This is how I look at it, but I'll admit I may have misunderstood the game!

Assuming you have a current banked score of B, the expected return for any given round is: $$E(return)=\frac{1}{6}(-B)+\frac{1}{6}(1+3+4+5+6)$$ $$E(return)=\frac{1}{6}(19-B)$$ So once you have a bank of 19 points, it is better to get out than take the chance.

I believe this will maximize your average score in the long run. However, when it comes to games, sometimes things are more complicated than simple optimization. For a 2 player game, I think I would follow the advice of my analysis above. However, for a 20 player game, it is clear that you will need some luck, since 2nd place is the first loser, you want to give yourself a chance at a very high score, not just try to avoid a very low one. Intuitively, I expect that this means you need to push your luck passed the 19 score mark, but I'll have to think harder about how to quantify that for a game of n people.

Running a simulation, I find that the mean is, indeed, optimized near a threshold of 19. However, as I also suspected, the lucky game (mean + 2$$\sigma$$) is actually optimized out around 29 or 30. So if you need to beat 19 other players wait till then. This is how I look at it, but I'll admit I may have misunderstood the game!

Assuming you have a current banked score of B, the expected return for any given round is: $$E(return)=\frac{1}{6}(-B)+\frac{1}{6}(1+3+4+5+6)$$ $$E(return)=\frac{1}{6}(19-B)$$ So once you have a bank of 19 points, it is better to get out than take the chance.

I believe this will maximize your average score in the long run. However, when it comes to games, sometimes things are more complicated than simple optimization. For a 2 player game, I think I would follow the advice of my analysis above. However, for a 20 player game, it is clear that you will need some luck, since 2nd place is the first loser, you want to give yourself a chance at a very high score, not just try to avoid a very low one. Intuitively, I expect that this means you need to push your luck passed the 19 score mark, but I'll have to think harder about how to quantify that for a game of n people.