I am trying to understand the implications of the kelly criterion for a real world portfolio. What I get as a result is that if I have free choice on any bet at any odds and chances I should, in total, invest more than I have. (Result by integrating over all probabilities/odds that allow positive expected value) In fact, I should invest infinitely much money. The wikipedia page states that taking out credit to buy a bet would be formalized by the loss formula so the infinity result is not exactly interpretable as taking out a loan, if I can.
One obvious fix is to limit the odds and probabilites to realistic values but that seems quite arbitrary. Intuitively I would expect the Kelly criterion to give a finite sum for all bets with positive expected value, at least it does so for any given odds b in wikipedias terminology.
What I get as a result is that if I have free choice on any bet at any odds and chances I should, in total, invest more than I have.
If you invoke infinities or indefinite sets of bets, it shouldn't surprise you that regular results might not apply: if you decide to invest in a bet of n at 99% odds of doubling, wouldn't it be even better to invest n at 99% odds of tripling? Or even better than that, invest n at 99.9% odds of tripling? Or no, invest n+1 at 99.9% odds of tripling! I'm not sure why you'd expect anything useful from a KC or a variant with su...
So the jackpot in the Ohio lottery is around 25 million, and the chance of winning it is one in roughly 14 million, with tickets at 1 dollar a piece. It appears to me that roughly a quarter million tickets are sold each drawing; so, supposing you win, the probability of someone else also winning is 1 - (1 - 1/14e6)^{250000}=2%, which does not significantly reduce the expectation value of a ticket. So, unless I'm making a silly mistake somewhere, buying lottery tickets has positive expected value. (I find this counterintuitive; where are all the economists who should be picking up this free money? But I digress.)
I pointed this out to my wife, and said that it might be worth putting a dollar into it; and she very cogently asked, "Then why not make it 100 dollars?" Why not, indeed! Is there any sensible way of deciding how much to put into an option that has a positive expected value, but very low chance of payoff?