Reset-Mobility Chess

[Time]

Regarding the fifty-move rule, my recollection is that the TCEC nicks windfishballad and the variant creator Aloril found a solution to this problem, if you look through the relatively short message history of the #rmob-live TCEC Discord sub-channel. The relevant discussion seems to start here, and it seems that the issue of three-folds has also been discussed. It seems that the exact technical way to solve this is not important (until the issue of perpetuals has been considered - if it is to be considered an issue, which we might need more empirical data mainly from opening theory and endgame theory to answer), because it is rather obvious that a fair solution should exist.

As for perpetuals, I briefly looked in some books (Play the French 3rd edition p. 99 (one perpetual), Instructive Modern Chess Masterpieces pp. 27-28 (three perpetuals) and p. 36 (one perpetual), and Secrets of Attacking Chess p. 12 (one perpetual) and p. 24 (one perpetual)). In all of these examples, the perpetual checks result in a negative r-mobility score for the player forcing it under my suggested modification. Most often, this player only has checks in the last segment. In one of Stohl's lines in Adams-Agdestein 1994, the side giving perpetual (white) has non-checks, but in the worst of these for black, black has more legal moves than white in the worst position for white with white to move.
[pgn][Event "Oslo m"]
[Site "Oslo"]
[Date "1994.??.??"]
[Round "2"]
[White "Adams, Michael"]
[Black "Agdestein, Simen"]
[Result "1-0"]
[WhiteElo "2660"]
[BlackElo "2595"]
[Annotator "Tisdall,J"]
[PlyCount "79"]
[EventDate "1994.??.??"]

1. e4 Nf6 2. Nc3 d5 3. e5 Ne4 4. Qf3 Nxc3 5. dxc3 c6 6. Bf4 Bf5 7. O-O-O e6 8.
Qg3 h6 9. h4 Qa5 10. Kb1 b5 11. Nf3 c5 12. h5 Na6 13. Nh4 Bh7 14. Qg4 b4 15.
Bd2 c4 16. Ng6 Rb8 17. Rh3 Be7 (17... Qa4 18. b3 (18. Bxc4 dxc4 19. Qxc4 Rg8
20. cxb4 Qb5 21. Qxb5+ Rxb5 22. Nxf8 Rxf8 23. Ra3 Nxb4 24. Bxb4 Rxb4 {-G30.5} {
"White has at least a perpetual and even probably more than that." - Stohl p.
27} 25. Rxa7 {G25.5} Be4 26. Rdd7 Bxg2 {-G27.5, reset} (26... Rc4 27. Re7+ Kd8
28. b3 Rxc2 29. f3 Bh7 30. Ka1 g5 31. Red7+ Kc8 32. Re7) 27. Rac7 {G28.5} Bd5 {
-G21.5} 28. Re7+ Kd8 29. c3 {G30.5, reset} Rf4 {-G20.5} (29... Rb8 30. Red7+ Ke8
) 30. Red7+ Ke8 31. b4 Rxf2 (31... Rc4 32. Ra7 Rxc3 33. Re7+ Kd8) 32. b5) 18...
bxc3 19. Rxc3 Bb4 20. bxa4 Bxc3+ 21. Kc1 {G30.5} Bb2+ 22. Kb1 Bxg6 (22... f5)
23. hxg6 {G31.5 reset} Ke7 (23... O-O {-G34.5} 24. Bxc4 dxc4 25. Qxc4 Rfd8 {
-G41.5} 26. gxf7+ Kxf7 27. Qxa6) 24. Bxc4 (24. c3 {G33.5} Ba3+ 25. Ka1) 24...
dxc4 25. Qxc4 Rhd8 26. c3 Ba3+ 27. Ka1 Nc5 28. a5) (17... b3 18. cxb3 cxb3 19.
axb3 Rxb3 20. c4 Nb4 21. Bxb4 Rxb4 22. Bd3 Rxb2+ 23. Kxb2 Qa3+ 24. Kb1 Qb3+ 25.
Kc1 (25. Ka1) 25... Qc3+ 26. Kb1) 18. Bxc4 dxc4 19. Qxc4 Qc5 20. Qxc5 Nxc5 21.
Nxh8 Ne4 22. Be1 bxc3 23. b3 Bb4 24. Rf3 Nd2+ 25. Bxd2 Rd8 26. Nxf7 Rxd2 27.
Rc1 Ba3 28. Nd6+ Kd7 29. Rxc3 Bxc1 30. Kxc1 Rxf2 31. g4 Rg2 32. Rc4 Re2 33. Rc5
Rg2 34. Nb5 Rxg4 35. Rc7+ Kd8 36. Rxa7 Rg5 37. Nd4 Kc8 38. Nxe6 Rxh5 39. Rxg7
Be4 40. Rg8+ 1-0

[/pgn]
Searching online, I quickly found Carlsen-Nepomniachtchi, WCh 2021, round 4, where the situation is better for the side forcing a perpetual under different modifcation suggestions to r-mobility rules:
[pgn][Event "WCh 2021"]
[Site "Dubai UAE"]
[Date "2021.11.30"]
[Round "4"]
[White "Carlsen, M."]
[Black "Nepomniachtchi, I."]
[Result "1/2-1/2"]
[ECO "C42"]
[WhiteElo "2855"]
[BlackElo "2782"]
[PlyCount "73"]
[EventDate "2021.??.??"]

1. e4 e5 2. Nf3 Nf6 3. Nxe5 d6 4. Nf3 Nxe4 5. d4 d5 6. Bd3 Bd6 7. O-O O-O 8. c4
c6 9. Re1 Bf5 10. Qb3 Qd7 11. Nc3 Nxc3 12. Bxf5 Qxf5 13. bxc3 b6 14. cxd5 cxd5
15. Qb5 Qd7 16. a4 Qxb5 17. axb5 a5 18. Nh4 g6 19. g4 Nd7 20. Ng2 Rfc8 21. Bf4
Bxf4 22. Nxf4 Rxc3 23. Nxd5 Rd3 24. Re7 Nf8 25. Nf6+ Kg7 26. Ne8+ Kg8 27. d5 a4
28. Nf6+ Kg7 29. g5 a3 {-G34.5} 30. Ne8+ Kg8 {-G30.5} 31. Nf6+ Kg7 32. Ne8+ (
32. Kg2 a2 33. h4 {G25.5} Rd2 {G36.5} 34. Ne8+ Kg8 35. Nf6+) 32... Kg8 33. Nf6+
{The game was drawn here, since 33...Kg7 would have been a three-fold
repetition. We continue the game a few more moves in order to explore
r-mobility scoring with a rule modification for perpetual checks.} (33. h4)
33... Kg7 $8 {-G34.5 Note that it would be wrong under rule modifications to
claim a draw at the point where Nepomniachtchi did, since the three-fold in
question does not involve a check.} 34. Kg2 (34. Ne8+ {Different scoring
scenarios: (a) Standard r-mobility: White achieves G2.0, having claimed a
three-fold after 33...Kg7. (b) Checks in the last reset segment by the side
giving a check in the position repeated thrice are disregarded. Black claims a
three-fold after 34.Ne8+, referring to the first of the two three-fold having
occurred if black plays 34...Kg8. (FIDE rules would here stipulate that black
can only claim a three-fold of 34...Kg8 is a three-fold. One could imagine a
scenario where black can only make moves not leading to a three-fold position.
For simplicity, one can assume that black can still claim a three-fold for the
previous three-fold position, since this is most in the spirit of the
intention of the rule modification.) The game ends -G30.5. (c) GM Larry
Kaufman's suggestion (as I understand it): The positions where a check is
given thrice are disregarded for rmobility purposes. Here, one counts
two-folds, so that all the checks leading up to the three-fold which would
have been three-folds themselves eventually, are also disregarded: (The
optional case that it would apply also to two-folds without a check given is
problematic since, if one assumes no fifty-move rule, which e.g. a tablebase
generation does, "all" positions become two-folds.) Black claims a three-fold.
The game ends -G30.5. (d) All moves in the last segment by the side giving a
check in the perpetual position are disregarded. The game ends in -G30.5. Even
if black generally had a worse mobility than white in the last segment, he
would have "won" the game with his best mobility value in the last segment.}) (
34. h4 h5 35. Kg2 Nh7 36. Nxh7 (36. Ne8+ Kf8) 36... Kxh7 37. Rxf7+ Kg8 38. Rd7
(38. Rf6 Rxd5 39. Rxg6+ Kh7 40. Rxb6 a2 41. Ra6 Rxa6 42. bxa6 Ra5 43. Kf3 Rxa6
44. Ke4 Kg6 45. Kd3 Kf5 46. Kc2 Kg4 47. Kb2 Kxh4 48. g6 Rxg6) 38... Rd2 39. d6
a2) 34... a2 35. h4 {G25.5} (35. Ne8+) 35... Rd2 {-G36.5} 36. Ne8+ Kg8 {-G32.5}
37. Nf6+ 1/2-1/2
[/pgn]
Again, one probably needs more data (openings, endgames, and top level games, and above all engine analysis) to know exactly which solution is best. It might be that white's (whom I assume has an objective advantage r-mobility-wise, with rule modifications or otherwise) potential for perpetuals in openings and middlegames is not that big, and that "status quo" rulewise is best.

[Lazy_Frank]

I have two very strong arguments against r-mobility chess:

1. R-mobilty chess changes the scoring system -> changes the main goal of the game.
2. "Goal of reset-mobility is to restrict your opponent mobility in legal moves as much as possible".
Approach "restrict opponent mobility" can be false in chess.
As we can see from Stockfish games, Stockfish doesn't care to restrict opponent mobility.
Instead Stockfish cares about how to maximize own mobility as much as possible, only at the end (when position is winning) restrict opponent king (only piece).

[lkaufman]

I have two very strong arguments against r-mobility chess:

1. R-mobilty chess changes the scoring system -> changes the main goal of the game.
2. "Goal of reset-mobility is to restrict your opponent mobility in legal moves as much as possible".
Approach "restrict opponent mobility" can be false in chess.
As we can see from Stockfish games, Stockfish doesn't care to restrict opponent mobility.
Instead Stockfish cares about how to maximize own mobility as much as possible, only at the end (when position is winning) restrict opponent king (only piece).

Your second point is the reason that I favor using the relative mobility of the two sides (preferably measured when no one is in check) for tiebreaking calculations rather than the mobility of just one side. Good point. Regarding the first point, either the scoring system or the initial position has to be changed to avoid 100% draws in top engine competitions or in correspondence chess using such engines. There are arguments for both approaches.