 1169106 0 Nathan Gao 202320 0

You can click here to view a table of outcomes and points gained/lost from all the matches with all the players in this tournament.

The "Outcome" column above, shows whether the match had an expected (player with the higher rating wins the match) or an upset (player with the higher rating loses the match) outcome. Based on this outcome, and using both the player's initial rating, we apply the point exchange table from above and show the ratings points earned and lost by Nathan Gao in the "Gain" column. Matches are separated out into two tables for wins and losses where points are gained and lost respectively. We get the following math to calculate the Pass 1 Rating for Nathan Gao:

Initial Rating Gains/Losses Pass 1 Rating
1728 + 0 + 7 + 0 + 0 - 7 + 0 + 20 + 0 + 8 + 0 + 10 + 0 + 0 - 1 + 0 + 0 + 0 + 0 =1765

You can click here to view a table of pass1 calculations for all the rated players in this tournament.

Pass 2 Rating
The purpose of this pass is solely to determine ratings for unrated players. To do this, we first look at the ratings for rated players that came out of Pass 1 to determine an “Pass 2 Adjustment”. The logic for this is as follows:

  1. We calculate the points gained in Pass 1. Points gained is simply the difference between the Pass 1 Rating and the Initial Rating of a player:

    ρi2 = Pi1 - Pi0
    where,

    Symbol Universe Description
    Pi0 Pi0+ the initial rating for the i-th player. We use the symbol P and the superscript 0 to represent the idea that we sometimes refer to the process of identifying the initial rating of the given player as Pass 0 of the ratings processor.
    Pi1 Pi1+ the Pass 1 rating for the i-th player.
    ρi2 ρi2 the points gained by the i-th player in this tournament. Note here that we use the superscript 2 to denote that this value is calculated and used in Pass 2 of the ratings processor. Further, ρi2 only exists for players who have a well defined Pass 1 Rating. For Players with an undefined Pass 1 Rating (unrated players), will have an undefined ρi2.
    i i[1,684] the index of the player under consideration. i can be as small as 1 or as large as 684 for this tournament and the i-th player must be a rated player.

  2. For rated players, Pass 1 points gained, ρi2, is used to calculate the Pass 2 Adjustment in the following way:
    1. If a player gained less than 50 points (exclusive) in pass 1, then we set that player's Pass 2 Adjustment to his/her Initial Rating.
    2. If a player gained between 50 and 74 (inclusive) points in pass 1, then we set the player's Pass 2 Adjustment to his/her Final Pass1 Rating.
    3. If a player gains 75 or more points (inclusive) in pass 1, then the following formula applies:
      • If the player has won at least one match, and lost at least 1 match in the tournament, then the player's Pass 2 Adjustment is the average of his/her Final Pass 1 Rating and the average of his/her opponents rating from the best win and the worst loss, represented using the formula below:

        αi2 = Pi1 + Bi + Wi 2 2

        where αi2 is the Pass 2 Adjustment for the current player, Pi1 is the Pass 1 Rating, Bi is the rating of the highest rated opponent against which the current player won a match, and Wi is the rating of the lowest rated opponent against which the current player lost a match.
      • If a player has not lost any of his/her matches in the current tournament, the mathematical median (rounded down to the nearest integer) of all of the player's opponents initial rating is used as his/her Pass 2 Adjustment:

        αi2 = {Pk0}

        where Pk0 is the initial rating of the player who was the i-th player's opponent from the k-th match.
        Symbol Universe Description
        i i[1,684] the index of the player under consideration. i can be as small as 1 or as large as 684 for this tournament and the i-th player must be a rated player.
        q q[1,3743] the index of the match result under consideration. q can be as small as 1 or as large as 3743 for this tournament and the q-th match must be have both rated players as opponents.
        g g[1,5] the g-th game of the current match result under consideration. q can be as small as 1 or as large as 5 for this tournament assuming players play up to 5 games in a match.
        Pk0 Pk0+ initial rating of the i-th player's opponent from the k-th match.


      • Therefore, the Pass 2 Adjustment for Nathan Gao is calculated as follows:
        • Given the initial rating of 1728,
        • and the Pass 1 rating of 1765,
        • the Pass 1 gain is 1765 - 1728 = 37.
        • Since the Pass 1 gain of 37 is less than 50, the Pass 2 Rating (also referred to as Pass 2 Adjustment) is reset back to the initial rating.
        • Therefore the Pass 2 Adjustment for Nathan Gao is 1728.

        You can click here to view a table of Pass 2 Adjustments for all the rated players in this tournament.

  3. After calculating the Pass 2 Adjustment for all the rated players as described above, we can now calculate the Pass 2 Rating for all the unrated players in this tournament (which is the main purpose of Pass 2). Pass 2 Rating is calculated using the following formula:
    1. If all of the matches of an unrated player are against other unrated players, then the Pass 2 Rating for that player is simply set to 1200. You can click here to view these players who received a 1200 Pass 2 Rating. Not all of Nathan Gao's matches were against unrated players. So this rule does not apply to him.
    2. For unrated players with wins and losses, where at least 1 of the opponents has an initial rating, the Pass 2 Rating is the average of the best win and the worst loss (using the Pass 2 Adjustment of all rated players) as defined by this formula here:

      Pi2 = Bi2 + Wi2 2

      where Pi2 is the Pass 2 Rating for the i-th player, Bi2 is the largest Pass 2 Adjustment (best win) of the opponenet against whom the i-th player won a match, and Wi2 is the smallest Pass 2 Adjustment (worst loss) of the opponent against whom the i-th player lost a match.
    3. For unrated players with all wins and no losses, where at least 1 of the opponents has an initial rating, the Pass 2 Rating is calculated using the following formula:
      Pi2 = Bi2 + k=0Mi-1 I(Bi2-αk2)
      where the function I(x) is defined as, \begin{equation} I(x)=\left\{ \begin{array}{ll} 10, & \text{if}\ x >= 1, x <= 50 \\ 5, & \text{if}\ x >= 51, x <=100 \\ 1, & \text{if}\ x >= 101, x <= 150 \\ 0, & \text{otherwise} \end{array}\right. \end{equation}
      where,
      Symbol Universe Description
      Pi2 Pi2+ the pass 2 rating, of the i-th player in this tournament only applicable to unrated players, where Pi0 is not defined
      Bi2 Bi2+ the largest of the Pass 2 Adjustments of opponents of the i-th player against whom he/she won a match.
      αk2 αk2+ the Pass 2 Adjustment of the player who was the opponent of the i-th player in the k-th match
      I(x) I:+ a function that maps all integers to one of the values from -- 0, 1, 5, 10.
      Mi Mi+ total number of matches played by the i-th player in this tournament
      k k[0,Mi-1]+ The index of the match of the i-th player ranging from 0 to Mi-1
    4. For unrated players with all losses and no wins, where at least 1 of the opponents has an initial rating, the Pass 2 Rating is calculated using the following formula:
      Pi2 = Wi2 + k=0Mi-1 I(Wi2-αk2)
      where I(x) is defined above and,

      Symbol Universe Description
      Pi2 Pi2+ the pass 2 rating, of the i-th player in this tournament only applicable to unrated players, where Pi0 is not defined
      Wi2 Wi2+ the smallest of the Pass 2 Adjustments of opponents of the i-th player against whom he/she lost a match.
      αk2 αk2+ the Pass 2 Adjustment of the player who was the opponent of the i-th player in the k-th match
      I(x) I:+ a function that maps all integers to one of the values from -- 0, 1, 5, 10.
      Mi Mi+ total number of matches played by the i-th player in this tournament
      k k[0,Mi-1]+ The index of the match of the i-th player ranging from 0 to Mi-1
    5. For the rated players, all the work done in Pass 1 and Pass 2 to undone and they have their ratings reset back to their initial ratings while the unrated players keep their Pass 2 Adjustment as their final Pass 2 Rating. Since Nathan Gao is a rated player, his Pass 2 Adjustment of 1728 will be ignored, along with him Pass 1 Rating of 1765 and his Pass 2 Rating will be set to his initial rating of 1728 with which he came into this tournament.


Click here to see detailed information about the Pass 2 Ratings of all the other players in this tournament.

Pass 3 Rating
Any of the unrated players who have all wins or all losses are skipped in Pass 3. Since Nathan Gao has an initial rating of 1728, he is not an unrated player, and therefore this rule does not apply to him. You can click here to view list of all the players that are skipped in this Pass 3.

Pass 3 Rating is calculated using 2 steps described below:
  1. In the first part of Pass 3, we apply the point exchange table described in Pass 1 above except this time by using all the players' Pass 2 Ratings. Looking at Nathan Gao's wins and losses and applying the point exchange table, gives us the following result:
    Nathan Gao's Wins
    Winner Loser
    Point Spread Outcome Gain Player USATT # Rating Player USATT # Rating
    25 EXPECTED 7 Nathan Gao 202320 1728 Tarun Sankuratri 96756 1703
    379 EXPECTED 0 Nathan Gao 202320 1728 Andrea Tang 202987 1349
    420 EXPECTED 0 Nathan Gao 202320 1728 Jason M Liu 263284 1308
    103 UPSET 20 Nathan Gao 202320 1728 Alex Luo 202029 1831
    241 EXPECTED 0 Nathan Gao 202320 1728 Diyan Shah 219347 1487
    7 EXPECTED 8 Nathan Gao 202320 1728 Chelsea Elston 34201 1721
    34 UPSET 10 Nathan Gao 202320 1728 Blake Osborn 229351 1762
    675 EXPECTED 0 Nathan Gao 202320 1728 Joshua Dinu Joseph 1157745 1053
    1138 EXPECTED 0 Nathan Gao 202320 1728 William Sun 230594 590
    602 EXPECTED 0 Nathan Gao 202320 1728 Vincent Liu 1157718 1126

    Nathan Gao's Losses
    Winner Loser
    Point Spread Outcome Loss Player USATT # Rating Player USATT # Rating
    249 EXPECTED -0 Husen Saidov 215307 1977 Nathan Gao 202320 1728
    254 EXPECTED -0 Andy Hou 218172 1982 Nathan Gao 202320 1728
    24 EXPECTED -7 Rignesh Achar Padamanur 215565 1752 Nathan Gao 202320 1728
    326 EXPECTED -0 Muskan Gupta 213603 2054 Nathan Gao 202320 1728