# Progressive Dice Game

You have a fair, regular 6-sided dice.

The game is played for $n$ turns. Each turn you make a roll and gain that many points the rolled side is showing, then do one of the following:

• Upgrade the dice evenly placing your rolled points to the sides of the dice;
Same side can't be assigned with two extra points unless all sides received one point, same goes that a side can't be assigned with three extra points unless all sides received two points, and so on.
• Downgrade the dice evenly taking points off the sides like when upgrading, but ignore the sides showing zero because sides showing $0$ cannot be further downgraded to a negative value. After the downgrade you take a REROLL;
Which effectively restarts your whole turn, so you can try to get a better value.

What is the optimal way to play to maximize your expected score at the end of the game?

## Optimal Play - My Guess

I concluded that you should "invest" points and then bank that investment in the last few turns to maximize your points?

If you upgrade the first $t$ turns, then bank the rest of the turns, you will expect the following amount of points on average: $$f(t) = 3.5\times\left(\frac{7}{6}\right)^{t}\times(n-t)$$

Which boils down to, that if you want to maximize your points, you should upgrade till the last $7$ turns and then bank those turns, which makes sense in a way.

But this approach almost completely ignores the third action; the rerolls.

## Rerolls?

I haven't fully considered the third action, since it downgrades the dice which means that the more turns you have left to roll after the downgrade, the more you will lose overall.

But that isn't always the case, for example rerolling a $1$ is pretty clear and won't have any negative effects on the dice, effectively providing you a $0$-side free reroll next time.

Those were my initial thoughts, but it turns out when playing for not too big $n$ number of turns, you can always reroll anything but the highest side to force the highest side on being rolled by only downgrading anything but the highest side and only banking and upgrading the highest side, effectively maximizing your score?

But sometimes you will get a case when rerolling a high but not the highest value, will downgrade the highest value, so this idea needs to be examined in detail to extract the best possible use of those rerolls.

• Can you be more specific about the meaning of "[increment] its sides distributing the points one by one" and "downgrade the dice evenly taking one by one point"? Are you incrementing/decrementing all sides by one? Just the smallest/largest side? – mjqxxxx May 17 '16 at 15:51
• @mjqxxxx You do one side by one point, then repeat for another side, till you use up all your points that were rolled, if you do all sides and have points left to spent, you do the same with the remaining points again. (Example; rolled $4$ = do $4$ sides, each by $1$ point, Example; $8$ = do all sides by one point, then do two sides by one point) – Vepir May 17 '16 at 15:53
• And in the downgrade you remove as many points as you'd rolled, and then effectively get a new turn, in which you have a new choice whether to bank, upgrade or downgrade? – joriki May 17 '16 at 16:30
• You can't maximise your score, since this is a random variable. I suspect what you mean is to maximise the expected value of the score? – joriki May 17 '16 at 16:31
• And for the downgrade: Does this also start over once you've taken a point off all sides? If there are sides with $0$, does the downgrade start over once you've taken a point off all non-zero sides, or do the zero sides count towards the points despite not being further downgraded? – joriki May 17 '16 at 16:34