Why in the middle of the shoe? The goal is to find a single index that maximizes the expected value given the following parameters: number of decks, cards remaining, cards removed, and a card counting system. This index can be integer or decimal, floored or rounded. Below is the table that I calculated with my program for Hi-Lo and for TKO.
An important fact is that numbers for integer or decimal values with one digit of precision can be generated with the new version of CVDATA. Maybe Norman can help with the verification.
a) Hi-Lo
b) TKOCode:+----------+-------+-----+----------+-----------+-----+----------------------+ | Play | Decks | CR | TC Index | RC Index | IRC | Total EV >= TC Index | +----------+-------+-----+----------+-----------+-----+----------------------+ | Ins | 1 | 20 | 2 | 1 | 0 | 0.58171980020663483 | +----------+-------+-----+----------+-----------+-----+----------------------+ | Ins | 2 | 26 | 2 | 3 | 0 | 0.97386472171957084 | +----------+-------+-----+----------+-----------+-----+----------------------+ | Ins | 3 | 39 | 3 | 5 | 0 | 0.90293467695143381 | +----------+-------+-----+----------+-----------+-----+----------------------+ | Ins | 4 | 52 | 3 | 6 | 0 | 0.83027321442020385 | +----------+-------+-----+----------+-----------+-----+----------------------+ | Ins | 5 | 65 | 3 | 8 | 0 | 0.75983020092833753 | +----------+-------+-----+----------+-----------+-----+----------------------+ | Ins | 6 | 78 | 3 | 9 | 0 | 0.69413408108466679 | +----------+-------+-----+----------+-----------+-----+----------------------+ | Ins | 7 | 91 | 3 | 11 | 0 | 0.63388417844782019 | +----------+-------+-----+----------+-----------+-----+----------------------+ | Ins | 8 | 104 | 3 | 13 | 0 | 0.57910212499303571 | +----------+-------+-----+----------+-----------+-----+----------------------+ | Play | Decks | CR | TC Index | RC Index | IRC | Total EV >= TC Index | +----------+-------+-----+----------+-----------+-----+----------------------+ | Ins | 1 | 20 | 1.4 | 1 | 0 | 0.58948461336093605 | +----------+-------+-----+----------+-----------+-----+----------------------+ | Ins | 2 | 26 | 2.4 | 3 | 0 | 0.98281795951625006 | +----------+-------+-----+----------+-----------+-----+----------------------+ | Ins | 3 | 39 | 2.7 | 5 | 0 | 0.90816953949120360 | +----------+-------+-----+----------+-----------+-----+----------------------+ | Ins | 4 | 52 | 2.9 | 6 | 0 | 0.83159288103950568 | +----------+-------+-----+----------+-----------+-----+----------------------+ | Ins | 5 | 65 | 3.0 | 8 | 0 | 0.75983020092833709 | +----------+-------+-----+----------+-----------+-----+----------------------+ | Ins | 6 | 78 | 3.0 | 9 | 0 | 0.69413408108466634 | +----------+-------+-----+----------+-----------+-----+----------------------+ | Ins | 7 | 91 | 3.1 | 11 | 0 | 0.63441572887679987 | +----------+-------+-----+----------+-----------+-----+----------------------+ | Ins | 8 | 104 | 3.1 | 13 | 0 | 0.58054269141619064 | +----------+-------+-----+----------+-----------+-----+----------------------+ | Play | Decks | CR | TC Index | RC Index | IRC | Total EV >= TC Index | +----------+-------+-----+----------+-----------+-----+----------------------+ | Ins | 1 | 20 | 1.36 | 1 | 0 | 0.58949259373855689 | +----------+-------+-----+----------+-----------+-----+----------------------+ | Ins | 2 | 26 | 2.40 | 3 | 0 | 0.98281795951624784 | +----------+-------+-----+----------+-----------+-----+----------------------+ | Ins | 3 | 39 | 2.73 | 5 | 0 | 0.90817947158494905 | +----------+-------+-----+----------+-----------+-----+----------------------+ | Ins | 4 | 52 | 2.88 | 6 | 0 | 0.83169540860503743 | +----------+-------+-----+----------+-----------+-----+----------------------+ | Ins | 5 | 65 | 2.97 | 8 | 0 | 0.75994376091156390 | +----------+-------+-----+----------+-----------+-----+----------------------+ | Ins | 6 | 78 | 3.03 | 9 | 0 | 0.69421662302300535 | +----------+-------+-----+----------+-----------+-----+----------------------+ | Ins | 7 | 91 | 3.07 | 11 | 0 | 0.63455736758509551 | +----------+-------+-----+----------+-----------+-----+----------------------+ | Ins | 8 | 104 | 3.10 | 13 | 0 | 0.58054269141618986 | +----------+-------+-----+----------+-----------+-----+----------------------+
Code:+----------+-------+-----+----------+-----------+-----+----------------------+ | Play | Decks | CR | TC Index | RC Index | IRC | Total EV >= TC Index | +----------+-------+-----+----------+-----------+-----+----------------------+ | Ins | 1 | 20 | -3 | -1 | -4 | 0.62207920068091205 | +----------+-------+-----+----------+-----------+-----+----------------------+ | Ins | 2 | 26 | -2 | -1 | -8 | 1.03126899891434554 | +----------+-------+-----+----------+-----------+-----+----------------------+ | Ins | 3 | 39 | -1 | -1 | -12 | 0.96589993725524714 | +----------+-------+-----+----------+-----------+-----+----------------------+ | Ins | 4 | 52 | -1 | -1 | -16 | 0.89567516597613084 | +----------+-------+-----+----------+-----------+-----+----------------------+ | Ins | 5 | 65 | -1 | -2 | -20 | 0.82406581226090381 | +----------+-------+-----+----------+-----------+-----+----------------------+ | Ins | 6 | 78 | -1 | -2 | -24 | 0.75620947896421276 | +----------+-------+-----+----------+-----------+-----+----------------------+ | Ins | 7 | 91 | -1 | -2 | -28 | 0.69325066039581429 | +----------+-------+-----+----------+-----------+-----+----------------------+ | Ins | 8 | 104 | -1 | -2 | -32 | 0.63555057036254503 | +----------+-------+-----+----------+-----------+-----+----------------------+ | Play | Decks | CR | TC Index | RC Index | IRC | Total EV >= TC Index | +----------+-------+-----+----------+-----------+-----+----------------------+ | Ins | 1 | 20 | -2.7 | -1 | -4 | 0.62411302793561507 | +----------+-------+-----+----------+-----------+-----+----------------------+ | Ins | 2 | 26 | -1.6 | -1 | -8 | 1.04217654345964661 | +----------+-------+-----+----------+-----------+-----+----------------------+ | Ins | 3 | 39 | -1.3 | -1 | -12 | 0.97122453130055963 | +----------+-------+-----+----------+-----------+-----+----------------------+ | Ins | 4 | 52 | -1.1 | -1 | -16 | 0.89657584952559255 | +----------+-------+-----+----------+-----------+-----+----------------------+ | Ins | 5 | 65 | -1.0 | -2 | -20 | 0.82406581226090447 | +----------+-------+-----+----------+-----------+-----+----------------------+ | Ins | 6 | 78 | -0.9 | -2 | -24 | 0.75632462454726679 | +----------+-------+-----+----------+-----------+-----+----------------------+ | Ins | 7 | 91 | -0.9 | -2 | -28 | 0.69476873014568830 | +----------+-------+-----+----------+-----------+-----+----------------------+ | Ins | 8 | 104 | -0.9 | -2 | -32 | 0.63837723117899026 | +----------+-------+-----+----------+-----------+-----+----------------------+ | Play | Decks | CR | TC Index | RC Index | IRC | Total EV >= TC Index | +----------+-------+-----+----------+-----------+-----+----------------------+ | Ins | 1 | 20 | -2.67 | -1 | -4 | 0.62411302793561529 | +----------+-------+-----+----------+-----------+-----+----------------------+ | Ins | 2 | 26 | -1.60 | -1 | -8 | 1.04217654345964506 | +----------+-------+-----+----------+-----------+-----+----------------------+ | Ins | 3 | 39 | -1.26 | -1 | -12 | 0.97135441954627266 | +----------+-------+-----+----------+-----------+-----+----------------------+ | Ins | 4 | 52 | -1.11 | -1 | -16 | 0.89658797329402284 | +----------+-------+-----+----------+-----------+-----+----------------------+ | Ins | 5 | 65 | -1.01 | -2 | -20 | 0.82407138460075480 | +----------+-------+-----+----------+-----------+-----+----------------------+ | Ins | 6 | 78 | -0.95 | -2 | -24 | 0.75658940195274205 | +----------+-------+-----+----------+-----------+-----+----------------------+ | Ins | 7 | 91 | -0.90 | -2 | -28 | 0.69476873014568685 | +----------+-------+-----+----------+-----------+-----+----------------------+ | Ins | 8 | 104 | -0.87 | -2 | -32 | 0.63854885119060589 | +----------+-------+-----+----------+-----------+-----+----------------------+
Sincerely,
Cac
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