{- |
We play the following game:
We roll a die until we stop or we get three spots.
In the first case we own all spots obtained so far,
in the latter case we own nothing.
What is the strategy for maximizing the expected score?
-}moduleNumeric.Probability.Example.DiceAccumwhereimportqualifiedNumeric.Probability.Random asRndimportqualifiedNumeric.Probability.Distribution asDistimportqualifiedNumeric.Probability.Transition asTransimportqualifiedNumeric.Probability.Monad asMonadExtimportNumeric.Probability.Trace (Trace )importNumeric.Probability.Example.Dice (Die ,)typeScore =Intdie ::Fractionalprob =>Dist.T prob Die die :: forall prob. Fractional prob => T prob Score
die =forall prob a. Fractional prob => Spread prob a
Dist.uniform [Score
1..Score
6]roll ::Fractionalprob =>Trans.T prob (MaybeScore )roll :: forall prob. Fractional prob => T prob (Maybe Score)
roll =forall b a. b -> (a -> b) -> Maybe a -> b
maybe(forall (m :: * -> *) a. Monad m => a -> m a
returnforall a. Maybe a
Nothing)(\Score
score ->forall a b c. (a -> b -> c) -> b -> a -> c
flipforall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmapforall prob. Fractional prob => T prob Score
die forall a b. (a -> b) -> a -> b
$\Score
spots ->-- where is my beloved 'toMaybe' ?ifScore
spots forall a. Eq a => a -> a -> Bool
==Score
3thenforall a. Maybe a
Nothingelseforall a. a -> Maybe a
Just(Score
score forall a. Num a => a -> a -> a
+Score
spots ))continue ::Score ->Boolcontinue :: Score -> Bool
continue Score
scoreInt =letscore :: Rational
score =forall a b. (Integral a, Num b) => a -> b
fromIntegralScore
scoreInt ::Rationalinforall a. Num a => T a a -> a
Dist.expected (forall prob a. Fractional prob => Spread prob a
Dist.uniform (Rational
0forall a. a -> [a] -> [a]
:forall a b. (a -> b) -> [a] -> [b]
map(Rational
score forall a. Num a => a -> a -> a
+)[Rational
1,Rational
2,Rational
4,Rational
5,Rational
6]))forall a. Ord a => a -> a -> Bool
>Rational
score -- | optimal strategystrategy ::Fractionalprob =>Trans.T prob (MaybeScore )strategy :: forall prob. Fractional prob => T prob (Maybe Score)
strategy Maybe Score
s0 =forall b a. b -> (a -> b) -> Maybe a -> b
maybe(forall (m :: * -> *) a. Monad m => a -> m a
returnforall a. Maybe a
Nothing)(\Score
score ->ifScore -> Bool
continue Score
score thenforall prob. Fractional prob => T prob (Maybe Score)
roll Maybe Score
s0 elseforall (m :: * -> *) a. Monad m => a -> m a
returnMaybe Score
s0 )Maybe Score
s0 -- | distribution of the scores that are achieved with the optimal strategygame ::Fractionalprob =>Dist.T prob (MaybeScore )game :: forall prob. Fractional prob => T prob (Maybe Score)
game =forall prob a. (Num prob, Ord a) => [T prob a] -> T prob a
Trans.compose (forall a. Score -> a -> [a]
replicateScore
18forall prob. Fractional prob => T prob (Maybe Score)
strategy )(forall a. a -> Maybe a
JustScore
0)-- MonadExt.compose (replicate 8 turn) (Just 0){- too inefficient
game :: Fractional prob => Dist.T prob Score
game =
 let turn score =
 if continue score
 then roll score >>= \s -> if s==0 then return 0 else turn s
 else return score
 in turn 0
-}walk ::Int->IO(Trace (MaybeScore ))walk :: Score -> IO (Trace (Maybe Score))
walk Score
n =forall a. T a -> IO a
Rnd.run forall a b. (a -> b) -> a -> b
$forall (m :: * -> *) a.
Monad m =>
Score -> (a -> m a) -> a -> m [a]
MonadExt.walk Score
n (forall prob a.
(Num prob, Ord prob, Random prob) =>
T prob a -> Change a
Rnd.change (forall prob. Fractional prob => T prob (Maybe Score)
roll ::Trans.T Double(MaybeScore )))(forall a. a -> Maybe a
JustScore
0)