Let’s consider a max-expectation bettor on a double-or-nothing bet with an 80% probability of paying out.
My expected value per dollar in this bet is $1.60, whereas the expected value of a dollar in my pocket is $1. So I maximize expected value by putting all my money in. If I start with $100, my expected value after 1 round is $160. The expected value of playing this way for two rounds is $100x1.6x1.6 = $256. In general, the expected value of this strategy is 100 ⋅1.6n.
The Kelly strategy puts 60% of its money down, instead. So in expectation, the Kelly strategy multiplies the money by .6⋅1.6+.4=1.36.
So after one round, the Kelly bettor has $136 in expectation. After two rounds, about $185. In general, the Kelly strategy gets an expected value of 100⋅1.36n.
So, after a large number of rounds, the all-in strategy will very significantly exceed the Kelly strategy in expected value.
I suspect you will object that I’m ignoring the probability of ruin, which is very close to 1 after a large number of rounds. But the expected value doesn’t ignore the probability of ruin. It’s already priced in: the expected value of 1.6 includes the 80% chance of success and the 20% chance of failure: .8⋅2+.2⋅0. Similarly, the $256 expected value for two rounds already accounts for the chance of zero; you can see how by multiplying out 100⋅(.8⋅2+.2⋅0)2 (which shows the three possibilities which have value zero, and the one which doesn’t). Similarly for the nth round: the expected value of 100⋅1.6nalready discounts the winnings by the (tiny) probability of success. (Otherwise, the sum would be $2^n instead.)
Let’s consider a max-expectation bettor on a double-or-nothing bet with an 80% probability of paying out.
My expected value per dollar in this bet is $1.60, whereas the expected value of a dollar in my pocket is $1. So I maximize expected value by putting all my money in. If I start with $100, my expected value after 1 round is $160. The expected value of playing this way for two rounds is $100x1.6x1.6 = $256. In general, the expected value of this strategy is 100 ⋅1.6n.
The Kelly strategy puts 60% of its money down, instead. So in expectation, the Kelly strategy multiplies the money by .6⋅1.6+.4=1.36.
So after one round, the Kelly bettor has $136 in expectation. After two rounds, about $185. In general, the Kelly strategy gets an expected value of 100⋅1.36n.
So, after a large number of rounds, the all-in strategy will very significantly exceed the Kelly strategy in expected value.
I suspect you will object that I’m ignoring the probability of ruin, which is very close to 1 after a large number of rounds. But the expected value doesn’t ignore the probability of ruin. It’s already priced in: the expected value of 1.6 includes the 80% chance of success and the 20% chance of failure: .8⋅2+.2⋅0. Similarly, the $256 expected value for two rounds already accounts for the chance of zero; you can see how by multiplying out 100⋅(.8⋅2+.2⋅0)2 (which shows the three possibilities which have value zero, and the one which doesn’t). Similarly for the nth round: the expected value of 100⋅1.6n already discounts the winnings by the (tiny) probability of success. (Otherwise, the sum would be $2^n instead.)