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Talk:Pig the dice game/Player: Difference between revisions
→Strategy: not "winning", but "more likely to win"
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To do better than this, we would have to engage in "discovering how our opponent is different from us". Here, we might start with the opponent using a "like us" strategy and then look for evidence that the opponent is using some other strategy, and then attempt to build a model of that strategy and play against our models of our opponent instead of just against an opponent who is "just like us". It's not at all clear, though, that there's any justification for doing that for this game. (And, of course, this approach is also intractable in the general case (where it's not trivial and irrelevant) because the number of implementations open to our opponent is infinite.) --[[User:Rdm|Rdm]] 12:02, 15 September 2012 (UTC)
: By "winning strategy" I meant the choice of roll or hold that maximizes the chance of winning at a given board state, that's what the P() function is. Instead of thinking it as function, consider P as a matrix with board state as its subscripts, and the elements need to satisfy the relations given in the code. An optimal strategy exists (for both sides) if there is a unique set of P values that satisfy all the interrelations; if no such solution exists, or there are multiple sets of solutions, then this game has no global optimal strategy. But, if a unique solution does exist, then there's no need to consider what strategy the opponent would choose: you just do what gives you best winning chance. If your opponent is smart enough, he'd do the same, otherwise it's all the worse for him.
: I believe this particular game has a unique solution, and I do have ''a'' solution of the P matrix, although proving it's unique is not easy. The holding choice pattern is pasted [http://pastebin.com/SWhU4mL7 here], where each row starts with two numbers, your current score and current holding score; the horizontal axis is the opponent's current score. Wherever there's a dot, it means you're better off hold; otherwise you should roll. For example, on line "0 21: ", there are 11 dots at the begining of the line, which means: if your total score is 0, and have already rolled 21 points this turn, then your best choice is hold if opponent has 0-10 points, but better keep rolling if he has 11 or more. The graph has many odd features, but generally makes sense. --[[User:Ledrug|Ledrug]] 13:13, 15 September 2012 (UTC)
I should at least include a strategy of random play in my solution so that I can gauge any other strategy against it.--[[User:Paddy3118|Paddy3118]] 11:28, 15 September 2012 (UTC)
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