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BlindKungFuMaster and I are having a debate (http://chat.stackexchange.com/rooms/34484/discussion-between-blindkungfumaster-and-jeff-y) and it got me wondering: Given any position that can be agreed as "about equal", what is the probability that a given player would make a game-losing move on any single move, as a function of his or her rating? I am asking independent of the opponent's strength, i.e. whether or not the opponent actually exploits the mistake to win the game.

Clearly, games are lost, even by Grandmasters, even by the very best. So (presuming Chess is a theoretical draw from the start position) even they make game-losing moves on occasion. So the probability in question is clearly not zero even at Elo 2800. How high would that probability be at, say, 2000, 1800, 1500, 1200 ratings, approximately? At what rating would said probability be approximately 50%? Is this something that can be definitively calculated in some fashion based on the definition of Elo and the average number of moves per game? Or would any answer be pure guesswork?

Update:

I've extracted all games from ChessBase's BIG99 database of 1,114,429 games where one player's Elo is 2500 or more and the other player's Elo is 2100 or less. There are 945 such games. After filtering out the upsets (surprisingly there are 79 win upsets and 102 draw upsets), below is the chart of ply-count vs. Elo. The darker series is where the loser played black, the lighter series is where the loser played white. Seems to be a horn-of-plenty type of shape.

Plot of ply-count vs. Elo for losses against 2500+

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  • I'm now thinking that maybe an analysis of the length of games lost by players against opponents more than 400 points above them (my understanding of Elo's "100% chance of loss" point) might be a start point for this calculation...
    – Jeff Y
    Commented Jan 21, 2016 at 11:26
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    Not all equal positions are created equal. There are dangerous equal positions where a misstep can easily be fatal and harmless equal position where you have a wide choice of decent moves. Against strong players the equal positions you reach will rather be of the former variety. And of course in your analysis the "game losing move" will most often have occurred in a position that is already quite bad.
    – BlindKungFuMaster
    Commented Jan 22, 2016 at 8:38
  • I like the plot, but what is its vertical axis?
    – thb
    Commented Jan 23, 2016 at 21:50
  • The plot, as my text above states, is "ply-count vs. Elo", so vertical axis is "length of game" in units of ply-count. ("Ply" is a move by one side, sometimes called a "half-move".)
    – Jeff Y
    Commented Jan 24, 2016 at 12:11

3 Answers 3

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Coincidentally I already answered exactly this question in response to a similar question.

Edit: This similar question was about frequencies of blunders in games, which made the analysis somewhat misleading when directly applied to this question. Originally I looked for blunders from equal positions per game move, which made the results a bit confusing because there was the unknown variable of how many equal positions you actually get per game move. So I redid the analysis for blunders per equal position which is a lot more appropriate in this context.

I happen to have a dataset with 25000 games with stockfish evaluations after every move. This allows to actually look for blunders in equal positions, which is what I did.

Blunders from an equal position (-1.00 < eval < 1.00) are relatively rare, even among weaker players. That is not particularly surprising, because we tend to leave the equality region in little steps during the opening and the blunders come when we are under real pressure and low on time.

I ran the analysis dependent on the strength of the opponent as well, to show that stronger opponents actually lead to more blunders even in equal positions. The stronger opponents in the analysis are more than 100 Elo points higher rated, the weaker 100 points lower rated. Players of all strength blunder more often against stronger opponents from equal positions than against weaker opponents.

Elo: 1500: 100cp Blunder every 26.4655172414 equal positions.
Elo: 1500: 100cp Blunder every 26.1266149871 equal positions against stronger players.
Elo: 1500: 100cp Blunder every 33.3684210526 equal positions against weaker players.

Elo: 1600: 100cp Blunder every 28.8888888889 equal positions.
Elo: 1600: 100cp Blunder every 28.3083832335 equal positions against stronger players.
Elo: 1600: 100cp Blunder every 37.12 equal positions against weaker players.

Elo: 1700: 100cp Blunder every 34.7788649706 equal positions.
Elo: 1700: 100cp Blunder every 34.0448933782 equal positions against stronger players.
Elo: 1700: 100cp Blunder every 39.7709923664 equal positions against weaker players.

Elo: 1800: 100cp Blunder every 34.9866156788 equal positions.
Elo: 1800: 100cp Blunder every 33.1406015038 equal positions against stronger players.
Elo: 1800: 100cp Blunder every 45.3865546218 equal positions against weaker players.

Elo: 1900: 100cp Blunder every 40.1570101725 equal positions.
Elo: 1900: 100cp Blunder every 38.315761729 equal positions against stronger players.
Elo: 1900: 100cp Blunder every 49.9418282548 equal positions against weaker players.

Elo: 2000: 100cp Blunder every 44.4308207705 equal positions.
Elo: 2000: 100cp Blunder every 41.5676238036 equal positions against stronger players.
Elo: 2000: 100cp Blunder every 56.3524305556 equal positions against weaker players.

Elo: 2100: 100cp Blunder every 52.5946657886 equal positions.
Elo: 2100: 100cp Blunder every 49.5823737821 equal positions against stronger players.
Elo: 2100: 100cp Blunder every 61.1668806162 equal positions against weaker players.

Elo: 2200: 100cp Blunder every 61.3163636364 equal positions.
Elo: 2200: 100cp Blunder every 56.0916284881 equal positions against stronger players.
Elo: 2200: 100cp Blunder every 75.2474916388 equal positions against weaker players.

Elo: 2300: 100cp Blunder every 69.6490486258 equal positions.
Elo: 2300: 100cp Blunder every 60.9148185484 equal positions against stronger players.
Elo: 2300: 100cp Blunder every 90.0941176471 equal positions against weaker players.

Elo: 2400: 100cp Blunder every 78.8800318852 equal positions.
Elo: 2400: 100cp Blunder every 67.7366828087 equal positions against stronger players.
Elo: 2400: 100cp Blunder every 100.431924883 equal positions against weaker players.

Elo: 2500: 100cp Blunder every 97.320568252 equal positions.
Elo: 2500: 100cp Blunder every 84.8542336549 equal positions against stronger players.
Elo: 2500: 100cp Blunder every 114.45814978 equal positions against weaker players.

Elo: 2600: 100cp Blunder every 110.2421875 equal positions.
Elo: 2600: 100cp Blunder every 97.9315068493 equal positions against stronger players.
Elo: 2600: 100cp Blunder every 127.470948012 equal positions against weaker players.

Elo: 2700: 100cp Blunder every 95.7817109145 equal positions.
Elo: 2700: 100cp Blunder every 78.6981818182 equal positions against stronger players.
Elo: 2700: 100cp Blunder every 167.296875 equal positions against weaker players.

So for a table of estimated probability of blunder on any given single move:

Elo 1500-1599:  0.0378
Elo 1600-1699:  0.0346
Elo 1700-1799:  0.0288
Elo 1800-1899:  0.0286
Elo 1900-1999:  0.0249
Elo 2000-2099:  0.0225
Elo 2100-2199:  0.0190
Elo 2200-2299:  0.0163
Elo 2300-2399:  0.0144
Elo 2400-2499:  0.0127
Elo 2500-2599:  0.0103
Elo 2600-2699:  0.0091
Elo 2700-2799:  0.0104

An approximation formula: p = (0.323 - 0.0850 * Elo / 1000) ^ 2

chart of probability

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  • Thank you for the data and the Crafty-analysis link. Especially interesting at the link is that outright game-losing moves are more common than moves that lose 4 or more pawns of eval. All these are focusing on single-move blunders however, when as you've noted, games are often lost in more of an incremental manner (multiple small blunders accumulating).
    – Jeff Y
    Commented Jan 22, 2016 at 11:38
  • Well, that's specified in your question:"Given any position that can be agreed as "about equal", what is the probability that a given player would make a game-losing move on any single move". Incremental losses are a different question.
    – BlindKungFuMaster
    Commented Jan 22, 2016 at 11:41
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    Apologies if the wording of the question fails to convey the theoretical nature of what I'm asking, but as the second paragraph about "everyone loses games" implies, I'm asking about "game-losing" in a theoretical sense. At some point the game necessarily switches, at a single move, from theoretically-drawn to theoretically-lost. E.g. a 3-pawn blunder (as measured by an engine) may not be the (theoretical) game-loser, it may be the 0.1-pawn blunder on the following move that actually tips it.
    – Jeff Y
    Commented Jan 22, 2016 at 12:07
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    The idea that a 1500 will be able to average 45 moves between 1-pt blunders against a Magnus or a 3300-rated computer doesn't sound right. I am about 1500 ELO and I know I can only predict moves in GM games perhaps 3 times in 4? If I pick the right move 75% of the time, I have a 95% chance of not choosing the right move after 10 or 11 moves.
    – Tony Ennis
    Commented Jan 22, 2016 at 13:43
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    @BlindKungFuMaster I can't agree with this idea that somehow stronger opponents provoke weaker moves or more blunders from a player. We'll have to agree to disagree on it, pending more data...
    – Jeff Y
    Commented Jan 22, 2016 at 13:47
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If we're talking about a pure game-losing move then the percentage is quite high.

Bear in mind one could make 20 sub-optimal moves that individually would not be fatal. But together, it's just too much.

The only way I know to figure this out is to analyze players' games using a strong engine at tournament controls. If the played move exceeds some threshold (say, 1 point) then the move is counted as a loser.

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    Thanks for answering. But I was looking for more of a calculated answer, or an explanation of why it couldn't be calculated. Of the 20 sub-optimal moves, only one might be the "straw" that flips the game from a theoretical draw to a theoretical loss; doesn't matter which one as to the probability I'm asking about.
    – Jeff Y
    Commented Jan 21, 2016 at 11:16
  • @JeffY Also note that there's no difference between a 1200 ELO player and a 2400 ELO player in this analysis. Both will lose against the finest play. (Think a dream-team of the previous 10 world champions and their seconds, plus the top 5 computers on the best hardware)
    – Tony Ennis
    Commented Jan 21, 2016 at 14:18
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    Right, but the question is, "how long can they hold on?" To a theoretically-drawn position, that is. On average. Before they make their first move that converts to a theoretically-lost position. Presumably the 2400 hangs on longer. Because the 2400 has a lower probability of making "game-losing" moves.
    – Jeff Y
    Commented Jan 21, 2016 at 15:22
  • "How long do they hold on" is a different question because the game losing move doesn't usually come in an equal position anymore. Instead we tend to leave equality in little steps the smaller the stronger we are. Which means the final step to a lost position will be from "almost lost" to "lost" for a 2400, "pretty bad" to "lost" for 2100, "bad" to "lost" for 1800 and "uncomfortable" to "lost" for 1500. But "equal" to "lost" is rare for all of them.
    – BlindKungFuMaster
    Commented Jan 22, 2016 at 9:00
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    Reminder: "theoretically-drawn" is very different than "equal position". I.e. your "pretty bad" still means theoretically-drawn. So "hold on to theoretical draw" is what I mean as to "how long".
    – Jeff Y
    Commented Jan 22, 2016 at 11:02
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if they dont punish your bad move then was it really bad at all ?

i would say the normal statistics apply

what is the chance a player with rating x can beat a player of rating y

unless you are grandmasters the winner is the one who made the next to last mistake

so on a given move the chance of making the fatal mistake would seem to be exactly what the ratings would indicate and nothing else

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