Difference between revisions of "Handicap"

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Playing with '''handicap''' means to give one of the [[player]]s (preferably the weaker one) an advantage at the start of the game. The point of this is to make the game more even, so that it will be challenging for both players.
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Playing with '''handicap''' means to give one of the [[player]]s (preferably the weaker one) an advantage at the start of the game. The point of this is to make the game more even, so that it will be challenging for both players.  
  
In [[Hex]] there is no standard way of playing with handicap, and because of this it is not very common to do so. This ought to be changed.
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There are several ways a handicap could be implemented in Hex. One of the main issues is that the advantage given to one player should be predictable and somewhat quantifiable. A system that is reasonably well-motivated and seems to be gaining popularity is the Demer handicap system, described below. Some other proposals for handicapping systems are also discussed below.
  
There are several ways a handicap could be implemented.
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== Demer handicap system ==
  
== Non-rhombic board ==
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The Demer handicap system is based on the idea of giving the weaker player a certain number of free moves at the beginning of the game. By selectively using the swap rule, the handicap can be given in increments of 0.5 moves. This system was proposed by Eric Demer.
One natural way is to play with an ''m × n'' [[board]] where ''m'' is distinct from ''n'', and let the weaker player have the shortest distance between his sides. Unfortunately, this doesn't work very well, since there exists an easy, explicit winning strategy for the player with shortest distance.
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[[Image:Rhomboid.png]]
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=== Measuring advantage by number of moves ===
  
Here is the winning strategy. Suppose the board is an m by m+1 grid. The diagram shows m=4. The board can be partitioned into two triangular regions of m cells on each side. Now pair each cell in one triangle with a cell in the other triangle, as shown. The pairing is like a mirror image which is shifted slightly. The winning strategy for black is to answer each white move by playing in the corresponding cell in the other triangular region. If black has already occupied the corresponding cell, then it does not matter where black plays.
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The idea behind measuring the handicap in terms of fractions of moves is the following. Consider a game in which Red goes first and the swap rule is not used. Compare this to a game in which Blue goes first and the swap rule is not used. How much better is the first game for Red? The only difference between the two games is that Red plays one extra move at the beginning of the first game. We therefore say that Red has a 1 move advantage in the first game, compared to the second game. It is also clear that the second game is equally good for Blue as the first is for Red. Compared to a theoretically fair game, it therefore makes sense to say that a game without the swap rule gives exactly a 0.5 move advantage to the player who moves first.
  
Suppose the board is filled with stones, and white has a win despite the fact that black followed this strategy. That implies the winning white path must cross the red line at least once. Consider the highest point at which the winning white path crosses the red line. This crossing cannot occur between two white stones on the same horizontal row, since for each such pair of cells, black must have occupied one of them. That implies the crossing from left to right must "go down" from B to A' or C to B' etc. Let us call this pair of cells Y (on the left side of the red line) and X' on the right side. The corresponding cell to Y we will call Y'.
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If the swap rule is used, the game is very close to fair. (In theory, it gives a slight advantage to the second player, but this advantage is small and we will ignore it.) In summary, we now have a game that gives a 0.5 move advantage to Red (Red goes first without swap), and a game that gives a 0 move advantage (the swap rule is used). We can increase any player's advantage by 1 move by giving that player an extra move at the beginning. In games where the swap rule is used, the extra move should be given just after the swap decision has been made (since swapping when there are already two pieces on the board would give a large advantage to the second player).  
  
For white to have a winning path which crosses at this point, there must be a continuous chain of white stones from cell Y to the left white border row. But since black followed the above strategy, this implies there is a continuous chain of corresponding black stones from Y' down to the bottom black border row. Therefore the white chain is blocked from connecting to the right. This contradicts the assumption that white has a win, so black must have a win.
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We therefore arrive at the following handicap system.
  
For other shape grids, m by m+k where k>1, the same strategy can be used, as long as the two triangular regions are adjacent to each other. For cells which lie outside these regions on the left or right, it does not matter how black responds to any white moves in these regions.
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=== Description of the Demer handicap system ===
--[[User:Twixter|David]] 17:10, 8 Oct 2006 (CEST)
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== Start with pieces on the board ==
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For the purpose of the following description, we assume that the the [[Conventions#Swapping|swap-pieces convention]] is used, i.e., when a player swaps, the player keeps the same color, but the board position is mirrored. If the [[Conventions#Swapping|swap-sides convention]] is used instead, the method remains the same but the description must be adjusted accordingly.
Alternatively one can use the kind of handicap used in [[Go]]: The weaker player places a certain number of [[piece|pieces]] on the board as his [[first move]]. A 1-piece handicap is the same as playing ordinary Hex without the [[swap option]]. With a handicap of two or more pieces, the first player either places the stones as he likes, or according to some pre-defined rules.
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The drawback of this option is that even a 1-piece handicap gives a very big [[advantage]]. At least this is true on [[Small boards|smaller boards]] (such as the 10 × 10 board). On larger boards, such as 19 × 19 this may be a good solution, and weak players may even be allowed to place three or four pieces against the strongest players.
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* 0 move advantage for Red (no handicap): Red starts and the swap rule is used. Symbolically: (Red, Blue swaps, Red, Blue, ...) or (Red, Blue, Red, Blue, ...)
  
== First to win N games ==
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* 0.5 move advantage for Red: Red starts and the swap rule is not used. Symbolically: (Red, Blue, Red, Blue, ...)
Another possibility in order to handicap games is to play "First to win N games" to win the match. For the weaker player let the N be less than the stronger's, in order to handicap the match.
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For example, "If you win 5 games, you win the match, but if I win 3, I win the match". --[[User:Gregorio|Gregorio]] 13:33, 13 Oct 2006 (CEST)
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* 1 move advantage for Red: Red gets one additional move before Blue's first non-swap move. Explicitly: Red plays the first piece, then Blue decides whether to swap or not. If Blue swaps, Red gets two consecutive moves. If Blue does not swap, Red gets one additional move. Symbolically: (Red, Blue swaps, Red, Red, Blue, ...) or (Red, Red, Blue, Red, Blue, ...)
  
== No swap rule / fixed openings ==
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* 1.5 move advantage for Red: Red plays the first two pieces and the swap rule is not used. Symbolically: (Red, Red, Blue, Red, ...)
  
It is possible to play Hex without the swap rule. Since this gives a large advantage to the player moving first, it can be used as a means of giving an advantage to the weaker player. A more fine-grained approach is to not use swap, but place the first piece according to how big a handicap the red player needs. The more central the piece, the larger the handicap.
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* 2 move advantage for Red: Red gets two additional moves before Blue's first non-swap move. Symbolically: (Red, Blue swaps, Red, Red, Red, Blue, ...) or (Red, Red, Red, Blue, Red, Blue, ...)
  
For very large handicaps, one could experiment by having a central red piece plus the first blue piece in a bad position as part of the setup (with red to move), placing two red pieces near the edges and letting blue go first etc. By setting up a position beforehand and deciding who is to move, you can create positions arbitrarily balanced towards the one or the other player.  
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For the integral handicaps, i.e., those where the swap rule is used, it is also possible to give the advantage to Blue. This can be done as follows:
  
The price is that one gets a slightly different game, so that it's possible that a player might become especially good at certain common handicap positions. But this should be of no more concern than it is in Go (and much less than in Shogi or Chess). I think it would be worthwhile to work out a standard ladder of handicap positions, sorted according to their bias to red.
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* 0 move advantage for Blue (no handicap): Red starts and the swap rule is used. Symbolically: (Red, Blue swaps, Red, Blue, ...) or (Red, Blue, Red, Blue, ...)
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* 1 move advantage for Blue: Blue gets one additional move before Red's second move. Explicitly: Red plays the first piece, then Blue decides whether to swap or not. If Blue swaps, Blue gets an additional move. If Blue does not swap, Blue gets two consecutive moves. Symbolically: (Red, Blue swaps, Blue, Red, ...) or (Red, Blue, Blue, Red, ...)
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* 2 move advantage for Blue: Blue gets two additional moves before Red's second move. Symbolically: (Red, Blue swaps, Blue, Blue, Red, ...) or (Red, Blue, Blue, Blue, Red, ...)
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The system can theoretically also be used for larger handicaps (2.5 moves, 3 moves, etc.), but such large handicaps probably do not make much sense on small board sizes. For example, on an 11 × 11 board, Red only needs 3 pieces to connect her edges by [[bridge]]s and [[edge template]]s.
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=== Sign convention ===
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By convention, a handicap that benefits Red is specified as a positive number, and a handicap that benefits Blue is specified as a negative number.
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=== Relation of handicap to player strengths ===
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One may ask what the appropriate handicap amount is, given two players' [[Elo rating]]s. There are currently no reliable statistics on this, as handicap games are rare (or even non-existent) on game servers where Elo-rated players play. A very ballpark estimate, based on limited anecdotal evidence, is that a 0.5 move handicap corresponds to a difference of about 250 Elo points on 11 × 11 boards. This means that a 0.5 move advantage increases the odds of winning by a factor of approximately 4. On larger boards, the effect of handicap moves is probably somewhat smaller.
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=== Drawbacks ===
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A drawback of the Demer handicap system is that even a 0.5 move handicap gives the player a relatively large advantage, especially on [[Small boards|smaller boards]].
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=== Handicap 0.5 vs. no swap ===
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Sometimes players choose to play without the swap rule, but without the intention to give an advantage to the weaker player. This typically happens because the players are novices and either don't know about the swap rule, or do not perceive the swap rule as making much difference at their level of play. Some game servers, such as [[PlayOK]], actually offer "no swap" as a game option, but then alternate player colors in subsequent games.
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Although playing without the swap rule is equivalent to playing with handicap 0.5 in the Demer handicap system, such games should not be marked as handicap 0.5 in game records. Instead, the notation "no swap" or "N/S" can be used.
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== Other suggestions for handicap systems ==
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Various other methods for handicapping games have been suggested. The potential advantage of these methods is that they might be able to produce more fine-grained handicaps than the Demer system (i.e., handicaps in increments smaller than 0.5 moves). The disadvantage is that without large-scale testing, it would be difficult to quantify these handicaps, i.e., to figure out exactly how many fractional "moves" each handicap corresponds to.
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=== Non-rhombic boards ===
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A seemingly natural way to give an advantage to a player is to decrease the distance between the player's edges, i.e., to play on an ''m × n'' board where ''m'' is distinct from ''n''. Unfortunately, this doesn't work very well, since there exists an easy, explicit winning strategy for the player with the shorter distance. See [[Hex_theory#Winning_strategy_for_non-square_boards|winning strategy for non-square boards]].
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However, the idea of a non-rhombic board can perhaps be combined with Demer handicaps to arrive at more fine-grained handicaps (i.e., handicaps of less than 0.5 moves). For example, it may make sense to give Red a 1.5 move advantage in exchange for slightly decreasing the distance between Blue's edges. But doing so would require careful calibration, and there is no obvious way to quantify the resulting advantage or disadvantage.
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 +
See [[parallelogram boards]] for an analysis of how much headstart the player with the longer distance needs, for various small non-rhombic board sizes.
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=== Fixed openings ===
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 +
Another possible way to give more fine-grained handicaps is to play without the swap rule, but to place Red's first piece in a pre-defined position (rather than allowing Red to place the piece freely). A piece placed in the center of the board would give Red an advantage of 0.5 moves, whereas a fairly-placed piece (i.e., a piece that Blue would be equally likely to swap or not to swap) would give 0 moves of advantage. If the initial piece is placed somewhere between these two extremes, handicaps between 0 and 0.5 moves can be achieved, although it is difficult to quantify the exact advantage conferred by any particular opening move.
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 +
This method could also be adapted to give handicaps greater than 0.5. For very large handicaps, one could experiment by having a central red piece plus the first blue piece in a bad position as part of the setup (with Red to move), placing two red pieces near the edges and letting Blue go first, etc. By setting up a position beforehand and deciding who is to move, one can in principle create positions that are arbitrarily balanced towards one or the other player.
 +
 
 +
The price is that one gets a slightly different game, so that it's possible that a player might become especially good at certain common handicap positions. But this should be of no more concern than it is in Go (and much less than in Shogi or Chess). It might be worthwhile to work out a standard ladder of handicap positions, sorted according to their bias to Red.
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 +
Some evidence (from the win rate evaluations of AI bots) suggests that c3 and the 2-2 obtuse corner (shown below) give Red approximately a 0.25 move advantage, at least for board sizes 13×13 to 19×19.
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<hexboard size="13x13"
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  coords="show"
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  contents="E *:(c3 b12 k11 l2)"
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  />
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=== First to win N games ===
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When playing matches, rather than individual games, a possible handicap method is to play "First to win N games" to win the match, with different values of N for each player. The weaker player would be expected to win fewer games than the stronger player, to win the match. This method may make sense if the players are relatively similar, but not equal, in strength. For example, for players whose [[Elo rating]] differs by 100, the odds of winning are approximately 7 : 4, so it may make sense to play "you win the match if you win 7 games, but I win the match if I win 4 games".
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(However, the specific numbers one should use for such a handicap are more complicated than that.  For example, with 7 : 4 game-odds, "you win the match if you win 5 games, but I win the match if I win 3 games" both is faster and makes the match-odds closer to 1 : 1.)
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== Progressive board size ==
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For beginners, it is often difficult to win against experienced players even with a large handicap. [[User:Mason|Mason Mackaman]] suggested the following procedure to keep the game enjoyable: Play with handicap 0.5 (i.e., without swap), starting from the smallest board size (even 1x1, if it is available). Each time the beginner wins, increase the board size by 1. The players will typically move through the smallest board sizes very quickly, and then slow down somewhere near 6x6 or 7x7. This ensures that the beginner is always on the verge of winning, and can clearly measure their improvement. This method might be more fun than starting with a large handicap on a typical board size. The steps up in difficulty are also potentially more fine-grained than the half-stone decrements of the Demer system, though no testing has been done to confirm this.
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[[category:Rules and Conventions]]

Latest revision as of 03:15, 7 October 2024

Playing with handicap means to give one of the players (preferably the weaker one) an advantage at the start of the game. The point of this is to make the game more even, so that it will be challenging for both players.

There are several ways a handicap could be implemented in Hex. One of the main issues is that the advantage given to one player should be predictable and somewhat quantifiable. A system that is reasonably well-motivated and seems to be gaining popularity is the Demer handicap system, described below. Some other proposals for handicapping systems are also discussed below.

Demer handicap system

The Demer handicap system is based on the idea of giving the weaker player a certain number of free moves at the beginning of the game. By selectively using the swap rule, the handicap can be given in increments of 0.5 moves. This system was proposed by Eric Demer.

Measuring advantage by number of moves

The idea behind measuring the handicap in terms of fractions of moves is the following. Consider a game in which Red goes first and the swap rule is not used. Compare this to a game in which Blue goes first and the swap rule is not used. How much better is the first game for Red? The only difference between the two games is that Red plays one extra move at the beginning of the first game. We therefore say that Red has a 1 move advantage in the first game, compared to the second game. It is also clear that the second game is equally good for Blue as the first is for Red. Compared to a theoretically fair game, it therefore makes sense to say that a game without the swap rule gives exactly a 0.5 move advantage to the player who moves first.

If the swap rule is used, the game is very close to fair. (In theory, it gives a slight advantage to the second player, but this advantage is small and we will ignore it.) In summary, we now have a game that gives a 0.5 move advantage to Red (Red goes first without swap), and a game that gives a 0 move advantage (the swap rule is used). We can increase any player's advantage by 1 move by giving that player an extra move at the beginning. In games where the swap rule is used, the extra move should be given just after the swap decision has been made (since swapping when there are already two pieces on the board would give a large advantage to the second player).

We therefore arrive at the following handicap system.

Description of the Demer handicap system

For the purpose of the following description, we assume that the the swap-pieces convention is used, i.e., when a player swaps, the player keeps the same color, but the board position is mirrored. If the swap-sides convention is used instead, the method remains the same but the description must be adjusted accordingly.

  • 0 move advantage for Red (no handicap): Red starts and the swap rule is used. Symbolically: (Red, Blue swaps, Red, Blue, ...) or (Red, Blue, Red, Blue, ...)
  • 0.5 move advantage for Red: Red starts and the swap rule is not used. Symbolically: (Red, Blue, Red, Blue, ...)
  • 1 move advantage for Red: Red gets one additional move before Blue's first non-swap move. Explicitly: Red plays the first piece, then Blue decides whether to swap or not. If Blue swaps, Red gets two consecutive moves. If Blue does not swap, Red gets one additional move. Symbolically: (Red, Blue swaps, Red, Red, Blue, ...) or (Red, Red, Blue, Red, Blue, ...)
  • 1.5 move advantage for Red: Red plays the first two pieces and the swap rule is not used. Symbolically: (Red, Red, Blue, Red, ...)
  • 2 move advantage for Red: Red gets two additional moves before Blue's first non-swap move. Symbolically: (Red, Blue swaps, Red, Red, Red, Blue, ...) or (Red, Red, Red, Blue, Red, Blue, ...)

For the integral handicaps, i.e., those where the swap rule is used, it is also possible to give the advantage to Blue. This can be done as follows:

  • 0 move advantage for Blue (no handicap): Red starts and the swap rule is used. Symbolically: (Red, Blue swaps, Red, Blue, ...) or (Red, Blue, Red, Blue, ...)
  • 1 move advantage for Blue: Blue gets one additional move before Red's second move. Explicitly: Red plays the first piece, then Blue decides whether to swap or not. If Blue swaps, Blue gets an additional move. If Blue does not swap, Blue gets two consecutive moves. Symbolically: (Red, Blue swaps, Blue, Red, ...) or (Red, Blue, Blue, Red, ...)
  • 2 move advantage for Blue: Blue gets two additional moves before Red's second move. Symbolically: (Red, Blue swaps, Blue, Blue, Red, ...) or (Red, Blue, Blue, Blue, Red, ...)

The system can theoretically also be used for larger handicaps (2.5 moves, 3 moves, etc.), but such large handicaps probably do not make much sense on small board sizes. For example, on an 11 × 11 board, Red only needs 3 pieces to connect her edges by bridges and edge templates.

Sign convention

By convention, a handicap that benefits Red is specified as a positive number, and a handicap that benefits Blue is specified as a negative number.

Relation of handicap to player strengths

One may ask what the appropriate handicap amount is, given two players' Elo ratings. There are currently no reliable statistics on this, as handicap games are rare (or even non-existent) on game servers where Elo-rated players play. A very ballpark estimate, based on limited anecdotal evidence, is that a 0.5 move handicap corresponds to a difference of about 250 Elo points on 11 × 11 boards. This means that a 0.5 move advantage increases the odds of winning by a factor of approximately 4. On larger boards, the effect of handicap moves is probably somewhat smaller.

Drawbacks

A drawback of the Demer handicap system is that even a 0.5 move handicap gives the player a relatively large advantage, especially on smaller boards.

Handicap 0.5 vs. no swap

Sometimes players choose to play without the swap rule, but without the intention to give an advantage to the weaker player. This typically happens because the players are novices and either don't know about the swap rule, or do not perceive the swap rule as making much difference at their level of play. Some game servers, such as PlayOK, actually offer "no swap" as a game option, but then alternate player colors in subsequent games.

Although playing without the swap rule is equivalent to playing with handicap 0.5 in the Demer handicap system, such games should not be marked as handicap 0.5 in game records. Instead, the notation "no swap" or "N/S" can be used.

Other suggestions for handicap systems

Various other methods for handicapping games have been suggested. The potential advantage of these methods is that they might be able to produce more fine-grained handicaps than the Demer system (i.e., handicaps in increments smaller than 0.5 moves). The disadvantage is that without large-scale testing, it would be difficult to quantify these handicaps, i.e., to figure out exactly how many fractional "moves" each handicap corresponds to.

Non-rhombic boards

A seemingly natural way to give an advantage to a player is to decrease the distance between the player's edges, i.e., to play on an m × n board where m is distinct from n. Unfortunately, this doesn't work very well, since there exists an easy, explicit winning strategy for the player with the shorter distance. See winning strategy for non-square boards.

However, the idea of a non-rhombic board can perhaps be combined with Demer handicaps to arrive at more fine-grained handicaps (i.e., handicaps of less than 0.5 moves). For example, it may make sense to give Red a 1.5 move advantage in exchange for slightly decreasing the distance between Blue's edges. But doing so would require careful calibration, and there is no obvious way to quantify the resulting advantage or disadvantage.

See parallelogram boards for an analysis of how much headstart the player with the longer distance needs, for various small non-rhombic board sizes.

Fixed openings

Another possible way to give more fine-grained handicaps is to play without the swap rule, but to place Red's first piece in a pre-defined position (rather than allowing Red to place the piece freely). A piece placed in the center of the board would give Red an advantage of 0.5 moves, whereas a fairly-placed piece (i.e., a piece that Blue would be equally likely to swap or not to swap) would give 0 moves of advantage. If the initial piece is placed somewhere between these two extremes, handicaps between 0 and 0.5 moves can be achieved, although it is difficult to quantify the exact advantage conferred by any particular opening move.

This method could also be adapted to give handicaps greater than 0.5. For very large handicaps, one could experiment by having a central red piece plus the first blue piece in a bad position as part of the setup (with Red to move), placing two red pieces near the edges and letting Blue go first, etc. By setting up a position beforehand and deciding who is to move, one can in principle create positions that are arbitrarily balanced towards one or the other player.

The price is that one gets a slightly different game, so that it's possible that a player might become especially good at certain common handicap positions. But this should be of no more concern than it is in Go (and much less than in Shogi or Chess). It might be worthwhile to work out a standard ladder of handicap positions, sorted according to their bias to Red.

Some evidence (from the win rate evaluations of AI bots) suggests that c3 and the 2-2 obtuse corner (shown below) give Red approximately a 0.25 move advantage, at least for board sizes 13×13 to 19×19.

abcdefghijklm12345678910111213

First to win N games

When playing matches, rather than individual games, a possible handicap method is to play "First to win N games" to win the match, with different values of N for each player. The weaker player would be expected to win fewer games than the stronger player, to win the match. This method may make sense if the players are relatively similar, but not equal, in strength. For example, for players whose Elo rating differs by 100, the odds of winning are approximately 7 : 4, so it may make sense to play "you win the match if you win 7 games, but I win the match if I win 4 games".

(However, the specific numbers one should use for such a handicap are more complicated than that. For example, with 7 : 4 game-odds, "you win the match if you win 5 games, but I win the match if I win 3 games" both is faster and makes the match-odds closer to 1 : 1.)

Progressive board size

For beginners, it is often difficult to win against experienced players even with a large handicap. Mason Mackaman suggested the following procedure to keep the game enjoyable: Play with handicap 0.5 (i.e., without swap), starting from the smallest board size (even 1x1, if it is available). Each time the beginner wins, increase the board size by 1. The players will typically move through the smallest board sizes very quickly, and then slow down somewhere near 6x6 or 7x7. This ensures that the beginner is always on the verge of winning, and can clearly measure their improvement. This method might be more fun than starting with a large handicap on a typical board size. The steps up in difficulty are also potentially more fine-grained than the half-stone decrements of the Demer system, though no testing has been done to confirm this.