How rebates on losses can give players the edge
By Henry Tamburin
I don’t believe that Johnson’s gigantic windfall at the blackjack tables was due solely to luck. Neither did several contributing writers to my Blackjack Insider Newsletter (BJI). They undertook a detailed mathematical analysis of Johnson’s deal with the casinos and arrived at some interesting conclusions
In my column last month, I summarized the deal that high-stakes blackjack player Don Johnson made with three Atlantic City casinos before he beat them out of $15 million. If you remember, his deal allowed him to play a six-deck game with liberal rules. Also, he could wager up to $100,000 per hand; he had to deposit $1million in front money with the casino; and he got back a 20% rebate on his losses, which he could take anytime he wanted (as long as his losses totaled at least $500,000).
By the way, Johnson stated afterwards that he wasn’t counting cards. This was backed up by a spokesperson from the New Jersey Division of Gaming Enforcement, who was quoted as saying, “Johnson’s play was closely monitored…and there was no indication of cheating.”
After Johnson beat the Tropicana for $5.8 million in April, the casino reported a stunning $1.86 million monthly loss at their blackjack tables. The Tropicana’s former CEO blamed the loss on “bad luck at the tables” from “the single-largest winner in our history.”
As I mentioned in my article last month, I don’t believe that Johnson’s gigantic windfall was due solely to luck. Neither did several contributing writers to my Blackjack Insider Newsletter (BJI). They undertook a detailed mathematical analysis of Johnson’s deal with the casinos and arrived at some interesting conclusions, which I’m about to share with you.
It may not be obvious to you how the 20% rebate on losses can give the player an advantage. On the surface, it would seem that the casino would simply make 20% less money in the long run, but still be profitable. Therefore, let me explain how this works with a simple coin toss.
Suppose you’re playing heads or tails against the casino. Here are the rules. You bet $1 on each toss; when you lose, you pay the casino $1; when you win, the casino pays you 99 cents; however, the casino will give you back 20% of your losses if you agree to make at least 100 bets. With these rules, who has the advantage?
Dan Pronovost, who analyzed and wrote two articles on loss-rebates in the BJI (www.bjinsider.com/loss_rebate), prepared a spreadsheet to analyze the above coin-toss game. Below is a table showing the player’s edge, assuming he took the 20% rebate on losses after a specified number of coin flips.
For example, if he took the rebate after one coin flip, he would have a 9.5% edge. If he waited until two coin-flips and then took the 20% rebate, his edge would be 4.55%, and so forth. What the data shows is that a player would have an initial edge over the casino in this coin-flip game, but this edge would gradually decrease as he continued to play before taking the rebate on losses.
When the 320th coin flip is reached, the player’s edge disappears. And if he continues to play this game, the edge will shift in the casino’s favor. Since the rules specified that you could take your 20% rebate on losses after you made at least 100 bets, a smart player would take the rebate after every cycle of 100 bets to gain the largest edge over the house. (Shame on the casino for offering this game, without first doing the math!)
Coin Flip Game
|# Flips Before Rebate||Player’s Edge|
BJI contributing writer Alan Krigman was the first to analyze the loss rebate deal that Johnson was granted. In his column in the July issue of the BJI, by way of an example, Krigman analyzed the effect of a 20% rebate on losses for a $100 wager on a 12-number bet on a double-zero roulette bet. The standard house edge for this roulette bet can be calculated by multiplying the probability of winning times the payoff, minus the probability of losing times the amount lost. Since you would be betting on 12 out of 38 numbers, the probability of winning the bet is 12/38. If the bet wins, the payoff is $200 (2-1 payoff). Therefore, the player’s edge is 12/38 x $200 minus 26/38 x $100, which equals –5.26%.
Suppose the casino gave you a 20% rebate on your loss after your first $100 wager, meaning you could win $200 but your loss would be only $80. If you plug in $80 as your loss in the above equation, you arrive at a player’s edge of 8.42% (which is 1.6 times greater than the normal house edge in roulette). Krigman calculated that if you continued to play before you took your rebate, the player’s edge would gradually decrease, and that, after five spins, the edge would shift to the casino.
Pronovost confirmed this result in his article in the July issue of BJI (see table below). Notice that the player’s edge gradually decreases as the number of spins increases. After spin #6, the player’s edge is a negative percentage, meaning the edge has now shifted in the casino’s favor. If a player were to continue to play, the house edge would gradually increase as the number of spins increased, and eventually it would bottom out at -4.21% (which is 80% of the normal house edge for this roulette bet).
|# Spins||PLAYER’S EDGE|
Thus far, I’ve demonstrated, with a simple coin toss and a bet on 12 numbers in roulette, how a rebate on losses can give the player the edge. The key is to get the casino to apply the loss-discount after the fewest number of wagers possible.
The mathematical analysis of a rebate on losses for the game of blackjack is more complicated because you don’t always win or lose one unit (you could get a 3-2 payoff for a blackjack, win double when you double down or pair split, or lose half your wager if you surrendered your hand). Krigman and Pronovost used slightly different mathematical approaches to analyze loss-rebates for blackjack, but both came to nearly the same conclusion. Here’s what they found.
Krigman assumed a –0.35% house edge for the game that Johnson was playing. Using that as the starting point, he used combinatorial analysis and estimated that Johnson would have a 9.68% edge if he took his rebate after one hand, and he would continue to have the edge for “close to 500 hands.” If he continued to play more than 500 hands, the edge shifted to the house. Pronovost used “Risk-of-Ruin” computer simulations to show that Johnson had the edge up to 900 hands (the difference in results was due to a different set of playing rules assumed by Pronovost; using the same playing rules, Krigman’s result increased to 750 hands with a player’s edge).
After Krigman and Pronovost completed their analysis, we learned from an interview that BJI contributing writer Mark Gruetze did with Johnson (July issue of BJI) that the casino stipulated that Johnson could get his rebate only after he lost a minimum of $500,000, and that he had to deposit $1,000,000 as “front money” with the casino. Pronovost reanalyzed the effect of the 20% rebate on losses with these two new factors in a subsequent article in the August issue of the BJI. The results were somewhat surprising.
First, the casino did not ensure itself any protection by setting the minimum loss-limit at $500,000 before Johnson could get his 20% rebate. For example, if Johnson was getting his rebate on losses after every couple of hours (200 rounds), Pronovost estimated that his hourly win rate dropped by only 9%. In addition, setting the amount of Johnson’s front money at “only” one-million dollars actually helped him, assuming he would have to stop playing when he lost the million and would get his 20% rebate. Overall, by carefully minimizing the number of rounds of blackjack he played before asking for his rebate on losses, Pronovost estimated that Johnson could make $50,000 an hour or more!
The bottom line is this: Krigman and Pronovost have proven that high-stakes players can get the edge over the casino when they negotiate a liberal deal on rebate on losses as Johnson did, and play as few hands as possible. This makes the strategy of getting a rebate on losses another way for a smart blackjack player, albeit a high-stakes one, to get the edge over the casino, even without card counting.
Henry Tamburin is the editor of Blackjack Insider Newsletter (www.bjinsider.com), Lead Instructor for the Golden Touch Blackjack Course (www.goldentouchblackjack.com), and host of www.smartgaming.com. For a free three-month subscription to his blackjack newsletter, go to www.bjinsider.com/freetrial.com. To receive his free Casino Gambling Catalog, call 1-888-353-3234 or visit www.smartgaming.com.