Can you develop an alternative game where the I can win what I actually rolled? for example, I rolled a 18, but my bet was on lessthan 64000, so I missed out on a potentially big payout. In this alternative game, if I roll a 16, then I get paid the 1998.853x multiplier or whatever multiplier that would be mathematically correct. I'd like to play this style.
If they paid 64000x for every zero, 31981.573x for every 1, etc. using the current payouts, the expected return per 1 BTC bet would be 9.3 BTC.
So to make a game with a small positive (1.9%) expectation for the house, but with the same proportions of the payouts, we need to scale the current multipliers back by 9.3 / 0.981. I left the 'losing' return at 0.005, and slightly adjusted the 3rd decimal place of the payouts to make it work. Then we get:
lessthan 1 6750.942x
lessthan 2 3373.527x
lessthan 4 1686.764x
lessthan 8 843.382x
lessthan 16 421.691x
lessthan 32 210.846x
lessthan 64 105.423x
lessthan 128 52.712x
lessthan 256 26.356x
lessthan 512 13.178x
lessthan 1000 6.747x
lessthan 1500 4.498x
lessthan 2000 3.374x
lessthan 3000 2.249x
lessthan 4000 1.687x
lessthan 6000 1.125x
lessthan 8000 0.844x
lessthan 12000 0.563x
lessthan 16000 0.422x
lessthan 24000 0.282x
lessthan 32000 0.211x
lessthan 32768 0.206x
lessthan 48000 0.141x
lessthan 52000 0.130x
lessthan 56000 0.121x
lessthan 60000 0.113x
lessthan 64000 0.106x
64000 or more 0.005x
Which gives pretty much the correct house edge:
>>> (
1 * 6750.942 +
1 * 3373.527 +
2 * 1686.764 +
4 * 843.382 +
8 * 421.691 +
16 * 210.846 +
32 * 105.423 +
64 * 52.712 +
128 * 26.356 +
256 * 13.178 +
488 * 6.747 +
500 * 4.498 +
500 * 3.374 +
1000 * 2.249 +
1000 * 1.687 +
2000 * 1.125 +
2000 * 0.844 +
4000 * 0.563 +
4000 * 0.422 +
8000 * 0.282 +
8000 * 0.211 +
768 * 0.206 +
15232 * 0.141 +
4000 * 0.130 +
4000 * 0.121 +
4000 * 0.113 +
4000 * 0.106 +
1536 * 0.005) / 65536
0.9810327911376953
Would people be willing to play a game where you need to roll less than 6000 to 'win'? It means that less than one ticket in ten is a winning ticket.