In particular, take note of the following points:

- The Optimal Solitaire Yahtzee Player
is designed to maximize its
*expected final score*, while playing*Solitaire*Yahtzee with*fair*dice. The employed strategy is*neither*aimed at maximizing the likelihood of breaking a high-score,*nor*aimed at beating the final score of opponents. To optimize performance for such purposes requires a different strategy. - When assessing a given game state,
the Optimal Solitaire Yahtzee Player
takes
*all legal future developments*of the game into account. Because the dice rolls are not under control of the player, there is no guarantee that in any particular game the indicated*expected final score*will be obtained, or even approached. For instance, the final scores from one million simulated games by the Optimal Solitaire Yahtzee Player ranged from 73 to 845. Also see Backgrounds. - The
*Law of Large Numbers*does guarantee that, when games are played with*fair*dice, the actually obtained*average*final score*taken over a large number of independent games*will be close to the theoretically*expected*final score (i.e. 254.5896...) with great probability. However, the number of games to play for obtaining a predetermined accuracy of approximation with a predetermined confidence level can be enormous. For instance, two independent simulations of one million games yielded average final scores of approximately 254.51 and 254.65 respectively. - The (conditional) expected final scores as given by
the Optimal Solitaire Yahtzee Player are based on
the assumption that
*all future choices in the game are made optimally*. It is imaginable that there are situations where one choice is best under optimal play, but where another choice is safer if you do not know how to proceed optimally.

© 1999, Tom Verhoeff (TUE, Math/CS)

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