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Understatement of the week: Grandmasters are good at chess. I am not.

I was wondering because of this how likely it would be through complete random play to beat a grandmaster, and whether or not it was more likely for me to beat one through random play than out of my own skill set. (I am about 1400 FIDE, and therefore probably actively discount a lot of moves which would actually be quite good. So my questions are: 1. How likely am I to beat a GM through random play? 2. Is this more likely than through my own skill?

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    Maybe it depends on what you mean by "random", but truly random moves really, really suck. I would put the odds in "infinite monkey" territory. Also see discussion at What would be the elo of a computer program that plays at random? – itub Aug 14 '18 at 12:44
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    For question 2, the answer is clearly "no". At 1400 elo you are far from GM-level but still much better than random play. – Evargalo Aug 14 '18 at 13:05
  • Astronomically unlikely. – Inertial Ignorance Sep 28 at 23:52
  • Out of the average of 35 possible moves, there might be 3 acceptable moves. In order to beat a grandmaster, you would have to, in theory, be within the 10% the entire game. So the answers are 1) 0% and 2) no. – Mike Jones Dec 10 at 5:00
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The ELO system is very good at predicting what percent win chance that a player has to beat another player. There is a table that we can consult to see the exact percent chance that one player has to beat another. Let's take a look.

ELO statistics

A quick summary of the table is...

  • Difference of 0 rating points = 50% chance of higher rated player winning
  • Difference of 100 rating points = ~65% chance of higher rated player winning
  • 200 is ~75%
  • 300 is ~85%
  • 400 is ~92%
  • 500 is ~96%
  • 600 is ~98%
  • 700 is ~99%
  • 735 is 100%

Grandmasters tend to be around ELO 2500-2700 FIDE. Let's assume that a typical grandmaster is 2600.

With that in mind, let's take a shot at answering your questions.

1) Can you (FIDE 1400) beat a grandmaster?

The difference in rating between you and the grandmaster is 1200 points. So according to the statistics, no. You'd have to be 735 ELO points away from the grandmaster or closer to stand a chance. So you'd need to be ELO 1865 to have a 1% chance of winning (winning 1 time every 100 games).

2) Can a player playing randomly (let's assume ELO 100) beat a grandmaster?

Now we are talking about a rating difference of 2500 points. Definitely not within our 735 point threshold. The answer here is definitely "no".

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    The "100%" on the table is, of course, rounded off; an 800 point upset is not impossible, just unlikely. If we use the formula 1/(1+10^((R1-R2)/400))), then a 2400 point difference is a 1 in 1,000,001 chance. – D M Aug 15 '18 at 14:55
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The chance is absolutely zero. Not just against grandmasters, but against anybody.

In fact, it would be hard to win a game with random play against someone actively trying to lose, after all you would just as likely remove your pieces from the vicinity of the opponent's king than play the checkmating move, if the opponent finally managed to get to a position where mate is possible.

Everybody who knows the rules and has some sort of goal in mind plays much better than random play.

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    It's not zero. It's a very tiny number and would almost certainly never happen, but that's not the same as an absolutely zero chance. – D M Aug 15 '18 at 14:57
  • @DM I agree. Pick any game a GM has lost, and consider all moves played by the opponent. A player picking a random legal move each turn clearly has some very small but strictly >0% chance of playing the exact same moves. – jafe Dec 9 at 13:58
  • @jafe: I was assuming that the GM knew that the opponent was playing random moves. That makes the chance a lot smaller still. – RemcoGerlich Dec 9 at 14:31
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Let's try to quantify why random moves are so bad. In an average chess position there might be 35 legal moves. Often there is only one good move in a position, but sometimes there are several. Let's say, to keep the math simple, that 10% of moves are "good", and if you always play a "good" move, you can beat a GM. Let's say that a typical game in which you beat a GM takes 40 moves. Therefore your probability of playing such a winning game at random is 10-40.

Now let's imagine that you have 10 billion people each play 100 thousand random games. That's a total of 1015 games. That's still almost nothing compared to the 1040 games that it would take to have a reasonable chance of having one win against a GM!

(Note that it is not nearly as unlikely as having a monkey type the entire works of Shakespeare at random. It would be maybe like typing the first line of the works of Shakespeare at random! Still, pretty unlikely. :-)

  • I like the idea of your math. But there's a factor you perhaps didn't consider, that has to do with the odds of a game reaching a particular length. A 20 move loss for a GM is rare; it might happen one loss in a thousand, maybe. But a one in a thousand chance of a 10^20 chance means it's a 10^23 chance, which is still way better than 10^40. – D M Aug 15 '18 at 14:58
  • You are right, this is a very simple model just to show that "big numbers" are involved, but it could easily be wrong by 20 orders of magnitude. Even GMs commit terrible blunders sometimes, as this 12-move loss by Karpov shows. (Although if a GM knew that he was playing against a random mover, who just got lucky so far, he might not be so quick to resign!) – itub Aug 15 '18 at 15:05
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    I bet the GM would refuse to believe the moves were random even if you told him they were - and given the odds, that would be perfectly reasonable of him :) – D M Aug 15 '18 at 15:35
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Funnily enough, there is some way to calculate this probability, and it's not vanishingly small. Reference: Elo World, a framework for benchmarking weak chess engines. This is more of a parody paper than anything, but the gist is that one can easily code weak chess engines and get them to play against strong ones millions of times.

To illustrate, some of the weak engines used are:

  • random_move. Self-explanatory, and directly relevant to this question.
  • same_color. When playing White, put pieces on White squares; when playing Black, put pieces on Black squares. If all pieces are already on the "right" color, make a random move.
  • generous. Move so as to maximize the number of possible captures the opponent can make on our pieces. The more valuable the piece we're giving away, the more weight we give to that move.
  • reverse_starting. "This player thinks that the board is upside-down, and as white, tries to put its pieces where black pieces start."

(I did say it's a parody paper, right?)

There're also a few serious engines. The strongest one playing is Stockfish1m. This is Stockfish, the strongest traditional chess engine, when asked to play a move after 1 million nodes (this means move after it has searched exactly 1 million positions). Exactly how strong this is I don't know. I know I can beat Stockfish 1 node fairly easily, but at 1 million nodes it's obviously a lot stronger, and it's likely strong enough to crush most humans.

If we assume that Stockfish1m is about as strong as a grandmaster, then there's an interesting table on the last page. Unsurprisingly there are many engines that are worse than random_move, and unsurprisingly Stockfish1m is the strongest engine. random_move is a full 2200 elo weaker than Stockfish1m, but it has a nonzero chance of winning a tournament between all these engines. In fact, p(random_move winning tournament) is 0.00000462. Small, but not zero.

Caveat: This isn't a direct head-to-head probability of random_move beating Stockfish1m.

Caveat #2: I am not completely sure where the line between the parody and the serious work is. It's possible the author is spoofing the use of Markov probabilities here, because p(random_move winning) = 0.00000462 sounds awfully large. Naively, I'd have expected something several orders of magnitude smaller.

Edit: I've convinced myself that the above is part of the satire. Here's a back-of-the-envelope calculation for the probability of random_move winning. If we assume that the grandmaster will never resign (and they presumably won't, since a random mover can always blunder a game) then the game might take 80+ moves to end in a mate, probably longer. Chess has a branching factor of about 35. Therefore, the random move needs to make "hit" roughly 35^(80) times in a row to win, making the odds around 10^(-123) against.

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I don't understand... Losing to a random player by checkmate is very hard. There is no way a GM would lose to a random player. No chance. You are more likely to win the world championship than the random player.

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Epsilon! [That is a math joke!] Say zero if you are not a mathematician:)

Playing randomly, you are guaranteed to lose. [Again random as mathematicians and most English speakers would use that word.]

If you played your best then you might win, as it has happened. But the probability is extremely small.

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For question 2, it might be best to combine yourself and random play. As you stated, you are blind to some good moves, and a GM could probably beat you with rook odds. So you might as well let random computer play until you are a queen up, then play at your strength to convert that to a win.

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