The Incredible Loss-Discount: by Dan Pronovost
Introduction
For the last couple of months, the big news in the advantage-gambling world has been Don Johnson, and his amazing high-roller winning streak of over 10 million dollars against three US casinos. This, of course, is big news for the While I was working with Henry to get this BJI issue ready and reading Alan Krigman’s excellent article about Mr. Johnson’s achievement, I became more curious about the 20% loss-discount that the casinos purportedly offered him. On the surface, it would seem that if you are playing any casino game where the house has an edge over you, then it doesn’t matter if they offer you some money back on your losses: you’re still losing money in the long run. However, the amazing thing is… Why loss discounts are a winning method To understand how this can be true, let’s begin with what a loss-discount means in practice. Casinos love whales, which is what they call their players who bet massive amounts of money at the tables. These gamblers make $10,000 bets, or higher, on every wager, and often end up losing millions in the long run. Suppose a player is betting, on average, $50,000 per hand, and makes fifty wagers an hour at a game where the casino has a 0.5% edge over the player (which is typical of blackjack games played perfectly). In theory, the casino is earning 50 * $50,000 * .005 = $12,500 every hour that the player plays. Mr. Johnson claims his average bet size was $100,000, and he was playing heads-up blackjack where over 100 hands per hour is quite common..Therefore, the casino’s theoretical profit every hour that Mr. Johnson plays is $50,000. To entice such whales to their casinos, it is not uncommon to offer them all kinds of high-roller perks. While room suites, food, and airfare reimbursement are common perks for high rollers, savvy high rollers can ask for, and often receive, an agreement to get back part of their losses. If, and when, they wind up losing money at the end of their trip or playing session, they will get some percentage of their losses credited back to them, so they owe the casino less money (but they keep all their winnings if they end up ahead). Mr. Johnson claims he managed to get a 20% loss-discount for his action. Our casino insiders say that this is extraordinarily high and rare, but not unheard of, given the size of his action at the tables. Furthermore, Mr. Johnson was probably not card counting, or was otherwise playing in a manner where the casino was certain they had a good and healthy base advantage over him at the tables. So, how can a loss-discount possibly be used to get an edge over the casino? The trick comes from understanding that the casino applies the loss discount after a certain amount of time or wagers, To understand why this works, let’s review the example that Mr. Krigman used in his article with roulette. Suppose you are playing double zero roulette and bet on a column (12/38 chance of winning, getting double your money back), and the casino offers you a 20% loss-discount. Normally, your edge in this game is the probability of winning times the amount you will win minus the probability of losing times the amount wagered. In the above roulette example, the math is 12/38 * 2 – 26/38 * 1 = -5.26%. In other words, the casino will make, on average, a 5.26% profit from every such wager you make. Now, suppose you make one such bet, but you get the casino to agree to apply your 20% loss-discount after one wager, meaning instead of losing one unit, you only lose 0.8. Then your edge is 12/38 * 2 - 26/38 * 0.8 = 8.4% in The mathematical answer to the above question of what happens with a loss discount over time is fairly complicated. However, the general trend is that the player’s edge goes down, and it will approach the casino’s normal edge in the game, minus the loss discount. Let’s look at the outcome of two successive column wagers in the above roulette game:
Multiplying the Probability times the Profit/Loss yields the Net Value for each outcome. Adding up the Net Value for each outcome and dividing by the number of rounds yields the Edge per round. Notice that our edge dropped from 8.4% with a one-round 20% loss-discount, to 4.1% per round with a two-round loss discount. The math is more complicated with more rounds, but it is easy to solve with a computer program or spreadsheet:
You can see that after five bets, the casino’s edge returns in its favor, and it will continue to increase and approach 5.26% * .80 = 4.21% when the loss-discount is applied after many rounds. Loss-discounts in blackjack Mr. Krigman goes further in his article and states that with a 20% loss-discount in a good blackjack game, "the player would start with an edge and stay the favorite for about 650 decisions; thereafter, advantage would flip to the house." This is an incredible result, and you should be picking your jaw off the ground. Unlike roulette, where the casino gets the edge back after only five rounds, it takes over 500 rounds in blackjack! That would typically be five hours of play for a heads-up blackjack player, and well in the rational range of time that a casino could apply a loss-discount. To be clear… the player could be using basic strategy, The math behind this for blackjack is much more complicated, since you don’t just win and lose one-unit bets in blackjack. You can get a blackjack and be paid 3 to 2, you could double-down and double some of your bets, split some pairs, and even surrender some hands (the game Mr. Johnson played included surrender). To accurately analyze this situation, this requires some estimation to compute the "even money equivalent" to convert the different blackjack bets to a simple win/lose ratio, with corresponding win and loss return rates. Then, you still have to use some combinatorial programming to compute the overall net value (with a loss discount) for any given number of rounds. I looked at this problem, and realized that you could view it as a variant of a Risk-of-Ruin simulation, a unique feature of my own company’s blackjack simulator, Blackjack Audit. Basically, you start with a fixed number of rounds of blackjack, and simulate many such "sessions" of play under the exact blackjack rules Mr. Johnson experienced. Then, look at the percentage of such sessions that had positive and negative ending bankrolls, the respective average win, or loss, and use that data to compute the effective win-rate, applying a 20% loss-discount. Repeat for different number of rounds of blackjack, and find the number of rounds of blackjack where the casino’s edge hits zero. Sounds complicated but by using the Risk of Ruin software, it’s quite easy to do. I set up a simulation with the following blackjack rules: 6 decks, 2/3 penetration, DAS, S17, resplit aces, late surrender, $5 flat betting with perfect basic strategy. I ran an ROR simulation with one-million playing sessions for each number of rounds below. Here are the results:
The "% win" column is the percentage of sessions with an ending bankroll of $0 or more, and the "$ win" column is the average of the ending bankrolls for these positive sessions. The "% lose" column is the percentage of sessions with a negative ending bankroll, and the "$ lose" column is the average of the ending bankroll for these negative sessions. The return column (win rate per round) is computed as: [(% win) * ($ win) – (% lose) * ($ lose) * 80% ] / $5 / (# rounds). It represents the profit or loss per blackjack round, in bet units. These results show that the player has the edge over the casino up to 990 rounds of blackjack, and at that point, the edge reverts to the casino. This is a bit higher than Mr. Krigman’s result, but not significantly so, given the even-money approximations and assumed game statistics involved. Krigman and I exchanged data, and using some of the exact game statistics for the blackjack rules I used, and his result increased to around 750. More importantly, both our results prove conclusively
Conclusions Since loss-discounts are generally only given to high rollers (i.e. big bettors), and assuming that any traditional means of winning like card counting is not used or necessary, the potential for profit is massive. Mr. Johnson is not admitting up front if this was his plan going in, but the reality is that he probably had a healthy edge over the casino, assuming that the casino applied his loss-discount on a nightly or even weekend frequency. In Gruetze’s BJI interview with Johnson, we asked how frequently he could have applied his 20% loss-discount, and he claimed far less than even 100 hands. Suppose he really could repeatedly get his loss-discount settled after every couple of hours of play, say every 200 rounds of blackjack, and then he would be winning 0.003604 bet units with every bet. So, if he averaged one hundred $100,000 wagers per hour for two hours, then that equals over $35,000 profit per hour! While getting a 20% loss-discount is probably very rare, don’t forget that this method does not require any obvious advantage playing strategies like card counting. A savvy high roller could, for example, start a playing history with a casino where they explicitly, and intentionally, use a bogus progression betting system (it won’t win you money, but it won’t increase the casino’s edge either.) After a weekend of play (without a loss discount), the casino bosses would be drooling to get more of this player’s action (regardless of whether they won or lost.) The player could then demand a 20% loss-discount, and based on his poor prior playing method, he’d get it ("He can’t win! He’s using a stupid win/loss betting method, and betting more money than makes sense! Give him the loss protection anyway. We want his action.") The high roller can also slow down his play, but keep his action high, which only works in his favor. The less bets he makes, but with larger wagers, the better off the loss-discount works in his favor. One obvious challenge with this advantage play method is the monstrous bankroll that is required. Even if he played only 200 rounds per session (betting $100,000 a hand), he would need a 2.8 million dollar bankroll to have no more than a 5% chance of going broke in any given blackjack session. This strategy is only for the richest of players, with very deep pockets! Kudos to Mr. Johnson for negotiating the sweet deal with the casinos that ensured himself an edge, and for beating them at their own game for millions, using solely basic strategy. Note: For the curious, all of the above math was computed using Blackjack Audit, a blackjack simulator and calculator from DeepNet Technologies.
Dan will study these variables in a follow-up article next month… try and guess in advance whether these factors help or hinder Mr. Johnson’s edge. The results will probably surprise you!
| |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

©2015, DeepNet Technologies. No material to be copied without express permission of DeepNet Technologies. |