In this question, I refer to two separate games. The first is a game where you roll and accumulate your score until a six appears:
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 rolled 6, then you lose everything and the game ends. Otherwise, add the score from the die to your total points and continue/stop the game. When should one stop playing this game?
The other is a game which I asked before, where you roll and accumulate your score until a repeat appears:
I keep rolling a die, and my score is the sum of all my rolls. However, if I roll a value I had rolled before, I lose all. What is the optimal strategy?
From what I gather in the first link, the greedy approach in the accepted answer is not technically correct since the calculated gain does not account for future rolls, although for some reason it is optimal. In the second link (the question I asked), I did not account for future rolls which hindered my analysis of when to stop (I am yet to figure out what the answerer means by "beating" the minimum sum of $1+2+3=6$. I am quite confused by these concepts, and therefore have two questions:
1) Sadly this is a repeat of one part the question I asked before — what exactly is meant by long-term gain of future rolls and how do you calculate it? Is it through some recursive probability calculation that accounts for our decision after each roll? Do we let the number of rolls tend to infinity?
2) More importantly, when is a greedy approach optimal, and how do we prove it? For the first link, I don’t really understand it (despite the thorough debate about it).