Simple game of dice - what is the optimal strategy?

Question

The game is pretty simple, whoever reaches 50 first wins, but there are a few important details:

• 2 players, one die. Players take turns.
• Rolled values are simply added up
• If you roll a 6, however, the turn ends and the current turn's sum is discarded
• If you decide to stop your turn, the current turn's sum is added to the total
• When a player reaches 50 or more, he has won, unless he was the starting player, in which case the other player can still try to top the result. He has to get to at least one point more, there is no draw. (This adds most complications. The starting player could e.g. try to get to ~60 instead of stopping at 50 - to make a win more likely when the other player stands at 48)

What is the optimal strategy? It obviously depends on the current turn sum. But it also depends on both player's totals.

Answer 1

In terms of expected value, the optimal strategy is to

roll as long as your total is less than 15.
This is because, if you currently have $P$ points pending, the expected value of the next roll is $\frac{1}{6}(1+2+3+4+5-P)$.
The expected value of playing this way is just over $6.15$ point per turn.

I don't know the strategy for endgames: the probabilities at play look weird enough that calculating the optimal "push" for a given score would have to happen on a case-by-case basis.

Answer 2

Here is programming solution for this problem.

+------------+-------------------------+
| # of Rolls | Expected Turn to Finish |
+------------+-------------------------+
|          1 |                   20.53 |
|          2 |                   12.55 |
|          3 |                   10.14 |
|          4 |                    9.14 |
|          5 |                    8.79 |
|          6 |                    8.66 |
|          7 |                    8.92 |
|          8 |                    9.27 |
|          9 |                    9.52 |
+------------+-------------------------+

The table above shows if you roll

6 times without thinking the outcome of the rolls at the end, your chance to win is the highest with $8.66$.

But if we consider stopping rolling after some sum values we got, let's see if something changes or not: https://pastebin.com/mSPgCi9m

I put # as the number of rolls to stop rolling, stop as stopping rolling after some values for your roll you got, and expected values for those.

As a result,

regardless of # of rolls, just roll until you get at least total roll value of $17$ then stop rolling, otherwise continue rolling seems the optimal way to play this game, which makes the game to win as expected around $8.362$ turn.

with the little tweak as @Jaap Scherphuis suggested, I tried a couple of tricks for the end value where you get close to 50 with some threshold value, and found that having

at least total roll value of $16$ then stop rolling, with little tweak where you continue rolling if you are $47$, gets us $8.345$.

still little improvement though :)