I've been trying my hand at card counting lately, and I've been doing some thinking about how a perfect gambler would act at the table. I'm not sure how to derive the optimal bet size.
Overall, the expected value of blackjack is small and negative. However, there is high variance in the expected value. By varying his bet size and sitting out rounds, the player can wager more money when expected value is higher and less money when expected value is lower. Overall, this can result in an edge.
However, I'm not sure what the optimal bet size is. Going all-in with a 60 percent chance of winning is EV+, but the 40 percent chance of loss would not only destroy your bankroll, it would also prevent you from participating in future EV+ situations. Ideally, one would want to not only increase EV, but also decrease variance.
Objective: Given a distribution of expected values, develop a function that transforms the current expected value into the percentage of the bankroll that should be placed at risk.
I'm not sure how to begin. Even if I had worked out the distribution of expected values. Are other inputs required (i.e. utility of marginal dollar won, desired risk of ruin)? Should the approach perhaps be to maximize expected value after one playing session? Why not a month of playing sessions, or a billion? Is there any chance the optimal betting size would produce behavior similar to the behavior predicted by prospect theory?
I eagerly await an informative discussion. If you have something against gambling, just pretend we're talking about how much of your wealth you plan on investing in an oil well with positive expected value.
This is a solved problem.
That was easy. Thank you, sir.
it's only a solved problem when Wikipedia's not blacked out. :'(
EDIT: I love you guys.
Here is the URL to add to AdBlock Plus to avoid the Wikipedia blackout.
Here's Google's cache of the page.
Here is the 1956 paper that introduced the Kelly Criterion.
There is an excellent relatively recent book on kelly and shannon and the background of their calculations, what they ultimately did with them, and other interesting things, I highly recommend. Fortune's Formula by William Poundstone.
If you're talking about a perfect gambler at a real casino blackjack table, then you should be aware that one of the pit manager's jobs is to spot card counting using the correlation between of a player's bet sequence and the EV of the hand. The casino is in the business of maintaining the house edge; if a player is thought to be counting cards, they will be politely shown the door.
When I was younger I actually was able to make a decent living in Vegas because I could card count pretty well. It really comes down to the dealer. Some do a bad job of hiding the cards from view others do a great job. It isnt luck, it is a science.