R-Mobility
R-mobility chess is a chess variant that completely removes draws from engine chess, and greatly reduces the number of draws in over the board games. In its simplest form, the side with the position with the fewest legal moves since the last pawn move or capture loses. Here not being in check counts as an extra ½ a legal move, and the first position achieving the eventual minimum takes precedence.
See OTB protocol below for practical play using a physical board and a score sheet.
Contents
Overview
The concept of r-mobility, also known as reset-mobility, is a generalization of mate and stalemate: it counts the number of legal moves available in a position and encourages you to force your opponent to have as few legal moves as possible. In classical chess tournaments where r-mobility is used as a tiebreak, it offers a way to quantify the advantage a player had in drawn games.
The goal of r-mobility is to achieve a position after the last pawn move or capture where your opponent has as few legal moves available as possible. Not being in check is counted as an additional ½ move. A pawn move or capture resets the goals achieved so far, and only positions after it are considered.
When fewer legal moves are available to a player, this results in a worse scoring.
Being in check is considered worse than not being in check.
- Mate is 0 legal moves and in check, and thus is the worst possible result for the side being mated and the best possible result for the side doing the mating.
- Stalemate is 0 legal moves and not in check, and thus is the 2nd worst possible result for side being stalemated and the 2nd best possible result for the side doing the stalemating.
- 1 legal move in check is 3rd worst/best result.
- 1 legal move not in check is 4th worst/best result and so on.
The first side to achieve a best goal gets it.
A short syntax for the best goal achieved so far is the G-score ±Gn.m, where n is the number of legal moves and m is 0 if in check and 5 if not in check. A minus sign indicates that the best goal was achieved by black. A game still ends according to the chosen rules of classical chess, only that now also for classical draws a winner can be decided based on the best achieved goal. Here, if a game ends due to the 50-move rule, then the position triggering the 50-move rule is ignored unless it is stalemate. Once a game has ended the game result according to r-mobility is the best G-score achieved since the last pawn move or capture:
- ±G0.0 = mate
- ±G0.5 = stalemate
- ±G1.0 = 1 legal move and in check
- ±G1.5 = 1 legal move and not in check
The reason the position triggering the 50-move is ignored is that otherwise one can make an unprotected adjacent check and get a good r-mobility result without the opponent being able to capture the checking piece.
Point scoring
A single game of r-mobility chess can always be scored as 1-0 or 0-1, see the Full points only version below. However, for tournaments it makes sense to award more points for mates than for G3.5, for example, and so a formula that shares the full point between the players, awarding between 1 and 0.5 points to the r-mobility winner, depending on how low an r-mobility was achieved, can be employed. One such example formula is given in TCEC below.
Tiebreaks
The most practical application of r-mobility is as a tiebreak criterion in classical chess tournaments. Here the points from classical chess, together with higher priority tiebreakers, still decide the tournament ranking, but for players that remain tied, r-mobility can be used to rank tied players. For example, the sum of all their r-mobility points may act as a tiebreak criterion, and this has in the past been employed at TCEC. Alternatively, as the first (additional) tiebreak the number of stalemates achieved minus the number of stalemates suffered can be used, with the second and third tiebreaks the analogous differences in G1.0 and G1.5, and so on.
50-move rule
The application of the 50-move rule, or not, is completely independent from the rules of r-mobility chess. In theory, especially in engine games, it could be ignored completely. Then a game only ends with mate, stalemate or 3-fold repetition. However, for purely practical purposes it in general makes sense to employ either the FIDE or the ICCF 50-move rules, or an even more general 50-move rule. Note that the ICCF rules allow for so-called "cursed wins" in positions with 7 pieces or fewer, where a mate can be forced, but it would run afoul of the 50-move rule. An even more general 50-move rule would allow "cursed wins" in any position, as long as the mate can be proven. Moreover, for r-mobility chess moves that reset the 50-move counter could be extended from just pawn moves and captures to also include moves that improve the current G-score. Finally, if a game does end due to a 50-move rule, then the final position is ignored for the purposes of computing the best achieved G-score, unless it is stalemate, as explained above.
Basic endgames
KPvK
| |
| Position after 1. Ke6 |
Mate or stalemate unless the pawn can be captured. Diagram for common stalemate result.
KBvK
| |
| Position after 1. Bf5 |
Usually G1.5. For example:
1. Bf5 (diagram)
Subgoal is to get positions where the lone king is forced further towards edge.
The game continues with
1... Kh5 2. Kf4 Kh6 3. Kg4 (diagram)
Now black king has only 1 legal move and is not in check (G1.5).
Players can continue game until a threefold happens or 50-move rule kicks in, but unless black blunders, white will never achieve a better result (stalemate, or one legal move while in check) and thus players might as well agree on the game result here.
| |
| Position after 3. Kg4 |
Stalemate is possible if the the lone king is near any corner and the other king is nearby in a suitable position.
1. Kf6 (diagram) Kf8 2. Bd7 Kg8 3. Be6+ Kh7 4. Bf7 Kh8 5. Kg6
| |
| Position after 1. Kf6 |
KNvK
| |
| Position after 1. Nd5 |
Usually G3.5 (diagram) but G1.5, G2.5 and stalemate are possible depending position. Rarely G2.0, G3.0 and G1.0.
KvK
| |
| Position after 1. Ke5 |
Usually G5.5, example:
1. Ke5 (diagram) Kg6 2. Ke6
Threefold will soon follow, as neither side is able to improve on the result.
G3.5, G8.5, G2.5, G1.5 and G4.5 results also possible depending on position.
With G8.5 or -G8.5 position (diagram) if either side tries to go closer to other king, will then lose to G5.5, for example in diagram position after
1. Kd5 Kf5
white has 5 legal moves and thus black wins game with -G5.5.
| |
| Position after 1... Kxg5 |
Thus both players have to keep distance and threefold follows soon and result stays at -G8.5.
KNNvK
Stalemate unless a knight can be captured or mate in 1 is available.
TCEC
The TCEC rules include r-mobility points as a tiebreak criterion. The only time it was needed so far was in the Top 4 engine bonus of Season 18, see the final standings at [1].
A script to add the RMobilityResult tag pair to all the games in a PGN file is at [2].
The script needs the python-chess library.
Usage: python parallel_tag_rmobility.py input.pgn output.pgn
At present the script ignores possible game continuations after a game has been adjudicated as draw, apart from scoring cursed wins as ±G0.0 for tablebase draw adjudications. That is, apart from cursed wins the best G-score of the game as played on TCEC is used. Recall that at present a game ends in a draw at TCEC either by 3-fold repetition, by the TCEC drawrule adjudication, by 6-men TB draw adjudication or by the 50-moves rule. In future, once r-mobility tablebases become available, these will be used to score adjudicated draws correctly.
The r-mobility tiebreak criterion used at TCEC is the number of r-mobility points scored over all games played in the event. For a game with the final r-mobility result ±Gn.m, the awarded points for the winner/loser are computed using the formula ½±½^(1+2*(n.m)), yielding the following values:
| Result | Winner | Loser |
|---|---|---|
| G0.0 | ½+1/2=1.000000 | ½-1/2=0.000000 |
| G0.5 | ½+1/4=0.750000 | ½-1/4=0.250000 |
| G1.0 | ½+1/8=0.625000 | ½-1/8=0.375000 |
| G1.5 | ½+1/16=0.562500 | ½-1/16=0.437500 |
| G2.0 | ½+1/32=0.531250 | ½-1/32=0.468750 |
| G2.5 | ½+1/64=0.515625 | ½-1/64=0.484375 |
| G3.0 | ½+1/128=0.507812 | ½-1/128=0.492188 |
| G3.5 | ½+1/256=0.503906 | ½-1/256=0.496094 |
| G4.0 | ½+1/512=0.501953 | ½-1/512=0.498047 |
| G4.5 | ½+1/1024=0.500977 | ½-1/1024=0.499023 |
| G5.0 | ½+1/2048=0.500488 | ½-1/2048=0.499512 |
| G5.5 | ½+1/4096=0.500244 | ½-1/4096=0.499756 |
| G6.0 | ½+1/8192=0.500122 | ½-1/8192=0.499878 |
| G6.5 | ½+1/16384=0.500061 | ½-1/16384=0.499939 |
| G7.0 | ½+1/32768=0.500031 | ½-1/32768=0.499969 |
| G7.5 | ½+1/65536=0.500015 | ½-1/65536=0.499985 |
| G8.0 | ½+1/131072=0.500008 | ½-1/131072=0.499992 |
| G8.5 | ½+1/262144=0.500004 | ½-1/262144=0.499996 |
| G9.0 | ½+1/524288=0.500002 | ½-1/524288=0.499998 |
| G9.5 | ½+1/1048576=0.500001 | ½-1/1048576=0.499999 |
| G10.0 | ½+1/2097152=0.500000 | ½-1/2097152=0.500000 |
| G10.5 | ½+1/4194304=0.500000 | ½-1/4194304=0.500000 |
Full points only version
The result of a game of r-mobility chess can also always simply be scored as 1-0 or 0-1, depending on which side achieved the best goal. Due to the likely inherent advantage of the side to move first, a komi as used in Go can be employed. Here a value of ¼ can be added to the G-score that white needs to achieve in order to win. For example if G3.75 is used as komi, then white has to get G0.0, G0.5, G1.0, ... or G3.5 to win, otherwise black wins the game. In this case the result would also always be 1-0 or 0-1, and there are no draws.
OTB games
For over the board games one could limit the application of the r-mobility rules to 1-3 legal moves, and use the standard FIDE rules for when a game ends. Here an alternative for the 50-move rule would be to start counting the moves for the 50-move rule only from the last record on the score sheet.
OTB protocol:
- The G-score of a position is the number of legal moves, plus 0.5 if not in check.
- If a move leads to a position with a G-score ≤ 3.5, we record it on the score sheet, but only if this is a new lowest G-score since the last pawn move/capture.
- A finished game is a draw if there are no recorded G-scores since the last pawn move/capture, otherwise the side with the most recent G-score loses. The best possible result is of course mate, G0.0.
Whole game mobility
Whole game mobility chess is a precursor of r-mobility chess and was invented by Aloril in January 2017. For whole game mobility (wg-mobility) all positions in a game count, and not just those after the most recent pawn move or capture. Tablebases for 2-5 pieces have been generated successfully for wg-mobility, and the obtained statistics, including records for longest proven stalemate, G1.0, G1.5 and so on, are available at [3]. R-mobility is often the same, but there are crucial differences and some games are longer and more natural for r-mobility.
References
- A talkchess thread on r-mobility from December 2020: [4].
