TL;DR - somewhere in the range of 15-23% of the time? This rough estimate only involves cases where the underpromotion is immediately possible on the player's move (so not anything like the example of Hjartarson having to see an underpromotion when calculating a variation). Also, the script only returns cases where underpromoting would have made a real difference if played.
I wrote a script that went through the top games in my big database (sorted by white elo), from Carlsen's elo to an elo in the high 2300s. Out of these 335,480 games, the script returned 206 instances where underpromotion was the best move (and where it would have actually mattered). E.g., if the second best move was still clearly winning, or if they both led to a mate for the same side, these cases wouldn't be counted.
- Note that the script isn't perfect -- I found one example where underpromotion was only as good as promotion to a queen. However, it seems in most of the positions, underpromotion is best and it would have had a significant impact.
Of these 206 instances, the script recorded 48 cases where the player didn't do the underpromotion. This is a failure rate of around 23%, although note that some of these games were blitz and online.
Here are some of the highlights:
Karpov-Korchnoi, Dortmund 1994 (white to move):
8/Qb3P1k/4q3/P2p4/2pPpbp1/1r2N1Pp/7P/5BK1 w - - 0 61
- In this one, Karpov obviously considered the option of promoting to a knight. However, after doing so White should not capture Black's queen! The way to a draw is very study-like. One line: 61.f8=N+! Kh6 62.gxf4! Qc8 63.Qb8!! Qxb8 64.Nf5+, and somehow the knights give a perpetual. This is just amazing.
1q3N2/1b6/7k/P2p1N2/2pPpPp1/1r5p/7P/5BK1 w - - 1 64
Popov-Mamedyarov, world blitz 2018 (black to move):
3R4/3b1kp1/5p2/8/6P1/4QP2/3p1K1P/2q5 b - - 7 54
- At the time, Mamedyarov was rated 2817 in classical, and so this might be the highest rated player missing an underpromotion ever. Of course, just a blitz game though.
Alekseenko-Ding Liren, World Cup 2019 (black to move). Note - this is the example I mentioned where my script was wrong:
8/1p6/p1p2kp1/P3p2p/1PP1Pb1P/R1n2K2/2Bp4/8 b - - 6 50
- My script was incorrect here, as underpromotion is not any better than regular promotion to a queen (although at the same time, it's not any worse).
Hou Yifan-Dzagnidze, Geneve FIDE GP Women 2013 (black to move):
4r3/1p3p2/p2n1P2/6R1/P4PP1/2P5/3kp1KP/R7 b - - 1 36
Here Black's best saving chance by a significant margin would have been ...e1=N+, although the game would have likely still been lost.
I also ran the script on the first 380,000 games in the big database (up until somewhere in the 1970s). Interestingly, 65 positions were found where underpromotion was best, but the player missed it in only 10. 10/65 ~= 15% is a lower failure rate than the 48/206 ~= 23% one mentioned earlier (though not by that much), which might seem surprising. One would assume that high elo players in the (mostly) current era would do better than players of the past. But there are two factors to consider here:
- As mentioned before, some of the games in the "high elo collection" were fast time controls.
- Selection bias: the high elo players tend to have high elo opponents, who would try to not go into a position if they saw that there was a strong underpromotion. So for some cases where an underpromotion was missed, it was probably the case that the player's opponent also didn't see it -- which means the position wasn't as trivial as avoiding a stalemate or something like that.
Anyway, two highlights from the search through these old games:
The first earliest case: Gossip-Schiffers, Germany 1889 (white to move):
rnb1r1k1/pP2qppp/8/1n6/3Q1P2/2P1BN2/P1P3PP/R3K2R w KQ - 0 14
And here's a game between two famous players: Pachman-Flohr, Interzonal 1948 (white to move):
8/4P3/1p2K3/5n2/1N1pk2p/8/7P/8 w - - 4 58