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Balanced counts do not exist.

Posted by Pete Moss on 13 May 1998, at 2:36 a.m., in response to unbalanced counts, posted by slightly experienced card counter on 9 May 1998, at 7:13 p.m.

There's a landmark work by Definetti that contains these words in bold face in the introduction:

PROBABILITY DOES NOT EXIST.

He then goes on to write two volumes on probability! He could have been a Zen master. Maybe he was.

In that same spirit, let's first see why "balanced counts" (in some sense) do not exist, then we will see why they are not well suited to betting by running count.

Card counting systems give each denomination of card a number called a "tag". By convention, if a card in play is bad for the player before a round begins, the denomination is given a positive tag. Cards that are good for the player before the round begins are given negative tags. The rationale for "good card, negative tag" is that we choose to add tags as we see cards removed from play, rather than subtracting them. Let's call the negative of a card tag the card's "value". Good card, positive value. Bad card, negative value. By keeping a running sum of the tags of the cards we see removed, we are keeping a running sum of the values of the cards remaining, plus some arbitrary offset that depends on where we started our running count. If we choose to start our running count with the sum of values in the original deck or shoe, then the running count is at all times exactly the sum of the values remaining.

None of this so far suggests there is any advantage or disadvantage in using a set of tags under which the sum of values in the original deck is zero. That's what a "balanced" count is. After you've seen just one card, the counting system is likely not to be "balanced" for the deck you are then playing. The counting system is balanced, for the remaining deck, only when the sum of values remaining is zero. In that sense, a "balanced count" does not exist.

The basis of card counting is that when there are a surplus of positive valued cards in the deck, the player has an advantage and should bet more. How then do we determine when there is a surplus that warrants an increased bet of some specified magnitude?

One method is called "true count". One estimates the average value for a remaining card by dividing the sum of values remaining by the number of cards remaining. The average is usually expressed per units of 52-card "decks". Notice that I'm defining true count to be the average value of a card remaining, so it does not depend on your initial running count.

The average value or true count is used as a basis for betting, and perhaps playing decisions. That can be done regardless of whether the count was originally balanced or not. "Balanced" has nothing at all to do with the mechanics of calculating true count. The choice of tags does make a big difference in what true count constitutes an advantage though. Using a balanced count like Hi-Lo, a true count of zero is usually disadvantageous. Using other sets of tags, a true count of zero might be very favorable (as blackjack favorability goes). A K-O true count of zero [sic] is quite favorable, for example.

The reason true count works as well as it does is that for a reasonable set of card tags, the player's advantage/disadvantage is roughly a linear function of the true count.

Another method is to use just the running count for making decisions. Some sets of tags are better for that than others. When the true count is zero, the running count will be zero plus some "pivot" number. If you started your running count at the sum of the values in the original deck, the pivot will be a convenient zero. (The K-O book uses +4 as the default pivot, which has cost me an untold number of keystrokes when writing these little essays.) Because the true count of zero corresponds exactly with a running count equal to the pivot, while no other running count exactly pegs the true count, one is best served when relying on running count to use a system like Red Seven or K-O for which a true count of zero corresponds to a transition point you are keenly interested in -- either to roughly a break-even situation (Red Seven) or to a clearly advantageous situation (K-O). Red Seven will tell you with some accuracy whether you have an advantage or not. K-O will tell you with some accuracy whether you have a situation that is in the "major wager" range.

- Pete


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