 for  USATT#: 14272

##### Introduction
This page explains how Marcos Madrid (USATT# 14272)'s rating went from 2106 to 2190 at the 1999 NA Teams 2 held on - 28 Nov 1999. 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 Marcos Madrid for this tournament.

Initial Rating Pass 1 Pass 2 Pass 3 Final Rating (Pass 4)
2106 2158 2106 2169 2167

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 721 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 ${X}_{i}^{3}$, 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 1999 NA Teams 2 held on - 28 Nov 1999 for Marcos Madrid, and its source tournament are as follows:
Initial Rating From Tournament Start Day End Day
2106 1998 North American Teams Cham n/a 29 Nov 1998

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

Winner Loser
Point Spread Outcome Gain Player USATT # Rating Player USATT # Rating
110 EXPECTED 4 Marcos Madrid 14272 2106 Pete May 8146 1996
366 EXPECTED 0 Marcos Madrid 14272 2106 Ross Novak 8591 1740
370 EXPECTED 0 Marcos Madrid 14272 2106 Wang Lim 58181 1736
31 EXPECTED 7 Marcos Madrid 14272 2106 Raymond C. Mack 7967 2075
202 EXPECTED 1 Marcos Madrid 14272 2106 Jeffrey Lau 18582 1904
0 0 Marcos Madrid 14272 0 Jung Lee 20463 0
86 EXPECTED 5 Marcos Madrid 14272 2106 John Jarema 6975 2020
126 EXPECTED 3 Marcos Madrid 14272 2106 Anthony Wai Hung Kwong 12257 1980
191 EXPECTED 1 Marcos Madrid 14272 2106 Merr W. Trumbore 10675 1915
119 EXPECTED 3 Marcos Madrid 14272 2106 Wei Zhang 18436 1987
136 EXPECTED 3 Marcos Madrid 14272 2106 Andre Scott 49583 1970
0 0 Marcos Madrid 14272 0 Guy Seltensperger 20372 0
15 EXPECTED 7 Marcos Madrid 14272 2106 Rafael A. Flores 63553 2091
240 EXPECTED 0 Marcos Madrid 14272 2106 Brian Farkas 11794 1866
26 UPSET 10 Marcos Madrid 14272 2106 Samson Dubina 9051 2132
180 EXPECTED 2 Marcos Madrid 14272 2106 Lennox Douglas 3212 1926
277 EXPECTED 0 Marcos Madrid 14272 2106 Duc Dinh 18425 1829
90 EXPECTED 4 Marcos Madrid 14272 2106 Wennin Chiu 15820 2016
96 EXPECTED 4 Marcos Madrid 14272 2106 William Brown 4476 2010
54 EXPECTED 6 Marcos Madrid 14272 2106 Lee Bahlman 60104 2052
277 EXPECTED 0 Marcos Madrid 14272 2106 William Horowitz 60907 1829
622 EXPECTED 0 Marcos Madrid 14272 2106 Herbert Hodges 12872 1484
78 UPSET 16 Marcos Madrid 14272 2106 Hiroyuki Hikawa 11762 2184
414 EXPECTED 0 Marcos Madrid 14272 2106 Alex Groyzburg 13007 1692
38 EXPECTED 6 Marcos Madrid 14272 2106 Kavitha Ganapathy 13118 2068
5 UPSET 8 Marcos Madrid 14272 2106 Jo-Annie Gagnon 2500 2111
14 EXPECTED 7 Marcos Madrid 14272 2106 Samir Gaagoa 10015 2092

Winner Loser
Point Spread Outcome Loss Player USATT # Rating Player USATT # Rating
66 UPSET -16 Wally Green 10195 2040 Marcos Madrid 14272 2106
83 UPSET -16 Erica Pilon 2587 2023 Marcos Madrid 14272 2106
46 UPSET -13 Kelvin Siu 10803 2060 Marcos Madrid 14272 2106

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 Marcos Madrid 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 Marcos Madrid:

Initial Rating Gains/Losses Pass 1 Rating
2106 - 16 + 4 + 0 + 0 + 7 + 1 + 0 + 5 + 3 + 1 + 3 + 3 + 0 + 7 + 0 + 10 + 2 + 0 + 4 + 4 + 6 + 0 + 0 + 16 + 0 + 0 + 6 + 8 + 7 - 16 - 13 $=\mathrm{2158}$

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:

${\rho }_{i}^{2}={P}_{i}^{1}-{P}_{i}^{0}$
where,

 Symbol Universe Description ${P}_{i}^{0}$ ${P}_{i}^{0}\in \mathrm{{ℤ}^{+}}$ 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. ${P}_{i}^{1}$ ${P}_{i}^{1}\in \mathrm{{ℤ}^{+}}$ the Pass 1 rating for the $i$-th player. ${\rho }_{i}^{2}$ ${\rho }_{i}^{2}\in ℤ$ 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, ${\rho }_{i}^{2}$ 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 ${\rho }_{i}^{2}$. $i$ $i\in \left[1,\mathrm{721}\right]\cap ℤ$ the index of the player under consideration. $i$ can be as small as $1$ or as large as $\mathrm{721}$ for this tournament and the i-th player must be a rated player.

2. For rated players, Pass 1 points gained, ${\rho }_{i}^{2}$, 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:

$\mathrm{{\alpha }_{i}^{2}}=⌊\mathrm{\frac{\mathrm{{P}_{i}^{1}}+\mathrm{\frac{\mathrm{{B}_{i}}+\mathrm{{W}_{i}}}{2}}}{2}}⌋$

where ${\alpha }_{i}^{2}$ is the Pass 2 Adjustment for the current player, ${P}_{i}^{1}$ is the Pass 1 Rating, ${B}_{i}$ is the rating of the highest rated opponent against which the current player won a match, and ${W}_{i}$ 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:

$\mathrm{{\alpha }_{i}^{2}}=\mathrm{⌊\stackrel{\sim }{\mathrm{\left\{\mathrm{{P}_{k}^{0}}\right\}}}⌋}$

where ${P}_{k}^{0}$ is the initial rating of the player who was the i-th player's opponent from the k-th match.
Symbol Universe Description
$i$ $i\in \left[1,\mathrm{721}\right]\cap ℤ$ the index of the player under consideration. $i$ can be as small as $1$ or as large as $\mathrm{721}$ for this tournament and the i-th player must be a rated player.
$q$ $q\in \left[1,\mathrm{6396}\right]\cap ℤ$ the index of the match result under consideration. $q$ can be as small as $1$ or as large as $\mathrm{6396}$ for this tournament and the q-th match must be have both rated players as opponents.
$g$ $g\in \left[1,5\right]\cap ℤ$ 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.
${P}_{k}^{0}$ ${P}_{k}^{0}\in \mathrm{{ℤ}^{+}}$ initial rating of the i-th player's opponent from the k-th match.

• Therefore, the Pass 2 Adjustment for Marcos Madrid is calculated as follows:
• Given the initial rating of 2106,
• and the Pass 1 rating of 2158,
• the Pass 1 gain is 2158 - 2106 = 52.
• Since the Pass 1 Gain of 52 is between 50 and 74 (inclusive), the Pass 2 rating (also referred to as Pass 2 Adjustment) is set to be the Pass 1 rating.

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 Marcos Madrid'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:

$\mathrm{{P}_{i}^{2}}=⌊\mathrm{\frac{\mathrm{{B}_{i}^{2}}+\mathrm{{W}_{i}^{2}}}{2}}⌋$

where ${P}_{i}^{2}$ is the Pass 2 Rating for the i-th player, ${B}_{i}^{2}$ is the largest Pass 2 Adjustment (best win) of the opponenet against whom the i-th player won a match, and ${W}_{i}^{2}$ 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 $I\left(x\right)$ 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
${P}_{i}^{2}$ ${P}_{i}^{2}\in \mathrm{{ℤ}^{+}}$ the pass 2 rating, of the i-th player in this tournament only applicable to unrated players, where ${P}_{i}^{0}$ is not defined
${B}_{i}^{2}$ ${B}_{i}^{2}\in \mathrm{{ℤ}^{+}}$ the largest of the Pass 2 Adjustments of opponents of the i-th player against whom he/she won a match.
${\alpha }_{k}^{2}$ ${\alpha }_{k}^{2}\in \mathrm{{ℤ}^{+}}$ the Pass 2 Adjustment of the player who was the opponent of the i-th player in the k-th match
$I\left(x\right)$ $I:ℤ↦\mathrm{{ℤ}^{+}}$ a function that maps all integers to one of the values from -- 0, 1, 5, 10.
${M}_{i}$ ${M}_{i}\in \mathrm{{ℤ}^{+}}$ total number of matches played by the i-th player in this tournament
k $k\in \mathrm{\left[0,\mathrm{{M}_{i}}-1\right]\cap {ℤ}^{+}}$ The index of the match of the i-th player ranging from 0 to ${M}_{i}-1$
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 $I\left(x\right)$ is defined above and,

Symbol Universe Description
${P}_{i}^{2}$ ${P}_{i}^{2}\in \mathrm{{ℤ}^{+}}$ the pass 2 rating, of the i-th player in this tournament only applicable to unrated players, where ${P}_{i}^{0}$ is not defined
${W}_{i}^{2}$ ${W}_{i}^{2}\in \mathrm{{ℤ}^{+}}$ the smallest of the Pass 2 Adjustments of opponents of the i-th player against whom he/she lost a match.
${\alpha }_{k}^{2}$ ${\alpha }_{k}^{2}\in \mathrm{{ℤ}^{+}}$ the Pass 2 Adjustment of the player who was the opponent of the i-th player in the k-th match
$I\left(x\right)$ $I:ℤ↦\mathrm{{ℤ}^{+}}$ a function that maps all integers to one of the values from -- 0, 1, 5, 10.
${M}_{i}$ ${M}_{i}\in \mathrm{{ℤ}^{+}}$ total number of matches played by the i-th player in this tournament
k $k\in \mathrm{\left[0,\mathrm{{M}_{i}}-1\right]\cap {ℤ}^{+}}$ The index of the match of the i-th player ranging from 0 to ${M}_{i}-1$
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 Marcos Madrid is a rated player, his Pass 2 Adjustment of 2158 will be ignored, along with him Pass 1 Rating of 2158 and his Pass 2 Rating will be set to his initial rating of 2106 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 Marcos Madrid has an initial rating of 2106, 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 Marcos Madrid's wins and losses and applying the point exchange table, gives us the following result: