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Keith Carson

USATT#: 221111

Introduction
This page explains how Keith Carson (USATT# 221111)'s rating went from 945 to 1078 at the 2021 Arizona Sizzler Open held on 31 Jul 2021 - 1 Aug 2021. 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 Keith Carson for this tournament.

Initial Rating Pass 1 Pass 2 Pass 3 Final Rating (Pass 4)
945 1043 945 979 1078


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 56 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 2021 Arizona Sizzler Open held on 31 Jul 2021 - 1 Aug 2021 for Keith Carson, and its source tournament are as follows:
Initial Rating From Tournament Start Day End Day
945 2021 Arizona State Championship 23 May 2021 23 May 2021

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 Keith Carson's match results and applying the point exchange table, gives us the following result:

Keith Carson's Wins
Winner Loser
Point Spread Outcome Gain Player USATT # Rating Player USATT # Rating
488 EXPECTED 0 Keith Carson 221111 945 Arush Khare 1168953 457
632 EXPECTED 0 Keith Carson 221111 945 Ira Shilimkar 1171503 313
0 0 Keith Carson 221111 0 Chema Estreller 271028 0
149 UPSET 30 Keith Carson 221111 945 Jon Redman 14969 1094
632 EXPECTED 0 Keith Carson 221111 945 Ira Shilimkar 1171503 313
0 0 Keith Carson 221111 0 Chema Estreller 271028 0
99 UPSET 20 Keith Carson 221111 945 Juli Blass 86407 1044
139 UPSET 30 Keith Carson 221111 945 Steve Glasberg 263095 1084
139 UPSET 30 Keith Carson 221111 945 Steve Glasberg 263095 1084
233 UPSET 45 Keith Carson 221111 945 Mark Morton 218433 1178

Keith Carson's Losses
Winner Loser
Point Spread Outcome Loss Player USATT # Rating Player USATT # Rating
294 UPSET -50 Niko Garcia 1150969 651 Keith Carson 221111 945
256 EXPECTED -0 Jeron Triggs 96237 1201 Keith Carson 221111 945
0 -0 Jeremy Nichols 270797 0 Keith Carson 221111 0
384 EXPECTED -0 Aditya Bhaskar Pandit 218523 1329 Keith Carson 221111 945
25 EXPECTED -7 Carl Novick 1171501 970 Keith Carson 221111 945

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 Keith Carson 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 Keith Carson:

Initial Rating Gains/Losses Pass 1 Rating
945 - 50 + 0 + 0 + 0 + 30 + 0 + 0 + 20 + 30 + 0 + 30 + 0 + 45 + 0 - 7 =1043

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[1,56] the index of the player under consideration. i can be as small as 1 or as large as 56 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[1,56] the index of the player under consideration. i can be as small as 1 or as large as 56 for this tournament and the i-th player must be a rated player.
        q q[1,264] the index of the match result under consideration. q can be as small as 1 or as large as 264 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 Keith Carson is calculated as follows:
        • Given the initial rating of 945,
        • and the Pass 1 rating of 1043,
        • the Pass 1 gain is 1043 - 945 = 98.
        • Since the Pass 1 Gain of 98 is greater than 74, and Keith Carson has won some matches and lost some matches, the Pass 2 Rating is calculated to be the average of the Pass 1 Rating and the average of the Best win and the worst loss:

          1043 + 1178 + 651 2 2 =978    (mod 1)


        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 Keith Carson'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 Keith Carson is a rated player, his Pass 2 Adjustment of 978 will be ignored, along with him Pass 1 Rating of 1043 and his Pass 2 Rating will be set to his initial rating of 945 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 Keith Carson has an initial rating of 945, 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 Keith Carson's wins and losses and applying the point exchange table, gives us the following result:
    Keith Carson's Wins
    Winner Loser
    Point Spread Outcome Gain Player USATT # Rating Player USATT # Rating
    488 EXPECTED 0 Keith Carson 221111 945 Arush Khare 1168953 457
    632 EXPECTED 0 Keith Carson 221111 945 Ira Shilimkar 1171503 313
    0 0 Keith Carson 221111 0 Chema Estreller 271028 0
    149 UPSET 30 Keith Carson 221111 945 Jon Redman 14969 1094
    632 EXPECTED 0 Keith Carson 221111 945 Ira Shilimkar 1171503 313
    0 0 Keith Carson 221111 0 Chema Estreller 271028 0
    99 UPSET 20 Keith Carson 221111 945 Juli Blass 86407 1044
    139 UPSET 30 Keith Carson 221111 945 Steve Glasberg 263095 1084
    139 UPSET 30 Keith Carson 221111 945 Steve Glasberg 263095 1084
    233 UPSET 45 Keith Carson 221111 945 Mark Morton 218433 1178

    Keith Carson's Losses
    Winner Loser
    Point Spread Outcome Loss Player USATT # Rating Player USATT # Rating
    294 UPSET -50 Niko Garcia 1150969 651 Keith Carson 221111 945
    256 EXPECTED -0 Jeron Triggs 96237 1201 Keith Carson 221111 945
    0 -0 Jeremy Nichols 270797 0 Keith Carson 221111 0
    384 EXPECTED -0 Aditya Bhaskar Pandit 218523 1329 Keith Carson 221111 945
    25 EXPECTED -7 Carl Novick 1171501 970 Keith Carson 221111 945

    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 for Pass 3 Part 1.

    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 Pass 2 Rating, we apply the point exchange table from above and show the rating points earned and lost by Keith Carson in the "Gain" column. Matches are divided up into two tables for wins and losses where points are "Gain"ed for the wins and "loss"ed for losses. Putting all the gains and losses together, we get the following math to calculate the rating for Keith Carson in this first part of Pass 3:

    Pass 2 Rating Gains/Losses Pass 3 Part 1 Rating
    945 - 50 + 0 + 0 + 0 + 30 + 0 + 0 + 20 + 30 + 0 + 30 + 0 + 45 + 0 - 7 =1043


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

  2. Given the Pass 3 Part 1 rating calculated above, the second part of Pass 3 looks very similar to the part of Pass 2 that deals with rated players where we calculate their Pass 2 Adjustment.
    1. First, we calculate the points gained in Pass 3 Part 1. Points gained is simply the difference between the Pass 3 Part 1 Rating and the Pass 2 Rating of a player:

      ρi3 = pi3 - Pi2
      where,

      Symbol Universe Description
      Pi2 Pi2+ the Pass 2 Rating for the i-th player.
      pi3 pi3+ the Pass 3 Part 1 rating for the i-th player. (Note that since this is an intermediate result, we are using a lower case p instead of the upper case P that we use to indicate final result from each pass of the ratings processor.
      ρi3 ρi3 the points gained by the i-th player in this tournament in Pass 3.
      i i[1,56] the index of the player under consideration. i can be as small as 1 or as large as 56 for this tournament.

  3. Pass 3 points gained, ρi3, is then used to calculate the Pass 3 Part 2 Rating in the following way:
    1. If a player gained less than 50 points (exclusive) in Pass 3 Part 1, then we set that player's Pass 3 Part 2 Rating to his/her Pass 2 Rating.
    2. If a player gained between 50 and 74 (inclusive) points in Pass 3 Part 1, then we set the player's Pass 3 Part 2 Rating to his/her Pass 3 Part 1 Rating.
    3. If a player gains 75 or more points (inclusive) in Pass 3 Part 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 3 Part 2 Rating is the average of his/her Pass 3 Part 1 Rating and the average of his/her opponents rating from the best win and the worst loss, represented using the formula below:

        αi3 = pi3 + Bi3 + Wi3 2 2

        where αi3 is the Pass 3 Part 2 Rating for the current player, pi3 is the Pass 3 Part 1 Rating, Bi3 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 rating is used as his/her Pass 3 Part 2 Rating:
        αi3 = {pk3}

        where pk3 is the Pass 3 Part 1 Rating of the i-th player's opponent from the k-th match.

      • Therefore, the Pass 3 Part 2 Rating for Keith Carson is calculated as follows:
        • Given the Pass 2 Rating of 945,
        • and the Pass 3 Part 1 rating of 1043,
        • the Pass 3 Part 1 gain is 1043 - 945 = 98.
        • Since the Pass 3 Gain of 98 is greater than 74, and Keith Carson has won some matches and lost some matches, the Pass 3 Part 2 Rating is calculated to be the average of the Pass 3 Part 1 Rating and the average of the Best win and the worst loss:

          1043 + 1178 + 651 2 2 =978.75


      The Pass 3 Part 2 rating ends up becoming the final Pass 3 rating (also referred to as the Pass 3 Adjustment) except as follows:
      • In the cases where the Pass 3 Part 2 rating is less than the players' initial rating Pi0, the Pass 3 rating is reset back to that players initial rating. Keith Carson's Pass 3 Part 2 Rating came out to 979. Since this value is greater than Keith Carson's initial rating of 945, his Pass 3 Adjustment is set to his Pass 3 Part 2 Rating of 979.

      • It is possible for the admin of this tournament to override the Pass 3 Adjustment calculated above with a value they deem appropriate. Keith Carson does not have a manually overridden value for his Pass 3 Adjustment, therefore the value remains at 979.
      You can click here to view a table of Pass 3 Part 2 Ratings for all the players in this tournament along with any manually overridden values.

Pass 4 Rating
Pass 4 is the final pass of the ratings processor. In this pass, we take the adjusted ratings (Pass 3 Adjustment) of all the rated players, and the assigned rating of unrated players (Pass 2 Rating), and apply the point exchange table to the match results based on these ratings to arrive at a final rating. Looking at Keith Carson's match results and applying the point exchange table, gives us the following result:

Keith Carson's Wins
Winner Loser
Point Spread Outcome Gain Player USATT # Rating Player USATT # Rating
522 EXPECTED 0 Keith Carson 221111 979 Arush Khare 1168953 457
399 EXPECTED 0 Keith Carson 221111 979 Ira Shilimkar 1171503 580
256 EXPECTED 0 Keith Carson 221111 979 Chema Estreller 271028 723
115 UPSET 25 Keith Carson 221111 979 Jon Redman 14969 1094
399 EXPECTED 0 Keith Carson 221111 979 Ira Shilimkar 1171503 580
256 EXPECTED 0 Keith Carson 221111 979 Chema Estreller 271028 723
65 UPSET 16 Keith Carson 221111 979 Juli Blass 86407 1044
105 UPSET 20 Keith Carson 221111 979 Steve Glasberg 263095 1084
105 UPSET 20 Keith Carson 221111 979 Steve Glasberg 263095 1084
199 UPSET 40 Keith Carson 221111 979 Mark Morton 218433 1178

Keith Carson's Losses
Winner Loser
Point Spread Outcome Loss Player USATT # Rating Player USATT # Rating
48 UPSET -13 Niko Garcia 1150969 931 Keith Carson 221111 979
222 EXPECTED -1 Jeron Triggs 96237 1201 Keith Carson 221111 979
263 EXPECTED -0 Jeremy Nichols 270797 1242 Keith Carson 221111 979
501 EXPECTED -0 Aditya Bhaskar Pandit 218523 1480 Keith Carson 221111 979
9 UPSET -8 Carl Novick 1171501 970 Keith Carson 221111 979

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 players' Pass 3 Adjustment, we apply the point exchange table from above and show the ratings points earned and lost by Keith Carson in the "Gain" and "Loss" columns. 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 4 Rating for Keith Carson:

Pass 3 Rating Gains/Losses Pass 4 Rating
979 - 13 + 0 + 0 + 0 + 25 + 0 + 0 + 16 + 20 - 1 + 20 + 0 + 40 + 0 - 8 =1078

You can click here to view a table of Pass 4 calculations for all the players in this tournament.