Quote from: dooglus on March 06, 2018, 04:38:41 PM

Quote from: suchmoon on March 06, 2018, 03:51:05 AM

Quote from: dooglus on March 06, 2018, 03:33:24 AM

This likely means that the "real leverage" player does worse than the fixed offsite investor in the much more common case where the player wins and loses more equally.

Shouldn't "real leverage" still do better in that scenario if the players lose slightly more than they win as expected due to house edge?

I kind of ran out of steam at that point. I can't imagine that "real leverage" is better than "offsite investing" in every case. Shouldn't there be a trade-off?

Back to the example:

A has 10 onsite and 90 offsite.
C has 10 onsite with 10x "real" leverage.

Suppose a whale plays for a long period with 50% chance of winning each bet. All bets are the maximum, aiming to win 1% of the bankroll. The payout when he wins is 1.98x (1% house edge), and he wins and loses the same amount of bets (as he is expected to do, long term). Let's call the effective bankroll "B".

50% of the time the player wins 1% of the bankroll: B/100. When this happens, A's effective bankroll is multiplied by 99/100, and C's is multiplied by 90/100.

The other 50% of the time the player loses his stake. He's aiming to profit by B/100, with a payout multiplier of 1.98x, so he's risking and losing B/98. When this happens, A's effective bankroll is multiplied by 99/98, and C's is multiplied by 108/98.

Since the whale wins and loses the same number of bets, we can pair these bets. Each pair consists of one win and one loss.

For each pair, A's effective bankroll is multiplied by 99/100 and 99/98, for a net growth factor of 99/100 * 99/98 = 9801/9800 ~= 1.0001x

And C's effective bankroll is multiplied by 90/100 and 108/98, for a net growth factor of 90/100 * 108/98 = 9720 / 9800 ~= 0.9918x

And so we see that my intuition was correct, and that "real" leverage is worse than this "offsite investment" thing. The "offsite investment" has a (small but) positive expected bankroll growth whereas the "real leverage" expects to lose almost 1% of the bankroll for every pair of (1 win + 1 lose) bets.

This is interesting... What would the gambler have to do so that investor A has negative bankroll growth?