The object of the dice game is to be the first player to reach a score of at least 100 points.
Each player’s turn consists of repeatedly rolling a die.
After each roll, the player has two choices: roll again, or stop.
- If the player rolls 1, nothing is scored in that turn and it becomes the opponent’s turn.
- If the player rolls a number other than 1, the number is added to the player’s turn total and the player’s turn continues.
- If the player stops, the turn total (the sum of the rolls during the turn), is added to the player’s score, and it becomes the opponent’s turn.

What's your strategy?

Interesting question.
I have to do some paid work, so I have time for just some preliminary thoughts.

1) Your strategy at any given decision point depends on how many points (P) you had at the start of the turn, how many points your oppponent had (Q), and how much you have rolled so far this turn (T).  If your opponent had 99 points at the start of the roll, you clearly keep rolling and hope you don't get a 1.

2) If you and your opponent started with zero points, I guess that you want to maximize the expected number of points scored on the turn.  (No proof, just a guess).  The break-even calculates as 20.   A roll will lose T points 1/6 of the time, and gain an expected 4 points 5/6 of the time.  So I guess that you roll again if T = 19 or less, and take your score if T = 21 or more.   If T = 20, I guess that you end your turn, on the theory (unproven) that you are ahead and you should be conservative.

More on this later ...

Posted by Steve Herman on 2005-04-21 20:41:24