Re: Optimal Betting Question?

Posted by Brh on August 25, 1997 at 05:26:08: In Reply to: Optimal Betting Question? posted by DvBj on August 24, 1997 at 17:32:49:

DvBj,

You ask some good questions, before I answer them, I want to clear up my own definitions, so there is no confusion.

I bet a constant times my advantage on a given hand times my bankroll up to a maximum bet. The reason for the MAX is
to have a spread that is reasonable given the casino's heat factor. I also have a minimum bet, betting this whenever
there is no advantage or the casino has the edge.

The correct 'Kelly' bet at any level is Bankroll*ev/var, where ev is the unit gain per hand, and var is the unit variance per hand. 'Advantage' seems to mean different things to different people. Some people define advantage as ev/(av bet), and others just call ev, the advantage. It appears that the definition Wong uses in 'Prof BJ' is ev, since his Kelly 'constant' is 0.78 which is 1/var.

1) If I did not have a maximum bet, and did not bet at all when there was no advantage for me would the constant have to
be 1/variance to be pure kelly or 1 to be pure Kelly?

The constant is always 1/var, this comes out in the math, and is basically they way the bets are adjusted for risk.

Would I be better off betting a higher minimum, even when I do not have the advantage, thus allowing a higher
maximum in terms of casino heat? When I say "better off", I mean would I grow the bankroll faster with out increased risk
of ruin.

No. We now know that for a given bankroll, and a given 1-M spread (determined presumably by heat considerations), there is an 'optimal' spread which will grow the bankroll at the fastest possible rate. What you suggest will always increase the risk. If you want to do this, then you will need a bigger bankroll.

The optimal spreading strategy can be reasonably summarised as :


1/. If you play all hands, the max bet of M units should be
    made at a relatively low TC, eg for Hi-Lo about +3, and
    you Kelly bankroll will be around 1000 units.


2/. If you leave when the shoe goes negative, ie for Hi-Lo
    whenever the TC hits -1, then you max bet should be a
    little higher, around Hi-Lo +5 (this seems to be the
    spread usually recommended by authors), with a Kelly
    bankroll of around 600 units.


3/.  If you back-count and only play high positive counts,
     then the max bet TC is very high, and the spread tends 
     to be very flat.

For example, play only Hi-Lo TC equal to +1 or greater, then TC_max is +8, and the Kelly bankroll is around 250 units. If you only play +2 or greater, then the TC_max is to high to matter, and you just play pure Kelly all the way up, with a bankroll of around 150 units.

Note though, as the spread gets flatter, for aggressive Wonging, the required bankroll becomes smaller, which means in $ terms, the real bets get larger. So despite the fact that you may never actually make a max bet, the intermediate bets will be larger than the 'max' bet you would have made in case (1) above. This scenario is the closest to your suggestion. Yes, you can increase your min bet and max bet to win faster, but you must only do it by back-counting, and your 'min' bet will be at some positive true count.

An example of how this works. Lets say you have a bankroll of $5000, then for a play all spread of 1-10, your unit bet will be $5000/1000 = $5, and you max bet will be $50 at +3.

If you wong out at -1, with a 1-10 spread, then your unit bet will be $5000/600 = 8.3 ~ $10, and you max bet of 83 ~ $85 units will be at +5. Note though, at +3 , you will bet around 60 units, which is more than the $50 max above.

If you play only +1 or greater, your unit bet is $5000/250 = $20, and your max bet of $200 will be at +8. The bet at +3 will be $60 and at +5 it will be $100.

I hope this helps you get the idea,

Cheers,
Brett.


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