# A mathematical formula for expected no. moves to win?

I was wondering if there was a formula for calculating an expected no of moves to win between two players of significant difference in rating? Outside my mathematical abilities, but think it may have too many parameters effecting to get a meaningful answer.

I doubt there's a pretty answer for this; Elo difference only gives a rough estimate of the probabilities of a win on one side.

Instead of computing this analytically, you could estimate it empirically, either by looking at the length of relevant games in a large database, or by simulating a large number of games with engines at the appropriate strength settings and collecting statistics. But again the answer you get may not be particularly meaningful.

Thanks. A passing wondering.  That is another way of getting a figure but wouldn't say much at all.

Expected outcome has to be based on some distribution, and we don't know the distribution so there cannot be an answer to your question.  HOWEVER, you can obtain an average and median of the number of moves between such-and-such players based on past games.  Now, here's the leap of faith, with those 2 numebrs you can usually pick a distribution (say normal) and then start from there : )

mgx9600 wrote:

Expected outcome has to be based on some distribution, and we don't know the distribution so there cannot be an answer to your question.  HOWEVER, you can obtain an average and median of the number of moves between such-and-such players based on past games.  Now, here's the leap of faith, with those 2 numebrs you can usually pick a distribution (say normal) and then start from there : )

Wouldn't a normal distribution entail a positive probability of games with a negative number of moves? And you don't say what to do with the games the favourite draws or loses. If you count the moves zero or negative in this case you'd get a multimodal distribution.

The number of moves can also shift by say 40 moves, depending on when the losing side chooses to resign.

RML11,

this is an intriguing thought. However, I do not see an accurate, calculatable method for approaching this thought - other than what is said in the forum. I problem I can't overcome in this idea is "inaccuracy", due to player inconsistency. What I mean is that many players mix up openings, and statistically some openings lead to shorter games/others longer games. Additionally, there are a few "jerks" who are proficient in endgames, but instead march all pawns into Queens before winning; statistically this is a nightmare. Some people will play the next 50 or so moves hoping they stalemate, while others will simply resign when they see this pawn march as an idea.

Finally, it is common to see lower rated players (but sometimes even high rated GMs too) miss mates, perhaps mate in 1 or mate in 10 (you get the idea). Problematically, the chance of a player seeing/missing mate is not based on rating, but a ability to find these mates - nothing more, but difficult to measure, although tactics-rating may be a closer (but not perfect) factor to calculate chance of not missing a forced mate sequence. Perhaps, this may be calculated for individual players, based on past games - but even this may be inaccurate due to other confounding variables.

However, what we can do is record the number of moves played for as many games as possible - therefore calculating the average number of moves a chess game takes (at any level/all players). This has already been done, mathematically 40 moves is the average chess game length.

This is why many time controls "extend" the time after 40 moves. However, what this forum is describing is closer to an individual "statistic" calculated from ONE player versus anyone, regardless of rating. I do not recall which, but another chess.com forum about a month ago suggested that chess.com profile "stats" section include more data on thoughts similar to this. This person also wanted statistics such as: How many games did I lose due to connection, checkmate, or resignation? When I play a certain opening, what is the average number of moves that the game takes? ...And so on

This grand idea had included many questions in similar nature to these above. That forum was intended to get the attention of chess.com, to motivate some action into creating features such as this, with primarily positive comments on that forum. I however, do see this as a practical reality due to the massive data/tedious work required for statistics that few chess.com members would view.

I think it is a bold, interesting thought to consider - but perhaps not practical. To wrap up this long post, the average number of moves a chess game takes is 40 moves (algebraic notation, so 1. e4 e5 counts as one move towards this 40 figure, not two moves). However, to calculate what this forum suggests may not be possible WITH accuracy because of too many problems such opening repertoire, or how determined the player is to not resign. Of course, with a lot of mathematical prowess it is perhaps possible to calculate these less common statistics, but with how much accuracy is another debate.

Pardon the grammar/other errors in my earlier post, as I am typing this on a smart phone. Most notably in the penultimate paragraph I mean to say it is "not" practical. (this is not witty Shakespeare writing of liotes, just spell check messing it up)

I hope this attends fully to this great insight, even if not likely to be calculated in the near future.

Thank you for thoughts everyone, KeSetoKaiba for your extended one. Too many parameters. really. The closest may be something that is very focused on two specific players with sufficient games. You could then bring in important factors such as early late/resigning tendency. Still, I don't think very helpful overall. Mathematicians may enjoy the challenge.   Thank you.

RML11 wrote:

I was wondering if there was a formula for calculating an expected no of moves to win between two players of significant difference in rating? Outside my mathematical abilities, but think it may have too many parameters effecting to get a meaningful answer.

This is an excellent question. Not sure of the answer. Somewhat related to your question, I've wondered what is the average length of games based on skill-level, especially for high-level play.

It seems that when playing level goes up for nearly equal players (grandmaster level and better), chess games tend to be longer than the 40-move average.

In the recent chess.com computer championship (here), in the Top-2 tournament (20 games between the top 2 engines), the average game length was 199.5 ply (100 moves)!

If this represents "very-high level" play, than good chess games can be longer and even more complicated than what many experts expected.

That's another twist on it, the expected maximum of moves. Even those who "never resign" and hang out to the bitter end will show their lack against stronger players who will presumably finish in shorter order nevertheless. A lesser opponent will muck around
lengthening the game. But that idea is at odds with the interesting data from the computer chanmpionship. But that is in very cloely rated situation. Mmm?
vickalan wrote:
RML11 wrote:

I was wondering if there was a formula for calculating an expected no of moves to win between two players of significant difference in rating? Outside my mathematical abilities, but think it may have too many parameters effecting to get a meaningful answer.

This is an excellent question. Not sure of the answer. Somewhat related to your question, I've wondered what is the average length of games based on skill-level, especially for high-level play.

It seems that when playing level goes up for nearly equal players (grandmaster level and better), chess games tend to be longer than the 40-move average.

In the recent chess.com computer championship (here), in the Top-2 tournament (20 games between the top 2 engines), the average game length was 199.5 ply (100 moves)!

If this represents "very-high level" play, than good chess games can be longer and even more complicated than what many experts expected.