I've been thinking recently it would be interesting to look at the evolution of cubeless win probability over the course of a game. On chess.com after every match you get a nice graph at the end of the engine's evaluation over the course of the game, and it's immediately obvious when e.g., one side made a blunder and it all went downhill from there, or one player started ahead managed to hold their small lead to the end, and so on. It would be especially interesting for backgammon because (unlike chess) the win probability will tend to converge to either 0 or 1 (if you have 75% win probability now, in the future their win probability is usually going to go up rather than down). And it would be easy to see when you're sneaking up on a double and maybe time it a little early or later than optimal, or when you roll a joker and immediately lose your market.

It looks like Mark Higgins and Rick Jankowski have done some work toward modeling this (treating CPW as a random walk, though I admit I haven't read their work closely enough to really get it yet) but I'd be curious to see some real data. Has anyone seen this before? I don't currently have a convenient way of automatically evaluating games and spitting out the CPW at each move but may try to set something up in the future.

(If Marc happens to see this it would be a neat addition to Galaxy).

"It would be especially interesting for backgammon because (unlike chess) the win probability will tend to converge to either 0 or 1 (if you have 75% win probability now, in the future their win probability is usually going to go up rather than down)"

I don't think this is correct. Of course, eventually the win probability will hit 0 or 1 (when the game ends) -- however, that is exactly the same as in chess. Meanwhile, it's not true that win probabilities ought to drift in the way you say: actually, they should be a random walk (at least if derived correctly using Bayes-Rule).

To read more about this: http://www.math.caltech.edu/~2016-17.../Lecture27.pdf

PS I do agree that it would be interesting to see how win probabilities evolve over time!

This is the part that isn't clear to me (but would be super obvious if we had some data to look at!).

For example, if I currently have an 80% win probability, what's the distribution of my win probability after the next roll? Is it really just as likely to go down as it is to go up (which is what a random walk would suggest)? Or is my win probability more likely to increase than decrease after the next roll? I'm not sure this follows directly from the Bayesian statistics.

We can make up an extremely simplified example where the random walk assumption doesn't hold up -- e.g., if our opponent has one roll left to bear off and we have one checker left on the 4-point, we win immediately with 34/36 rolls and lose the rest, so our current win probability is 94%. After our roll, 34/36 times our win probability goes up (to 100%), and 2/36 times our win probability goes down (to 0%). So in this toy example our win probability is more likely to go up than down because it's already high. This is obviously a silly example but it's just to show that Bayes' theorem doesn't necessarily imply win probability follows a random walk.

Also, the idea of doubling before you lose your market is based on the expectation that your win probability is more likely to go up than it is to go down, no? -- if CPW really were a random walk, it seems like the optimal strategy would be to wait until you lose your market before you double. (See this previous thread where I eventually wrapped my head around this idea).

My apologies for answering a bit hastily! (I actually meant to edit my answer just after posting it but lost internet...)

So what I meant to say is that you cannot *expect* the probability to go up or down following a roll. (At least, if your 'expectations' are the expected value.) If the probability is 75% this roll, then the expected probability next roll is 75%. You are quite right, however, that the distribution could be asymmetric and so the *median* scenario could involve your position improving or becoming worse.

Turning to your example, I note that your winning chance (before you roll) is 34/36. After you roll, your winning chance is either 0 or 1; and your expected winning chance is

0 x 2/36 + 1 x 34/36 = 34/36

and so your expected winning chance is just your winning chance now. Anyway, that's what I recall hearing about Bayes etc. -- but please let me know if I have mis-understood something.

Regarding your question (which I guess was more about the probability that your roll gets better): I suspect that we get a fairly symmetric distribution when looking at rolls that aren't too close to the end. Though I agree that it would be interesting to look at the data here.

My apologies for answering a bit hastily! (I actually meant to edit my answer just after posting it but lost internet...)

So what I meant to say is that you cannot *expect* the probability to go up or down following a roll. (At least, if your 'expectations' are the expected value.) If the probability is 75% this roll, then the expected probability next roll is 75%. You are quite right, however, that the distribution could be asymmetric and so the *median* scenario could involve your position improving or becoming worse.

Turning to your example, I note that your winning chance (before you roll) is 34/36. After you roll, your winning chance is either 0 or 1; and your expected winning chance is

0 x 2/36 + 1 x 34/36 = 34/36

and so your expected winning chance is just your winning chance now. Anyway, that's what I recall hearing about Bayes etc. -- but please let me know if I have mis-understood something.

Regarding your question (which I guess was more about the probability that your roll gets better): while I do agree that the distribution could be asymmetric, I still don't see why the *median* roll should drift in the way you say. For example, suppose your winning chance is 75%. I do agree that the median roll could either improve or worsen your position. However, I'm not seeing why it should generally improve your position (though maybe I'm being slow again!)

There are many BG apps that can evaluate your winning chances precisely after every move, as well as telling you the correct cube action for both sides. You can download Gnu BG for free. There is much more fluctuation in your winning chances than in chess due to the luck of the dice.

Thanks for the thoughtful response. To test this out I manually transcribed the win probabilities for each move of three 1-point matches I recently played on Galaxy (to avoid the cube aspect for now), a total of 127 rolls.

[Link to results]

As you can see, even though this is a tiny sample, it’s already clear that at higher win probabilities your win probability is more likely to further increase than it is to decrease after each roll, and vice versa. Still haven’t fully grokked the mean-vs-median issue you mentioned, but I’ll keep thinking about it. (I agree that the mean will stay the same across all possible rolls, but it’s interesting that the median can change. I wonder if there are some implications for thinking about approaching doubling decisions, which is why I started thinking about this in the first place).

That looks very interesting -- do keep us updated when you have a chance to check with a larger sample size!

BTW, here are the three traces I manually transcribed that led to the data above: