Re: Just-Dice.com : Invest in 1% House Edge Dice Game

Umm... you're right in the first part, but to the last paragraph I could add: given any fully specified Martingale (odds, betting size progression, ending conditions), I can construct an equivalent single bet strategy. The only reason to employ a Martingale over a single bet strategy is that the former seems easier to risk-manage intuitively for the human mind.

I disagree, but will allow you to prove me wrong...

Suppose I have 127 BTC and want 128. I propose betting 1 BTC at 49.5% (2x payout). If it wins, stop, else double and repeat until it wins. So bet:

1, 2, 4, 8, 16, 32, 64

until one of them wins. If none wins, I've lost all 127 BTC. If any wins, I've got the 128 BTC I wanted.

This strategy will work so long as any one of the maximum of 7 bets wins. ie. unless all 7 lose. The probability of all 7 losing is 0.505^7 = 0.008376, so I have a 100 * (1 - 0.008376) = 99.1624% chance of success.

It's pretty likely that I'll end up betting less than the full 127 BTC. My expected total stake is less than 127. So my expected loss is less than 1.27 BTC.

What's your equivalent single bet strategy? How much does it expect to risk? How much does it expect to lose?

I would like t follow up on this since I think it is a very important point.

What this means is that a player can improve his expected return from -1% to -0.05% (!) using the above (truncated) martingale strategy.
This in turn means that the current 0.5% max winnings is at least 10 times the optimal Kelly bet against a martingaling player if I'm not mistaken. And that is dangerous.

I reply since I was originally doubting it, maybe I can tell you how I see this point now.

The expected loss is less than 1.27BTC because on average you bet much less than 127BTC (in the above case it is roughly 7BTC on average). If you compute the average bet size, which is much smaller than the worst case one, you can see that the expected return is -1% and this is the only possible expected return. However, if you specify that your strategy can lose at maximum 127BTC and wins at maximum 1BTC, you can find different ways of betting the money with different average bet sizes and thus different expected losses. Betting all the 127BTC is probably the worst way :-)