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Hey Mes, by optimal I thought that meant the statistically fastest way to grow your bankroll while minimizing risk of ruin. By definition how could you be better than optimal?
a late response, but what Kelly is designed to maximize is expected utility, given a logarithmic utility of money function. That is sometimes expressed as maximizing the "expected growth rate" which is a bit unintuitive.
Here's how to envision what logarithmic utility of money means.
Linear money is how we normally think about poker decisions. You are getting 2:1 money odds and we have 40% equity, therefore, good decision. And we moderate our risk by making sure we only ever put a fairly small fraction of our bankroll on the table.
Linear money utility decisions are how you would maximize your bankroll, period. But a consequence of linear utility is that you would be indifferent to flipping a coin for your entire bankroll with no overlay, and want to flip for your entire bankroll with any overlay whatsoever. So for example. Your entire net worth is 100k. I offer to flip a coin. You can be assured that the coin is absolutely and perfectly fair. If you win, I give you $100,001, if I win, you give me $100,000.
Linear utility says this is a good bet, you earn $1 on average.
Linear utility is obviously way, way, way out there into degen land on the gambool scale.
Logarithmic utility says that you are indifferent to bets where winning will in crease your bankroll by a factor of x, and losing will leave you with bankroll * 1/x. So you have a 100k bankroll, and I offer a flip for your whole bankroll where if you win, you get 100k, and if you lose, you pay me 50k. So you double your roll on a win, and halve it on a loss.
Or alternately, you win $10k, or lose $9091. You should be indifferent. Or alternately, you win $900k, or lose $90k. You should be indifferent.
If this sounds too gambooly for you, then kelly is too aggressive. If it sounds pretty close to accurate, then you are a fairly typical aggressive advantage gambler, which is why Kelly is popular.
When I say Kelly specifies the optimal amount to bet, what I mean is that betting the kelly% maximizes the expected logarithm of your total bankroll.
Betting more than this amount will increase your linear EV, but slightly decrease your log EV versus betting the kelly amount. But betting a bit more than the kelly amount is better than not betting at all.
That's why I mean it can be advantageous to flip for more than kelly would suggest -- if you don't have the optino to bet the kelly amount, but you can bet something like 1.5x the kelly amount, then it's probably still a good bet.
utility adjusted EV is my own formulation that I use to determine whether a bet is good/bad. It's worth playing with the formula I gave in the last post, until you understand what's going on.