Re: Challenge: What's the best way to win 1 BTC with 1 BTC?

Quote from: blockage on January 29, 2015, 04:24:18 PM

Strategy 1:
You're always playing 9900x using a martingale sequence, i.e. your wager for each round is

Code:

ceiling((2 - PLAYER_BR)/9899,1e-8)

you run out of funds after 6861 bets. You lose all of them with a probability of

Code:

0.9999^6861 = 0.5035187605260536

or put differently you have a 49.64812% chance of winning.

Strategy 2:
You're constantly playing 20203 satoshi but increase the payout each round by exactly 1x starting at 4951x which will leave you with 1.00004850 BTC profit. You run out of funds after 4949 games and your last payout will be 9899x. You lose all of them with a probability of

Code:

prod(1-0.99/i, i=4949..9899) = 0.5035283796177312

leading to a 49.64716% chance of winning.

I Hope the math is correct Tongue

How about this:

bet 0.01088928 BTC @ 1.0664%
if you win, your profit is ((99 / 1.0664) - 1) * 0.01088928 = 1.00002474 BTC
if not, martingale at the same chance of winning, to keep the same net profit
you can afford 64 bets
chance of winning = 100 * (1 - ((100 - 1.0664) / 100) ** 64) = 49.649475445976265%

Edit: here are the 64 bets:

chance multiplier 1.0664% 92.83570893x bet # stake total lost profit ----- ---------- ----------- ---------- 1 0.01088902 1.00000087 2 0.01100759 0.01088902 1.00000081 3 0.01112745 0.02189661 1.00000065 4 0.01124861 0.03302406 1.00000001 5 0.01137110 0.04427267 1.00000036 6 0.01149492 0.05564377 1.00000036 7 0.01162009 0.06713869 1.00000051 8 0.01174662 0.07875878 1.00000040 9 0.01187453 0.09050540 1.00000048 10 0.01200383 0.10237993 1.00000031 11 0.01213454 0.11438376 1.00000032 12 0.01226667 0.12651830 1.00000004 13 0.01240025 0.13878497 1.00000078 14 0.01253527 0.15118522 1.00000019 15 0.01267177 0.16372049 1.00000049 16 0.01280975 0.17639226 1.00000021 17 0.01294924 0.18920201 1.00000063 18 0.01309024 0.20215125 1.00000022 19 0.01323278 0.21524149 1.00000024 20 0.01337687 0.22847427 1.00000007 21 0.01352254 0.24185114 1.00000091 22 0.01366978 0.25537368 1.00000026 23 0.01381863 0.26904346 1.00000022 24 0.01396910 0.28286209 1.00000011 25 0.01412121 0.29683119 1.00000014 26 0.01427498 0.31095240 1.00000051 27 0.01443042 0.32522738 1.00000047 28 0.01458755 0.33965780 1.00000020 29 0.01474640 0.35424535 1.00000075 30 0.01490697 0.36899175 1.00000041 31 0.01506929 0.38389872 1.00000021 32 0.01523338 0.39896801 1.00000024 33 0.01539926 0.41420139 1.00000057 34 0.01556694 0.42960065 1.00000032 35 0.01573645 0.44516759 1.00000045 36 0.01590780 0.46090404 1.00000005 37 0.01608102 0.47681184 1.00000003 38 0.01625613 0.49289286 1.00000036 39 0.01643314 0.50914899 1.00000007 40 0.01661208 0.52558213 1.00000001 41 0.01679297 0.54219421 1.00000009 42 0.01697583 0.55898718 1.00000020 43 0.01716068 0.57596301 1.00000020 44 0.01734755 0.59312369 1.00000086 45 0.01753644 0.61047124 1.00000016 46 0.01772740 0.62800768 1.00000067 47 0.01792043 0.64573508 1.00000031 48 0.01811557 0.66365551 1.00000070 49 0.01831283 0.68177108 1.00000065 50 0.01851224 0.70008391 1.00000077 51 0.01871382 0.71859615 1.00000078 52 0.01891759 0.73730997 1.00000032 53 0.01912359 0.75622756 1.00000088 54 0.01933182 0.77535115 1.00000024 55 0.01954233 0.79468297 1.00000076 56 0.01975512 0.81422530 1.00000015 57 0.01997024 0.83398042 1.00000073 58 0.02018769 0.85395066 1.00000016 59 0.02040752 0.87413835 1.00000072 60 0.02062973 0.89454587 1.00000001 61 0.02085437 0.91517560 1.00000025 62 0.02108146 0.93602997 1.00000085 63 0.02131101 0.95711143 1.00000028 64 0.02154307 0.97842244 1.00000067 0.99996551

Edit: and the calculations showing that it works for the first 5 bets:

>>> 0.01088902 * (99/1.0664 - 1) 1.0000008712228057 >>> 0.01100759 * (99/1.0664 - 1) - 0.01088902 1.0000008112303074 >>> 0.01112745 * (99/1.0664 - 1) - 0.01088902 - 0.01100759 1.0000006493023257 >>> 0.01124861 * (99/1.0664 - 1) - 0.01088902 - 0.01100759 - 0.01112745 1.0000000137959488 >>> 0.01137110 * (99/1.0664 - 1) - 0.01088902 - 0.01100759 - 0.01112745 - 0.01124861 1.0000003597824456

Edit2: I can make ever increasing longer sequences which do slightly better, but it appears never to beat 49.65222222222% chance of doubling.