Quote from: kidgorgeous on January 08, 2013, 05:32:51 AM

Hmmm, what I think you're seeing is actually satoshidice fudging the payout multiplier and to make nice even numbers

I wasn't using the published multipliers, because as you say they're rounded. I was calculating the actual multipliers using a formula which I derived like this:

Code:

(lessthan * payout + (65536 - lessthan) * 0.005) / 65536 = 0.981 (multiply both sides by 65536)
(lessthan * payout + (65536 - lessthan) * 0.005) = 0.981 * 65536 (subtract (65536 - lessthan) * 0.005 from both sides)
(lessthan * payout) = 0.981 * 65536 - (65536 - lessthan) * 0.005 (divide both sides by lessthan)
so, payout = (0.981 * 65536 - (65536 - lessthan) * 0.005) / lessthan

If you compare the values that gives you for the payout multiplier with the published values, you find it's very close in every case except for the "lessthan 1", where the site is actually a little over generous (but that's deliberate - they state the hose edge is slightly smaller for that bet):

Code:

>>> for (lessthan, actual) in [(64000, 1.004), (60000, 1.071), (56000, 1.147), (52000, 1.235), (48000, 1.338), (32768, 1.957), (32000, 2.004), (24000, 2.670), (16000, 4.003), (12000, 5.335), (8000, 8.000), (6000, 10.666), (4000, 15.996), (3000, 21.326), (2000, 31.987), (1500, 42.647), (1000, 63.968), (512, 124.933), (256, 249.861), (128, 499.717), (64, 999.429), (32, 1998.853), (16, 3997.701), (8, 7995.397), (4, 15990.789), (2, 31981.573), (1, 64000.000)]:
... print "lessthan %5d payout %9.3f (error %9.6f)" % (lessthan, actual, actual - (0.981 * 65536 - (65536 - lessthan) * 0.005) / lessthan)
...
lessthan 64000 payout 1.004 (error -0.000424)
lessthan 60000 payout 1.071 (error -0.000052)
lessthan 56000 payout 1.147 (error -0.000199)
lessthan 52000 payout 1.235 (error -0.000060)
lessthan 48000 payout 1.338 (error 0.000435)
lessthan 32768 payout 1.957 (error 0.000000)
lessthan 32000 payout 2.004 (error 0.000152)
lessthan 24000 payout 2.670 (error -0.000131)
lessthan 16000 payout 4.003 (error 0.000304)
lessthan 12000 payout 5.335 (error -0.000261)
lessthan 8000 payout 8.000 (error -0.000392)
lessthan 6000 payout 10.666 (error 0.000477)
lessthan 4000 payout 15.996 (error 0.000216)
lessthan 3000 payout 21.326 (error -0.000045)
lessthan 2000 payout 31.987 (error 0.000432)
lessthan 1500 payout 42.647 (error -0.000091)
lessthan 1000 payout 63.968 (error -0.000136)
lessthan 512 payout 124.933 (error 0.000000)
lessthan 256 payout 249.861 (error 0.000000)
lessthan 128 payout 499.717 (error 0.000000)
lessthan 64 payout 999.429 (error 0.000000)
lessthan 32 payout 1998.853 (error 0.000000)
lessthan 16 payout 3997.701 (error 0.000000)
lessthan 8 payout 7995.397 (error 0.000000)
lessthan 4 payout 15990.789 (error 0.000000)
lessthan 2 payout 31981.573 (error 0.000000)
lessthan 1 payout 64000.000 (error 36.859000)

Quote from: kidgorgeous on January 08, 2013, 05:32:51 AM

What happens with your strategy if you try a martingale on the less than 2 (ignoring mins and max bets for hypothetical purposes)?

Let's not ignore the mins and maxes. They're currently 0.01 and 0.1844 (though they seem to change every few seconds now).

Here's a strategy to win 319 BTC betting on lessthan 2 in 91761 bets, starting at min bet and working up to max bet:

Code:

win on turn 1 : bet = 0.01000000, spent = 0.01050000, win = 319.81523000, lose = 0.00000001, back = 0.00000000, profit = 319.80473000 (99.997% to lose all)
win on turn 2 : bet = 0.01000033, spent = 0.02100033, win = 319.82573031, lose = 0.00000001, back = 0.00000001, profit = 319.80472999 (99.994% to lose all)
win on turn 3 : bet = 0.01000066, spent = 0.03150098, win = 319.83623095, lose = 0.00000001, back = 0.00000002, profit = 319.80472999 (99.991% to lose all)
win on turn 4 : bet = 0.01000099, spent = 0.04200197, win = 319.84673192, lose = 0.00000001, back = 0.00000003, profit = 319.80472998 (99.988% to lose all)
win on turn 5 : bet = 0.01000131, spent = 0.05250328, win = 319.85723322, lose = 0.00000001, back = 0.00000004, profit = 319.80472998 (99.985% to lose all)

win on turn 91757 : bet = 0.18437485, spent = 5580.49620770, win = 5896.59719089, lose = 0.00042187, back = 3.70328891, profit = 319.80427210 ( 6.080% to lose all)
win on turn 91758 : bet = 0.18438062, spent = 5580.68108832, win = 5896.78164963, lose = 0.00042190, back = 3.70371078, profit = 319.80427209 ( 6.079% to lose all)
win on turn 91759 : bet = 0.18438638, spent = 5580.86597470, win = 5896.96611411, lose = 0.00042193, back = 3.70413268, profit = 319.80427209 ( 6.079% to lose all)
win on turn 91760 : bet = 0.18439215, spent = 5581.05086685, win = 5897.15058433, lose = 0.00042196, back = 3.70455461, profit = 319.80427209 ( 6.079% to lose all)
win on turn 91761 : bet = 0.18439792, spent = 5581.23576477, win = 5897.33506029, lose = 0.00042198, back = 3.70497657, profit = 319.80427209 ( 6.079% to lose all)

It risks 5581.23576477 to win 319.80427209 and has a 6.079% chance of losing everything.

That's a payout of 1.057x and has the same risk as "lessthan 61552" (if it existed):

Code:

>>> 319.80427209 / 5581.23576477 + 1
1.0572999037433028
>>> 65536 * (1 - 6.079/100)
61552.06656

If satoshidice offered "lessthan 61552", they'd payout 1.044x on it:

Code:

>>> lessthan = 65536 * (1 - 6.079/100); (0.981 * 65536 - (65536 - lessthan) * 0.005) / lessthan
1.04417121836437

So again it's better to martingale a series of high-multiplier bets than to play a single large low-multiplier bet.

Why??