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I'm looking for positions where the game-theoretic perfect move is actually a mistake for any human to play (because it would lead to complications that they can't be expected to handle), and conversely - the most recommended move for a human who wants to maximize their odds, is one that is a game-theoretic error (e.g., turns a forced mate to a draw).

I'm thinking about something along the following lines - black is in a material disadvantage, but managed to get to a position where they have the following moves:

  1. One which gives perpetual check, securing a draw.

  2. One which leads to a forced mate against white; but the mating sequence is too difficult for a human to find, and any deviation would free white to exercise their advantage and win.

While the game-theoretic correct play (and the one that will likely be played by a strong AI) is 2, but any human would be advised to play 1.

It doesn't have to be this exact scenario, but the closer to it, the better. It can be either a constructed position or one from an actual game.

PS. I'm pretty sure I've seen such examples in the past, but a search didn't come up with any results...

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  • An interesting question. "Best move for a human to play" seems a little bit vague, as it may depend on the humans involved.
    – bof
    Commented May 28, 2023 at 1:56
  • 1
    This is a bit tricky in this day in age where we have end game tablebases (eg:Syzygy) and also Quite powerful engines Stockfish et al. With these tools , positions we use to have difficulty playing are now understood at intermediate to high levels.
    – Dheebs
    Commented May 28, 2023 at 3:29
  • @Dheebs I actually think conversely it might now be easier to find such positions. Before it would have been really hard to prove that the "human-good" move is drawing and the alternative would have been winning, but now with tablebases we can tell for sure. And I think those are a good place to start, search through material configurations with long distances to simplification and choose one that a human won't reasonably win against decent defense (maybe something like BB vs N), then try to construct a position with an alternative approach that leads to a position with more winning chances.
    – koedem
    Commented May 31, 2023 at 9:04

3 Answers 3

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I'll attempt to back up my comment with a slightly constructed example: https://syzygy-tables.info/?fen=3kr3/8/8/1B6/2p1N3/2N5/8/4K3_w_-_-_0_1

3kr3/8/8/1B6/2p1N3/2N5/8/4K3 w - - 0 1

The only winning move in this position is Bxe8, leading to the famously difficult to win NN vs P endgame. Instead, a human might simply take the pawn and then try to win with BNN vs R.

I'm sure someone else can come up with a better example (as here you could argue that a human should instead try NN vs P), but I think this kind of approach might be one way to go.

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There are many such positions, mainly because engines know only to play the best moves, and sometimes the best move is equivalent to "resign". You might not find them very interesting as a result.

4r2Q/4kp1p/2p3pP/8/p7/PqP5/1P2RP2/1K6 b - - 19 42

Stockfish recommends 1...Qe6, but Black is so clearly busted after 2. Rxe6+ that he might as well resign. Granted, Black is also busted after 1...Kd6 (only move that doesn't lose the rook with check) 2. Qd4+ followed by collecting the rook, but at least White has to play 2. Qd4+. If White just grabs the rook, Black draws with 2...Qd1+ - Qb3+ perpetual.

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  • 2
    I suspect this is not really an example of what the OP is asking for. Here you give an example where it's practically better to play a theoretically-quickly-losing move than a theoretically-slowly-losing move. I think the question is looking for a case where it's better to play a theoretically-losing move than a theoretically-drawing move (or better to play a theoretically-drawing move than a theoretically-winning move). Ignore considerations of how quickly the game ends - what examples are there where the practically best thing is actually to throw away a win (or to throw away a draw)? Commented May 30, 2023 at 10:55
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This answer will be a bit tangential, for which I apologize.

Well known opening which leads to quite irrational positions (for both humans and engines) is anti-moscow gambit in semi-slav, which was quite topical in correspondence play until recently.

Its first branching point is reached after:

[fen ""]
[Startply "16"]

1.d4 d5 2.c4 c6 3.Nf3 Nf6 4.Nc3 e6 5.Bg5 h6 6.Bh4 dxc4 7.e4 g5 8.Bg3 b5

Where white can choose between 9. Be2, 9. h4 and 9. Ne5. White has in exchange for a pawn marveolous center and lead in development, though black has a lot of surprising defensive resources at his disposal. In all three options recently black found safe paths to equality.

As an example let me offer my game against ICCF IM Josef Kloster:

[fen ""]
[Startply "1"]
[Event "GER-SLO2020"]
[Site "ICCF"]
[Date "2020.02.01"]
[Round "-"]
[White "Sirk, Matija"]
[Black "Kloster, Josef"]
[Result "1/2-1/2"]
[WhiteElo "2330"]
[BlackElo "2472"]
[Board "7"]
[WhiteTeam "Slovenia"]
[BlackTeam "Germany"]

1.d4 Nf6 2.c4 e6 3.Nf3 d5 4.Nc3 c6 5.Bg5 h6 6.Bh4 dxc4 7.e4 g5 8.Bg3 b5 9.Be2 Bb7 10.h4 g4 11.Ne5 Nbd7 12.O-O Rg8 13.Nxg4 b4 14.Na4 Nxe4 15.Bxc4 h5 16.Ne3 Qf6 17.Qxh5 O-O-O 18.Rad1 Nxg3 19.fxg3 Qh6 20.Qf3 c5 21.Ba6 Bxa6 22.Qc6+ Kb8 23.Qxa6 Qxe3+ 24.Rf2 Ne5 25.Qb5+ Ka8 1/2-1/2 

Even with assistance of engine this game was very though to play, since in positions like this you have to second-guess engine at every move - the only thing you can be sure about is that neither yourself neither engine have very good understanding of the position.

Interesting was also this game between Nakamura and Ding, where black main mistake was 16. ... Rg8:

[fen ""]
[Startply "1"]
[Event "Sinquefield Cup"]
[Site "Saint Louis, MO USA"]
[Date "2016.08.14"]
[EventDate "2016.08.04"]
[Round "9"]
[Result "1-0"]
[White "Hikaru Nakamura"]
[Black "Ding Liren"]
[ECO "D44"]
[WhiteElo "2791"]
[BlackElo "2755"]

1. d4 d5 2. c4 c6 3. Nf3 Nf6 4. Nc3 e6 5. Bg5 h6 6. Bh4 dxc4 7. e4 g5 8. Bg3 b5 9. Be2 Bb7 10. h4 g4 11. Ne5 Nbd7 12. Nxd7 Qxd7 13. Be5 Qe7 14. b3 cxb3 15. axb3 a6 16. Qc1 Rg8 17. O-O Nh5 18. d5 Qxh4 19. g3 Qg5 20. dxc6 Qxe5 21. cxb7 Rb8 22. Nd5 exd5 23. Qc8+ Ke7 24. Rxa6 Nxg3 25. Bxb5 Ne2+ 26. Bxe2 f6 27. Re6+ Qxe6 28. Qxb8 1-0

So maybe the best move for humans after all is 6. Bxf6, which leads to more rational if drawish positions.

On endgame side very hard to win for humans but rather easy for engines (as long as they are equiped with tablebases) are endgames with two knights versus pawn. Not all of them are won (see Troitsky line), but even won ones often lead to draws. Here Karjakin scored a very hard win 4 years ago:

[fen ""]
[Startply "1"]
[Event "Isle of Man Masters"]
[Site "Douglas IMN"]
[Date "2018.10.24"]
[EventDate "2018.10.20"]
[Round "5.10"]
[Result "1-0"]
[White "Sergey Karjakin"]
[Black "Samuel Sevian"]
[ECO "C84"]
[WhiteElo "2760"]
[BlackElo "2634"]
[PlyCount "183"]

1. e4 e5 2. Nf3 Nc6 3. Bb5 a6 4. Ba4 Nf6 5. O-O Be7 6. Re1 b5 7. Bb3 O-O 8. a4 b4 9. a5 d6 10. d3 Be6 11. Nbd2 Qc8 12. h3 Rb8 13. Nc4 h6 14. Be3 Rd8 15. Qe2 Bf8 16. Nfd2 g6 17. Ba4 Qb7 18. Qf3 Nh7 19. Qg3 Bg7 20. f4 exf4 21. Bxf4 Nf6 22. Be3 Nd7 23. Qf2 Nc5 24. Bxc6 Qxc6 25. Bd4 Bxd4 26. Qxd4 b3 27. c3 Nd7 28. Qe3 d5 29. exd5 Bxd5 30. Qxh6 Re8 31. Ne3 Ne5 32. c4 Bxg2 33. Nxg2 Nxd3 34. Re3 Qc5 35. Kh1 Nf2+ 36. Kh2 Qd6+ 37. Kg1 Rxe3 38. Qxe3 Nxh3+ 39. Kh1 Kg7 40. Re1 Rh8 41. Qe5+ Qxe5 42. Rxe5 Ng5+ 43. Kg1 Rd8 44. Rd5 Rxd5 45. cxd5 Kf6 46. Nxb3 Ne4 47. Ne3 Ke5 48. Kg2 f5 49. Nc1 f4 50. Nc2 Kxd5 51. Nb4+ Kc4 52. Nxa6 Kb5 53. Nxc7+ Kxa5 54. Nd3 Nd2 55. Ne6 Kb5 56. Nexf4 Nc4 57. Ne6 Nxb2 58. Nxb2 g5 59. Nd4+ Kc5 60. Nf5 g4 61. Ng3 Kd4 62. Kf2 Kc3 63. Nd1+ Kd3 64. Ke1 Kc4 65. Kd2 Kd4 66. Nc3 Kc4 67. Nce2 Kd5 68. Kc3 Kc5 69. Nf4 Kc6 70. Kc4 Kd6 71. Nd3 Kc6 72. Ne5+ Kd6 73. Kd4 Ke6 74. Nc4 Kf6 75. Ne3 Ke6 76. Nef5 Kd7 77. Kd5 Kc7 78. Nd4 Kd7 79. Ne6 Ke7 80. Nc5 Kf7 81. Kd6 Kf6 82. Nce4+ Kf7 83. Kd7 Kf8 84. Nd6 Kg7 85. Ke6 Kg6 86. Nde4 Kg7 87. Ke7 Kg8 88. Kf6 Kh7 89. Nf5 Kg8 90. Ke7 g3 91. Nf6+ Kh8 92. Kf8 1-0

Related are endgames of king+rook versus king+bishop, which are often objectively drawn, but practically very hard for defending side due to myriad of tactical complications. So while objectively accepting draw offer in these positions is best, practically attacking side has great chances for victory - so it's humanely better to play it until 50 move rule or sweet victory.

For the last example a lot of very lost positions (K+R against K, K+Q against K+R, etc.) are played very differently by tablebases and by humans. Human player will try to trick his opponent when defending, setting stalemate traps. On the other hand tablebase will play moves that prolong the game before mate the longest, even if they are simple to counter. Try playing K+R against K against computer and against human player and I think you will find mating human player a bit harder. So optimal defence in lossing position is different for human than for machine - machine plays to prolong game against best attack, while human is trying to trap the winning side with suboptimal but tricky moves.

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