How can a player legally beat a casino? When this question is asked of the typical gambler-in-the-street, the answer is almost universally “card counting blackjack.” When asked about card counting, this gambler talks about how smart you have to be and mentions the MIT team, but has no idea of what is actually involved in being a card counter. This myopic view of beating casinos is not unique to the player side. Austere policies are put in place in many casinos to protect themselves against the army of card counters. Advanced advantage play is about all the other ways players are beating the house. It begins with the axiom that every game can be legally beat, the puzzle is to find out how to do it. This blog gives solutions to some of these puzzles.

Advantage play is the act of legally exploiting procedural or structural weaknesses in some aspect of casino games or operations in a way that generates an edge over the casino. Advantage players (APs) play table games, side bets, slot machines and video poker. They take advantage of promtions, loss rebates, casino hosts and marketing. They look for casinos that misunderstand or have incorrectly implemented the rules of new games. They seek out dealers who are error prone. They manipulate the elements of games to create situ
ations where errors are more likely to occur. They disassemble automatic shufflers and exploit the most subtle of weaknesses. They find casinos that use cards with minor defects in their designs and exploit these irregularities. They operate alone and in teams. They relentlessly scout. They are well-funded with huge bankrolls. They communicate efficiently. They will travel anywhere at any time. Advanced advantage players are the real professionals beating casinos today. This is the best of times for these players. The money has never been easier.

A casual examination of the gambling section of any bookstore shows a wide selection of books on blackjack and poker. The blackjack books are all on the subject of card counting. The poker books are not about advantage play as I’ve defined it; the poker player is beating other players, not the house. There may be some books on roulette, baccarat or craps, but none of these will give legitimate ways to beat the house. Systems based on betting progressions, patterns, flow or timing, are not mathematically sound. In craps, extraordinary claims about dice control demand extraordinary proof, and these books fall woefully short. If you look really hard, you will walk out of the store with one half-way decent book that tells you how to beat one game using a barely profitable method that has been known for fifty years.

What you will not find among these books are any that cover advantage play against today’s modern casino table game pit, full of novelty table games and side bets. These games include titles like Three Card Poker, Ultimate Texas Hold’em and Mississippi Stud. Side bets include wagers like Lucky Ladies in blackjack and the Dragon 7 in EZ Baccarat. Novelty games have encroached into the modern casino to the point that in many casinos they comprise at least half the table games. In Nevada, there are over 700 approved proprietary games. Other jurisdictions are also saturated. And every single one of these approved games has unique vulnerabilities and game protection problems; ways it can be beaten that are unique to its design and procedures.

The sheer quantity of proprietary games makes a full treatise on advantage play  impossible. But that does not mean that the effort has never been undertaken. The book “Beyond Counting” by James Grosjean covers many of the most popular and profitable proprietary games and side bets. However, there are two reasons why "The Book" (as it is often called) may not be the book. First, The Book is extremely dense; it can be difficult for the average reader to wade through its mathematics and pick out the valuable nuggets. Second and more importantly, "Beyond Counting" is simply not available to the general public. It cannot be purchased, loaned or obtained through any normal channels. You’ve got to know a guy who knows a guy in order to get The Book.  However, the geography that The Book covers is far too important to have its information be so inaccessible. This blog is, in part, my endeavor to go “Beyond Grosjean."

Though I work as a consultant to the casino industry, I do not take sides when it comes to using the information provided here. If an AP can use what’s here to beat the house, good for them. If casino management learns about a method that can be used against one of their games and takes proactive or defensive actions, good for them.  If a game developer learns about a problem with his game and puts in a fix, good for them. Knowledge has intrinsic value and serves the greatest good when in the public domain. As a professor for over 25 years, with over twenty research articles published in academic journals, I am firmly committed to this premise. Lead with knowledge and good will follow.

On the other hand, this blog has generated a lot of vitriol directed squarely my way. Casino management wonders why I tell people how to beat their games when I am working for their industry. Game inventors and the companies that license games don’t want me to show the weaknesses of their games; you can’t sell something that is broken. APs don’t want their opportunities brought into the open and burnt out; the less casino management knows about these vulnerabilities, the more opportunities for the APs. Mostly, this wrath falls like water off of a duck’s back. Once in a while, I take it personally.

The following are choice comments by some of my biggest fans:

  • In the hacking community there are numerous people that do what you do. Post proof of concept exploits and then how to patch the problem. You know people use your information for gain. By posting this you are going to get a bunch of desperate people in trouble. Bad choice on your part to include content of this sort.
  • When were you an AP? You mean when you were failing as an AP so you got back at the AP world by turning into complete scum.
  • You are a failed AP who has gone to the other side. Little of your work is original. You focus on minutia that should be of no concern to the casino’s bottom line. You are regarded by the industry (casinos and APs) as a clown not to be taken seriously.
  • Your analysis is meaningless. The people who are on the casino floor (who are the first people who sweat the money and thus cause the casino to lose money from their sweat) have no concept of mathematics.
  • Seriously, anything at all would be better than this type of uninformed, fear-mongering. It creates havoc for people on both sides of the game. I know you are capable of much better than this Eliot.

They like me, they really like me.

Most of my advantage play analysis is centered on four types of vulnerabilities that are common to proprietary games and side bets: card counting, hole-carding, edge sorting and collusion. I am going to give a short introduction describing each of these methods. There are individual posts that present the same content in much greater detail.

Card Counting

While blackjack card counting has been around since the early 1960’s, the understanding that card counting principles apply to other games is not as well-known. Any time multiple rounds are dealt from a shoe or single-deck between shuffles, card counting applies to all wagering opportunities at that game. For example, all proprietary variations of blackjack can be card counted, including games like Spanish 21 and Blackjack Switch. I have made a conscious decision to not do any research on card counting blackjack or its variants. If it’s blackjack you’re looking for, you’ll have to go somewhere else.

Because of its extraordinary world-wide popularity, one of the games that demands card counting investigation is baccarat. Baccarat is typically dealt from an eight-deck shoe with the cut card placed just a few cards from the end of the shoe. It has been known for 50 years that card counting baccarat yields very little long-term profit. Now there are new variations of baccarat, like EZ Baccarat, 7-Up Baccarat and others. These also deserve consideration to determine if card counting is a potential advantage play tool.

Mostly, however, I apply card counting principles to side bets. These ad-on wagers have a huge range of vulnerabilities. In some cases, they are worth pennies, in other cases they are worth thousands of dollars. Worldwide, there are professional teams who are aggressively targeting side bets. Even tried-and-true side bets like the baccarat pairs bet are falling victim to powerful counting schemes. New and highly vulnerable side bets are being introduced and burnt out with clockwork regularity. If you study just a few of these side bets, you will be able to quickly gain an intuition for how card counting applies to many others.


Hole-carding is applicable to those games where the player gets to make a decision after receiving his cards. Hole-carding also requires that the dealer receive one or more face-down cards that will be exposed by the end of the hand. Hole-carding can apply to games dealt from a shoe or deck, like blackjack and baccarat, and it can also apply to proprietary games.  There is no doubt that hole-carding is the number one method that APs use to beat table games.

When the “blackjack pays 6/5 on single deck” revolution swept the Las Vegas valley in 2003, single deck blackjack had a renaissance. Overnight, dealers found themselves dealing pitch games instead of the shoe games they had been dealing for years. While blackjack card counters complained widely about the degradation of the game by the evil casino industry, advanced APs were rejoicing. Blackjack hole-carding opportunities were appearing everywhere. Closing one door had opened another. I do not cover hole-carding blackjack in this blog. This is simply my “anything but blackjack” philosophy to advantage play research.

Hole-carding can be tough to research. Determining the edge from hole-carding often requires a huge amount of computational effort. Once that optimal edge is determined, there is a more significant back-end issue: giving a strategy to get that edge. Sometimes I provide optimal hole-carding strategy. Other times, I provide a spread sheet with output from the computational work and shrug my shoulders, leaving the creation of a strategy to the reader. There are also times when all I give is the edge with no indication how to get there.

Hole-carding opportunities arise for a variety of reasons. Regardless of why, this method of advantage play must be fully understood and fully addressed as an advantage play and game protection problem.

Edge Sorting

Marking cards and then using those marks to alter the odds of a game is an act of cheating.. However, if the cards in use are marked through natural wear or because of some manufacturing or design defect, then that information can be freely used in an attempt to beat the house. What is truly remarkable is that the majority of cards have significant irregularities in their card-back designs. These defects are typically most visible along the edges or borders of the cards, hence the method of exploiting these design irregularities is called “edge sorting.”

Edge sorting can be used against most games. In a recent and now infamous case, poker player Phil Ivey beat Crockford’s Casino for more than $12 million in early 2012 by edge sorting baccarat. Edge sorting is also commonly used by individuals and teams to beat blackjack. Edge sorting has been used against proprietary games as well. I confess to having no personal experience as an edge sorter. In this sense, everything I write about edge sorting is either anecdotal or theoretical.

Edge sorting works best on games where there is a distinguishable subset of cards the AP wants to be able to identify. For example, in baccarat it would be the high cards, {6, 7, 8, 9}. In blackjack, it’s the ten-valued cards and aces. In poker-style proprietary games, edge sorters want to be able to identify the qualifying cards. For example, in Three Card Poker, the cards the AP wants to identify are {Q, K, A}, while in Caribbean Stud its {K, A}.

The edges that can be obtained by edge sorting are significant and the profit potential is huge. For many games, including baccarat, Three Card Poker and Caribbean Stud, optimal edge sorting strategy is trivial and can be learned in a matter of minutes. The downside for the would-be edge sorter is that it is tough to get and keep every card sorted. For this reason, errors creep into play that can be costly. I have made no attempt in my research to quantify the costs of these potential errors.

The biggest problem for the edge sorter is that the procedural game protection fix is trivial. All the casino has to do is include a turn at the appropriate place in their shuffle procedure and this method is effectively defeated. Then again, it would be nice if card manufactures could create a quality product.


There are some proprietary games where players can get an edge by sharing information about their hands with each other. This is not as easy as it sounds. There are three sub-tasks that must be fully integrated to produce an edge. First, the players have to occupy a critical number of seats at the table. Some games may require six or seven teammates; others can be beat with as few as two players. Second, the players have to find a way to share key information about their hands. Each player not only has to signal his hand, he also needs to be able to evaluate the signals from every other player into his own strategic decision-making. Finally, the players need an effective strategy to use this information to gain an edge. These strategies can get extraordinarily complex.

Collusion is a method I used on occasion in an opportunistic way when I was a player. For example, in placing an insurance wager on the first hand of single-deck blackjack, I might ask to see other player’s cards. But I have never used this method against a proprietary game and I have no practical experience in that sense. What I am certain about is that AP teams are out there colluding to crush some of the most vulnerable proprietary games.

The method of analysis for collusion usually produces a theoretical edge that is far larger than APs could ever achieve in practice. Correct strategy using full information for the cards that several other players hold is usually achievable only with computer analysis. A simplified and human-feasible strategy will give back a sizable portion of the AP’s edge. For the most part, I am satisfied to report the edge attainable by perfect collusion without giving a practical strategy. In some cases I have worked out a strategy, but this is mostly a pedantic effort. The teams that are beating games through collusion have much bigger targets than any games I’ve personally examined.

There are several newer games that have the possibility of being dealt to the last few cards in the deck, forcing the feasible hands for the dealer into a very narrow range. Games where the dealer’s cards are essentially known can be thoroughly crushed through collusion. At present, I know three different proprietary games with live placements that have the “down-to-the-last-card” issue. Because of the complexity of these games and the analysis, I have not considered these games and I doubt I will do so anytime soon.

There is no easy fix for collusion as a game protection problem. A sign that says that “players cannot show their cards to each other” will do little good against the signaling methods these teams work out. Surveillance may watch for a long time, knowing that it just doesn’t look right, but collusion is a very tough conclusion to reach. For this reason, it is far more important to understand the principles that make collusion work than to become an expert in any one game.

Loss Rebates

In my opinion, exploiting loss-rebate opportunities constitutes the most significant revenue stream for many of the biggest AP-teams working today. The lesson that Don Johnson gave to the industry when he beat three Atlantic City casinos out of over $15 million by exploiting loss rebates on blackjack has not been quickly learned.

In this blog you can find extensive analysis of loss rebate advantage play, including details on baccarat, blackjack, craps, roulette and video poker.  In particular, the Loss Rebate Theorem gives good approximations for the win/loss-exit points, average playing time and expected profit available. Applying this theorem requires knowing only the loss rebate percentage, house edge and standard deviation of the game.

I have not looked at beating proprietary games through loss rebates. Loss rebate players ignore games where the table limits and potential for win/loss are not high enough to generate significant theoretical profit. Theirs is the playground of the big boys. They want to make that $50,000 blackjack wager or that $40,000 complete bet in roulette.


I recognize that my research has not made me the most popular guy in some circles. There are large groups who for various and conflicting reasons wish I was not doing this work. The opposite is also true. There are many who genuinely appreciate what I am doing. They see that there is great value in having this information easily available, in a format that is readable and understandable without requiring a significant mathematical foundation. They come from all corners of the gaming community, not just the casino side, and they have a variety of reasons for their appreciation. As one surveillance operator wrote to me, “you just make me look good.”

No matter the game or method, I’ve tried to give it a fair and mathematically sound assessment. I try to apply scientific and academic research principles to what I do. I attempt to independently arrive at results from at least two directions. For highly sensitive results, I seek secondary confirmation. I have no personal agenda towards any games. Wherever the needle lands are the results I report. My goal is scientific journalism for a popular audience. I try to avoid subjectivity and editorializing except when the topic calls for an educated opinion. If I’ve failed in any way to give credit or to recognize another’s accomplishments, it was not my intention to do so. If my opinions strike some as arrogant or ill-informed, I am not beyond changing my opinion based on a well-presented contrarian argument. I have been hungry my whole life and I am still hungry.

I recognize that much of what I do is in the land of the trivial or well-known, that it has been done before or it has been done better.  I make no claims to be the first or the best. When my work is replicating the research of others, I do my best to cite sources. When others have given alternative methods or viewpoints, I do my best to include those results. To the extent that work I’ve produced is already public and freely available, I am thankful. For the rest, I’ve done my best to innovate alongside the best minds out there.  Many top APs are extraordinarily brilliant and creative individuals and I am humbled to play in the same playground.

At the end of the day, it is clear which side of the tracks I am on. I work for the casino industry. I give seminars. I speak at conferences. I write articles. I consult with staff and management. When an AP contacts me about analyzing a game, my reply is “yes, but I’m going to publish my findings.” They rarely want to go forward. On the other hand, when casino surveillance or tables games management requests advantage play analysis, they are usually happy to see my findings go public. They recognize that the information I provide to them will help raise their level of game protection. At the same time they know that I will send a message out to the AP community that yet another game is understood and under scrutiny.

Advanced advantage play is a journey. It starts with the desire to beat the house. Unfortunately, this journey ends for most with ordinary blackjack card counting. The modern AP is far more sophisticated. He is hungry to innovate, analyze and implement.  He is not satisfied with small or tortuous gains. He wants it all. And he wants to get away with it without the casino even knowing they got hit: another lucky idiot walks out the door.  In this era of rampant innovation, the task of understanding who these individuals are and what they are doing has never been more difficult or more urgent. This blog is my invitation to you, whoever you are, to participate in the journey. Begin with the axiom that every game can be beaten. Then pick a game, learn a method to beat it and take another step.

Received his Ph.D. in Mathematics from the University of Arizona in 1983. Eliot has been a Professor of both Mathematics and Computer Science. Eliot retired from academia in 2009. Eliot Jacobson