The game Thane describes is a member of the "Pig" family, aka "jeopardy dice games". Many variants have been solved by Neller and Presser; their two papers and a summary of variants are at http://cs.gettysburg.edu/projects/pig/pigCompare.html The thing that makes this type of game hard to analyze is that the "probability of winning from a given state" calculation is based on your roll/hold decisions, which are in turn based on which would give you a higher probability of winning, in a loopy way. (If you and your opponent both bomb your turns, you end up back in the same state.) As Neller and Presser say, "There is no known general method for solving equations of the form x = max(Ax+b, A'x+b')." The method they use is to find the probabilities by some sort of iterative approximation, but I think there are subtleties I haven't thought about, so I won't try to give more details than that. Doubtless if someone pointed this variant out to them, they could report perfect play and the winning probabilities from every position in short order. It doesn't look like they report komi -- which is, I suppose, the 2nd-player-bonus to leave the odds as close to 50-50 as possible? In that case it's not clear which player will have the advantage once komi is used, though we expect it will be small in either case. --Michael Kleber On 8/25/07, Thane Plambeck <tplambeck@gmail.com> wrote:
Target is selling a game called Toss Up. It comes with 10 identical 6-sided dice and a rulebook. Each die has 3 green faces, 2 yellow faces, and 1 red face.
http://www.areyougame.com/Interact/item.asp?itemno=PA7367
On a player's turn, he starts by rolling all 10 dice. Dice that come up as green are added to the player's score, one point per die, and are put aside. The player then has the option to roll again with the remaining dice. If at least one is green, all the just-rolled greens are then put aside, he gets that many more points, and he has the option to roll again, etc.
If ever the player fails to roll a green and also has at least one red, he is bankrupted. His total accumulated score becomes zero and all the dice are passed to the next player. If alternatively he manages to score ten points, he gets to start over with all ten again. Finally, a player can also decide to stop at any time, passing the dice on to the next player at any point.
Whoever is the first to score 100 points wins.
In a two person game, it's presumably got to be advantageous to be the first to roll. But suppose the second player is given a "fair" initial positive score X (ie, a komi) to compensate for that.
What is X?
-- Thane Plambeck tplambeck@gmail.com http://www.plambeck.org/ehome.htm
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