When I first mentioned Pig in the Toss Up context, I said:
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 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.
Now I have thought about the subtleties a little, and I'm confused. This is all stuff which is the same for both games. Their way to solve the game is to know all values of P(i,j,k), your probability of winning where (i,j,k) = (#pts for you, #pts for other player, #pts you've accumulated so far on this turn, which you'd risk by rolling again). Once you know these probabilities, they imply a set of roll/don't roll decisions, which manifest in the set of equations relating the P(i,j,k)s as max()s. Neller and Presser use "value iteration" to find a solution to that set of equations. But I don't understand how they conclude that their solution is indeed optimal play. I do believe that if you knew the correct roll/don't roll decisions, you could solve for the probabilities. But it seems to me that you can have some set of P()s and sub-optimal set of roll/don't roll decision which are compatible with one another, but suboptimal globally. Can anyone help me out here? --Michael Kleber -- It is very dark and after 2000. If you continue you are likely to be eaten by a bleen.