1. Gambling Statistics By Race Against
2. Gambling Statistics By Race Statistics
3. Gambling Statistics By Race Results
4. Gambling Statistics By Race In America

According to The American Gaming Association (AGA), the gambling industry in the US is worth $261 billion and supports 1.8 million jobs in 40 states (1); however, gambling has had a difficult (and illegal) history in the USA and it is just up until now that the path is being cleared.

Several anti-gambling laws have been in place since the great depression and gaming has been heavily regulated ever since. Some of those regulations haven’t been updated since the 60’s. Despite this, the widespread use of the internet (and online casinos) have made the betting scene change notably and laws have started to shift (2).

Today, the industry faces a rapid growth and increased revenue moving away from the stigma it once held, making it easier for people to access online gaming and betting.

Gambling Revenues in the USA

National gambling statistics. The following reports are available for download: 2019. National Gambling Statistics, Financial Year ending 31 March 2019 - Audited.

Race groups within offense categories (tables 1, 3, 4, 5 and 6) and changes in arrest rates by offense and demographic groups (table 7). The report also contains graphs displaying trends in arrest rates for individual offenses between 1990 and 2010 and graphs of age-specific arrest rates for 1990, 2000 and 2010. Everything about betting race horses involves numbers. There are numbers on the horses. The are speed figures. There are distances. Most importantly, there are statistics. Some statistics in horse racing hold true year after year. An example is that favorites win one out of every three races. Mathematics is the science behind statistical analysis. A betting odd opportunity should be considered valuable if the probability assessed for an outcome is higher than the implied probability estimated by the bookmaker. Read more on the math behind.

USA gambling revenues increased to over $160 billion in 2018. Now that sports betting is legal, most gaming experts and financial analysts predict the total revenues for US betting each year will top $200 billion. Whether it’s a lottery ticket, slot machines, sports bets, bingo, or poker, Americans love to gamble.

As each year passes, US state governments expand legalized and regulated games of chance, which encourages more gambling. 2018 was no different. Below is a breakdown of the revenue generated by each form of betting each year.

US commercial casinos generated $41.2 billion in 2017, a 3.7% increase year-to-year from 2016 and a new record for gross gaming revenues for US commercial casinos. New casinos in Maryland and New York added to the increased revenues in 2017. MGM National Harbor in Maryland was the most successful new casinos, but Del Lago Resorts, Rivers Casino, and Resorts World Catskills in New York state also contributed.

In all, commercial casinos generated $51,395,562,664 in revenues in 2018. The total represents a 3.5% increase over 2017. Commercial casinos sustained 737,450 jobs and paid $34.334 billion in worker income. The Las Vegas Strip generated $6.59 billion in gaming revenues in 2018, making it the top destination. Atlantic City came in second with $2.51 billion, which shows a strong 5-year bounce back from its $2.1 billion in 2014. Chicagoland, Baltimore-Washington DC, and New York City finished 3rd, 4th, and 5th among commercial casino markets.

The National Indian Gaming Council still has not released full tribal casino statistics, so the NIGC’s 2016 gross gaming revenues are the most recent official revenue statistics. In 2016, America’s tribal casinos generated $31.195 million in gross gaming revenues. That is up nearly $1.3 billion from 2015, which itself increased $1.4 billion from 2014. If one projected similar growth year-to-year from 2016 to 2017, one might expect to see gross gaming revenues in the range of $33.3 billion for 2017. To avoid speculation, we list the 2016 official statistics. The NIGC releases the previous year’s figures in June or July each year, so check back for the latest results.

Tribal casinos generated $32.801 billion in 2018 — a 3.5% increase over 2017. At its current pace, the tribal casino industry should overtake commercial casino revenues by the year 2030. Like the commercial gambling industry, a handful of tribal casinos launched sportsbooks.

The Mississippi Band of Choctaw Indians opened sports betting at their three land-based casinos, while the Pueblo of Santa Ana opened a sportsbook at Santa Ana Star Casino near Albuquerque, New Mexico. The bookmakers’ success could open the door for many other tribal sportsbooks across the United States.

Nevada poker room revenues showed less than 1% growth, as gross gaming revenues were $118.46 million. The 2017 poker rake was $117.7 million. The Nevada figures were a minor miracle, considering that several Nevada poker rooms closed in the past year. MGM Resorts closed the Mirage poker room in February 2017, while the Hard Rock Las Vegas poker room closed in March 2017 and Luxor cardroom closed in May 2017. (The cardrooms for the Linq, Eastside Cannery, The Plaza, and Aliante Casino closed in 2016.)

Atlantic City’s seven poker rooms added $28.38 million in gross gaming revenues in 2017. Pennsylvania’s live poker revenues equaled $60 million. MGM National Harbor in Maryland collected $44.5 million in poker rake alone last year, which was the vast bulk of Maryland poker revenues in 2017. New York’s new live poker rooms, on the other hand, contributed only $6.7 million to the nation’s live poker revenue total. It is difficult to calculate accurately the live poker revenues for America’s 6,100+ live poker rooms. Tribal casinos contain many card rooms and they do not have to report their earnings to states. Thus, it is likely that some poker revenues might be listed under tribal casino statistics.

Nevada poker rooms increased their total rake from $118.45 million to $120.1 million. On the other hand, Atlantic City poker revenues dropped from $28.38 million in 2017 to $27.028 million in 2018. Pennsylvania’s live poker revenues remained in the $60 million range. MGM National Harbor remained high with over $45 million in poker rake, while the New York state live card rooms continued with a week haul with a little over $7 million in revenues. As always, it’s difficult to gauge total poker revenues, because tribal casinos include their poker rake with their other casino games.

Ultimately, the official US poker room revenues remained steady at $1.9 billion. 2018 showed the US poker industry holding steady after two years of decline. Fewer Las Vegas casinos have card rooms than they did ten years ago, but 2018 did not have any major Vegas poker rooms close. 2016 and 2017 showed a marked decline. MGM Resorts closed the Mirage poker room in February 2017, Hard Rock Las Vegas’s poker room closed in March 2017, and the Luxor cardroom closed in May 2017. In 2016, the cardrooms for The Plaza, the LINQ, Eastside Cannery, and Aliante Casino closed. Still, the United States has over 6,100 poker rooms, so the revenue stream remained steady throughout 2018.

US online gambling produced $247.5 million in revenues in 2017. New Jersey’s online gambling industry continues to grow. In 2017, the combined revenue of Atlantic City’s online casinos and poker sites was $245 million. That’s a 21% increase from 2016 when New Jersey’s iGaming niche generated $195 million. Delaware added only $2.4 million in online gambling revenues, which was an 18% decline.

Mike Lawton of the Nevada Gaming Control Board said online poker revenues are included in total poker revenues for the state, so it is hard to get official statistics for Nevada online poker. Given the fact, only two Nevada online poker sites exist and state regulators try to hide the small revenue stream, one can assume Nevada’s iPoker stats are tiny. Nevada has about twice the online gamblers as Delaware, so $5 million is a safe assumption. Those poker revenues are not included in our figures, though, because they are included in Nevada’s overall poker stats.

Gambling Statistics By Race Against

New Jersey grabbed the lion’s share of legal US online gambling revenues in 2018 with $298.7 million. Five years into its experiment with online gambling, New Jersey’s online casino and poker portals continue to grow apace. Delaware and Nevada both have regulated online poker, but despite the Multi-State Internet Gambling Association (MSIGA), their revenues were negligible. Pennsylvania will be a huge factor in 2019, but its online poker and casino industry was still getting off the ground in 2018.

Interstate online gambling faces a severe test in 2019. The U.S. Department of Justice declared in January 2019 that interstate online poker is illegal under federal law, striking a blow to the MSIGA pact between Delaware, Nevada, and New Jersey. The New Hampshire Lottery sued because its online lotto ticket sales are endangered by the 2019 DOJ opinion, while New Jersey and Pennsylvania filed their own lawsuit to protect their online poker and casino industries. Whatever happens with the legal cases, US online gambling revenue growth should be significant in the coming year, because Pennsylvania’s iGaming industry launches.

Like the tribal casino revenues, compiling an official list of lottery revenues takes a bit longer each year, because of the patchwork of state-run lotteries and multistate lottery associations. In 2016, US lottery ticket sales were $80.5 billion. The projected figures for 2017 are expected to exceed $85 billion, due to upticks in the sale of scratchcards and the increases in the Powerball and Mega Millions jackpot sizes. Scratch-off tickets are the biggest contributor, with the Powerball and Mega Millions multi-state lottery association games contributing the second and third-most to the revenue pool.

New York state had the biggest lottery ticket sales, with over $10 billion. California, Florida, Massachusetts, and Texas were next in line. Each of those four states had between $5 billion and $6.5 billion in lottery sales.

The combined revenues of US lotteries for 2018 sit around the $72 billion mark. The U.S. Census Bureau releases the official lottery revenue statistics each year. So far, the Census Bureau has not released official 2018 lottery revenues. The 2017 lottery total was $71.826 billion, while the 2016 lottery statistics were $72.649 billion. That shows a regression of nearly $800 million from 2016 to 2017, though that number reflects a statistical anomaly in lottery drawings more than a loss of interest in the state and multistate lotteries.
State politicians are leery of allowing legal online gambling, though, because they fear it would harm lottery sales. Departing Michigan Gov. Bill Snyder vetoed an online poker and casino bill in December 2018, because he said lottery taxes are higher and iGaming would hurt lottery sales (thus tax revenues). Given the recent trends, it’s a safe bet that US lottery revenues in 2018 were around $72 billion for the year. Mega Millions and Powerball drawings continue to dominate national attention, while their revenues increase year to year.

The revenue figures for US legal sportsbooks came from Nevada and Delaware in 2017. Nevada bookmakers won $248 million from sports bettors in 2017, which was a record year. New forms of betting on eSports helped, though the growing impact of William Hill USA on the Nevada sports betting scene helped. Even hockey betting was up in 2017, thanks to the inclusion of the Las Vegas Golden Knights to the NHL. The Golden Knights’ surprising first-year performance helped drive local sports betting, though it was general NFL, Super Bowl, MLB, NBA, and March Madness betting which generated the most revenues.

Delaware, which had legalized sports lotteries, generated the remaining $6 million in sports betting revenues. Delaware’s sports lotteries require players to make parlay bets of 3 or more games. Delaware sportsbook revenues should increase significantly in the wake of the US Supreme Court’s repeal of the PASPA sports betting ban. Delaware plans to open legal sportsbooks at Dover Downs, Delaware Park, and Harrington Raceway on June 5, 2018. New Jersey sportsbooks at the Atlantic City casinos and Monmouth Park Racetrack in Oceanport should follow suit quickly. US sports betting revenues should have a big jump in 2018.

Given the landmark US Supreme Court decision to strike down the Professional and Amateur Sports Protection Act (PASPA) as unconstitutional, US sports betting revenues increased significantly. Several states (New Jersey, Delaware, Mississippi, West Virginia) legalized sports betting in the months after the May 14 SCOTUS decision. Led by New Jersey, the new sports gambling jurisdictions generated $129.6 million in a little over 6 months of activity.

Meanwhile, Nevada benefited from the additional media coverage given to sportsbooks, so it generated $301.0 million. In all, the legal US sports betting industry generated $430.6 million in revenues – a $182 million increase from 2017. The number should increase significantly in 2019, when New Jersey and Mississippi feature a full year of sportsbook operations. Pennsylvania bookmakers also launched in late 2018 and early 2019, which should boost numbers greatly.

Pari-mutuel racebooks generated $300 million in revenues in 2018. Pari-mutuel wagering is the legal term for betting on horse racing and dog racing. Bettors receive winning for the thoroughbreds, harness racers, or greyhounds which finish in the top three of a particular race (win-show-place). Pari-mutuel wagering also includes a variety of trifecta bets and parlay bets, which pay out more, but hit for the bettor less often.

Dog racing now is allowed in only 5 US states, as the Florida electorate voted to ban greyhound racing (Amendment 13) in the coming years. Most greyhound tracks continue to act as off-track betting facilities (OTBs), offering simulcasting and wagers at over 200 race venues worldwide. Churchill Downs, site of the annual Kentucky Derby, continues to generate almost 60% of the racebook revenue ($175 million) in the United States. Besides the world-famous Louisville-area horse track, Churchill Downs Incorporated owns racetracks and even a few land-based casinos across the United States.

The US pari-mutuel racebook industry must find new ways to drive customers or die a slow death in America. Race betting declined significantly in the past generation due to competition from the lottery, tribal casinos, online casinos, and mobile betting. After the US Supreme Court repealed the federal ban on sports betting (PASPA) in May 2018, sports betting should grow significantly in the coming 5 to 10 years. US pari-mutuel racebooks face another major competitor, so they’ll need to find innovative ways to bet on horse and dog racing or that $300 million total will decline.

For generations, charitable bingo halls have generated cash for civic organizations, veterans groups, religious groups, fraternal organizations, firefighters, and other charity organizations. Charitable bingo remains a major focus, but organizations also hold raffles, poker nights, and sell pull-tab games to the public.

As lottery betting and tribal casinos have increased over the past two decades, charitable gaming and bingo participation has decreased in many states. Charitable gaming is a major source of fundraising for nonprofit organizations in the United States. The numbers haven’t declined across the board, as Michigan’s poker nights (millionaires clubs) increased greatly from 2002 to 2012, due to making it easier to attain a charitable gaming license. Even in Michigan, though, the decline since 2012 has been significant. Meanwhile, Minnesota increased its charitable gaming revenues significantly, as part of a plan to fund the Minnesota Vikings’ football stadium.

Gambling Statistics By Race Statistics

As the US population has increased in the past 16 years, the number of charitable organizations holding bingo nights and raffles has increased. That means the overall decline in charitable gaming revenues was slight, from $2.2 billion to $2.15 billion, though the decline in revenue-per-venue and real money value due to inflation is stark.

Each US state organizes and regulates charitable gaming in their own way. Groups pays taxes to the states, but otherwise remain coy about their fundraising. Different states have different terms for their games (Millionaire’s clubs, pickle games, fish games), which adds to the confusion. This makes a completely accurate total of charitable gaming revenues more an estimate than an official tally. Charitable gaming revenues stayed in the $2.1 billion range for 2018.

Gambling Statistics By Race Results

What Is Online Gambling in the US?

The gambling industry in the United States is the sum total of all forms of legal betting. US gambling includes commercial and tribal casinos, state and multistate lotteries, sportsbooks and racebooks, real money online casinos and poker sites, as well as charitable gaming and bingo halls.

US gambling statistics do not include unregulated online gambling, local bookies, organized poker games in your neighborhood, office pools, fantasy leagues, or March Madness brackets. Because such gaming is unregulated and untaxed, it is hard to get accurate statistics for them. The American Gaming Association and other groups estimate yearly betting turnover of illegal gambling, but the AGA’s data is an educated guess.

In this US gambling report, Online United States Casinos stick to the facts. Most of the statistics below are compiled from state regulatory agencies. Each month, gaming commissions, gaming control boards, and state lotteries across the United States produce gambling statistics. We’ve collected the latest gaming data and compiled it into several gambling categories, which we present below.

Is Gambling Legal in the US?

Many forms of gambling are legal in the United States, but no form of gambling is legal everywhere. Because the United States is a constitutional republic with a federated system of government, the USA is a patchwork of state and federal gaming laws. That makes gambling legality complicated in the United States. What it does mean is Americans who love to bet can find places to live with permissive gambling laws, while Americans who dislike gambling can find states where gambling is 100% banned.

US gambling laws are complicated because each state deals with its own gaming interests and social mores. Nevada is the most pro-gambling states in the United States, as Las Vegas and Reno are gambling destinations. At the same time, Nevada is one of a handful of states which bans lotteries — because it would compete with the billion-dollar resorts on the Las Vegas Strip. Kentucky bans casino betting, though it is the center of the United States horse racing (and horse betting) industries.

As a general rule, legalized gambling has expanded greatly in the past 25 to 30 years. After the Indian Gambling Regulation Act of 1988, tribal casinos expanded to 28 states. California and Oklahoma are two US states with huge tribal gaming industries, but dozens of others exist. The expansion of tribal gaming caused US states to liberalize their commercial casino industries, so states like Illinois, Ohio, Arizona, and Kansas expanded casinos greatly. Pennsylvania legalized casino betting at racetracks and now collects more gaming tax revenues than any states besides Nevada.

Lottery gambling has grown exponentially, too. The Powerball and Mega Millions have the biggest lottery jackpots in the world. State lotteries’ scratch-card tickets generate the most tax revenue. Meanwhile, online and mobile casino gambling and poker betting has grown in popularity. While many Internet gaming sites are unregulated, legal online/mobile gambling exists in 4 states and is expected to be legalized in other US states in the coming years.

Utah and Hawaii are the only two US states which have a 100% ban on all forms of gambling. In Utah or Hawaii, you can’t visit a casino, buy a lottery ticket, make a sports bet, or even play in a bingo hall. All 48 other U.S. states have some form of legalized gambling. In the USA, many southern states restrict gambling significantly, because of the social conservative values in many states. Despite that general statement, all southern states can’t be pegged as anti-gambling.

Alabama bans gambling (even lotteries) in anything but tribal casinos, while Mississippi has dozens of casinos on the Gulf Coast and Tunica County. South Carolina bans most forms of gambling, while Florida allows expanded gambling for the Seminole Tribe and even 8 counties throughout the state. Texas bans all but lottery betting and horse racing, while its neighbors, Oklahoma and Louisiana, allow casino betting; Texans flock to those casinos.

Types of Gambling

Gambling Statistics By Race In America

Readers might be wondering about the types of gambling that takes place inside the United States. Below is a glossary list of the betting opportunities Americans have, along with a quick description of each form of gambling. We provide revenue data for each type below.

• Commercial Casinos: Owned by private companies and publicly-traded companies alike. Commercial casinos can be land-based casinos, riverboat casinos, airport casinos, racetrack-casinos (racinos), or casino cruises. Many have Class III or Las Vegs-style slot machines, while others use Class II Video Lottery Terminals (VLTs) or Video Gambling Terminals (VGTs).
• Tribal Casinos: Owned by Native American tribal gaming authorities and based on Indian reservation lands. A landmark 1986 US Supreme Court case (Cabazon v. California) stated Native American reservations are sovereign lands (and thus able to house casinos) if they were recognized by the US Department of the Interior’s Indian Affairs Bureau by 1934 or before. The Cabazon case led to the US Congress passing the 1988 Indian Gaming Regulatory Act, which says Indian tribes which reach a gaming compact with the states they are inside can have casino gambling with Class II slot machines. States can tax tribal casinos for the cost of regulation but can tax more, if the state gives special concessions like statewide monopolies on certain gaming types.
• Card Rooms: Card rooms can exist inside or outside a land-based casino. Many Las Vegas Strip casinos and Atlantic City have their own poker rooms. In certain states, commercial card rooms or poker clubs exist. The Bicycle Club, Hollywood Park Casino, and Cameo Club in Los Angeles are a few examples of private card rooms. Tribal casinos like WinStar World Casino, Choctaw Casino, Mohegan Sun, and Foxwoods have poker rooms.
• Charitable Games: Charitable gaming takes many forms. In most places, charitable gaming organizations host bingo nights, but other forms of gambling are allowed. Not-for-profit charitable gaming organizations sometimes host Las Vegas Nights (Millionaire Clubs), raffles, bell jar competitions, pickle card contests, and pull-tab contests. Pickle cards and pull-tab tickets are similar in many ways to a lottery scratch-card. Bell Jar gaming is a kind of raffle.
• Bingo: Bingo is the most popular and widespread form of charitable gaming. Church groups, veterans’ groups, police and firefighter organizations, medical research groups, and civic organizations all host bingo nights. Organized bingo nights might be found in a dedicated bingo hall, a community center, or a VFW outpost; online bingo has become widely popular as well.
• Lotteries: State lotteries have existed since the early days of the United States. During the Revolutionary War, the Continental Congress funded the war partly with lottery betting. In the latter half of the 20th century, many US states enacted state lotteries to produce public school funding and scholarship funding. Since the 1980s, the multistate lottery associations, Powerball and Mega Millions, have grown to include 44 US states apiece. State lotteries also sell scratch-cards, the best revenue producer.
• Sportsbooks: Legal sportsbooks are found in land-based casinos in Las Vegas. For the past 25 years, sports lotteries are legal in Delaware, Oregon, and Montana. After the US Supreme Court struck down the PASPA federal ban on sports betting in the 46 other US states, any US state can legalize sportsbooks. Delaware and New Jersey plan to open sportsbooks in the coming weeks, while 5 other US states have sportsbook legalization bills in committee. Lawmakers in 13 other US states are discussing the legalization of sports betting.
• Horse Racing: Horse racing has been legal in many US states for generations because betting on horses is considered a less dangerous form of gambling — and one wealthier Americans enjoy. Horse racing, harness racing, and greyhound racing involve pari-mutuel wagering, in which one bettor’s win means other bettors lose. The horse bettor competes against other bettors and not the racebook, though the bookmaker sets the odds based on betting volume for each horse. Off-track betting facilities now exist, with simulcast horse races and betting on historical horse races. Because the horse betting industry has struggled, many states now allow slot machine gambling at horse tracks.
• Online Casinos: Online casinos, poker sites, and sportsbooks were a huge industry in the USA until 2006 when the US Congress passed the Unlawful Internet Gambling Enforcement Act. The UIGEA banned all forms of Internet gambling which were banned for interstate telephone lines under the 1961 Wire Act. From 2007 to 2011, this meant casino sites, online cardrooms, and bookmaker sites. A 2011 US Department of Justice opinion reversed the DOJ’s stance on online casinos and poker sites, so four US states now have legal online casinos and poker sites: New Jersey, Delaware, Nevada, and Pennsylvania. New York, California, and others might follow suit eventually. It is estimated that between 65% to 85% of casino revenue in the US comes from real money slots.
• Smartphone Betting: In most parts of the United States, mobile casino and poker sites are banned or unregulated. Now that sportsbooks are legal, live/in-play smartphone betting apps could become legal. Tom King of the Readyfire smartphone trivia apps says legal US sports betting will transform Android and iOS gaming apps in America.

Other Types of Wagering: Other forms of gambling takes place in the United States. 50 million Americans bet on Super Bowl office pools and March Madness brackets each year. Though it’s illegal, so many people engage in office betting that authorities look the other way in most cases. Fantasy football and fantasy baseball leagues involve small season-long wagers. In over a dozen US states, fantasy sports have been legalized, but in many other states, fantasy sports betting is allowed to happen. Some argue that daily fantasy sports gaming sites like FanDuel and DraftKings are sports wagering, but they exist in a gray area. Over a dozen US states have legalized DFS sites. Jai Alai is a legal form of sports betting in Connecticut and Florida.

Statistical inference might be thought of as gambling theory applied to the world around us. The myriad applications for logarithmic information measures tell us precisely how to take the best guess in the face of partial information.^[1] In that sense, information theory might be considered a formal expression of the theory of gambling. It is no surprise, therefore, that information theory has applications to games of chance.^[2]

Kelly Betting[edit]

Kelly betting or proportional betting is an application of information theory to investing and gambling. Its discoverer was John Larry Kelly, Jr.

Part of Kelly's insight was to have the gambler maximize the expectation of the logarithm of his capital, rather than the expected profit from each bet. This is important, since in the latter case, one would be led to gamble all he had when presented with a favorable bet, and if he lost, would have no capital with which to place subsequent bets. Kelly realized that it was the logarithm of the gambler's capital which is additive in sequential bets, and 'to which the law of large numbers applies.'

Side information[edit]

A bit is the amount of entropy in a bettable event with two possible outcomes and even odds. Obviously we could double our money if we knew beforehand for certain what the outcome of that event would be. Kelly's insight was that no matter how complicated the betting scenario is, we can use an optimum betting strategy, called the Kelly criterion, to make our money grow exponentially with whatever side information we are able to obtain. The value of this 'illicit' side information is measured as mutual information relative to the outcome of the betable event:

where Y is the side information, X is the outcome of the betable event, and I is the state of the bookmaker's knowledge. This is the average Kullback–Leibler divergence, or information gain, of the a posteriori probability distribution of X given the value of Y relative to the a priori distribution, or stated odds, on X. Notice that the expectation is taken over Y rather than X: we need to evaluate how accurate, in the long term, our side information Y is before we start betting real money on X. This is a straightforward application of Bayesian inference. Note that the side information Y might affect not just our knowledge of the event X but also the event itself. For example, Y might be a horse that had too many oats or not enough water. The same mathematics applies in this case, because from the bookmaker's point of view, the occasional race fixing is already taken into account when he makes his odds.

The nature of side information is extremely finicky. We have already seen that it can affect the actual event as well as our knowledge of the outcome. Suppose we have an informer, who tells us that a certain horse is going to win. We certainly do not want to bet all our money on that horse just upon a rumor: that informer may be betting on another horse, and may be spreading rumors just so he can get better odds himself. Instead, as we have indicated, we need to evaluate our side information in the long term to see how it correlates with the outcomes of the races. This way we can determine exactly how reliable our informer is, and place our bets precisely to maximize the expected logarithm of our capital according to the Kelly criterion. Even if our informer is lying to us, we can still profit from his lies if we can find some reverse correlation between his tips and the actual race results.

Doubling rate[edit]

Doubling rate in gambling on a horse race is ^[3]

${\displaystyle W(b,p)=\mathbb{E}[{\log }_{2}S(X)]=\sum _{i=1}^m{p}_i{\log }_2{b}_i{o}_i}$

where there are ${\displaystyle m}$ horses, the probability of the ${\displaystyle i}$th horse winning being ${\displaystyle p_i}$, the proportion of wealth bet on the horse being ${\displaystyle b_i}$, and the odds (payoff) being ${\displaystyle o_i}$ (e.g., ${\displaystyle o_i=2}$ if the ${\displaystyle i}$th horse winning pays double the amount bet). This quantity is maximized by proportional (Kelly) gambling:

${\displaystyle b=p}$

for which

${\displaystyle \underset{b}{max}W(b,p)=\sum _i{p}_i{\log }_2{o}_i-H(p)}$

where ${\displaystyle H(p)}$ is information entropy.

Expected gains[edit]

An important but simple relation exists between the amount of side information a gambler obtains and the expected exponential growth of his capital (Kelly):

${\displaystyle \mathbb{E}\log K_t=\log K_0+\sum _{i=1}^t{H}_i}$

for an optimal betting strategy, where ${\displaystyle K_0}$ is the initial capital, ${\displaystyle K_t}$ is the capital after the tth bet, and ${\displaystyle H_i}$ is the amount of side information obtained concerning the ith bet (in particular, the mutual information relative to the outcome of each betable event). This equation applies in the absence of any transaction costs or minimum bets. When these constraints apply (as they invariably do in real life), another important gambling concept comes into play: the gambler (or unscrupulous investor) must face a certain probability of ultimate ruin, which is known as the gambler's ruin scenario. Note that even food, clothing, and shelter can be considered fixed transaction costs and thus contribute to the gambler's probability of ultimate ruin.

This equation was the first application of Shannon's theory of information outside its prevailing paradigm of data communications (Pierce).

Applications for self-information[edit]

The logarithmic probability measure self-information or surprisal,^[4] whose average is information entropy/uncertainty and whose average difference is KL-divergence, has applications to odds-analysis all by itself. Its two primary strengths are that surprisals: (i) reduce minuscule probabilities to numbers of manageable size, and (ii) add whenever probabilities multiply.

For example, one might say that 'the number of states equals two to the number of bits' i.e. #states = 2^#bits. Here the quantity that's measured in bits is the logarithmic information measure mentioned above. Hence there are N bits of surprisal in landing all heads on one's first toss of N coins.

The additive nature of surprisals, and one's ability to get a feel for their meaning with a handful of coins, can help one put improbable events (like winning the lottery, or having an accident) into context. For example if one out of 17 million tickets is a winner, then the surprisal of winning from a single random selection is about 24 bits. Tossing 24 coins a few times might give you a feel for the surprisal of getting all heads on the first try.

The additive nature of this measure also comes in handy when weighing alternatives. For example, imagine that the surprisal of harm from a vaccination is 20 bits. If the surprisal of catching a disease without it is 16 bits, but the surprisal of harm from the disease if you catch it is 2 bits, then the surprisal of harm from NOT getting the vaccination is only 16+2=18 bits. Whether or not you decide to get the vaccination (e.g. the monetary cost of paying for it is not included in this discussion), you can in that way at least take responsibility for a decision informed to the fact that not getting the vaccination involves more than one bit of additional risk.

More generally, one can relate probability p to bits of surprisal sbits as probability = 1/2^sbits. As suggested above, this is mainly useful with small probabilities. However, Jaynes pointed out that with true-false assertions one can also define bits of evidence ebits as the surprisal against minus the surprisal for. This evidence in bits relates simply to the odds ratio = p/(1-p) = 2^ebits, and has advantages similar to those of self-information itself.

Applications in games of chance[edit]

Information theory can be thought of as a way of quantifying information so as to make the best decision in the face of imperfect information. That is, how to make the best decision using only the information you have available. The point of betting is to rationally assess all relevant variables of an uncertain game/race/match, then compare them to the bookmaker's assessments, which usually comes in the form of odds or spreads and place the proper bet if the assessments differ sufficiently.^[5] The area of gambling where this has the most use is sports betting. Sports handicapping lends itself to information theory extremely well because of the availability of statistics. For many years noted economists have tested different mathematical theories using sports as their laboratory, with vastly differing results.

One theory regarding sports betting is that it is a random walk. Random walk is a scenario where new information, prices and returns will fluctuate by chance, this is part of the efficient market hypothesis. The underlying belief of the efficient market hypothesis is that the market will always make adjustments for any new information. Therefore no one can beat the market because they are trading on the same information from which the market adjusted. However, according to Fama,^[6] to have an efficient market three qualities need to be met:

• There are no transaction costs in trading securities
• All available information is costlessly available to all market participants
• All agree on the implications of the current information for the current price and distributions of future prices of each security

Statisticians have shown that it's the third condition which allows for information theory to be useful in sports handicapping. When everyone doesn't agree on how information will affect the outcome of the event, we get differing opinions.

See also[edit]

References[edit]

1. ^Jaynes, E.T. (1998/2003) Probability Theory: The Logic of Science (Cambridge U. Press, New York).
2. ^Kelly, J. L. (1956). 'A New Interpretation of Information Rate'(PDF). Bell System Technical Journal. 35 (4): 917–926. doi:10.1002/j.1538-7305.1956.tb03809.x.
3. ^Thomas M. Cover, Joy A. Thomas. Elements of information theory, 1st Edition. New York: Wiley-Interscience, 1991. ISBN0-471-06259-6, Chapter 6.
4. ^Tribus, Myron (1961) Thermodynamics and Thermostatics: An Introduction to Energy, Information and States of Matter, with Engineering Applications (D. Van Nostrand Company Inc., 24 West 40 Street, New York 18, New York, U.S.A) ASIN: B000ARSH5S.
5. ^Hansen, Kristen Brinch. (2006) Sports Betting from a Behavioral Finance Point of View (Arhus School of Business).
6. ^Fama, E.F. (1970) 'Efficient Capital Markets: A Review of Theory and Independent Work', Journal of Financial Economics Volume 25, 383-417

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