Modified Proportional Betting

 

Optimal Kelly wagering requires precise bet sizing (which can result in wagers in fractions of a chip). Though powerful in theory, in practice, perfect Kelly betting is not realistic. A somewhat more practical approach to betting is a method referred to as "modified proportional betting." The wagering is still quasi-Kelly, but there are important differences. First, it's assumed that any win or loss for a playing session is small compared to the entire bankroll. This way, the bankroll need not be re-evaluated prior to every wager. Before a session begins, the bankroll can be evaluated once and bet sizes predetermined for the entire session.

Furthermore, bets are capped at a maximum level we'll call the "ceiling." Ideally, we'd like to bet proportionally (to our advantage) no matter how high the expectation rises. In practice, however, we cannot get away with spreading from, say, 1 to 100 units, even though the count may merit it. In the casino, we must have a ceiling at some point, if only because of the table limit. The range of the floor to ceiling levels is commonly called the "bet spread."

It's always somewhat problematic to develop a benchmark for determining the performance of a card-counting system (for purposes of comparing with other systems). For this book, we've chosen to use a modified proportional betting comparison, which places the respective systems on a similar scale in terms of risk of ruin.

Assuming a modified proportional betting scheme is the appropriate vehicle of comparison, two important variables must be determined. First, what is the bet spread? Second, how quickly does the wagering traverse the bet spread?

We've already touched on some of the interesting ramifications of choosing and attempting to implement a bet spread. Ideally, we'd like to use an infinite spread. But again, in practice this isn't possible. We believe a reasonable bet spread for which counters should strive is 1 to 5 in a single- or double-deck game, and 1 to 10 in 6- and 8-deck games.

Traversing the bet spread is a concept that also merits further attention. The issue here is how fast we change our wagering from the minimum (at a disadvantage) to the maximum (with the advantage). With a finite bankroll, we'll move up and down with our bets in direct proportion to our prevailing expectation (a la Kelly). The slope of the ramp is proportional to our starting bankroll, with a greater bankroll implying a steeper ramp (we'll make more maximum bets).

   
 
   
     
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