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# Playing It Smart: The maximum boldness theorem

7 January 2008

The "maximum boldness" theorem is a basic law of gambling when you're fighting an edge. Simply stated, starting with a given stake in a particular game, you have the best chance to reach a specified earnings level by making the biggest allowable bets consistent with that goal the fewest possible number of times.

An intuitive rationale for the theorem is that the more money you bet during a session, for instance by recycling intermediate earnings or compensating for interspersed losses, the greater the bite taken by edge. To see how this pans out, picture an even-money game with 49 percent chance of winning and 51 percent of losing. These probabilities give the house 2 percent advantage.

Make believe you have a \$1,000 stake and \$1,000 win goal. Bet the \$1,000 at once and your chance of joy is 49 percent. Instead, say you bet \$500 and win, then \$500 and win again. You earned \$1,000 but had to win twice in a row to get it; the chance of this is 49 percent multiplied by 49 percent or 24 percent. True, with \$500 bets, you could lose the first and still be able to attain your goal by winning the next three times in a row; the chance of a loss followed by three wins is 6 percent. Or, you could reach the \$1,000 goal in four bets all at \$500 with a victory followed by a defeat and two consecutive triumphs likewise a 6 percent probability bringing you up to 24 + 6 + 6 or 36 percent.

Extending this reasoning to other, longer combinations of \$500 wins and losses eventually gets you to 48 percent. Not quite as good as maximum boldness by doing or dying on \$1,000 in one fell swoop. As bets decrease, chances fall off further. This is shown in the accompanying table for games with 2 and 5 percent edge.

```Chance of doubling a \$1,000 bankroll in an even-money game with
2 and 5 percent house advantage, making bets of various sizes

bet	chance	chance
of success	of success
(2% edge)	(5% edge)
\$1,000	49%	48%
500	48%	45%
250	46%	40%
100	46%	27%
50	40%	12%
25	17%	2%
10	2%	under 1%

Enquiring minds, naturally, want to know whether and    if so    how maximum boldness applies to bets paying over 1-to-1. Envision a proposition paying 10-to-1 where the house has 2 percent edge. The probability of winning any coup would be 8.9 percent. The chance of doubling a \$1,000 stake rather than going belly-up, wagering \$100 per round, is slightly under 49 percent. Betting less, for instance \$25 per round, prospects drop to 46 percent. With \$10 bets, it's down to 40 percent. This demonstrates that the maximum boldness theorem holds regardless of payoff ratio. However, it also shows that the chance of reaching a goal before exhausting a bankroll falls off less rapidly with decreasing bet size, the higher the payoff ratio rises above 1-to-1.

The maximum boldness theorem has exceptions. For example, pretend you're willing to invest your whole stake to double your money at craps. Maximum boldness says to go for broke with one shot on Don't Pass. You'd have 49.3 percent chance of success. If table limits prevent your doing so, there may be circumstances when your chances would be enhanced putting less then the maximum on the line and reserving money to lay Odds if the shooter establishes a point. This, because Odds reduce the edge over the course of the action. Similarly in blackjack, holding back so you have money to split or double may improve your overall prospects.

Few solid citizens gamble with strict win goals and loss limits, such that maximum boldness would be appropriate. Of course, only dunderheads would dig deeper than the bust-out points that seemed sensible when they left home. But, rational exit strategies can be based on many criteria, monetary as well as involving time, excitement, pandering from unctuous hosts, flat-screen digital rather than tube-type analog TV sets as rewards for loyalty, and whatnot. Here's what that laudable lyricist, Sumner A Ingmark, advised when addressing such a surfeit of personal preferences: