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Proportional Betting Can Limit Losses while Enhancing Gains27 January 2004
Bettors who have or think they have an edge, such as blackjack card counters or sports handicappers, often size their wagers in proportion to their current bankrolls. Objectives range from minimizing chance of a big long-term loss to maximizing expected rate of profit growth. Although the goals differ, proportional betting is also an option for players facing a house advantage.
In practice, it's impossible to bet exact fractions of a bankroll indefinitely.
The "required" amount quickly involves cents as well as dollars, and
could eventually go above or below a table limit. Exact theoretical models,
though, show the idea.
Place bets on the nine at craps illustrate a situation where the house has an edge. These bets have 40 percent chance of winning and pay 7-to-5, giving the bosses 4 percent edge. Expected loss always betting $10 is $0.40 per decision, to be $40 behind after 100 tries, $80 after 200, $400 after 1,000, and so forth. A proportional strategy, starting with $500 and betting 2 percent of what's available per try (rounded to the nearest dollar), is expected to cost $51 after 100 rounds and $97 for 200; however, the gap narrows with extended play. At 600 coups, proportional betting is projected to be $238 rather than $240 in the hole; the expected loss after 1,000 rounds is $329 as opposed to $400.
The proportional effect grows as bets increase relative to a bankroll. Flat bettors' losses rise steadily; those of proportional players get closer and closer to the initial stake (presumably a loss limit) but, by definition, can't exceed it.
Of course, nobody gambles to lose the "expected" amount. Everyone
is familiar, anecdotally if not arithmetically, with the fact that volatility
swamps edge for sessions or casino visits of reasonable duration. Proportional
betting gets interesting, perhaps useful, when the actual frequency of wins
stays above or below the theoretical value. As it often does in the short haul.
Proportional betting accordingly limits losses while improving gains as play extends and luck flouts the law of averages. More, the usual betting progressions and regressions "work" when wins occur on big bets and losses on small. In proportional systems, instances of wins and losses matter. But when they happen is irrelevant. Finessing the details of bets not exact multiples of $5, two wins and two losses betting 2 percent on the nine with an initial $1,000 stake will yield $1015, regardless of the order.
How you might apply proportional concepts to your betting is something new to think about. But isn't thinking what it's all about? The bard, Sumner A Ingmark, thought so when he said:
Folks with preferences exotic,
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