We can approximate it with our pre-existing value function for a known stopping time/edge/max wealth, sampling from the posterior; for example, we might draw 1000 values from 𝒩(300,25), the Pareto, the Beta etc., calculate the value of each set of parameters, and taking the mean value. Since the value of information is always zero or positive, the value of the POMDP with known parameters must be an upper bound on its true value, since we can only do worse in each game by being ignorant of the true parameters and needing to infer them as we play; so if we calculate a value of $240 or whatever, we know the POMDP must have a value <$240, and that gives a performance target to aim for.
The Haskell version is the easiest to work with, but Haskell doesn’t have convenient distributions to sample from, so instead I generate 1000 samples from the prior distribution of parameters in Python (reusing some of the Gym code):
and then we can modify nshepperd’s Haskell code to allow the edge and wealth cap to be modifiable, and then we simply run memo over the 1000 possible sets of parameters, and take the mean. Using the raw samples turns out to be infeasible in terms of RAM, probably because higher wealth caps (which are Pareto distributed and can be extremely large) increases dimensionality too much to evaluate in my ~25GB RAM, so the wealth cap itself must be shrunk, to TODO $1,000? which unfortunately muddles the meaning of the estimate since now it’s a lower bound of an upper bound, or to reverse it, value of clairvoyant play assuming that many games have a lower wealth cap than they do.
import Data.Vector (Vector)
import qualified Data.Vector as V (generate, (!))
type Wealth = Int
type Bets = Int
type Edge = Double
type Cap = Int
type EV = Double
value :: (Wealth -> Bets -> EV) -> Wealth -> Bets -> Edge -> Cap -> EV
value _ 0 _ _ _ = 0
value _ w 0 _ _ = fromIntegral w
value next w b e cap = maximum [go x | x <- [0..min (cap - w) w]]
where
go x = e * next (min cap (w+x)) (b-1) + (1-e) * next (w-x) (b-1)
memo :: Wealth -> Bets -> Edge -> Cap -> EV
memo w b e cp = cached_value w b
where
cached_value w' b' = table V.! b' V.! w'
table :: Vector (Vector EV)
table = V.generate (b + 1)
(\b'' -> V.generate (cp + 1)
(\w'' -> value cached_value w'' b'' e cp))
main :: IO ()
main = do let evs = map (\(w,b,e,cap) -> memo w b e cap) parameters
print evs
print $ mean evs
where mean x = sum x / fromIntegral (length x)
maxCap :: Int
maxCap = 800
parameters :: [(Wealth, Bets, Edge, Cap)]
parameters = map (\(w,b,e,c) -> (w,b,e, max maxCap c)) [(25, 236, 0.72435085557, 205), (25, 237, 0.845435411107, 260), (25, 238, 0.413782110845, 202),
(25, 240, 0.345133535603, 23999), (25, 240, 0.400147029568, 2521), (25, 242, 0.661912880989, 201), (25, 242, 0.676168121514, 6740),
(25, 242, 0.701381545443, 1726), (25, 242, 0.849537381053, 203), (25, 242, 0.898462493046, 201), (25, 243, 0.54257657328, 201),
(25, 245, 0.77972062792, 48534), (25, 246, 0.813395396485, 203), (25, 247, 0.480134063147, 214), (25, 247, 0.533273012857, 222),
(25, 247, 0.754222974601, 201), (25, 249, 0.70782818171, 200), (25, 249, 0.748310452532, 202), (25, 251, 0.606478140487, 200),
(25, 251, 0.809355705263, 526), (25, 252, 0.524864167122, 200), (25, 252, 0.749206398756, 352321), (25, 252, 0.822818892461, 201),
(25, 253, 0.542598606499, 201), (25, 253, 0.600469152584, 200), (25, 253, 0.648598435209, 219), (25, 253, 0.649453264073, 2589),
(25, 253, 0.679414549454, 28311200), (25, 253, 0.86227613481, 202), (25, 254, 0.585056497076, 2575033),
(25, 254, 0.680452842612, 200), (25, 255, 0.877950348346, 201), (25, 256, 0.643298850757, 205), (25, 256, 0.712973552037, 204),
(25, 256, 0.718153926517, 201), (25, 256, 0.813126205309, 201), (25, 256, 0.8280445068, 208), (25, 257, 0.402688232643, 201),
(25, 257, 0.548591912297, 651), (25, 257, 0.601967601919, 206), (25, 257, 0.862541920316, 200), (25, 257, 0.863389650382, 200),
(25, 258, 0.474519595118, 213), (25, 258, 0.735744534702, 208), (25, 258, 0.738339372201, 360), (25, 258, 0.84277195943, 83144),
(25, 258, 0.93190614133, 2409618061), (25, 259, 0.471094149142, 745), (25, 260, 0.628797799899, 202), (25, 260, 0.710033767475, 9552510),
(25, 260, 0.813466187199, 200), (25, 261, 0.550748996178, 230), (25, 261, 0.629544256918, 259), (25, 261, 0.683386138653, 200),
(25, 261, 0.792209742213, 201), (25, 261, 0.794533480259, 207), (25, 262, 0.344887568623, 201), (25, 262, 0.450645011078, 473),
(25, 262, 0.738248174312, 225), (25, 262, 0.791527693933, 210), (25, 262, 0.832198167412, 200), (25, 263, 0.35093259949, 203),
(25, 263, 0.521818533732, 366), (25, 263, 0.64076936694, 59847), (25, 263, 0.684063131912, 201), (25, 263, 0.714065733976, 201),
(25, 263, 0.789350373174, 1705), (25, 263, 0.829406017035, 210), (25, 263, 0.902144916794, 255), (25, 263, 0.951257247224, 202),
(25, 264, 0.32472812884, 203), (25, 264, 0.604563170498, 250), (25, 264, 0.626423177116, 22769), (25, 264, 0.871339562605, 205),
(25, 265, 0.452014126574, 200), (25, 265, 0.54226419925, 201), (25, 265, 0.625113911686, 212), (25, 265, 0.654404133816, 206),
(25, 265, 0.678345789504, 420), (25, 265, 0.771116406164, 210), (25, 265, 0.771222624754, 205), (25, 265, 0.801480809161, 200),
(25, 265, 0.806182111956, 60957), (25, 265, 0.851715904261, 18981), (25, 266, 0.685689594858, 200), (25, 266, 0.711416415881, 210),
(25, 266, 0.711937019202, 851), (25, 266, 0.724721571466, 200), (25, 266, 0.816529965664, 654), (25, 266, 0.820884461992, 287),
(25, 267, 0.798462101302, 200), (25, 267, 0.814520382826, 200), (25, 267, 0.815795664692, 201), (25, 267, 0.877810894936, 200),
(25, 268, 0.37374747435, 7152), (25, 268, 0.66383704238, 201), (25, 268, 0.671499425192, 224), (25, 268, 0.72882798744, 205),
(25, 268, 0.788161778332, 534537579002), (25, 268, 0.793723223502, 584), (25, 268, 0.83414422735, 16291),
(25, 268, 0.933374294433, 219), (25, 269, 0.460012103336, 211), (25, 269, 0.575089962258, 203), (25, 269, 0.587666214938, 213),
(25, 269, 0.617907437269, 201), (25, 269, 0.711693869892, 203), (25, 269, 0.755193420929, 200), (25, 269, 0.764409970652, 2593),
(25, 269, 0.769445254497, 6962), (25, 269, 0.781046904571, 1030), (25, 269, 0.92721940969, 2062780), (25, 269, 0.937746457329, 201),
(25, 270, 0.43758799016, 201), (25, 270, 0.55945290829, 206), (25, 270, 0.594599037706, 579), (25, 270, 0.6205174976, 506),
(25, 270, 0.660258985896, 217), (25, 270, 0.688180575479, 308), (25, 270, 0.703263135667, 201), (25, 270, 0.730492162736, 46132),
(25, 270, 0.7898205196, 25541), (25, 270, 0.893976819617, 201), (25, 270, 0.921080018968, 200), (25, 271, 0.525815069976, 200),
(25, 271, 0.554615752325, 200), (25, 271, 0.629469767651, 200), (25, 271, 0.709135907244, 9875), (25, 271, 0.759981836553, 202),
(25, 271, 0.811129692174, 91717), (25, 271, 0.837347167701, 200), (25, 271, 0.873514454515, 200), (25, 271, 0.923252939562, 208),
(25, 272, 0.556207216343, 785), (25, 272, 0.575453865622, 215), (25, 272, 0.638921822998, 200), (25, 272, 0.642223697178, 219),
(25, 272, 0.66336333735, 213), (25, 272, 0.798971543861, 209), (25, 272, 0.826187121413, 216), (25, 272, 0.864733208766, 204),
(25, 273, 0.517135630277, 200), (25, 273, 0.560438221638, 221192794), (25, 273, 0.77368870442, 883), (25, 273, 0.846232723863, 785),
(25, 273, 0.9652451074, 200), (25, 274, 0.607659170171, 339), (25, 274, 0.611760970714, 201), (25, 274, 0.638249890182, 201),
(25, 274, 0.706185385091, 201), (25, 274, 0.760147689131, 200), (25, 274, 0.804365960436, 1051), (25, 274, 0.848798405341, 1117),
(25, 275, 0.333707647765, 200), (25, 275, 0.555403709258, 205), (25, 275, 0.655942931908, 206), (25, 275, 0.676684395133, 201),
(25, 275, 0.698400431321, 203), (25, 275, 0.796131451966, 200), (25, 275, 0.840528265848, 200), (25, 275, 0.856009338181, 200),
(25, 276, 0.497611590615, 211), (25, 276, 0.719196036042, 202), (25, 276, 0.72119258703, 212), (25, 276, 0.724848372617, 201),
(25, 276, 0.727773523116, 947132), (25, 276, 0.786181091729, 223), (25, 276, 0.835470024888, 213), (25, 276, 0.850785506884, 1650536293),
(25, 276, 0.872106280647, 205), (25, 277, 0.409025006548, 1245), (25, 277, 0.432733743192, 200), (25, 277, 0.571657665503, 45217),
(25, 277, 0.702562712047, 201), (25, 277, 0.730418085673, 575), (25, 277, 0.749515377892, 200), (25, 277, 0.775231917334, 200),
(25, 277, 0.790660296053, 204), (25, 277, 0.875858958177, 236), (25, 278, 0.379322394893, 309), (25, 278, 0.420790416182, 201),
(25, 278, 0.527244364126, 200), (25, 278, 0.615000956496, 205), (25, 278, 0.675667333791, 201), (25, 278, 0.696208960885, 201),
(25, 278, 0.790420764423, 204), (25, 278, 0.841240203947, 523), (25, 279, 0.341095083629, 201), (25, 279, 0.476230165726, 1798),
(25, 279, 0.518063098433, 209), (25, 279, 0.656454646158, 306), (25, 279, 0.684136349, 273), (25, 279, 0.709847569568, 119033),
(25, 279, 0.762133225185, 200), (25, 279, 0.776577127267, 200), (25, 279, 0.79611434817, 201), (25, 279, 0.808780479341, 425),
(25, 279, 0.850439803007, 204), (25, 279, 0.898230041029, 204), (25, 279, 0.945894882249, 616), (25, 280, 0.368073973138, 14001),
(25, 280, 0.464321107456, 1523), (25, 280, 0.617310083425, 504403), (25, 280, 0.657193497398, 204), (25, 280, 0.707839204221, 1265268),
(25, 280, 0.76379246649, 304), (25, 280, 0.772912128233, 201), (25, 280, 0.865579809939, 77454), (25, 281, 0.475947154647, 200),
(25, 281, 0.557771053261, 2188305), (25, 281, 0.56666775076, 12340), (25, 281, 0.638041977737, 200), (25, 281, 0.668510798451, 212),
(25, 281, 0.678366665681, 1294), (25, 281, 0.716116162808, 213), (25, 281, 0.748052894007, 203), (25, 281, 0.749911576065, 23865988060),
(25, 281, 0.759712569326, 204), (25, 281, 0.806434961546, 353082), (25, 281, 0.893617069228, 200), (25, 281, 0.931891968132, 203),
(25, 282, 0.454895506694, 11408), (25, 282, 0.549639923973, 462), (25, 282, 0.569017378489, 201), (25, 282, 0.577850694309, 200),
(25, 282, 0.584479523845, 334042071872), (25, 282, 0.599723920362, 200), (25, 282, 0.632612897763, 1293),
(25, 282, 0.681109488172, 200), (25, 282, 0.724133951717, 200), (25, 282, 0.754930901531, 201), (25, 282, 0.778740223985, 203),
(25, 282, 0.779781137188, 200), (25, 282, 0.781305596085, 202), (25, 282, 0.83569736595, 200), (25, 282, 0.850446466345, 1253),
(25, 282, 0.859549928516, 201), (25, 282, 0.871855706726, 200), (25, 283, 0.327637841131, 200), (25, 283, 0.404249604299, 201),
(25, 283, 0.561172940189, 200), (25, 283, 0.668083130397, 201), (25, 283, 0.678629207223, 203), (25, 283, 0.74122980107, 208),
(25, 283, 0.744679958097, 200), (25, 283, 0.760349422921, 203), (25, 283, 0.781697384113, 209), (25, 283, 0.784799938769, 202),
(25, 283, 0.789112337759, 236), (25, 283, 0.819466827649, 262), (25, 283, 0.839687812522, 200), (25, 283, 0.856859655325, 271),
(25, 284, 0.495523668627, 201), (25, 284, 0.56435102208, 205), (25, 284, 0.588093361148, 219), (25, 284, 0.593268824769, 200),
(25, 284, 0.640596228637, 213), (25, 284, 0.653491369691, 44714), (25, 284, 0.675009753521, 202), (25, 284, 0.773401673731, 200),
(25, 284, 0.805092875342, 213), (25, 284, 0.82735685621, 263), (25, 284, 0.840696528783, 16521), (25, 285, 0.362802762398, 320),
(25, 285, 0.367012291981, 20073), (25, 285, 0.521190733548, 200), (25, 285, 0.575799236698, 200), (25, 285, 0.637555604061, 201),
(25, 285, 0.642866106846, 200), (25, 285, 0.739442877608, 1154709), (25, 285, 0.740973872676, 205), (25, 285, 0.779312184025, 258),
(25, 285, 0.7798668215, 1286), (25, 285, 0.841286817662, 200), (25, 285, 0.913642856385, 858), (25, 286, 0.526160290087, 202),
(25, 286, 0.57102556171, 200), (25, 286, 0.605564882529, 200), (25, 286, 0.673079677685, 202), (25, 286, 0.675864181927, 200),
(25, 286, 0.723247596231, 224), (25, 286, 0.723347705484, 203), (25, 286, 0.767358570943, 207), (25, 286, 0.775537484183, 241),
(25, 286, 0.780033922166, 2962), (25, 286, 0.806301077907, 1553), (25, 286, 0.829016366841, 200), (25, 286, 0.839169441571, 200),
(25, 286, 0.852776222427, 28148), (25, 286, 0.853758532671, 200), (25, 286, 0.906829095884, 200), (25, 286, 0.922712587421, 202),
(25, 287, 0.498032911641, 200), (25, 287, 0.592863506249, 233), (25, 287, 0.622406587519, 208), (25, 287, 0.718903577193, 218),
(25, 287, 0.72138759488, 227), (25, 287, 0.749942677419, 240), (25, 287, 0.762366566525, 215), (25, 287, 0.770805576807, 200),
(25, 287, 0.797492731584, 200), (25, 287, 0.835165503936, 200), (25, 287, 0.946751084636, 863358347021),
(25, 288, 0.306016851261, 201), (25, 288, 0.468949036459, 4770), (25, 288, 0.549651576351, 201), (25, 288, 0.592965066535, 6236),
(25, 288, 0.706825000163, 216), (25, 288, 0.74286176604, 41482), (25, 288, 0.743238045326, 206), (25, 288, 0.777949156606, 213),
(25, 288, 0.781597068828, 418), (25, 288, 0.794578713285, 200), (25, 288, 0.828331653519, 418), (25, 288, 0.834874015356, 212),
(25, 288, 0.848009292304, 200), (25, 288, 0.852116433152, 200), (25, 288, 0.906593204881, 202), (25, 288, 0.920623087617, 220),
(25, 289, 0.524452337599, 1057), (25, 289, 0.552951317234, 208), (25, 289, 0.651146664053, 200), (25, 289, 0.658611767347, 1126709768047),
(25, 289, 0.691072913786, 201), (25, 289, 0.695521738775, 42204), (25, 289, 0.710204540486, 207), (25, 289, 0.716186135845, 3696),
(25, 289, 0.748287169753, 209), (25, 289, 0.750374480604, 201), (25, 289, 0.810573219838, 201), (25, 289, 0.824662154794, 200),
(25, 289, 0.881953772318, 200), (25, 289, 0.885424313568, 200), (25, 290, 0.478371525842, 2087), (25, 290, 0.509245836087, 200),
(25, 290, 0.559208272006, 202), (25, 290, 0.566118076472, 336), (25, 290, 0.592660738469, 200), (25, 290, 0.594018239861, 273),
(25, 290, 0.609100591562, 201), (25, 290, 0.613710234154, 235), (25, 290, 0.622454795175, 201), (25, 290, 0.642187814474, 201),
(25, 290, 0.650179222226, 202), (25, 290, 0.71534164315, 3261), (25, 290, 0.720180673012, 206), (25, 290, 0.737585832942, 200),
(25, 290, 0.775169473922, 229), (25, 290, 0.783784198305, 53137), (25, 290, 0.789135961744, 201), (25, 290, 0.800078889184, 200),
(25, 291, 0.405626786206, 34744), (25, 291, 0.525738841457, 211), (25, 291, 0.533871294974, 12996), (25, 291, 0.608310459901, 201),
(25, 291, 0.70893418817, 200), (25, 291, 0.748805670518, 200), (25, 291, 0.760314517315, 202), (25, 291, 0.760919201657, 215),
(25, 291, 0.763324684293, 200), (25, 291, 0.778634985402, 200), (25, 291, 0.805883746215, 200), (25, 291, 0.827079825705, 203),
(25, 291, 0.827672243803, 200), (25, 292, 0.353361504401, 9552), (25, 292, 0.413680391502, 240), (25, 292, 0.498283007005, 200),
(25, 292, 0.560682661174, 201), (25, 292, 0.660502227784, 201), (25, 292, 0.676014312494, 200), (25, 292, 0.729161823418, 200),
(25, 292, 0.754733824112, 230), (25, 292, 0.772617271317, 312), (25, 292, 0.778956063712, 212), (25, 292, 0.808767644485, 200),
(25, 292, 0.808779697891, 201), (25, 292, 0.844087667679, 202), (25, 292, 0.904903907681, 200), (25, 293, 0.524954771554, 4945),
(25, 293, 0.57444832542, 233), (25, 293, 0.600122035306, 200), (25, 293, 0.69001246376, 205), (25, 293, 0.691232039332, 207),
(25, 293, 0.691742268177, 211), (25, 293, 0.698366150359, 200), (25, 293, 0.711927031055, 1815880), (25, 293, 0.71619667357, 3964),
(25, 293, 0.728536730678, 200), (25, 293, 0.741604172773, 200), (25, 293, 0.743399425593, 201), (25, 293, 0.758084864431, 200),
(25, 293, 0.759978441853, 200), (25, 293, 0.780081691192, 1448778253), (25, 293, 0.788319084437, 208),
(25, 293, 0.801525808156, 200), (25, 293, 0.813155425067, 206), (25, 293, 0.819811779201, 377), (25, 293, 0.837649946204, 209),
(25, 293, 0.869752644338, 4592), (25, 293, 0.875398670372, 200), (25, 293, 0.918150841685, 200), (25, 293, 0.921450726956, 201),
(25, 294, 0.445036622233, 202), (25, 294, 0.496380029754, 2294499), (25, 294, 0.517533203971, 214), (25, 294, 0.597738939234, 213),
(25, 294, 0.597783619419, 102427), (25, 294, 0.600782734524, 263), (25, 294, 0.625741320644, 200), (25, 294, 0.632555212361, 202),
(25, 294, 0.633928301179, 217), (25, 294, 0.720710116312, 200), (25, 294, 0.741687972492, 235), (25, 294, 0.775184150825, 203),
(25, 294, 0.799975478422, 2129), (25, 294, 0.811533351383, 221), (25, 294, 0.827766846665, 47518), (25, 294, 0.834896019626, 200),
(25, 294, 0.855889683449, 200), (25, 294, 0.887096214635, 201), (25, 294, 0.898185143854, 200), (25, 294, 0.904470557778, 207),
(25, 295, 0.510000548384, 4948), (25, 295, 0.527335029277, 48990), (25, 295, 0.553592976478, 200), (25, 295, 0.587128450848, 200),
(25, 295, 0.685163151117, 221), (25, 295, 0.689665955815, 200), (25, 295, 0.693619741737, 200), (25, 295, 0.698808800461, 212),
(25, 295, 0.79024052726, 203), (25, 295, 0.805930533305, 233), (25, 295, 0.870408868052, 208), (25, 295, 0.879989559956, 209),
(25, 295, 0.917289124506, 201), (25, 296, 0.368675367618, 200), (25, 296, 0.422106669815, 87196), (25, 296, 0.570294648439, 201),
(25, 296, 0.576307309235, 5064), (25, 296, 0.577913079103, 223), (25, 296, 0.615569633698, 201), (25, 296, 0.619178871735, 43506),
(25, 296, 0.718168554371, 310), (25, 296, 0.747322660094, 202), (25, 296, 0.760729108487, 200), (25, 296, 0.822530768331, 205),
(25, 296, 0.836178694446, 200), (25, 296, 0.922082501039, 202), (25, 297, 0.575427625651, 325), (25, 297, 0.582330676039, 200),
(25, 297, 0.59032029128, 201), (25, 297, 0.590800056359, 200), (25, 297, 0.601983239354, 200), (25, 297, 0.604624445572, 349),
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