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
- 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?
(In reply to Nearly there
a) Thanks for the words of encouragement.
b) "The total solution is probably not something for this web site,
more for a highly specialised math web site." -- Boy, are you right about that!
c) A general comment. In the start of the game, we probably come
very close to an ideal strategy when we maximize our expected
points. But, of course, what we really want to maximize is the
probability that we will win the game, not our margin of victory.
Therefore, at some point we get more conservative (stop with something
less than 20) if we are ahead and more aggressive (keep rolling even
with 21) if we are behind. And I'm sorry, but I can't be any more
specific than that yet.
d) You ask, 'Why would
the opponent stop at 99?". Well here is one scenario. My
brother (who is not a mathemetician, but who is very lucky) is tied
with his opponent at 50 points apiece. He starts rolling while I
am out of the room, and so far has rolled 49 points on his turn.
I walk into the room and he asks me if he should roll again. I
consult my very complicated spreadsheet, and tell him that:
1) If he stops now, his probability of eventually winning is
94.08%. (It's right, but don't ask me to prove this).
2) If he rolls one more time, he will win immediately 5/6 of
the time. If he rolls a one, however, he will go back to
50 points, lose the roll, and have something less than
a 50% chance of winning the game. All together, if he
rolls again his chance of winning is something less than
My brother wisely pays attention, stops at 99, has a bad run of luck thereafter, and takes 3 more turns to win the game.