There was this casino, a major online operation based in Europe, and this casino had a problem. A little problem, sure, one lousy player on a hot streak. But he was getting to be a bigger problem every day.
The player — call him Ludwig — was a regular customer at roulette. That is to say, he was a regular loser, since few casino games are quite so elegantly designed to separate chumps from their chips as roulette. But Ludwig wasn't losing so regularly anymore.
After dropping 3,000 euros with his usual low-wager strategy, Ludwig had come back a week later a changed man. Multiple bets per spin, some quite sizable. He was up, he was down, but he kept putting more money in play. He spewed ten, twenty, even fifty bets all over the layout, like a house painter with a spray gun, trying to cover up all those unsightly black and red numbers. At one point, he had 10,000 euros riding on one spin of the wheel. Then 40,000. Then 100,000.
Common sense, or what passes for it in the gamblers' world, dictated that Ludwig should bust — and bust hard and fast, since many of his bets essentially canceled each other out. But he didn't. He kept coming out ahead. When his winnings crossed the one-million-euro mark (about $1.42 million at current rates), the casino's manager decided to bring in two experts. One was a private investigator, hired to look into Ludwig's background, in search of unsavory associates — mobsters, say, or computer hackers, or even a confederate inside the virtual casino.
The other was a different kind of gumshoe: Robert Hannum, a professor at the University of Denver. Although his doctoral work leaned heavily toward the theoretical side of statistics, Hannum now carries the title "professor of risk analysis and gaming" — a tribute to his many years studying the odd collision of mathematics, marketing and fantasy that fuels the commercial gambling industry. Hannum has been the official mathematician in residence for the MGM Grand and the Aria Resort & Casino; other casino operators call on him to run the numbers on tricky new games or review the math behind special betting promotions that, despite the best of intentions, might actually end up costing their casino some money. He is their go-to guy, the probability king, the wizard of odds.
The anxious manager at the European casino wanted to know one thing: Was Ludwig legit? Was it possible for someone to bet so crazily and come out ahead, spin after spin? Or was it more likely that something crooked was going on?
Hannum collected the data from the hot streak and subjected it to various statistical tests. He determined that the result of each spin had a degree of randomness one would expect from an honest wheel. Ludwig's action was a bit harder to track; in his first 1,600 rounds at the table, he had come out almost exactly even. But in the days that followed, he'd increased the size and number of his bets. After 3,200 rounds, he was up nearly 1.5 million euros.
The house edge on a standard American roulette wheel, one that has a zero and a double zero, is 5.26 percent. The European wheel has only one zero and an edge of 2.7 percent; on average, a player shooting a thousand bucks on a series of bets on red at a Monte Carlo roulette table will end up with $973. But that's just an average; some players will lose more, a few will come away winners. Hannum's challenge was to figure out, using what he describes as "some rather involved and sophisticated mathematics," the probability that Ludwig's bizarre betting pattern would yield such staggering results. To do so required an understanding of what statisticians call the volatility of the game and just how far from the expected win-loss range Ludwig had strayed.
After some vigorous number-crunching, Hannum told the casino manager to relax. Ludwig's run was unlikely — an 80-1 shot, it turned out — but hardly the one-in-a-million outcome that would almost have to be cheating. Reassured, the manager decided to do what good casino bosses always do when they take a hit: smile and encourage the lucky fella to keep playing.
Ludwig did. The wheel kept spinning and the numbers kept changing. A week after his streak reached its 1.5 million-euro high point, Ludwig was cleaned out. His rapid collapse was yet more proof, if any was needed, that it's hard to argue with probability.
It's hard to argue with Hannum, too. He has one foot in the academic world and the other in the high-stakes world of commercial gaming; his dealings with casino surveillance teams, game manufacturers and state gaming commissions have given him a unique perspective on legalized gambling in America. But consulting is only one aspect of his work. He's taught courses in gaming law and held classes in Las Vegas casinos, to give students a firsthand look at the industry. He's written papers examining how many shuffles are required to truly randomize a deck of cards and explaining how statistics get misused in gambling. His book Practical Casino Math, now in its second edition, has a tendency to disappear from library shelves and never return.
In recent years, Hannum has devoted much of his research to the study of poker — in particular, the issue of whether it's primarily a game of chance or one of skill. The question is more complicated than it sounds and has become a flashpoint in legal battles over the game. Starting with a Colorado trial two years ago, Hannum has emerged as a formidable expert witness for the defense in cases around the country in which poker-game organizers have been prosecuted for defying state laws.
Through the use of computer simulations and other methods, Hannum has shored up the case for poker as predominantly a game of skill. "Bob's work has been hugely impactful on the debate," says Anthony Cabot, a prominent gaming-law attorney in Las Vegas who collaborated with Hannum on Practical Casino Math and other projects. "He's done some research that's truly seminal in this area, and it always comes up and is cited in the discussion of whether poker is skill or chance."
This week, at the Hawaii International Conference on Social Sciences, Hannum will unveil what may be his most important contribution to the field to date: a presentation based on detailed research of a billion hands of live online poker. The timing couldn't be better, as legislators argue over what to do about the regulation of online poker sites in the wake of "Black Friday," on April 15. That was the day the Department of Justice announced indictments against owners of the three largest Internet poker sites, charging them with bank fraud and gambling violations. The move made the sites unavailable to American players and threw the entire online poker industry, estimated to draw as much as $6 billion in wagering a year, into chaos.
The recent crackdown "makes me feel the stuff I'm working on now is important," Hannum says, "because the government is going to have to deal with it. They're either going to have to make it clearly illegal or regulate it."
He hopes his research might be the winning hand. "I haven't seen a study that examines this many real games," he says. "I just think someone — and I hope it's us — can and should produce something that's pretty convincing on the question of skill versus chance in poker. I'm really hoping that's what will come out of this."
In his book Casino, a study of the waning mob influence in Vegas in the 1980s that's the basis for the Robert De Niro film, Nicholas Pileggi writes, "A casino is a mathematics palace set up to separate players from their money. Every bet made in a casino has been calibrated within a fraction of its life to maximize profit while still giving the players the illusion that they have a chance."
It's a passage that Hannum is fond of citing in his own work. And why not? It's the intricacy of that palace, which players storm again and again with fresh strategies and hubris — trying to claw their way to some kind of advantage, however illusory — that made Hannum realize that gambling could be a rich area of research.
Hannum grew up in the Philadelphia area and demonstrated an aptitude for math at an early age. By high school, he was sufficiently accomplished at it to earn cruel nicknames from classmates. He took his first course in statistics and probability as an undergraduate at the University of Dayton and knew it was the field he wanted to pursue. "I liked that you could take the mathematics and apply it to real-world situations," he says now.
After graduate work at Florida State and a stint teaching at Bucknell University, Hannum arrived at DU in 1979. The school, the students, the mountains all suited him just fine. He wrote well-received papers on Bayesian nonparametrics and income inequalities and other rarefied statistical issues and may well have continued in that vein, except for a fateful trip to Las Vegas in 1994.
"I was at an international conference on risk-taking and gambling," he recalls. "Up to that point, I wasn't aware of how much study was being done in the area of commercial gaming. When I saw all the cool applications, people doing high-level, scholarly studies — it was much more interesting than the usual fare."
The experience prompted Hannum to develop what he calls "applied probability" courses at DU, focusing on gaming operations. One course, "Risky Business," involved assigned reading on the theory of gambling before a week-long trip to Sin City to meet with casino executives, gaming regulators and other industry types. Some classes were held in a dead pit on the casino floor, where an employee would set up games and explain their finer points. Hannum discovered it was an excellent way to convey key concepts of probability to students.
"People relate to gambling for a number of reasons," Hannum notes. "The notion of odds and risk is universal. Uncertainty is in almost everything we do. And people find casinos are fun. It's kind of a sexy thing, and students like examining it."
The course attracted widespread media attention and led to spinoffs. So many DU law students wanted to take it that Hannum developed another class dealing with gaming law. A course on Colorado's limited-stakes gaming industry followed, with required field work in the mountain towns where gambling was legalized twenty years ago. This month he'll be teaching another popular class, "The Science of Poker," which requires as a prerequisite an introductory course in statistics and delves into principles, strategies, the history and language of the game, and what a course description lists as "pot odds, expected value, variance, effective odds, implied odds, game theory, deception, bluffing, earning rate, bankroll analysis" and more.
As various forms of legalized gambling swept the country, Hannum started fielding calls from reporters who were looking for a "math guy" to run down the odds behind the games. Hannum explained that, in certain forms of video poker or with a traditional game of blackjack using a single deck, a player pursuing an optimal strategy might have a reasonable expectation of almost breaking even. He was less encouraging about slot machines. In fact, he once testified in a tax case that a man who claimed to be a professional slot-machine player was aspiring to a profession that didn't exist, since there was no mathematical expectation of making a profit by playing slots.
That's not to say that strange things don't happen in casinos. Like Ludwig's roulette run. Or the $9 million a player at another casino won at blackjack, a windfall that Hannum was asked to analyze. (His verdict: unusual but not all that suspicious, given the size of the bets the player was making.) Or the case of Patricia Demauro, which makes Ludwig's hot hand seem positively tepid. In 2009, Demauro walked into an Atlantic City casino and proceeded to throw the dice at a craps table for four hours and eighteen minutes, 154 rolls in all, without crapping out. It was a world record, a one-in-1.56 trillion event — and only her second time playing craps. Call it beginner's luck.
From a purely mathematical perspective, of course, there's no such thing as luck, just the complexities of probability. Yet people's myths and superstitions about slots, roulette and other games that strongly favor the house are hard to break — for example, the notion, known as the gambler's fallacy, that a particular one-armed bandit is "overdue" for a big jackpot or that a particular number will come up soon on the roulette wheel because it hasn't shown up in dozens of spins.
"People think that the spins are not independent and that slot machines run hot and cold," Hannum says. "It's one of the most common mistakes made in thinking about gambling and the mathematics involved."
Even the sharpies who run the joint, Hannum discovered, sometimes make errors in their math. A few years ago, the operators of an Illinois riverboat casino figured they would offer payoffs of 2-1 on natural blackjacks one slow day of the week, instead of the customary 3-2 return — a bonehead move that effectively wiped out the house's slight edge and gave the players a 2 percent advantage. The place ended up losing $200,000 in one day on the promotion. Other innovative side bets and promotions have backfired, too, prompting casinos to consult with Hannum or other math professionals before embarking on such ventures.
"You might be surprised how many people there are in the industry who aren't well-versed in the mathematics," Hannum says. "They usually know the numbers for the traditional games, but there are a lot of new games and new machines out there."
He's developed a keen appreciation for the kind of calibrations that Pileggi wrote about. New variations of casino games and slot machines have to be sufficiently attractive in payouts to keep customers playing yet still afford the house a reliable return on investment. The trick is how to design a game that allows most players to lose slowly while a few might actually win.
"They can grind you into the dust, as long as they don't do it too quickly," he notes. "If people get crushed, they're not going to come back. The game developers have to play a balancing act between making money and not making the edge too big."
Perhaps because he knows the numbers all too well, Hannum doesn't consider himself much of a gambler. He prefers to spend his free time shooting hoops or coaching youth soccer or baseball. When he does venture into the green felt jungle, he'll occasionally check out new casino games — "I guess you can say, wink wink, I do play occasionally for research purposes," he says — but the odds are better of finding him in the poker room.
Poker fascinates Hannum. There's no house edge; casinos simply take a cut of the pot for hosting the game. But there are layers and layers of strategy involved, from knowing the odds of drawing certain cards to the psychology of bluffing, the elaborate considerations that go into betting decisions, and the art of reading other hands — and other players. And, of course, each variation of the game has its own dynamics and hazards.
"I like Texas Hold 'Em," Hannum says, "but I also like Seven-Card Stud. Unfortunately, that tends to be an East Coast game. It's harder to find in Las Vegas."
The huge rise in popularity of poker in the past decade — not just in casinos but in bars and basement rec rooms across the land — is generally attributed to the increasing television coverage of high-stakes Texas Hold 'Em tournaments. In particular, the 2003 $2.5 million victory at the World Series of Poker by the aptly named Chris Moneymaker, an accountant who won a seat at the Main Event by battling his way up from a $39 online satellite tournament, seemed to galvanize legions of back-room players into dreaming that they, too, had the stuff to go all the way. But with the explosion of poker tournaments and online play came a slew of prosecutions; in most states, poker operations for profit are lumped with other "games of chance" and are considered illegal outside of licensed casinos or card rooms.
Yet many state laws also hold that a game that is "predominantly" a contest of skill is exempted from the ban. To poker devotees, the game's reliance on skilled decisions seems obvious. But when Hannum first started looking into the relative roles of skill and chance in poker a decade ago, he found a surprising lack of data on the subject.
Hannum credits his friend and collaborator, gaming attorney Tony Cabot, with urging him to get into the skill question — a critical question, in many jurisdictions, as to whether the game was legal or not. But how, Hannum wondered, do you go about proving that poker is a game of skill? How do you approach the question, and what sort of hard statistical evidence would persuade a court of law?
"It was immensely difficult," he says now. "And it continues to be difficult."
The way Kevin Raley saw it, he and his friends were doing everything right. When the owners of a Greeley bar asked him and a small group of fellow poker enthusiasts if they'd like to organize a tournament, they gave considerable thought to how to keep the game legit under Colorado's gambling laws.
Raley's group charged a twenty-dollar entrance fee to players, just to make sure people would have "some skin in the game" and not play stupidly. But all of the money went back to the players in the form of prizes. And a newcomer had to be invited to play by one of the established regulars, who could vouch for the fresh face. Colorado's prohibition against gambling excludes games among players who have a "bona fide social relationship," and Raley believed his group met that criteria.
"We had a whole set of rules and bylaws set up," he says now. "You had to know someone who was in the game. You couldn't just walk in and start playing. But unscrupulous people in the group ended up inviting an undercover agent. That's what brought on the bust."
In 2008, state liquor authorities were looking into a licensing issue at the bar when they learned about the poker game. An undercover agent paid his entrance fee, played a few hands, and then the cops swooped in. Raley and his friends were charged with operating a professional gambling enterprise.
The group hired attorneys and refused to be stampeded into plea bargains. Raley was the first to go trial. On the day of trial, the prosecutor knocked the charge down to a petty offense, subject to no more than a $100 fine. Todd Taylor, Raley's attorney, wanted a jury trial just the same.
"It's the only time in my career that I've tried a petty offense," recalls Taylor, now a district court judge in Greeley. "The judge allowed me to have a jury of six, a two-day trial and an expert witness."
The star expert for the defense was Hannum. Raley had contacted the Poker Players Alliance, a grassroots advocacy group with more than a million members, for assistance — "looking specifically for an expert to testify that poker is a contest of skill," he says. The PPA informed him that one of the leading researchers on the subject lived an hour away from him; the organization also arranged to pay for Hannum's appearance. It was the beginning of the professor's tour of courtrooms in several states as a crusader in the poker wars.
By the time he testified for Raley, Hannum had read widely in the literature of poker mathematics and co-authored with Cabot a couple of key studies on the skill-versus-chance debate. Along with other forms of analysis, he ran computer simulations tracking the outcomes of a million hands of Seven-Card Stud and Texas Hold 'Em, inserting different player profiles in order to demonstrate the role that skill played in earnings. When all of the players were equally skilled, the individual earnings were quite similar. But insert one or more "unskilled" players, making random decisions rather than following a consistent strategy, and the results changed dramatically. The unskilled players lost big. The skilled players dominated the game.
Another simulation involved a billion hands of Texas Hold 'Em, with one skilled player facing off with one unskilled. The skilled player followed a very elementary but aggressive strategy: Always raise when allowed; otherwise, call. The unskilled player called, raised or folded in an utterly random fashion. The skilled player won 97 percent of the hands.
The results confirmed a fundamental truth of poker: Skill beats luck. Not every time, certainly, and not in every version of poker, but in the long run, a player making intelligent decisions in a game such as Hold 'Em or Stud will pound a novice into the ground. At a professional level, it's not about winning the most hands, but making the right decisions, based on the cards and the tells and the pot, and ultimately making the most of what you've got. That's why publications can rank with a fair degree of accuracy the top poker players in the world, the people who consistently land in the money at high-stakes tournaments. There are no lists of top players in roulette.
"There are multiple skills in poker, but they're all manifested in betting decisions," Hannum says. "Do you bet or do you fold? Skilled players fold when they see that will minimize their losses, and they'll extract more money from the other players when they have a good hand. Weak players can have great cards, too, but they can't do that as well."
Raley felt he had a strong case that his friendly little game involved "bona fide social relationships" — but having Hannum testify that poker was mostly a game of skill didn't hurt. "I believe it was pretty instrumental in what happened," he says. "The prosecution didn't have much of a case, and the jury found me not guilty."
Although they couldn't contest Raley's acquittal, Weld County prosecutors appealed the case, contending that Hannum shouldn't have been allowed to testify because existing case law in Colorado already defined poker as a game of chance. A district court judge agreed.
"Some states analyze the issue from the question of whether skill prevails or if chance does," Taylor explains. "But the district court in this case said it's not a question of predominance; poker relies to some extent on chance, so under Colorado law, it's gambling."
Raley appealed that decision, but the Colorado Supreme Court declined to review the case. These days he plays poker at casinos or at a friend's house, but he's hopeful that some day the state's antiquated approach to the game will change. "It's like pushing a snowball uphill," he says. "It's just a matter of time to get enough cases to get it over the top."
Hannum has been pushing the snowball in other courtrooms over the past two years. In South Carolina, his testimony helped to persuade a judge that poker is primarily a game of skill, but the judge ruled there, too, that the predominance test didn't apply; the case is on appeal. He's also testified in trials in Kansas and Pennsylvania, with mixed results. His appearances are often paid for by the Poker Players Alliance. "They put their resources behind these cases, and one of their resources happens to be me," he says.
According to Cabot, scientific evidence about poker as a game of skill has now been presented in more than a dozen cases across the nation. "Poker has been found to be a game of skill in five or six of those cases," he says. "In a couple of cases, the courts decided it was equal chance and skill. It's a difficult issue. There are so many types of games, and the way they're played can have an impact on their legality."
John Pappas, executive director of the PPA, expects the issue will ultimately be resolved through legislation rather than the courts. "There is legislation pending in a few states, based on the re-evaluation poker is receiving," he says. "It's only in the last few years that we've been able to get hard data that poker is a game of skill."
In his view, Hannum's efforts have been invaluable in making that case. "He's very convincing," Pappas says. "He doesn't come across as a pointy-head scholar, but as a down-to-earth academic who has a firm grasp of the facts."
In recent months, the PPA's energies have focused less on state cases and more on the federal government's efforts to protect American citizens from the depravity of online poker. The Black Friday shutdown stemmed from a 2006 law, the Unlawful Internet Gambling Enforcement Act. The UIGEA didn't outlaw the foreign-based poker sites, but it made it illegal for American banks to process financial transactions with them. The recent indictments allege that site executives attempted to disguise fund transfers and engaged in other activity to circumvent the law. But outrage over Black Friday has put pressure on lawmakers to repeal the UIGEA and come up with a workable framework for regulating and taxing the lucrative online business.
"This has been the awakening of a sleeping giant," Pappas says. "A lot of poker players were apathetic about getting involved politically because they were playing online. Now that the largest sites have been taken away from them, they're starting to recognize why state and federal regulation is so important."
And, of course, companies with a stake in traditional gaming outlets are trying to figure out how to get a piece of the action, too. "The brick-and-mortar casinos, the tribes — they all view this as a real opportunity for them," he adds. "Instead of facing opposition from antique gaming interests, we're seeing those interests wanting to get in on this."
Ironically, much of the hard data that could end up changing state and federal laws about poker is now coming from Internet poker sites. That includes Hannum's latest research project, a collaboration with two other DU professors, Terry Dalton and Matt Rutherford. Not yet published, their research draws on a well of information never available before: more than a billion hands of actual online poker games. Although he's reluctant to discuss the data in detail at this point, Hannum says they were able to view the hands in "God mode" — meaning they could see all the hole cards, whether the players folded or not.
"We can look at how those hands were played, with full knowledge of everyone's cards," he explains. "No one else has done a study with a huge database like that."
The process allowed the DU team to track how a highly skilled Hold 'Em player fares when given a killer starting hand, such as a pair of aces, compared with a weak player. Both will probably win the hand, but the skilled player is likely to win more money. And by comparing actual outcomes of hands with what would be expected to happen, based purely on the strength of the cards — in other words, if the betting decisions that drive real-life play were removed — Hannum believes their analysis makes a particularly persuasive case for poker as a game of skill.
His colleagues in Hawaii will get the first glimpse of the research this week, with a paper on the subject to follow. Hannum cautions that no single study is definitive, but the online data serves to reinforce what professionals have known all along: Good betting trumps good cards.
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And the evidence is mounting. A 2009 study of online Texas Hold 'Em games found that three-fourths of the hands never go to a showdown — and only half of the rest are won by the player at the table who would have had the best hand, because in many cases that player had already folded, subdued by a more skilled player's betting decisions. A recent working paper by Freakonomics author Steven Levitt and Thomas Miles found that 2010 World Series of Poker players rated as highly skilled earned an average return on investment of more than 30 percent, while the rest of the field averaged a 15 percent loss of their investment.
In the course of his work, Hannum has had the opportunity to confer with several players who've ranked in poker's elite, including Howard ("The Poker Professor") Lederer, Greg ("FossilMan") Raymer, Robert Williamson III, Phil Gordon and Mike Sexton. Some have been guest lecturers in his classes, wowing students not only with their grasp of the numbers but also with the finer mysteries of the game.
"They're all good at reading their opponents," Hannum says. "But I'm really amazed at how good these guys are at putting you on a range of hands. Not that they know you have a pair of queens, but they can narrow it to three or four or five possibilities. If you can do that well, you have a pretty good chance of winning at poker."
A pretty good chance, yes. But not a sure thing. If a billion hands of poker have taught the probability king anything, it's that there's no such animal as a sure thing.