# Handicap

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.

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. There are several ways a handicap could be implemented.

## Contents

## 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.

## 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.

### 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.)