Suppose we're playing a dice game where the goal is to roll dice with a sum of at least $s$. We have $n$ dice with $k$ sides.
The rules for rolling the dice are as follows:
- A player must roll at least once.
- A player must roll all the dice they can.
- Each time a player rolls their dice, they must submit at least one die, locking its value in place. Submitted dice may not be rerolled.
- A player may continue rolling until they either decide to stop or run out of dice.
- A player always submits the die or dice that provide the best a posteriori chance of winning.
For example, suppose I have 2d6, with a target sum of $s=7$.
I roll the dice and they land on $(1, 4)$. Since I have to submit at least one die, I choose to lock in the 4. I've not yet met my goal, so I roll my remaining die again, and get a 3. It's my last die, so I must submit it. The total value of my submitted dice is $4+3=7$, so I win.
I've been able to simulate the odds of each outcome and build a state transition matrix, but my attempts to find a closed form for the probability of winning (assuming perfect play) have been unsuccessful so far. Is there a closed form for it? Alternatively, is there a convenient expression for calculating this without doing a simulation? Is it substantially different if the dice have arbitrarily-valued sides (e.g. a d6 where, instead of 6, one side is -1)?