On Tue, 06 Sep 2005 07:53:14 -0400, Eric Sosman
<esosman@[EMAIL PROTECTED]
> wrote:
> Arthur J. O'Dwyer wrote:
>>
>> On Sun, 4 Sep 2005 videoguy505@[EMAIL PROTECTED]
wrote:
>>>[...]
>>> My question is, is there an optimal strategy for this game...
>>> especially regarding when to save a 5? I know that a later turn is
>>> obviously better, since you know the score to beat.
>>
>> No, that can't be right. Wouldn't you just try to score as high as
>> possible all the time, no matter when your turn was? I know I would. :)
>
> That's a slightly different goal. The strategy that
> maximizes your expected score may not be the same as the
> strategy that maximizes your likelihood of exceeding a
> specified score. For example, suppose an earlier player has
> already achieved a perfect score of 24, so your only hope
> of sharing in the pot is to match that score. A strategy
> that aims to maximize your expected score might tell you
> to be content if you roll 1-2-5-6-6-6 for a total of 23,
> but if you follow that strategy in this situation you are
> guaranteed to lose.
>
> It seems to me the goal of the Kth player should be to
> maximize the likelihood that his score exceeds the maximum
> of the K-1 preceding scores and the expected maximum of
> the N-(K-1) scores that will follow. Since the score that
> K actually achieves can influence the expected scores of
> the later players (by "raising the bar"), the calculation
> seems quite complicated. Perhaps it could be attacked by
> starting with the Nth player and computing his expected scores
> when he aims to exceed each of the possible preceding maxima,
> then work back to the (N-1)st player, and so on.
>
(I think that this method will analyze a slightly simpler problem, where
draws do not split the pot; they entirely go to the first tying player,
though I may have made an error in it...)
Let NDICE and NSIDES be fixed constants
Define f(total, pot, later_players) -> (avg_payoff, p_later_wins, avg_pot)
- total is the highest total of any previous qualifying player.
- pot is the number of chips in the pot
- later_players is the number of players who play after this player
- avg_payoff is the average win by this player in this situation
- p_later_wins is the probability this player or someone later wins
- avg_pot is the average end value of the pot
Define g(total, has1, has2, pot, later_players, dice_left) ->
(avg_payoff, p_later_wins, avg_pot)
- total is (points by highest previous player) - (points stored by this
player)
- has1 is 1 if this player has stored a 1, 0 if otherwise
- has2 is 1 if this player has stored a 2, 0 if otherwise
- pot is the value of the pot initially
Return values are as above
Define h(total, has1, has2, pot, later_players, (dice)) ->
(avg_payoff, p_later_wins, avg_pot, no_dice)
- (dice) is a sorted list of dice values
- no_dice is the number of dice taken by the player
f and g are cached; h is not. (If g is written properly, no value of h
will be evaluated by g more than once; it will only be reevaluated if the
player is looking for a strategy.)
g (A, B, C, D, E, F) ->
weighted_avg (sorted lists L of F numbers from 1 to NSIDES:
weight(multinomial(L)) : h(A, B, C, D, E, L))
h(...) uses rules:
Always take a 1 if there is one and you don't already have one.
Similar with 2.
After that, consider each possibility of taking top N dice.
Return values corresponding to highest avg_payoff.
(Rest of recursion omitted).
--
------------------------
Mark Jeffrey Tilford
tilford@[EMAIL PROTECTED]
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