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by Alan Krigman


Alan Krigman, and his poetic sidekick Sumner A. Ingmark, have been illuminating the dark recesses of casino gambling for more than a dozen years. Mr. Krigman is especially well known for sharing his insights into the mathematics underlying the various games (including blackjack), the influence of volatility and skewness as well as edge on bankroll during the course of a session, and the impact of betting as well as decision strategies on expected performance. A searchable archive of Mr. Krigman's prose and Mr. Ingmark's muse is online at

An Atlantic City blackjack buff won $4 million from Caesars, $5 million from Borgata, and $6 million from Tropicana Ė all in about six months. His net was less, about $12 million, because of some losing sessions along the way.

The player told the Press of Atlantic City newspaper that all three casinos courted his action by agreeing to return up to 20 percent of anything he lost on a visit. (Theyíre no longer making this offer!) Since the bosses are proficient at spotting card counters, itís fair to assume they wouldnít have wooed his business in this manner, or allowed him bet up to $100,000 per hand, if they had any inkling that he was card counter. Therefore, itís a fair assumption that he pretty much followed basic strategy.

The promise of 20 percent rebate on losses undoubtedly moderated his setbacks on the losing forays. However, this type of deal has broader implications, and it helps winning gamblers, too.

Pretend a casino says itíll refund 20 percent of the loss on your first wager. Suppose your game is double-zero roulette and you bet $100 on a 12-number column. Your chances are normally 12 out of 38 to pick up $200, and 26 out of 38 to drop $100. The edge has a dollar value of $200 x (12/38) - $100 x (26/38) which equals -$200/38 or -$5.26 (the minus sign because the house has the advantage). The 20% discount doesnít affect the probabilities of winning but it puts you up to collect the $200 when you win a bet against a possible setback of just $80 when you lose. The monetary equivalent of edge for the one spin becomes $200 x (12/38) - $80 x (26/38); this equals +$320/38 or +$8.42. The advantage is not only with you, itís 1.6 times greater than the amount the casino normally has over roulette aficionados.

Suppose, as is actually how this incentive works, youíre to get back 20 percent of any net shortfall over multiple spins (rather than round-by-round loss). Your effective edge starts at +8.42 percent in your favor and gradually shifts toward the house as you continue to play. This is because your rebate isnít based on the sum of your round-by-round losses, but rather on the net of your losses minus your wins. For column bets at double zero roulette, the casino regains the edge after five spins. The house advantage increases as the number of coups rises beyond five and gradually approaches 0.8 x -0.526 or -4.21 percent, which is 80 percent of the nominal house edge.

Imagine, instead of $100 on a column, you bet the money on a single spot. Your chances are normally one out of 38 to win $3,500 and 37 out of 38 to lose $100. The dollar value of the edge is $3,500 x (1/38) - $100 x (37/38), which again equals -$200/38 or -$5.26 for the house. With 20 percent forgiven on a loss, the formula becomes $3,500 x (1/38) - $80 x (37/38) which gives you a whopping +$14.21 dollar equivalent per spin. If you keep going, the house doesnít regain the edge on the gross wager until the 99th spin, after which it again begins to approach -4.21 percent.

The player who won the $12 million in Atlantic City with the guaranteed 20 percent rebate on net losses had an even sweeter situation because of the characteristics of blackjack. Standard rules in the establishments in question confront high limit bettors with a nominal house advantage as low as the range between -0.35 and -0.40 percent. The precise figure varies with options such as availability of late surrender, number of hands to which pairs can be resplit, ability to resplit aces, and how closely players adhere to the nuances of composition-dependent Basic Strategy.

At -0.35 percent nominal edge, the player would start with an edge of +9.68 percent and stay the favorite for about 650 decisions; thereafter, advantage would flip to the house and creep up toward 0.8 x -0.35 or -0.28 percent. At the -0.40 percent level, the bettor would be in the catbird seat for close to 500 hands, starting with an edge of +9.64 percent; beyond this, the casino would have an edge that slowly approached 0.8 x -0.40 or -0.32 percent.

A joint might be willing to risk losing big bucks to high rollers, and this can happen Ė give-back on losses or not. However, gaming executives whoíve done their homework Ė or know enough to hire someone who has Ė add constraints so solid citizens will not have an edge. It doesnít make sense for casinos to pursue heavy action if the laws of math predict it will be stacked against them.

Properly conceived rebate-on-loss programs, therefore, have strings attached requiring bettors to participate in at least a stated minimum number of coups to qualify for the refund. The number depends on the edge and volatility of the games in question and on the fraction of the net losses to be rebated. Moreover, that minimum is, or should be, well above the point where edge flips from the player to the house. Itís as the punterís poetizing pundit, Sumner A Ingmark, pronounced:

Sometimes lucky guesses underlie successes,

Oft, though, itís laborious to emerge victorious.


Editorís Note: For more details on loss rebates, read Dan Pronovostís article in the issue.

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