A bettor who can make an infinite number of expected profitable bets is going to outperform one who can only make a finite number of bets.

Agreed

(any number between 1 and 0 exclusive)^infinity=0

Agreed

i.e. for an infinite series of bets, the probability of ruin with naive EV maximization is 1.

Agreed

So, expected value is actually −1x your bet size.

Not so!

I think you’ll start to see where you’re wrong pretty fast if you examine finite examples.

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 .

The Kelly strategy puts 60% of its money down, instead. So in expectation, the Kelly strategy multiplies the money by .

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 .

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: . Similarly, the $256 expected value for two rounds already accounts for the chance of zero; you can see how by multiplying out (which shows the three possibilities which have value zero, and the one which doesn’t). Similarly for the th round: the expected value of *already* discounts the winnings by the (tiny) probability of success. (Otherwise, the sum would be $2^n instead.)

What’s the point of a loan if you need 100% collateral, and the collateral isn’t something like a house that you can put to good use while using as collateral?

If I can use bitcoin as collateral to get some Doge, hoping to make enough money with the Doge to get my bitcoin back later… couldn’t I just sell my bitcoin to buy the Doge, again hoping to make enough money to get my bitcoin back later?