# Jewdan's Handicap System

Path: <= Handicap =>
Keywords: Rules, Variant

## Description

This is a mathematically designed handicap system intended to even the odds as correctly as is possible between players of disparate ratings. This article is an explanation of the derivation of the method, followed first by the resulting formulae and then by tables showing the appropriate handicaps for 19x19, 13x13, and 9x9 boards given different counting methods. In this way, players that are more than nine ratings separated may be able to play fair games by playing appropriate handicaps on smaller board sizes, and players that wish to play on smaller (or larger) board sizes while still within nine ranks of each other can find proper equivalents.

The purpose of this page is two-fold. First, I think I have created a decent handicap system, and I would like to share it. Second, I need feedback to know whether it truly is decent. If you have feedback, please add a comment to the comments section I have added at the bottom.

Please try it out, and let me know what you think! I would also welcome corrections to the algorithm if something is fundamentally wrong.

## Disclaimer

I do not think that I can claim ownership of this handicap system in the sense that I created it entirely on my own. I did some research into handicap systems, took information I found useful, and compiled it into what you see here. The reason I put my name on it is that I reworked the formulae I found to work (theoretically) for any board size.

I assume that readers know about komi and the standard handicap system, so detailed exploration of these topics is left out. I discuss them insofar as I believe it to aid the reader in following my reasoning.

In places where I got information from another source, but the information seems like common sense, I did not bother to cite the source. If it is information that I could not have arrived upon myself, I have cited it in the text. Both cited and uncited sources that helped in the development of this system have been provided in the "Sources" section at the bottom of the article.

## Derivation

### Analysis of 19x19 Handicaps

Even game is a term that has three possible definitions. The first is a game in which each player has equal odds of winning the game. The second is a game played between two players of equal ability. The final puts the two together: the rules for a game in which two players of equal ability have equal chances of winning the game.

Statistics have shown that, without any special rules, the player who takes Black will beat White by a significant margin if they have equal ability. While it is impossible to calculate precisely what this margin is, trial and error have shown that currently the margin of victory is 6 points under territory counting and 7 under area counting. Because of this, White is given that many points as compensation (komi), plus an extra half-point. The reason for the half point is to acknowledge that White has managed to best Black if he can hold Black off well enough to tie him in score.

It is important to note that, even in an "even game", a handicap system is being used. A precise measurement of the handicap will be discussed in a moment.

When two players differ in ability, the weaker player places extra stones on the board. In a sense, White passes to Black a number of times equal to one less than the number of stones Black places on the board, so that Black gains an advantage to offset his weakness. In a situation in which Black takes two stones against White, Black gets one free move before White gets to counter Black's second. This is commonly interpreted as each of Black's free moves being twice as effective as the final one, since White gets to respond to that move. This thinking is used to derive the standard value of a stone, which is believed to be twelve points in territory counting, and fourteen points in area counting. Revisiting komi, this means that standard komi is defined as half a stone's value plus an additional half-point.

Using the value of stones as a standard, it is easy to assess the handicaps in standard Go. In an even game, Black takes a handicap of -0.5 stones (or White takes a handicap of half a stone). If Black does not give komi, he gets a handicap of half a stone. For each stone he takes after that, his handicap increases by one. In other words, if Black takes four stones against White without giving komi, his handicap is essentially 3.5 stones.

While this system is fine so long as it is used as represented here, the difficulty is that rating systems are based upon differences of whole stones. In other words, a three kyuu is one stone stronger than a four kyuu. This means that, if they played using the standard handicaps and they each played to the best of their abilities, White would be expect to win by half a stone (a Black win of half a stone - a whole stone = a White win by half a stone). This means that the handicap is insufficient. This deficiency stands throughout the system, so White always has half-a-stone's advantage more than the players' ranks dictate.

A good proposal for handling this is for Black to take an extra stone beyond what his rating indicates while continuing to give White half-a-stone's komi. This system, however, gives Black half-a-stone's komi in addition to the current standard handicap placement, which is mathematically equivalent while leading to improved system cohesion. This means that, for a 19x19 game, the following table lists the appropriate handicap stone placement and komi:

Rank Difference | 19x19 Stones | 19x19 Komi
0       |       0      |     6.5
1       |       0      |    -5.5
2       |       2      |    -5.5
3       |       3      |    -5.5
4       |       4      |    -5.5
5       |       5      |    -5.5
6       |       6      |    -5.5
7       |       7      |    -5.5
8       |       8      |    -5.5
9       |       9      |    -5.5

Note that, in a game between two players of the same rating, White gets 6.5 points komi, but when Black takes a handicap, he only gets 5.5 points komi. The reason for this is that the half point is not inherent to a half-stone value; it is given to White so she wins a tie. To make a game even between two players, regardless of the difference in rating, White deserves this half point, so it is subtracted from any reverse komi Black is given.

### Differences Among Board Sizes

The 19x19 handicap system described above is equivalent to other systems proposed by others, and thus is a good starting place. To develop handicap systems for smaller boards, starting with the handicap for an even game is the logical starting point. An analysis of various proposed handicap systems appears to conclude that half-a-stone komi for White in an even game is fair on all standard board sizes, which is reasonable, as the value of a stone in an even game is based upon current understanding of correct play as opposed to the board's size.

However, the impact a stone has on the board is dependent on the board's size. A simple transferral of the 19x19 handicaps to the lower sizes obviously will not work. As an extreme example, taking nine stones and a reverse half-a-stone komi is equivalent to a 108 point handicap, which is twenty-one more points than there are on a 9x9 board. This means that the size of the board is a factor for determining a handicap's value. According to an article on Wikipedia ( http://en.wikipedia.org/wiki/Go_handicaps), free moves are worth 2.5 times more on a 13x13 board than on a 19x19 board and 6 times more on a 9x9 board. This, too, seems reasonable, for though these factors are greater than the ratios between the board sizes (19x19 : 13x13 ≅ 2.14, 19x19 : 9x9 ≅ 4.46), the reduced space on the board means that some techniques can be used on a 19x19 board do not have the space to work on smaller boards and thus are removed from the stronger player's arsenal.

Thus, on a 13x13 board, a new handicap stone should be awarded every two-and-a-half ranks, and on a 9x9 board, a new handicap stone should be awarded every six ranks.

This is a fine starting point, but this ignores the fact that while the differences in player ratings on smaller boards will similarly get smaller, they will still appear. It is reasonable to assume that even though a shodan would give a three kyuu, four kyuu, and five kyuu the same two stones on a 13x13 board, those three players would not be able to use them equally because they are of differing strengths. This means that fine-tuning of the handicap is required through komi as it was on the 19x19 board, but to determine how to fairly set the komi for the smaller boards, a bit more work is required.

### Mathematically Determining Stones and Komi

The trick to determining what is a correct komi given a handicap requires determining precisely how much of an advantage a given number of stones grants Black on a board, comparing that to the amount by which he is expected to lose, then setting the komi so that it properly counteracts the excessive or deficient advantage his stones gives him.

The starting point is calculating the handicap value (please do not confuse the handicap value, as was discussed in the 19x19 Analysis section, with the number of stones Black places on the board, even though they are related). On a 19x19 board, the rating system is simple: a rating difference of one is equal to one stone. On smaller boards, because a free move confers a heftier advantage to Black, the handicap is inversely proportional to the board factor. In other words:

handicap = rating difference / board factor - 0.5

in which the unit for the handicap is stones. Remember that any stones that are not counted as free moves are considered as half a stone's advantage for Black, thus the formula subtracts half a stone. This means that, for a rating difference of two, Black needs a handicap of 0.3 stones on a 13x13 board for the game to be fair, from which it is logical to conclude that, though Black does not need to place any stones, he will need some amount of compensation.

As a sanity check, it is good to prove that this formula works for calculating the handicap level on a 19x19 board. Tautologically, a free move on a 19x19 board has as much impact on a 19x19 board as it does on a 19x19 board, so the board factor for a 19x19 board is 1. Any number divided by 1 is itself, so the formula can be treated as rating difference - 0.5 for 19x19 boards, which is how the handicap value for 19x19 boards was determined earlier.

Next, it is important to determine quantitatively the value of a handicap so that a precise komi value can be determined. This can be done by calculating the expected outcome between two players if they played without handicap stones or komi. If a positive value is used to represent a victory for Black, 0 to represent a draw, and a negative number to represent a victory for White, this value can be calculated by:

expected outcome = stone value * -handicap

in which the units are points. Returning to the example of two players two ranks apart playing on a 13x13 board, if they were to play evenly, it can be expected that White will win by about four points in both territory and area counting (territory: 12 * -(0.3) = -3.6; area: 14 * -(0.3) = -4.2). Intuitively, this would mean that Black would need four points in komi for the result to be a draw.

This formula holds true when compared to the handicaps assigned for even and one-rank-different 19x19 play. The expected outcome when Black gives no komi against an equal player is that he would win by half a stone, and this formula holds true: 12 * -(-0.5) = 6 points with territory counting, and 14 * -(-0.5) = 7 points with area counting. Furthermore, if Black gives no komi to a player one rank better, one would expect White to win by half a stone, and she would: 12 * -(0.5) = -6 points with territory counting, and 14 * -(0.5) = -7 points with area counting.

A formal definition for when to grant Black handicap stones is the next step. It is difficult to explain without the data as an example, so please refer to the following table:

Handicap Value Range | Handicap Stones
-0.5 ≤ n ≤ 0.5       | 0
0.5 < n ≤ 1.5       | 0 (would be 1 if it were considered a valid handicap stone value)
1.5 < n ≤ 2.5       | 2
2.5 < n ≤ 3.5       | 3
3.5 < n ≤ 4.5       | 4
4.5 < n ≤ 5.5       | 5
5.5 < n ≤ 6.5       | 6
6.5 < n ≤ 7.5       | 7
7.5 < n ≤ 8.5       | 8
8.5 < n ≤ 9.5       | 9

Note that handicaps stones jump to one above the whole number portion of the handicap value when the fractional portion of the handicap value raises above half a stone. Accounting for the fact that most handicap systems do not consider a handicap of "one stone" to be very meaningful, this table can be produced by a simple set of equations:

stones = 0 if ceiling(handicap + 0.5) = 1 OR ceiling(handicap + 0.5) otherwise

See the Constants, Functions, and Formulae section for an explanation of the ceiling function.

From this point, it is trivial to calculate how many free moves a handicap stone count has:

free moves = 0 if stones = 0 OR stones - 1 otherwise

At this point there is enough information to calculate what komi a given handicap level should have. Komi should account for the expected outcome and how any free moves help balance the expected outcome, awarding any remaining points to the player who is lacking. Finally, a half a point should be added so that White wins a draw (remember that komi is measured in points awarded to White, so a negative value is reverse komi). Thus, the formula for komi looks like this:

komi = round(expected outcome + free stones * stone value) + 0.5

The reason for rounding the komi before adding half a point is that, in the case of 13x13 Go, it is possible to get fractional points in the calculations. Save for White's half-point, players should only be awarded full points as komi. While this ruins the linearity of the 13x13 handicaps, it should be close enough in play that players should not notice the slight discrepancy.

With these values, if one adds the stone values gained by Black's free moves to the expected outcome and subtracts the komi, one will find that the conditions have been set such that White should win by half a point:

check = round(expected outcome) + free moves * stone value - komi = -0.5

Here is a final proof of the formulae by determining appropriate handicaps on all three board sizes when two players eight ranks apart play using territory counting:

19x19:

handicap = rating difference / board factor - 0.5 = 8 / 1 - 0.5 = 7.5
expected outcome = stone value * -handicap = 12 * -(7.5) = -90
stones = {ceiling(handicap + 0.5) = 1 ? 0 : ceiling(handicap + 0.5} = ceiling(7.5 + 0.5) = 8
free moves = {stones = 0 ? 0 : stones - 1} = 8 - 1 = 7
komi = round(expected outcome + free moves * stone value) + 0.5 = round(-90 + 7 * 12) + 0.5 = -5.5
check = round(expected outcome) + free moves * stone value - komi = round(-90) + 7 * 12 -(-5.5) = -0.5

13x13:

handicap = rating difference / board factor - 0.5 = 8 / 2.5 - 0.5 = 2.7
expected outcome = stone value * -handicap = -32.4
stones = {ceiling(handicap + 0.5) = 1 ? 0 : ceiling(handicap + 0.5} = ceiling(2.7 + 0.5) = 4
free moves = {stones = 0 ? 0 : stones - 1} = 4 - 1 = 3
komi = round(expected outcome + free moves * stone value) + 0.5 = round(-32.4 + 3 * 12) + 0.5 = 4.5
check = round(expected outcome) + free moves * stone value - komi = round(-32.4) + 3 * 12 - 4.5 = -0.5

9x9:

handicap = rating difference / board factor - 0.5 = 8 / 6 - 0.5 = 0.83...
expected outcome = stone value * -handicap = -10
stones = {ceiling(handicap + 0.5) = 1 ? 0 : ceiling(handicap + 0.5} = ceiling(0.83... + 0.5) = 2
free moves = {stones = 0 ? 0 : stones - 1} = 2 - 1 = 1
komi = round(expected outcome + free moves * stone value) + 0.5 = round(-10 + 1 * 12) + 0.5 = 2.5
check = round(expected outcome) + free moves * stone value - komi = round(-10) + 1 * 12 - 2.5 = -0.5

## Constants, Functions, and Formulae

stone value
12 under territory rules, 14 under area rules
board factor
1 for 19x19, 2.5 for 13x13, 6 for 9x9
ceiling
the mathematical function by which a number is rounded up if it has a fractional component; i.e., ceiling(2.1) = 3, ceiling(4) = 4
round
the mathematical function by which a number is rounded up if its fractional component is greater than or equal to 0.5 and rounded down otherwise; i.e., round(2.6) = 3, round(1.3) = 1, round(7) = 7, round(-0.5) = -1

handicap = rating difference / board factor - 0.5
expected outcome = stone value * -handicap
stones = 0 if ceiling(handicap + 0.5) is 1, ceiling(handicap + 0.5) otherwise
free moves = 0 if stones is 0, stones - 1 otherwise
komi = round(expected outcome + free moves * stone value) + 0.5

Therefore, given a rating difference between two players, a rule set, and a board size, it is possible to calculate the handicap stones given Black (the stones formula) and the komi given to White (the komi formula).

## Territory Counting Handicaps Table

Rank Difference | 19x19 Stones | 19x19 Komi | 13x13 Stones | 13x13 Komi | 9x9 Stones | 9x9 Komi
0       |       0      |     6.5    |       0      |     6.5    |      0     |    6.5
1       |       0      |    -5.5    |       0      |     1.5    |      0     |    4.5
2       |       2      |    -5.5    |       0      |    -3.5    |      0     |    2.5
3       |       3      |    -5.5    |       2      |     4.5    |      0     |    0.5
4       |       4      |    -5.5    |       2      |    -0.5    |      0     |   -1.5
5       |       5      |    -5.5    |       2      |    -5.5    |      0     |   -3.5
6       |       6      |    -5.5    |       3      |     1.5    |      0     |   -5.5
7       |       7      |    -5.5    |       3      |    -3.5    |      2     |    4.5
8       |       8      |    -5.5    |       4      |     4.5    |      2     |    2.5
9       |       9      |    -5.5    |       4      |    -0.5    |      2     |    0.5
10       |      ---     |     ---    |       4      |    -5.5    |      2     |   -1.5
11       |      ---     |     ---    |       5      |     1.5    |      2     |   -3.5
12       |      ---     |     ---    |       5      |    -3.5    |      2     |   -5.5
13       |      ---     |     ---    |       6      |     4.5    |      3     |    4.5
14       |      ---     |     ---    |       6      |    -0.5    |      3     |    2.5
15       |      ---     |     ---    |       6      |    -5.5    |      3     |    0.5
16       |      ---     |     ---    |       7      |     1.5    |      3     |   -1.5
17       |      ---     |     ---    |       7      |    -3.5    |      3     |   -3.5
18       |      ---     |     ---    |       8      |     4.5    |      3     |   -5.5
19       |      ---     |     ---    |       8      |    -0.5    |      4     |    4.5
20       |      ---     |     ---    |       8      |    -5.5    |      4     |    2.5
21       |      ---     |     ---    |       9      |     1.5    |      4     |    0.5
22       |      ---     |     ---    |       9      |    -3.5    |      4     |   -1.5
23       |      ---     |     ---    |      ---     |     ---    |      4     |   -3.5
24       |      ---     |     ---    |      ---     |     ---    |      4     |   -5.5
25       |      ---     |     ---    |      ---     |     ---    |      5     |    4.5
26       |      ---     |     ---    |      ---     |     ---    |      5     |    2.5
27       |      ---     |     ---    |      ---     |     ---    |      5     |    0.5
28       |      ---     |     ---    |      ---     |     ---    |      5     |   -1.5
29       |      ---     |     ---    |      ---     |     ---    |      5     |   -3.5
30       |      ---     |     ---    |      ---     |     ---    |      5     |   -5.5
31       |      ---     |     ---    |      ---     |     ---    |      6     |    4.5
32       |      ---     |     ---    |      ---     |     ---    |      6     |    2.5
33       |      ---     |     ---    |      ---     |     ---    |      6     |    0.5
34       |      ---     |     ---    |      ---     |     ---    |      6     |   -1.5
35       |      ---     |     ---    |      ---     |     ---    |      6     |   -3.5

## Area Counting Handicaps Table

Rank Difference | 19x19 Stones | 19x19 Komi | 13x13 Stones | 13x13 Komi | 9x9 Stones | 9x9 Komi
0       |       0      |     7.5    |       0      |     7.5    |      0     |    7.5
1       |       0      |    -6.5    |       0      |     1.5    |      0     |    5.5
2       |       2      |    -6.5    |       0      |    -3.5    |      0     |    2.5
3       |       3      |    -6.5    |       2      |     4.5    |      0     |    0.5
4       |       4      |    -6.5    |       2      |    -3.5    |      0     |   -1.5
5       |       5      |    -6.5    |       2      |    -6.5    |      0     |   -4.5
6       |       6      |    -6.5    |       3      |     1.5    |      0     |   -6.5
7       |       7      |    -6.5    |       3      |    -3.5    |      2     |    5.5
8       |       8      |    -6.5    |       4      |     4.5    |      2     |    2.5
9       |       9      |    -6.5    |       4      |    -3.5    |      2     |    0.5
10       |      ---     |     ---    |       4      |    -6.5    |      2     |   -1.5
11       |      ---     |     ---    |       5      |     1.5    |      2     |   -4.5
12       |      ---     |     ---    |       5      |    -3.5    |      2     |   -6.5
13       |      ---     |     ---    |       6      |     4.5    |      3     |    5.5
14       |      ---     |     ---    |       6      |    -3.5    |      3     |    2.5
15       |      ---     |     ---    |       6      |    -6.5    |      3     |    0.5
16       |      ---     |     ---    |       7      |     1.5    |      3     |   -1.5
17       |      ---     |     ---    |       7      |    -3.5    |      3     |   -4.5
18       |      ---     |     ---    |       8      |     4.5    |      3     |   -6.5
19       |      ---     |     ---    |       8      |    -3.5    |      4     |    5.5
20       |      ---     |     ---    |       8      |    -6.5    |      4     |    2.5
21       |      ---     |     ---    |       9      |     1.5    |      4     |    0.5
22       |      ---     |     ---    |       9      |    -3.5    |      4     |   -1.5
23       |      ---     |     ---    |      ---     |     ---    |      4     |   -4.5
24       |      ---     |     ---    |      ---     |     ---    |      4     |   -6.5
25       |      ---     |     ---    |      ---     |     ---    |      5     |    5.5
26       |      ---     |     ---    |      ---     |     ---    |      5     |    2.5
27       |      ---     |     ---    |      ---     |     ---    |      5     |    0.5
28       |      ---     |     ---    |      ---     |     ---    |      5     |   -1.5
29       |      ---     |     ---    |      ---     |     ---    |      5     |   -4.5
30       |      ---     |     ---    |      ---     |     ---    |      5     |   -6.5
31       |      ---     |     ---    |      ---     |     ---    |      6     |    5.5
32       |      ---     |     ---    |      ---     |     ---    |      6     |    2.5
33       |      ---     |     ---    |      ---     |     ---    |      6     |    0.5
34       |      ---     |     ---    |      ---     |     ---    |      6     |   -1.5
35       |      ---     |     ---    |      ---     |     ---    |      6     |   -4.5