You could add a twist if each student rolled their own two dice, versus the whole class rolling just 2 dice, that changes things a lot.

If the objective of each individual student is to win the game, and be the only winner, and there’s an absolute payoff to being #1, this is different than everyone trying to simply maximize their own score disregarding the scores of their classmates, the latter being more akin to playing multiple rounds of an entire game of CHANCE.

If we start our analysis in the 6th and final round, the choice is to either stand and gamble, or sit and take an automatic 0 points (as it is for each round).

I think the answer for the last round is:

If you are winning by more than the expected value of a return on standing, then you should definitely sit out the last round, otherwise you should gamble.

Expected value of gains is 6.75. In the last round, the game is only between the leader(s) and you. So if in the last round you are down to the leader by any amount, and the leader chooses to sit out, then you must gamble to win the game. If the leader(s) is up by any amount and chooses to gamble, you don’t have to gamble to win the game, but it’s much more likely that the leader will gain points in the last round, so you should gamble in that case also.

On the other hand, if you are winning the game by more than 6.75, then sit, or if you are winning the game by any margin and nobody else chooses to gamble (what?!) then sit.

In the 5th 4th, and prior rounds, things are much more complicated and it’s not just a game between you and the leader(s).

Other it could possible be this simple. In the 5th round: if you are up by more than 2 x 6.75, you should sit out that round, otherwise gamble.

In the 4th round, same idea as 5th round. If you up by more than 3 x 6.75, sit, otherwise gamble.

In the 3rd round, always gamble, because you cannot be up by more than 4 x 6.75 at this round.

In the 2nd round, always gamble.

In the 1st round, always gamble.

Things change if you don’t know the scores of the other players, then you have to compare your current score in each round with the expected maximum score of the other players, and you have to assume the other players are perfectly rational, because you can’t see what choices they are making ðŸ™‚

Definitely a fun problem.

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