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Best of Alan Krigman

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Proportional Betting Can Limit Losses while Enhancing Gains

27 January 2004

No betting scheme can create an advantage in a negative expectation game. However, solid citizens can shape their sessions by the way they wager, for instance to trade probabilities of wins and losses with the associated amounts.

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.

Assume you could bet even money on a proposition with 50 percent chance of success. Winning and losing twice each at a flat $10 per round, you'd break even in four tries. With a $100 stake and betting 10 percent of the money in your current bankroll on each of four rounds, again winning and losing twice, you'd finish with $98.01. You'd be $1.99 behind the flat bettor. Both ways, results are independent of the order in which the wins and losses ensued.

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.

While details vary with precise cases, placing the nine for $10 or 2 percent of a current bankroll starting at $500 typifies results. Say that instead of the "correct" 4/10 win rate, a table is cold and success averages only 3.5/10. Flat bettors are expected to lose less than proportional punters for the first 50 coups and more thereafter. It's $160 versus $146 after 100 tries, $320 versus $250 after 200, and $1,600 versus $484 after 1,000. On the hot side, were hits to average 4.5/10, flat bettors would earn more for the first 300 coups: $80 versus $70 for 100 rounds, $160 versus $150 for 200 rounds; the worm would then turn: $400 versus $465 at 500 and $800 versus $1,363 after 1,000.

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,
Esoteric or Quixotic,
Look for more than rules robotic.

Alan Krigman

Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.
Alan Krigman
Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.