Answer by richard1941 for When to stop rolling a die in a game where 6 loses...
Why is your summation divergent? For n sufficiently large, each term is less than 123456 * (11/12)^n, which certainly converges to something finite. Particularly, look up "gradient series" in a book on...
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Answer by Adrian for When to stop rolling a die in a game where 6 loses...
Here's an answer using value function iteration:Let $v(x)$ denote the value of state $x$. The Bellman equation for this problem is $$v(x) = \max\left\{u(x), \frac{1}{6} \left[u(0) +...
View ArticleAnswer by A.G. for When to stop rolling a die in a game where 6 loses everything
You should stop when the expected score from another toss becomes negative or zero. Let $X$ be the score; the expected gain for the next toss...
View ArticleAnswer by Mike Earnest for When to stop rolling a die in a game where 6 loses...
It's not hard to see that the optimal strategy in this game is "stop on $k$ or higher," for some positive integer $k$. This is because what is optimal is only determined by your current score, not your...
View ArticleAnswer by user381133 for When to stop rolling a die in a game where 6 loses...
I interpreted this such that you're in competition against another player.If playing second your strategy is simple - play until you've won.If playing first, you must accumulate enough points such that...
View ArticleAnswer by Chris Johns for When to stop rolling a die in a game where 6 loses...
The mathematics of this is actually very simple, every time you roll you have a 1 in 6 chance of losing regardless of previous outcomes so the optimal strategy depends on information not covered by the...
View ArticleAnswer by user14972 for When to stop rolling a die in a game where 6 loses...
I'm going to tackle the problem of trying to maximize the average score of a strategy, with the disclaimer that this is often not what you actually care about when playing actual games.Often, the trick...
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Answer by Timothy Shields for When to stop rolling a die in a game where 6...
The question is missing the concept of utility, a function that specifies how much you value each possible outcome. In the game, the utility of ending the game with a certain score would be the value...
View ArticleAnswer by Litteul for When to stop rolling a die in a game where 6 loses...
There is unfortunately many mistakes in the calculation of the expectation proposed:1) The expectation does not take into account the optionality of the player deciding to play or not.2) Also since the...
View ArticleAnswer by Parcly Taxel for When to stop rolling a die in a game where 6 loses...
Before deciding whether to stop or roll, suppose you have a non-negative integer number of points $n$.How many more rolls should you make to maximise the expected gain over stopping (zero)?Suppose that...
View ArticleAnswer by Djura Marinkov for When to stop rolling a die in a game where 6...
In the last round you can get $\frac{1+2+3+4+5}{6}$ or lose $p\frac 1 6$, whenever the second is more than the first you should stop. So once you have scored more than 15 you should stop. If you score...
View ArticleWhen to stop rolling a die in a game where 6 loses everything
You play a game using a standard six-sided die. You start with 0 points. Before every roll, you decide whether you want to continue the game or end it and keep your points. After each roll, if you...
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