Mastodawn

Matthew Conroy Sep 14, 2025

I added another problem and solution to my dice collection. It's problem 77. https://www.madandmoonly.com/doctormatt/mathematics/dice1.pdf

Here's the problem statement: "Suppose we play a game with a die. We roll once, and this first roll is the score. We may continue to roll and add to the score, but if the roll ever divides the score we start with (e.g., if our score is 15 and we roll a 1, 3, or 5), then we lose everything and end up with nothing. If instead we choose to stop, we win an amount proportional to the score. What strategy will yield the maximum expected value of our final score?"

This is an optimal stopping problem. I've been interested in optimal stopping problems for quite a while, as a result of these dice problems and other things. However, I haven't found an introductory text to the general theory of optimal stopping that really appeals to me, so I end up doing things "from scratch" whenever I solve such problems. I feel like I might be missing some useful tools, but I'm not sure that is the case for the simple problems I'm solving.

Anyone have any optimal stopping texts or papers they'd recommend?

#math #mathematics #optimalStopping #probability

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undrendo Sep 14, 2025

@matthewconroy Sounds like a very interesting problem! But I don't fully understand it. For example, how can the score we start with be 15? are we using a die of a special kind?

It sounds like it could be solved by brute force recursively exploring a decision tree, if we limit the max number of rolls. My favourite book, simply written and very clear, that explains the basics of decision theory (of which optional stopping is a particular instance) is Howard Raiffa's "Decision Analysis: Introductory Lectures on Choices under Uncertainty" (Addison-Wesley 1970) <https://archive.org/details/decisionanalysis00raif>.

Decision analysis; introductory lectures on choices under uncertainty : Raiffa, Howard, 1924- : Free Download, Borrow, and Streaming : Internet Archive

Includes bibliographies

Internet Archive

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Matthew Conroy Sep 14, 2025

@pglpm Hello! We can't start the whole game with a score of 15. We start with a single roll, and the first score is the result of that roll. We then continue to roll as long as we like, until we choose to stop or the roll divides our current score. So we might start with 5, then roll a 4, to get a score of 9, then roll a 6, to get a score of 15. Starting then from a score of 15, what can happen? A 1, 3, or 5 will end the game, and give us nothing, any other roll will increase our score.

So, "starting from a score of x" means you are playing the game and your score is x and you have to decide whether or not you want to roll.

The maximum number of rolls is not automatically bounded, but in my solution I argue that one does not benefit, on average, by rolling with a score of 20 or more, and so this puts an upper bound on the number of rolls one would want to make.

Thanks for the reference (and archive link!)!

Cheers!

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undrendo

@matthewconroy Ah now it's clear! As you phrase it in your explanation, "with a current score of x" is maybe clearer than "starting from a score of x" (it made me think of the initial score).

In principle the problem can be solved exactly (optimal sequence of decisions) by constructing a decision tree. Owing to the branching structure, I'm not sure about the maximum expected value "5 (x+6)/6". At every outcome/inference/chance branch, the utility of some branches keeps increasing, while the corresponding probabilities seem to have a lower bound. Maybe it's possible that this leads to infinite expected utilities. BUT I haven't sat down to properly think about the problem, so mine is mostly babbling :)