INTRODUCTION

This thesis examines the arbitration situations that arise in bookmakers.

Arbitration (surewins, surebets, surebets) is a situation that occurs between bookmakers due to the difference in the scores of the same sporting event. In such a situation, it is absolutely not important for the player how this sporting event will end, since the player receives his profit regardless of its outcome. Arbitration situations also arise when playing on the stock exchange and in the stock market. The essence of the arbitration situation: to find such coefficients for a sporting event in which, by betting certain amounts on each possible outcome, the player turns out to be in the black for any outcome of this event. Also in this paper, a betting method called "Kelly Criterion" is considered. The essence of this method is to help gamblers determine the amount of the bet on each individual event overrated by the bookmakers (in the gambler's opinion). That is, to get a theoretical advantage over the bookmaker's line.

The main objectives of this work:

Give a brief concept to bookmakers, show how they work;

Mathematically substantiate the arbitration situation (fork);

Give a short concept of the "Kelly criterion" betting method;

Substantiate this method mathematically;

Develop a program that will calculate the profit of the fork and implement it in the programming language in the Delphi environment.

Develop a program that will calculate the amount of a player's bet using the "Kelly Criterion" method and implement it in the Delphi programming language.

The number of bookmakers is growing, and with it the number of arbitrage situations is growing. Now almost all offices give players the opportunity to bet on sports events via the Internet, that is, without leaving their homes. This diploma will show that anyone with a minimum amount of knowledge about sports betting and possessing a set of some mathematical knowledge can make money.

BOOKMAKING

The history of the emergence of bookmakers

The passion for gambling has long been inherent in humans. All kinds of disputes and bets were concluded in ancient times. But what to say - many ancient myths and legends of different peoples testify that even the gods loved to argue! Gradually, people were found who began to benefit from other people's disputes, taking money bets from the disputants. The history of the emergence of the profession of a professional debater is not known for certain, but according to the preserved historical information, the first of them accepted bets on the Olympic Games in Ancient Greece, gladiator fights in Ancient Rome, tournaments of medieval knights. Today they are called bookmakers - from the English word bookmaker, which means "to do, keep a book." The fact is that each bookmaker had a special notebook where he entered all the data on the accepted rates. A game with bets and predicting a particular outcome of a sports competition or some other event has come to be called a sweepstakes. Traditionally, the first sweepstakes appeared on the racetracks of Ancient Rome. The aristocracy gladly bet on this or that rider. Later, cockfighting appeared, as a result of which sweepstakes for the lower class arose. Soon disputes over money became an integral part of sports and, unfortunately, around sports and entertainment. Ultimately, bookmaking as a profession was formed in England, where the world's first bookmakers were founded. The British have always been known for their love of horses and equestrian sports. In 1766, an auction of thoroughbred horses was organized in London by the groom Richard Tattesol, which was held in Hyde Park in specially designated premises. At the same time, a couple of premises were given to the Jockey Club, whose members were notable horse owners and organizers of horse races. Gathering gentlemen talked, watched the races and at the same time made bets on the best participants. Bookmakers of the 18th century accepted handicap bets, where participants of various classes competed. This kind of racing initially assumed different capabilities of the horses, therefore, in order to equalize the chances, the weak got a head start in time or distance. This situation caused serious controversy and eventually led to the emergence of the so-called "honest bookmakers". They were the British Fred Swindall and Leviathan Davis. Around the 1860s, they circulated “odds sheets” to Londoners where everyone could guess who would win the races. Bets began to be accepted with an assessment of the winning chances of one or another participant. Bookmakers accepted bets, disposed of winnings and received a good commission. Later, the winnings were calculated depending on the amount of the bet and the odds. Soon, in addition to simple ones, complex bets appeared - for example, it was possible to bet not only on the winner in a particular race, but also to try to guess the results of individual participants in the race, i.e. bet on which horse numbers will be in the top three.

Some researchers of the issue believe that the founders of bookmaking in the world were not the British, but the Frenchman Pierre Oller, a merchant who often attended horse races. Once he had an idea to organize a game-argument, in which any of the spectators could put a certain amount on the horse, which, in his opinion, will come first. In the years 1860-70. an enterprising Frenchman opened several ticket offices where it was possible to purchase a ticket with the name of the prospective winner, and a little later counters appeared, with the help of which it was possible to calculate the possible winnings. Oller took part of the amount that came from the proceeds for tickets as a commission, and divided the rest of the money between the spectators who managed to make the correct bet and guess the winner. Almost immediately, the state introduced a tax on tote.

Regardless of where exactly the founders of bookmaking appeared in the world, starting from 1880, sweepstakes spread faster than the epidemic of an infectious disease, covering all of Europe, reaching Russia and even America. The first American firm to accept bets on the winner of the horse and dog races opened in Philadelphia in 1866. Unfortunately, history did not retain its name, but later the successful undertaking spread to other states. The heyday of bookmaking came with the advent of team sports games - football, basketball, baseball, hockey. However, in a short time, sweepstakes were outlawed in most states of America, which led to the emergence of numerous underground establishments and the transition of bookmakers to an illegal position. Today, bookmaking in America is officially allowed only in some states - Nevada, centered in the famous Las Vegas and New Jersey, where Atlantic City is located. Despite the fact that only two states of the country can officially accept bets, the income from legal bookmaking in the United States is more than $ 240 billion annually. The UK is still the country with the largest number of officially permitted sweepstakes and bookmakers. The British quickly realized that betting could be a source of good income. Gradually, betting on dog races, tennis, football, cricket, boxing and other sports matches and fights arose. In addition, bets appeared, the subject of controversy in which socio-political and cultural events, weather and various incredible assumptions, for example, the new coming of Christ or the end of the world, became. The ratio of such a rate today is 1: 1,000,000,000. The popularity of bookmakers is enormous, which makes it possible to make good money on the excitement of the debaters. Today in England there are more than 4.5 thousand points where you can place bets. William Hill is rightfully considered the oldest bookmaker - these bookmakers began their activity back in 1934. The office accepts bets on the outcome of sports and is considered one of the most respected and reliable. Another leading bookmaker in England is Ladbrokes. She owns about 1900 locations in different parts of the country. Each year the turnover of each bookmaker in England is about 14 billion pounds, in addition, each of them pays to the treasury up to 15% of the total income. The absolute record among sweepstakes winnings was about £ 3 million, which was paid in 1994 in Littlewood, England. The lucky man managed to correctly guess the results of about 10 football matches. In England, there are also funny bets that can already be considered traditional, for example, guessing the weather for Christmas, when half bets on rain and half on snow. Among the members of the royal family, it became popular to bet on the length of the royal Christmas dinner at Sadringham Castle. In 2000, the Betfair sports betting exchange was opened in the UK by two Englishmen. Her activities have changed the concept of bookmaking as such. The betting exchange gives players the opportunity to bet on the result not with a bookmaker, but with other disputants, on their own terms and with their own odds. The office has more than 2 million clients, and its founders received an award from the Queen herself for their bold innovation. In the Soviet Union, gambling was condemned and considered a sign of moral decay, so it was officially possible to gamble for money only at the hippodrome. However, the winnings were scanty and could not satisfy the real racing fans. Therefore, underground bookmaking - illegal sweepstakes - was widespread, although a criminal offense. It was a cruel and dangerous business, where the loser could pay not only with his property, but also with his life. The first permitted bookmakers on the territory of the former USSR appeared in Moscow in 1991, after which the sweepstakes began to gain momentum with confidence.

What are forks?

Forks in sports betting is a unique strategy that provides guaranteed profits for the person placing the bet, regardless of the outcome of the sporting event. This income is achieved due to the difference in the coefficients between the two markets. In other words, in one bookmaker's office the player makes a 1X bet with a certain odds, and in another P2, while the odds are set in such a way that the client remains in profit for any outcome.

Forks are also called arbitrage or arbitration situations. Such a strategy is considered a win-win, since the player should remain in the black at any outcome of a sporting event, however, the strategy cannot be called absolutely safe and win-win, because there are pitfalls here.

Bookmaker offices	Game features
Fonbet	Can you make surebets in Fonbet? Of course, yes, although here it is worth playing carefully so as not to attract the attention of the company's security service. It is advisable not to maintain the balance at the same level here, not to place several bets on one event during the day. In this case, surebets in Fonbet will be a great way to make money for you.
League Betting	The Betting League is one of those bookmakers that do not block arbers. Perhaps the account will be cut with the wrong approach to the game, but the account will remain with you, so there is no need to worry about the safety of the money.
1xBet	The office has a low margin and a large number of sports events for betting. This is one of the best betting shops for arbers, as there are many opportunities to find suitable bets on which to earn
Leon	Many gamblers complain about the following - in Lyon there is not such an extensive line as they would like to see, however, the office is loyal to the arbitrage game and is the best office for arbers.
1xBet	Using special online calculators of surebets scanner in bookmaker, you can place bets in 1xBet. However, be careful to keep your account safe and sound without cutting, otherwise it will be possible to bet limited amounts on all sporting events.
Betfair	There are foreign surebet scanners that work with a foreign bookmaker Betfair. Many of these resources are paid, so you can get your money back at the office.

Now you know which bookmakers do not block surebets, although we strongly recommend playing only at legal bookmakers. Companies operating illegally can deceive the client at any time by blocking his account. Sometimes you don't even need a pretext for this, because there is always a formal reason - “suspicion of dishonest play on the part of the client”.

Forks in bookmakers are often cited as the only way to make money on betting, although this is also not entirely true. Professional cappers are able to predict matches with high quality and win on value bets without resorting to arbitration situations. For those who are not yet versed in a particular sport, there are special programs for determining the betting surebets.

You do not need to search for suitable matches on your own, you just need to turn on the program that will find arbitrations in the next fights, after which you just need to place a bet and get your profit.

Not all offices allow their clients to play according to the surebet strategy.

Features of the strategy

It is necessary to understand that the surebet strategy in bookmaker bets is not suitable for every gambler. Here are some features of the strategy that must be taken into account before trying it on your own gaming account:

You must be prepared for the fact that arbitrage situations will quickly lead to a cut in the bookmaker's account. Bookmakers do not like arbers, because they consider their game to be dishonest in relation to the company. Therefore, your account may be "cut", which means that you will not be able to wager large sums on certain sporting events.
The surebet game in bookmaker office presupposes the presence of a large game bank. To stay in a good profit, sometimes you need to make two big bets on opposite events. In no case should you borrow money for a strategy game, because later we will see that it can turn out to be a losing one for you.
There are bookmakers loyal to surebets, but this does not mean at all that you can safely play in them using this strategy. Often these companies do their best to keep arbitration situations as small as possible.
Thanks to special programs and the development of technology, nowadays it has become more and more difficult to find betting surebets online. If such situations arise, the bookmakers quickly react to them, correcting their mistake.
You can search for surebets in bookmakers yourself, but it will take a lot of time and effort, which is unacceptable for most people who are busy with other things.

You can really make money with this strategy, although not everything is as simple as it might seem at first glance. Some players once got burned at arbitration, and now they no longer even want to think about this strategy, bypassing it.

The strategy has its own characteristics, which should be known to everyone who plays on surebets.

Types of forks

Arb strategies can be used both in prematch and in live bets. Here are some examples of how arbitrage can form:

The simplest surebet, consisting of only two bets. The victory of 1 team - a bet in one bookmaker, another bet at another bookmaker - on 2X. Also, sometimes you can catch profitable odds for the total more and the total less.
More complex option. 1 bet - Handicap 1 (-0.5), 2 bet - a draw, 3 bet - Handicap 2 (-0.5), etc.
In some cases, the outcomes may overlap, and therefore sometimes there may be a double win at once or a return of one of the bets. For example, 1 - Handicap 1 (0), 2 - draw, 3 - Win of the 2nd team.
In tennis, you can often catch a fork, consisting of 4-6 outcomes. This is due to the fact that here you can place bets on the exact score for the sets with good odds.

At the same time, all this knowledge may be useless for you if you do not improve your skills in this area. It so happens that a player cannot place bets on all arbitrage options, since the bookmaker limits the size of the bet or refuses to accept it at all outside the express. In addition, the odds in the bookmaker also change, so you should do all the work as quickly as possible, clearly realizing what you are doing. That is why the surebets play is called a strategy for professionals.

Advantages and disadvantages

The main advantages of arbitration situations are the following:

You can develop a win-win strategy for betting on surebets. With any outcome of a sporting event, you receive financial profit.
There is no need to understand the basics of betting or watch a specific sport. You just need to search for surebets without delving into other features of betting.
You can find surebets for free. Today there are special services that allow you to detect arbitration situations, and if you wish, you can independently find suitable matches in the offices.
The profit of a surebet can be from 1 to 10% of the turnover, which is very good for large amounts of bets.
You can understand well the work of bookmakers and the general principles of the game in betting, which in the future will make it possible to make the most accurate predictions for sports.

The strategy is considered a win-win, although it can lead to a minus under certain circumstances

However, the strategy of surebets in betting can turn out to be unprofitable for the player under certain circumstances. Therefore, some of its disadvantages should be mentioned:

Fast cutting of the account, which means limiting the size of the bet on sports events. Special bookmaker services keep track of which strategy their clients play, making it easy for them to find arbers. Do not be surprised if, after several bets on arbitrage situations, you can put a maximum of 10-20 rubles on your next forecast.
Some programs for surebets in bookmakers are paid, so you will have to risk a lot to recoup the money spent on the strategy.
Sometimes bookmakers act much faster than players. When the program shows you the existing arbitration, the company is already fixing its mistake by overriding the fork. Thus, you can no longer benefit from it.
Bookmakers who are already aware of the presence of a surebet in their line may simply not accept a player's bet on one of the events. This is fraught with the fact that you will lose all the money bet on the opposite outcome if the event does not work. The company can limit the size of the bet on the opposite event, because of which the strategy will turn out to be a losing one.

Why do arbitration situations arise?

The reasons for arbs are quite simple and straightforward:

Competition between bookmaker companies, as a result of which each office tries to set too high odds for this or that event, thereby attracting the attention of customers. At the same time, for this reason, arbitrations that are beneficial to players can be formed.
Errors of bookmaker analysts, which lead to incorrect calculations and the setting of incorrect coefficients.
Intentional formation of an arbitration situation in order to "catch" a group of arbers.

There are special sites for surebets that allow you to identify such situations in a timely manner, track them in order to be able to make profitable bets in different offices.

Before playing this strategy, you need to carefully study all the information about surebets.

Do I need to become a surebet?

As you might have guessed, arbers are people who play according to strategy in arbitrage situations. There are many articles on how good it is to be a surebet, how it makes huge profits, how you can become rich in no time, etc. In reality, everything is not as simple as it might seem to a beginner who does not understand betting.

First of all, you will need to find suitable bookmakers for surebets, where your bets will be treated loyally. This is not easy to do, since almost all companies suppress such activities of players, considering it dishonest and illegal.

Create a separate game bank for bets on arbitration situations.
Open accounts in many bookmakers in order to be able to quickly place a bet in any company.
It is advisable to open accounts for loved ones who live with you in order to be able to play in bookmaker office after blocking your account or cutting the account.
Use special programs to identify arbitrations.

Now you know who the bettors are in bookmakers and how they conduct their business.

It remains to answer the question of whether it is worth playing with this strategy. The fact is that surebets in a bookmaker receive an insignificant percentage of the turnover, so they have to make rather large bets to stay in the black. If you have such an opportunity, then you can try this strategy. At the same time, be prepared for the fact that you may lose part or all of the capital due to your own inattention, bookmaker rules, etc.

Everyone must decide for himself whether he needs to become a surebet

How do bookmakers calculate surebets?

Bookmaker companies may not like the fact that one client or group of players takes a decent monthly profit from the office, which is why their attitude towards surebets is, to put it mildly, not very good.

How do bookmakers calculate surebets? This is a rather complex issue that requires more detailed consideration. There are several ways to recognize these players:

Bet amounts. This is one of the main answers to the question of how arbers are calculated in bookmakers. If a player regularly bets to the maximum, then something is wrong with him. For example, he bets all-in on some match of an unknown league, loses all the money, but after a few days or weeks he replenishes the account and makes another all-in bet. The player's profit directly depends on the turnover of funds, therefore, the amounts are usually large, which attracts the attention of the office's security service.
Another way that bookmakers calculate surebets is by observing players who place several separate bets on the same outcome. For example, a player places a bet on the England Second League match Newport - Port Vale 1X in the morning, and then in the evening repeats the bet on the same match, only at the changed odds. This is due to the fact that the odds can fluctuate throughout the day, and arbitrators try to benefit from this for themselves.
Search for surebets from bookmakers. If you think that only gamblers use special scanners for sports betting, then you are wrong. Bookmakers have much more opportunities to find an arbitration situation in a timely manner and correct it if necessary. Even if the bookmaker's search service for surebets did not work, then there are some other ways to determine arbitrage.
Maintaining the balance on the account at approximately the same level. If an ordinary player gradually increases his bank or loses it, then the movement of funds on the account is very rare. The bettor strives to balance his balance in several offices, so he sometimes replenishes the account for no apparent reason, even if there is money left on it, he also withdraws funds, leaving the same amount of money on the balance.

Now you understand how bookmakers catch arbers. This will help you avoid the unwanted fate of arbers and make profit from bets on arbitrage situations as long as possible. On various forums, you can also find a lot of useful information on how bookmakers calculate surebets, and what this means for players.

There are different tactics for playing with surebets, the main thing here is to avoid attention from the bookmaker's office

What fate befell the arbers?

What do arb sports betting lead to? Many people strongly praise such a strategy, but forget that bookmakers take certain measures in relation to the people who play on it. The bettors at bookmakers should be prepared for the following development of events:

Cutting the account to 0 rubles / dollars / hryvnia, etc. Thus, you will no longer be able to place bets on your account. If, in exceptional cases, the office can make concessions to the client and renew the maximum bet size for him, then in case of playing with surebets, the company categorically prohibits him from making further bets.
Blocking an account with the inability to withdraw own funds. This is done only by illegal offices that can easily block an account and confiscate all the player's money. That is why you need to look for bookmakers loyal to arbers who can cut the account, but not steal money from the account.
When a second account is opened from the player's side, it will also be cut to the maximum. Sometimes even opening accounts for relatives does not help, since the office closely monitors the activities of arbers.
Due to certain restrictions of bookmakers, betting on surebets on sports online can lead to partial or complete loss of play capital.
Under good circumstances, you can earn money on arbitration situations, which is what some professionals do. However, for this you need to know all the features of such a strategy in order to avoid close attention from bookmakers for as long as possible.

You can both earn and lose money on surebets.

How much can you earn from this?

How much can you really earn on surebets with Russian bookmakers? The question is broad, since for each it can be a different amount of money, depending on the initial capital and the amount of bets on each sporting event.

Some people manage to make good profit every month through arbitrage situations, without attracting the attention of bookmakers. This requires certain knowledge, skills and a bit of luck, since there are many players whose accounts have long been blocked due to unfair arbitrage play.

Some professional gamblers use surebets on sweepstakes to make money, but this strategy is considered too risky and not entirely understandable for beginners. The bettor can earn from 1 thousand rubles to several hundred thousand if he uses the strategy correctly, and in practice applies all the recommendations mentioned above.

conclusions

Arbitrage situations guarantee a profit for the player, but bookmakers can always surprise you by calculating the bet with odds of 1 or limiting its maximum size, so surebets cannot be called a 100% win-win strategy. At the same time, some offices allow you to play according to this strategy, if you do not violate other company rules and do not take out large sums of money from there every day.

There are free and paid arb scanners. Recently, free services have given very little benefit, since they offer arbitrage with a small percentage of guaranteed profit, and even bets on these events can be made only in unfamiliar and unreliable bookmakers.

There are paid and free services for finding surebets in betting shops

Certain paid resources are more useful, although here, too, you should be careful not to fall into the hands of the office's security service, because this will entail cutting the account at best. You can make money on surebets, but you need to do it professionally, never getting excited or greedy in situations where no one needs it.

(4 estimates, average: 3,00 out of 5)

FEDERAL STATE BUDGETARY EDUCATIONAL INSTITUTION OF HIGHER PROFESSIONAL EDUCATION

RUSSIAN STATE SOCIAL UNIVERSITY

Faculty of Information Technology

Department of Applied Mathematics

Fatin Sergey Vladislavovich

Arbitration situations in bookmakers

Bachelor's final qualifying work

Direction 010500.62 - Applied Mathematics and Informatics

Degree - Bachelor of Applied Mathematics and Computer Science

Head Associate Professor

Associate Professor Teveleva E.A.

______________________

Reviewer Ph.D.

_____________________

Admit to protection:

Head of the Department

applied mathematics

Associate Professor, Ph.D.

_______ / Komarova E.V. /

Moscow 2014

INTRODUCTION 3

CHAPTER I. BOOKMAKING 5

§1.1. History of the emergence of bookmakers 5

§ 1.2. Prospects for the development of bookmakers 9

§ 1.3. Bookmaker's line and main types of bets 11

§ 1.4. Profit margin. How bookmakers make money 13

§ 1.5. Basic rules of the game in a bookmaker's office 14

CHAPTER II. ARBITRATION SITUATIONS 21

§ 2.1. Real-life example of an arbitration situation 22

§ 2.2. Mathematical justification of the arbitration situation 24

§ 2.3. Calculation of Probabilities of Outcomes and Payout Ratios 25

§ 2.4. Condition of Arbitration Situation 27

CHAPTER III. KELLY CRITERION METHOD 30

§ 3.1. The essence of the "Kelly criterion" method 30

§ 3.2. Description of the Kelly Criterion Method and Its Properties 31

§ 3.3. An example of using the properties of the "Kelly criterion". Generalizing Kelly's Formula 35

CHAPTER IV. PROGRAM IMPLEMENTATION 37

§ 4.1. An example of using the program to calculate surebets 37

§ 4.2. Listing of the program for calculating surebets 40

§ 4.3. An example of using the program for calculating the bet amount using the "Kelly criterion" method 44

§ 4.4. Listing of the program for calculating the bet amount using the "Kelly criterion" method 47

CONCLUSION 50

REFERENCES 51

INTRODUCTION

This thesis examines the arbitration situations that arise in bookmakers. Arbitration (surewins, surebets, forks)- a situation that arises between bookmakers due to the difference in the ratings of the same sporting event. In such a situation, it is absolutely not important for the player how this sporting event will end, since the player receives his profit regardless of its outcome. Arbitration situations also arise when playing on the stock exchange and in the stock market. The essence of the arbitration situation: to find such coefficients for a sporting event in which, by betting certain amounts on each possible outcome, the player turns out to be in the black for any outcome of this event. Also in this paper, a betting method called "Kelly Criterion" is considered. The essence of this method is to help gamblers determine the amount of the bet on each individual event overrated by the bookmakers (in the gambler's opinion). That is, to get a theoretical advantage over the bookmaker's line.

The main objectives of this work:

Give a brief concept to bookmakers, show how they work;

Mathematically substantiate the arbitration situation (fork);

Give a short concept of the "Kelly criterion" betting method;

Substantiate this method mathematically;

Develop a program that will calculate the profit of the fork and implement it in the programming language in the Delphi environment.

Develop a program that will calculate the amount of a player's bet using the "Kelly Criterion" method and implement it in the Delphi programming language.

With regard to the practical application of these programs, at present, the interest in sports events is constantly increasing, and this determines and supports the development of bookmakers offering sports betting. The number of bookmakers is growing, and with it the number of arbitrage situations is growing. Now almost all offices give players the opportunity to bet on sports events via the Internet, that is, without leaving their homes. This diploma will show that anyone with a minimum amount of knowledge about sports betting and possessing a set of some mathematical knowledge can make money.

Arbitration situations between bookmakers - “Forks”. The profitability of your investment is GUARANTEED to be - 0.20% per day.
Arbitration situations between bookmakers - “Forks”. The profitability of your investment is GUARANTEED to be - 0.20% per day.

Current state

As of 03.03.2016, Ironbetting Company is a leader in its online business segment.
During the operation of the project (48 days), 262 clients (Investors) were attracted, which invested - 71,025.83 USD, and the company paid dividends during this time in the amount of 9,022.80 USD. USA, which averages - 0.26% per day.

Market

The gambling industry showed a 10% increase in income in 2015 compared to 2014, which makes it more attractive for investment. Not many other sectors of the economy showed similar growth rates in 2015. Taking into account the hundred-year dynamics of the gambling business, we can look to the future with confidence. The undisputed leader of online betting (sports betting) and bookmakers show positive dynamics of development, both in the world and in the post-Soviet countries. All these factors are the driving force behind our business. The higher the dynamics of opening new bookmakers, the greater the likelihood of arbitrage situations that arise when the odds are formed and, accordingly, the higher our profit and dividends of our Investors.

Problem or Opportunity

The product is created, tested and does not need any modifications.

Competitors

We will provide the main competitors of the sports arbitration market:
1.http: //sportarbs.com/
2.https: //smartbet.biz/
3.http: //www.arbitragetraining.com/

Benefits or Differentiators

The main advantage of Ironbetting Company over its competitors is undoubtedly many years of experience and a time-tested solid team of professionals. More than 5 years in the arbitration market, as well as developed, tested and implemented its own technical platform for finding arbitration situations between bookmakers. Our platform scans the lines of more than 150 bookmakers. All this gives us the opportunity to be leaders in this highly liquid business.

Federal State Budgetary Educational Institution

higher professional education

Faculty of Information Technology

Department of Applied Mathematics

Bachelor's final qualifying work

Arbitration situations in bookmakers

Fatin Sergey Vladislavovich

Direction 010500.62 -

Applied math

and informatics

Degree - Bachelor

Supervisor:

Associate Professor Teveleva E.A.

Moscow 2014

bookmaker arbitrage bet payout

Introduction

Chapter I. Bookmaking

1.1 History of the emergence of bookmakers

1.2 Prospects for the development of bookmakers

2.1 A real example of an arbitration situation

2.2 Mathematical justification of the arbitration situation

2.3 Calculation of Probabilities of Outcomes and Payout Ratios

2.4 Condition of the arbitration situation

3.1. The essence of the "Kelly criterion" method

3.2. Description of the "Kelly criterion" method and its properties

3.3. An example of using the "Kelly criterion" properties. Generalizing Kelly's formula

4.1 An example of using the program for calculating betting "surebets"

3 An example of using the program to calculate the bet amount using the "Kelly criterion" method

Conclusion

Introduction

This thesis examines the arbitration situations that arise in bookmakers. Arbitration (surewins, surebets, surebets) is a situation that occurs between bookmakers due to the difference in the scores of the same sporting event. In such a situation, it is absolutely not important for the player how this sporting event will end, since the player receives his profit regardless of its outcome. Arbitration situations also arise when playing on the stock exchange and in the stock market. The essence of the arbitration situation: to find such coefficients for a sporting event in which, by betting certain amounts on each possible outcome, the player turns out to be in the black for any outcome of this event. Also in this paper, a betting method called "Kelly Criterion" is considered. The essence of this method is to help gamblers determine the amount of the bet on each individual event overrated by the bookmakers (in the gambler's opinion). That is, to get a theoretical advantage over the bookmaker's line.

The main objectives of this work:

give a brief concept to bookmakers, show how they work;

to mathematically substantiate the arbitration situation (fork);

give a brief concept of the "Kelly criterion" betting method;

justify this method mathematically;

develop a program that will calculate the profit of the fork, and implement it in the programming language in the Delphi environment.

develop a program that will calculate the amount of a player's bet using the "Kelly Criterion" method, and implement it in the Delphi programming language.

Chapter I. Bookmaking

1 The history of the emergence of bookmakers

The passion for gambling has long been inherent in humans. All kinds of disputes and bets were concluded in ancient times. But what to say - many ancient myths and legends of different peoples testify that even the gods loved to argue! Gradually, people were found who began to benefit from other people's disputes, taking money bets from the disputants. The history of the emergence of the profession of a professional debater is not known for certain, but according to the preserved historical information, the first of them accepted bets on the Olympic Games in Ancient Greece, gladiator fights in Ancient Rome, tournaments of medieval knights. Today they are called bookmakers - from the English word bookmaker, which means "to do, to keep a book." The fact is that each bookmaker had a special notebook where he entered all the data on the accepted rates. A game with bets and predicting a particular outcome of a sports competition or some other event has come to be called a sweepstakes. Traditionally, the first sweepstakes appeared on the racetracks of Ancient Rome. The aristocracy gladly bet on this or that rider. Later, cockfighting appeared, as a result of which sweepstakes for the lower class arose. Soon disputes over money became an integral part of sports and, unfortunately, around sports and entertainment. Ultimately, bookmaking as a profession was formed in England, where the world's first bookmakers were founded. The British have always been known for their love of horses and equestrian sports. In 1766, an auction of thoroughbred horses was organized in London by the groom Richard Tattesol, which was held in Hyde Park in specially designated premises. At the same time, a couple of premises were given to the Jockey Club, whose members were notable horse owners and organizers of horse races. Gathering gentlemen talked, watched the races and at the same time made bets on the best participants. Bookmakers of the 18th century accepted handicap bets, where participants of various classes competed. This kind of racing initially assumed different capabilities of the horses, therefore, in order to equalize the chances, the weak got a head start in time or distance. This situation caused serious controversy and eventually led to the emergence of the so-called "honest bookmakers". They were the British Fred Swindall and Leviathan Davis. Around the 1860s, they circulated "odds sheets" to Londoners where everyone could guess who would win the races. Bets began to be accepted with an assessment of the winning chances of one or another participant. Bookmakers accepted bets, disposed of winnings and received a good commission. Later, the winnings were calculated depending on the amount of the bet and the odds. Soon, in addition to simple ones, complex bets appeared - for example, it was possible to bet not only on the winner in a particular race, but also to try to guess the results of individual participants in the race, i.e. bet on which horse numbers will be in the top three.

Today, bookmaking in America is officially allowed only in some states - Nevada, centered in the famous Las Vegas and New Jersey, where Atlantic City is located. Despite the fact that only two states of the country can officially accept bets, the income from legal bookmaking in the United States is more than $ 240 billion annually. The UK is still the country with the largest number of officially permitted sweepstakes and bookmakers.

The British quickly realized that betting could be a source of good income. Gradually, betting on dog races, tennis, football, cricket, boxing and other sports matches and fights arose. In addition, bets appeared, the subject of controversy in which socio-political and cultural events, weather and various incredible assumptions, for example, the new coming of Christ or the end of the world, became.

The ratio of such a rate today is 1: 1,000,000,000. The popularity of bookmakers is enormous, which makes it possible to make good money on the excitement of the debaters. Today in England there are more than 4.5 thousand points where you can place bets. William Hill is rightfully considered the oldest bookmaker - these bookmakers began their activity back in 1934. The office accepts bets on the outcome of sports and is considered one of the most respected and reliable. Another leading bookmaker in England is Ladbrokes. She owns about 1900 locations in different parts of the country. Each year the turnover of each bookmaker in England is about 14 billion pounds, in addition, each of them pays to the treasury up to 15% of the total income. The absolute record among sweepstakes winnings was about £ 3 million, which was paid in 1994 in Littlewood, England. The lucky man managed to correctly guess the results of about 10 football matches. In England, there are also funny bets that can already be considered traditional, for example, guessing the weather for Christmas, when half bets on rain and half on snow.

Among the members of the royal family, it became popular to bet on the length of the royal Christmas dinner at Sadringham Castle. In 2000, the Betfair sports betting exchange was opened in the UK by two Englishmen. Her activities have changed the concept of bookmaking as such. The betting exchange gives players the opportunity to bet on the result not with a bookmaker, but with other disputants, on their own terms and with their own odds. The office has more than 2 million clients, and its founders received an award from the Queen herself for their bold innovation.

In the Soviet Union, gambling was condemned and considered a sign of moral decay, so it was officially possible to gamble for money only at the hippodrome. However, the winnings were scanty and could not satisfy the real racing fans. Therefore, underground bookmaking - illegal sweepstakes - was widespread, although a criminal offense. It was a cruel and dangerous business, where the loser could pay not only with his property, but also with his life. The first permitted bookmakers on the territory of the former USSR appeared in Moscow in 1991, after which the sweepstakes began to gain momentum with confidence.

2 Prospects for the development of bookmakers

Now in Russia bookmaker offices are a widespread and respectable business, the number of operating companies is constantly growing, and their number is approaching 100. True, there are only about 10 large companies, they account for about 95% of the market. These companies operate on the Internet and have betting points throughout the country. The rest of the companies huddle on the Internet and try to attract players with the help of all kinds of bonuses. As practice shows, in Russia only 5-10% of players play via the Internet, in contrast to Europe where about 40% of players play via the Internet. If you have a great team of programmers and sports analysts on hand, then opening your own betting company will be the best investment. The market is constantly growing, the annual turnover increases at the initial stages by 20% per year, then by almost 5% annually. Despite the high competition, in my opinion, it is necessary to try to open a bookmaker's office. At the initial stage, you need a few things:

-Reserve capital;

-programmers (or a ready-made program that needs to be bought);

-people who are well versed in sports, sports betting (sports analysts);

-and a team of marketers, PR specialists.

As the practice of Russian bookmakers shows, the company starts working in a plus after 2-3 years of opening. It greatly depends on whether you are opening an office from scratch or buying an existing company and rebranding it. Recently, there has also been the following practice: the merger of two or more small bookmakers into one.

After you open a bookmaker's office, you just have to take the last and most difficult step, find and lure players (clients). As modern practice shows, players are very reluctant to change their bookmaker. Therefore, a staff of qualified PR managers and advertisers is needed. In order for a player to change his favorite bookmaker, you need:

-When registering a player on the company's website, he must receive cash bonuses, which he can receive by making, for example, 20 bets.

-Constantly hold drawings of valuable prizes, and players who have made bets higher than any amount should be allowed to participate in the drawing.

-The most important thing is to provide high-quality service, the best support service among the existing ones.

Western players will never bet in an office where any of the staff spoke rudely to them. In Russia, very few people understand this, so if you do everything correctly (even according to the prototype of a large Western firm), then those players who will start betting with you, and even if you are in some way worse than others (for example, you will have lower odds than competitors), they will remain with you.

A very good move is to make new betting points. Previously, they looked something like this, a small room where the cashier sits and accepts bets from the players. It is necessary to do as follows:

A large room that will combine a restaurant, a sports bar, and a betting point at the same time. One young company put a lot of emphasis on this point, and it was right. Now it literally for 3 years, controls about 10-12% of the gaming market in Russia.

1.3 Bookmaker's line and main types of bets

Bookmakers necessarily use mathematics in their work, and especially its sections: combinatorics, probability theory and statistics. The main job of bookmakers is to solve the main problem - to correctly build a line. A line is a list of sporting events and their possible outcomes, each outcome in which is assigned its own odds. The expression "correctly draw a line" for a bookmaker means that you need to correctly assess the odds for each outcome and offer such odds at which the bookmaker will remain in the black. It is not easy to achieve this balance, it is a difficult task on which hundreds of specialists are working.

The probability of any outcome varies from 0 to 100%, or from 0 to 1. Each event can have several outcomes. For example, in a football match, the first team can win, the second team, or there will be a draw. Each outcome has its own probability. The sum of the probabilities of all outcomes is always 100%. Bookmaker specialists are obliged to assess the probability of each outcome as best as possible.

To assess the likelihood, bookmakers use:

-statistics of teams' previous games;

-a list of injured players;

-motivating the team and individual players;

-relationships within the team and between individual players and the coach;

-the situation around future transfers of players to other teams or the expectations of the arrival of new players, this affects the morale of the team;

-the goals of the team in the tournament in which the match is held;

-the country, city, stadium where the match will take place (based on this: the duration of the flight, the aggressiveness of the fans and the level of support, lawn coverage, etc.);

-the loyalty of the referees appointed to the match;

-weather forecast for the time of the event;

and many other factors.

After such an in-depth analysis, the bookmaker deduces the value of the probability for each outcome of the sporting event. Then it translates the value into the range from 0 to 1 (50% = 0.5, 30% = 0.3, 7% = 0.07). After that, the unit (1) is divided by the values of the probabilities and the coefficients are obtained. For example, for 50% the ratio is: 1 / 0.5 = 2.0, for 30%: 1 / 0.3 = 3.33, for 7%: 1 / 0.07 = 14.29. The resulting odds equalize the chances of event outcomes. If the odds are determined correctly, then for the bookmaker it is absolutely irrelevant what outcomes of events the players will bet on. This is due to the fact that if many matches of exactly the same are played, then the outcomes will be evenly distributed: the first team will win every second match, a draw will occur in 3 out of 10 matches, and the second team will win only 7 times out of 100. But if the odds are calculated correctly, then the amount of winnings will never exceed the amount of money wagered on all outcomes of one event. The only exception can be the case in a bookmaker's office when a large amount of money is placed on one outcome and the so-called money imbalance occurs in the line. Professional players constantly notice situations when the odds in the line change closer to the beginning of the match. This is due to the fact that the bookmaker is adjusting the line due to the fact that much more money has been bet on one outcome than on the opposite one, and they want an even distribution of bets. In this case, they lure players with high odds on an outcome they don't bet on. And small odds on the outcome, on which the most money has already been bet, repel those who wanted to bet.

1.4 Profit Margin. What do bookmakers make money on?

In order to calculate the standard winnings (margin) of the bookmaker's office, you need to make the following calculations: imagine that 2 players bet on opposite outcomes and each risk 110 units to win 100 units. If this event takes place and the office pays out, the first player will lose 110 units, the second will win 100 units. In total, they will bet 110 * 2 = 220 units, and the winning player in the bet will take 110 units of the initial bet and 100 units of winning, that is, the office's commission will be 10 units. Thus, the office will accept 220 units from two players and retain 10 units for itself, which will be its reward: 10/220 = 4.54%.

The commission in our offices for moneyline (moneyline) is also considered, as a rule, consisting of 2-3 outcomes with different coefficients.

Suppose there is an event with two possible outcomes with odds A for the first outcome and B for the second. In order to determine the theoretical advantage of the bookmaker at each outcome, you need to use the formula:

Margin = 1- (K1 * K2) / (K1 + K2).

With odds of 1.6 and 2.1, we get that the bookmaker has a theoretical advantage of 9.19%, that is, we lose on average $ 9.19 on every $ 100 bet.

In the case of a three-way event with coefficients K1, K2, K3, the formula becomes a little more complicated:

Margin = 1- (K1 * K2 * K3) / (K1 * K2 + K2 * K3 + K3 * K1).

Thus, if we see the coefficients in the line 1.4 / 4.0 / 7.0, then by substituting these coefficients in the formula, we will get a theoretical advantage, which is equal to 9.7%.

1.5 Basic rules of the game in a bookmaker's office

The simplest, most popular and widespread type of bet is single or moneyline. In the line, you will be given an event and possible outcomes (in the case of football, we have three outcomes - one of the teams wins, loses or draws) and the odds will stand next to it. Your winnings will be formed as the product of your bet amount by the odds. Example: Manchester-Arsenal

Win 1 (2.7) Draw (2.75) Win 2 (2.85).

With a bet equal to $ 100 on the victory of Manchester you will receive $ 100 * 2.7 = $ 270 if you win, if you bet on a draw in case of a draw - $ 275 and if you win Arsenal you will receive $ 285.

Handicap (handicap) bets are also a very common type of bet. With this type of bet, some team is given a conditional handicap in points, balls, and so on. Bookmakers traditionally like to set odds in such a way that odds range from 1.6 to 2.5 (ideally 1.9-1.9 or so), which in itself makes a bet for the player similar to guessing / not guessing. In the case of a moneyline, for example, in football, when a clear favorite and an outsider meet, the odds for the three outcomes will be something like this: 1-X -2: 1.2-5.0-10.0, and with a very likely victory of the favorite, the payout small. Moreover, everything can be decided by one random goal, which can lead to a draw or even to the victory of an outsider. Psychologically, most players do not like both very low odds and very high odds, but once the bookmaker offers a handicap line for a given match and, for example, give a handicap of 1 or 2 goals to the favorite, the odds will become normal: Favorite Outsider 1 1.6 2.2, which suits both the players and the bookmaker more.

With the help of handicaps, the teams' forces are virtually equalized and the favorite team gets a head start over the outsider team. At the same time, when calculating bets, the number of the handicap is added or subtracted from the final outcome of the meeting between the teams, and the team that scored the larger number, taking into account the handicap, becomes the winner.

Example: Utah Dallas +5.5 1.95-1.85

With a $ 100 bet on the Utah team, the player will win only if Utah does not lose by more than 5 points, and if he wins, he will receive $ 195.

Also note that the handicap does not have to be an integer. It is not by chance that offices often use not a whole handicap (when possible), and this is done in order not to get an expense or return. In case of spending, the office will have to return the money to two parties of the players and it turns out that the office, having processed a certain amount of money, is forced to lose profit. Sometimes this is not so scary (in the case of a non-cash event), but it happens that the consequences are very negative for the offices.

A classic example of a big shortfall in profit was the main sporting event of the year in American football - the Super Bowl, when most of the offices were forced to set a head start, and having taken millions of dollars from the players in total, they had to return this money, since the match ended with just such a difference that was installed by the offices.

Some bookmakers offer players the purchase of glasses or the sale of glasses.

The essence of this is very simple. For example, you have a match in the Utah Dallas +5.5 1.95-1.85 line. At the same time, you think that the handicap for Utah is not enough and you like the handicap of +7.5 more. The bookmaker can offer you such a head start, but only with a reduced odds, say, 1.70. That is, you bought two glasses for Utah. You can also sell two points, and then the office will offer you an increased coefficient, and you will get the final line, say, Utah +3.5 2.05.

Buying and selling points is used by experienced players and, despite this, it is not very popular, so you need to have a good reason to use this type of bet for your own purposes.

The Asian handicap is a kind of attempt to bring the odds in the formed line as close as possible to guessing / not guessing (with the same or similar odds), in relation to low-performance football. Therefore, not only handicaps that are multiples of half a goal (0, 0.5, 1, 1.5, and so on) are used, but also equal to a quarter of a goal (-0.25, +1.75, and so on). If you are given a handicap with a quarter of a goal, then, in fact, the rate is divided into two halves, that is, a handicap of +0.25? is split into two handicaps: 0 and +0.5. The winnings on such a handicap are effectively equal to the sum of two separate bets with a handicap.

Example: Manchester Arsenal (-0.25) 2.10-1.7. This means that Manchester gives a quarter-ball handicap to Arsenal (there are corresponding odds next to it) and our quarter-ball handicap is split into two parts: zero handicap 0 and half-ball -0.5. If you bet on Manchester and Manchester won with a difference of 1 or more balls (1-0; 3-1 and so on), then our winnings will be $ 100 * 2.10 = $ 210; in case of a draw (0-0; 2-2 and so on) we get $ 0 + $ 50 * 2.1 = $ 55; if Arsenal wins, our bet loses.

Sometimes bookmakers charge a commission for the return of the handicap and, say, in the event of a draw in this match, we would still be charged a commission for an unplayed zero handicap in the amount of $ 50 * 5% = $ 2.50.

If we bet on Arsenal (+0.25) at a price of 1.70, then similarly to the above example, in case of winning Arsenal with any account, we would receive $ 100 * 1.7 = $ 170 (since we won a zero handicap, and half a ball odds); in case of a draw, we would receive a return on a zero handicap and win a half-ball handicap, that is, we would receive: $ 0 + $ 50 * 1.7 = $ 85. In any case, if Manchester wins with any score, our bet loses.

Total. This type of bet was invented in the early 80s and is now extremely popular too. The bookmaker offers a certain number and you need to guess whether the total number of goals scored or points in this event will be more or less than this number.

Example: Manchester-Arsenal Total 2.5: Over 1.90 Under 1.80.

The number 2.5 is a kind of separator, and if you bet $ 100 on Over, then in the case of an account that is greater than or equal to 3 goals, you will receive $ 190. If the total of goals scored is less than 3, then you will lose.

There are also varieties of totals, such as the individual total of the team, the individual total of the player.

Example: Manchester Individual Total Over 2 with Odds 2.7. If you bet $ 100 on this total, then if Manchester scores more than 2 goals (it doesn't matter to us how many goals Arsenal score), then you will receive $ 270; if Manchester scores exactly 2 goals, then the bet will be expended; if Manchester scores 1 goal or does not score, the bet will be lost.

The same applies to the player's individual total: if you see a total of more than 0.5 (2.6) in Messi's line, then if Messi scores, then you win $ 260 for the $ 100 wagered, if not, the bet will be lost.

Express. A fairly popular type of bet, especially among beginners and those who want to receive a significant payout in the event of a favorable outcome of the victory. Express is often called a paravoz. You have the opportunity to combine several events independent of each other into one bet, and the final multiplier odds are formed as the product of all the odds of the events included in the multiplier. The player will receive payment only if and only if all independent events included in the express win.

Example: You like 3 bets and you decided to put a total of $ 100 on them anyway: on Dynamo win with odds of 1.5, on Manchester win with odds of 2.7, on total over 2.5 in Leeds Tottenham match with odds of 1, eight.

If you bet separately equal amounts of money for each match, then if you win all three bets, you will receive: (1.5 + 2.7 + 1.8) * $ 33.33 = $ 200, or a net win of $ 100.

However, under the same conditions, having formed an express, your final odds will be 1.5 * 2.5 * 1.8 = 6.75 and the potential payout will be $ 100 * 6.75 = $ 675, or the net winnings will be $ 575, which is almost six times more than single bets.

Thanks to the express bets, the player gets the opportunity to make a very large final coefficient and hit the jackpot for a very small amount of money.

Also, most bookmakers do not accept matched expresses when, say, there is a favorite with a very strong attack and an outsider with a weak defense. With all this, if we assume that the favorite will win with this handicap, that is, he will break it, then the total in this meeting will also be greater. Therefore, by making a double express from the fact that the favorite breaks the handicap and more from the total, you link events into one bet, which in most cases can hardly be called independent. In general, the bookmakers prefer not to take unnecessary risks and prohibit express trains if they believe that in one way or another the events in the express depend on each other.

Futures or bets on future events are also a fairly common type of bet and they are in every bookmaker's office. The essence of betting is that the player is asked to guess who will become, for example, the champion next year or who will win some kind of tournament. As a rule, futures are considered rates that will play at least in 1-2 weeks, and in most cases in six months - a year.

Example: whether the Spartak team will become the Russian football champion in 2011/2012, the coefficient is 8.0.

Bets on some intervals in the game, halves, periods, sets, quarter, half. In many bookmakers you can bet on totals, take a bet with a handicap, moneyline not only for the entire match, but also for some of its components.

Example: Utah Dallas Total Over 40 After First Quarter at 1.91. You will only win if the total points scored by the two teams are over 40.

A kind of combination of double express and betting on intervals is a time-match in football, a simultaneous bet on the outcome of the first half and the match.

Example: Manchester Arsenal draw, Manchester win by odds of 5.00. In order for the bet to win, it is necessary that the first half ends in a draw, and in the end Manchester wins. If at least one of the two conditions is not met, then the bet will be considered lost.

Interactive betting or live betting is a very interesting and exciting type of bet, when the odds for a given event change over time during a particular game. As you can imagine, this is only possible online.

Example: At the start of Manchester Arsenal you can bet $ 10 at odds of 2.7 on Manchester to win. You will watch this match on TV and evaluate the strengths of the teams by directly observing the course of the game. If Arsenal scores, then the odds for Manchester's victory will increase, but assessing that the goal was random or understanding the teams' attitude, you can adjust your decisions in the course of the game and, for example, put another $ 5 on a draw with the current odds of 3.0, or put 20 dollars to win Arsenal at a factor of 2.0 if you don't believe Manchester can do something and so on.

Other bets: the system is a certain set of express bets with different variations; conditional bets, where one component of the bet depends on the other and the second bet is activated only if the first one won; passes are when you need to assess whether a team or player will be able to go up the standings or not, all these bets are interesting in their own way, but they are an assortment proposal and a pleasant addition to the above types of bets.

List: the analysts of the offices give extended offers for some box office matches, and in order to fully satisfy the tastes of their clients, they come up with more and more new opportunities for betting.

Examples: how many corners there will be, how many goals will be scored in the round, whether this player will score a hat-trick, will there be a golden goal, and so on.

The main types of coefficients.

All bookmakers pay at fixed odds, so it is always possible to know the amount of winnings and losses. The odds may change over time, but if you place a bet on an event with a given odds at a given moment in time, the bookmaker is obliged to pay you in accordance with this odds after the event occurs.

The most common are European or decimal odds: 1.3, 5.0, 2.15, and so on. If you win, the bookmaker will pay you the amount of the bet multiplied by this coefficient. If you lose, you will lose the entire bet amount. If you bet 100 rubles with the odds of 2.1 and win, then you will receive 100 * 2.1 = 210 rubles. If your bet loses, you will lose 100 rubles.

Each office has its own rules of payments in the event that the event did not happen, did not happen "in full" or a winner was not identified in the bet. For example, if the event is canceled (the match did not take place, and so on), most of the bookmakers will immediately pay you money with a coefficient of 1.0, that is, they will simply return the money. This procedure is called a return or expense. There is one more system of rates, it is used in bookmakers, which have many English clients. We are talking about fractional or English odds. Examples of such odds: 18/11, 9/2. In order to convert such coefficients into the European format, it is necessary to add the unit to the decimal value of the fraction. Ѕ = 1 + 0.5 = 1.5; 17/9 = 1 + 1.89 = 2.89; 14/3 = 1 + 4.67 = 5.67 and so on. In fact, fractional odds are closer in spirit to decimal odds, with the only difference that we are dealing with net profit, and not with the total return on money invested.

Chapter II. Arbitration situations

Arbitration (surewins, surebets, surebets) is a situation that appears between bookmakers due to the difference in the assessment of the same sporting event. In such a situation, it is absolutely not important for the player how this sporting event will end, since the player receives his profit regardless of its outcome.

1 A real example of an arbitration situation

Take the handball match Russia - Norway. Here we are dealing with a three-source line (1-X-2). Let's take the lines of three different bookmakers, Fonbet, Betcity, Favorite.

At the Fonbet bookmaker, let's take the odds for the victory of Norway (outcome - W1) - 2.40. This means that if we bet on Norway, and she wins, then for every 100 rubles that we bet, we will receive 240 rubles from the bookmaker's office. Of these, 100 rubles is our initial rate, and 140 rubles is our net profit.

Table 2.1. Office line "Fonbet"

No.DateEventP1HP2FC126.11 18:00 Germany - France 1.2812.003.90-3.5 +3.51.85 1.85226.11 20:00 Norway - Russia 2.4010.001.65 + 1.5 -1.51.75 1.95

In the Betcity bookmaker we will place a bet on the victory of Russia (outcome - P2) with the odds of 2.20. This means that if we bet on Russia and she wins, then for every 100 rubles that we bet, we will receive 220 rubles from the bookmaker's office. Of these, 100 rubles is our initial rate, and 120 rubles is our net profit.

Table 2.2. Office line "Betcity"

No.DateEvent1Х21ХХ2126 / 11 19: 00 Germany - France 1.4010.003.301.272.60226 / 11 20: 00 Norway - Russia 1.958.502.201.601.75

In the bookmaker Favorit we are interested in the odds for a draw (outcome -X), which is 11.5. This means that if we bet on a draw and no team wins, then for every 100 rubles that we bet, we will receive 1150 rubles from the bookmaker's office. Of these, 100 rubles is our initial rate, and 1050 rubles is our net profit.

Table 2.3. Office line "Favorite"

Time Team 1 Team 21X219: 00GermanyFrance1.3012.503.8421: 00RussiaNorway1.8511.502.06

If we place a bet only in one of these offices and on one event, then it can always happen that we did not guess the result of the game and, accordingly, lost.

The same can happen if we place bets in two offices on two events - since there are three outcomes: a victory for Norway, a victory for Russia and a draw, then the outcome that we did not bet on, with all the ensuing consequences, may happen. That is, the risk is inevitable.

However, what happens if you bet on all three outcomes at the same time? If we bet on all three outcomes at the same time in the same office, then, despite the fact that one of our bets will definitely win, as a payment we will receive an amount that does not cover the amount of the bets made - that is, in the end we will lose. This is due to the fact that the bookmaker's margin is included in the payout coefficients for three possible outcomes, which gives it the opportunity to make a profit. However, the odds of different bookmakers for various reasons may not be as "coordinated" as in the same bookmaker. And then we can get that situation, which is called an arbitration situation or a fork.

Let us show that the three outcomes selected above in three different bookmakers of our choice give us an example of an arbitrage situation. That is, by placing certain amounts on all three possible outcomes, we will receive a profit for any real outcome of the game. Let's say that we have 1000 rubles and we want to place bets on this amount. We divide this amount into bets in the following special way:

-we put 434.86 rubles on Norway,

-we will put 474.39 rubles on Russia,

-and bet 90.75 rubles on a draw.

Let's see what happens when each of the three possible outcomes is realized.

If Norway won, then since we were betting on it at a coefficient of 2.4, we will get 434.86 * 2.4 ~ 1043.66 rubles in our hands.

If Russia won, then since we were betting on it at a coefficient of 2.2, we will receive 474.39 * 2.2 ~ 1043.66 rubles in our hands.

If friendship (draw) won, then since we were betting on it at a coefficient of 11.5, we will receive 90.75 * 11.5 ~ 1043.63 rubles in our hands.

From this it is clear that no matter what happens, we will get our hands on approximately 43.63 rubles more than we bet on all three outcomes combined. That is, we got a profit of 4.36% from the turnover without the risk of losing, from one operation.

2 Mathematical justification of the arbitration situation

First, I will give a list of designations that will be used in the future: - the payout rate for the outcome of the team's victory 1- the payout rate for the outcome of the team's victory 2- the payout rate for the outcome of a draw.

The odds for other outcomes are indicated in the same way: - the probability of the first team winning - the probability of the second team winning - the probability of a draw.

The probabilities of other outcomes are indicated in the same way: - the amount wagered on the victory of the first team - the amount wagered on the victory of the second team - the amount wagered on the draw

Amounts wagered on other selections are indicated in the same way.

3 Calculating the probabilities of outcomes and payout ratios

Here are the formulas that relate the coefficients for the main outcomes of sporting events. They will be useful for dropouts new types of forks when they are listed by some formal procedures.

First, we will derive formulas that do not require complex mathematics. Let's say we know the probabilities of a win for the first team, a draw, and a win for the second team.

What coefficients can be calculated under these conditions? Under these conditions, you can calculate the theoretical payout ratios of the following lines:

X-12-2X (mirror image of line 1-X-2)

Let's calculate the coefficients for the line 1-X-2. If we bet on the 1st event, then on average we will receive income P1 * K1 * V, which should be equal to the amount of the bet V, provided that the bookmaker's margin is equal to zero. That is

and it means that

In the same way, we get the values of other coefficients:

Insofar as

then we obtain the condition for the odds at zero margin of the bookmaker's office:

/ K1 + 1 / KX + 1 / K2 = 1

The coefficients K1X, K12 and K2X are calculated in a similar way. The probability of an event is 1X = P1 + PX. Accordingly, the theoretical coefficient is

K1X = 1 / (P1 + PX) = 1 / (1 / K1 + 1 / KX) = (K1 * KX) / (K1 + KX)

Likewise:

K2X = 1 / (P2 + PX) = 1 / (1 / K2 + 1 / KX) = (K2 * KX) / (K2 + KX)

12 = 1 / (P1 + P2) = 1 / (1 / K1 + 1 / K2) = (K1 * K2) / (K1 + K2)

How to calculate theoretical odds 1-2 or money lines based on the same probabilities. In bets on money line in case of a tie, the amount of the bet is returned. Therefore, if we bet on outcome 1, we get on average

which should be equal to V.

Similar reasoning is valid for bets on outcome 2. Therefore:

* KP1 + PX = 1 * KP2 + PX = 1

KP1 = (1-PX) / P1 = (KX-1) * K1 / KX.

But KX = 1 / (1-1 / K1-1 / K2) = (K1 * K2) / (K1 * K2-K1-K2) = (K1 + K2) / (K1 * K2-K1-K2) P1 = (K1 + K2) / K2P2 = (K1 + K2) / K1

We will also need formulas that give expressions for the odds for the handicap -0.25 and +0.25 - they are also easily deduced from the odds K1, KX.

Let's designate KF1 odds on handicap -0.25. Then the win-loss balance formula will be:

since in case of a tie we get a refund of half of the bet.

Likewise

* KF1 + PX * KF1 / 2 + PX / 2 = 1

KF1 = (1-PX / 2) / P1 = (2 * KX-1) * K1 / (2 * KX) = K1 * (1-1 / (2 * KX))

Likewise

KF2 = (1-PX / 2) / P2 = (2 * KX-1) * K2 / (2 * KX = K2 * (1-1 / (2 * KX))

Now let's designate the KF1 odds for the handicap +0.25. Then the win-loss balance formula will be:

* KF1 + PX * KF1 / 2 + PX / 2 = 1

4 Condition of the arbitration situation

When a bet is placed on an event, it can lose, win, and it is also possible that we do not lose or win anything - that is, we have a return (of money). Each event has its own coefficient of winning: Ki> = 1, i = 1, N. If the coefficient Ki> 1, then when this outcome is realized, we will have a net profit Vi * (Ki-1), where Vi is the sum of our bet. If Ki = 1, then this is a case of a money back, such odds are not present in the lines of bookmakers (but are implied for outcomes that are not included in the bet condition). Let's say we bet on each outcome of the game the sum Vi, i = 1, N. As it will be clear from the further course of the analysis, in the presence of a surebet, we will be forced to place bets on all events (game outcomes) included in our list (which depends on the type of surebets). Since, making bets, the player wants to win money, that is, to receive more than he has bet, and wants it to be for any possible outcome of the game (this is the essence of the "fork"), then we get a system of inequalities of "profitability":

* Vi> V1 + V2 +… VN = V, i = 1, N (1)

It means that each (any) possible winnings for each outcome of the game (Ki * Vi) must cover all our expenses on all outcomes of the bet, including those that did not play, that is, the total costs equal to V. Naturally, the coefficients satisfying these conditions cannot be found in one bookmaker's office, there must be at least two such offices. We rewrite these inequalities as

where Di = Vi / V, part of the total amount assigned to this outcome. The question arises as to how to determine from this system of inequalities whether a given set of coefficients gives us the opportunity to profit from at least one variant of the distribution of the total bet amount by possible outcomes.

Since all Ki> 1> 0, the system of inequalities can be (dividing by Ki) rewritten as

> 1 / Ki, i = 1, N

Adding the right and left sides of all these inequalities, we obtain

D2 +… + DN> 1 / K1 + 1 / K2 +… + 1 / KN

But D1 + D2 +… + DN = V1 / V + V2 / V +… VN / V = (V1 + V2 +… VN) / V = V / V = 1,

therefore, we get a condition that the coefficients of events (game outcomes) must satisfy:

/ K1 + 1 / K2 +… + 1 / KN< 1 (2)

The condition was obtained without any assumptions about the method of dividing the total amount by outcomes, which means it is true for all options without exception. This condition is necessary for the fork to exist. Since if a surebet exists (all the original "profitable" inequalities are satisfied), then by virtue of the derivation the coefficients Ki, i = 1, N will satisfy the last relation.

It is necessary to check whether this condition is sufficient for the existence of a fork. To do this, it is necessary to show that when this relation (2) is fulfilled, there will always be such Vi (distribution of the total bet amount over the outcomes) that all "profitable" relations (1) will be satisfied with them. That is, it is possible to make a profit regardless of the outcome of the event. To do this, let us designate L = 1 / K1 + 1 / K2 +… + 1 / KN and divide the entire bet amount by the outcomes in proportion to 1 / Ki, i = 1, N.

For this we put Vi = (1 / Ki * V) / L. Indeed, adding all Vi, we get V, and, moreover, Vi are split in proportion to 1 / Ki. Let's check that with such a distribution of the total amount of bets by the outcomes, our profitable (surebet) relations (2) are fulfilled. Substituting Vi into each of the relations (1), we obtain:

* Vi = (Ki * 1 / Ki * V) / L = V / L> V

(since L<1 по условие, которому, как предполагается, удовлетворяю наши коэффициенты исходов).

That is, we got that under this condition on Ki (L<1) и предложенном распределении общей суммы ставки по исходам, мы при любом исходе игры получим прибыль, что и требовалось доказать. То есть условие 1/K1 + 1/K2 + … + 1/KN < 1 является необходимым и достаточным для получения прибыли независимо от исхода игры.

If we know that our odds satisfy condition (2), that is, they are surebets, then we can easily calculate the amounts that need to be bet on each outcome in order to be in the same plus regardless of how the event ends:

For two initiating events:

V1 = V * K2 / (K1 + K2) = V * K1 / (K1 + K2)

For three initiating events:

V1 = V * K2 * KX / (K1 * K2 + K1 * KX + K2 * KX)

V2 = V * K1 * KX / (K1 * K2 + K1 * KX + K2 * KX) = V * K1 * K2 / (K1 * K2 + K1 * KX + K2 * KX).

Chapter III. Kelly test method

1 The essence of the "Kelly criterion" method

A fundamental problem in gaming is finding positive expectation betting opportunities. A similar problem in investing is finding investment opportunities with "excess" risk-adjusted returns. Once such opportunities are identified, the player or investor must decide how much of their capital to stake. One approach is to value money using a utility function. It is defined for all non-negative real numbers, has real values and is non-decreasing.

Some examples:

U (x) = x ?, 0 ? ? < ? and U (x) = log x,

where log means loge and log 0 = - ?. Once the utility function is determined, the goal is to maximize the expected utility of capital. The essence of the "Kelly criterion" method is to find the value of the rate for each attempt, such that it maximized E, the expected value of the logarithm of capital X. The log x utility function was re-used by John Kelly in 1956, who showed that it has some remarkable properties. If all bets have positive expectation and are independent, Kelly's bets when playing on a single bet will be extremely simple: put a fraction of your current capital equal to your expectation. In practice, this estimate changes somewhat (usually decreases) in order to allow for the possibility of "waiting bets" having some negative expectation, with higher fluctuations arising from payouts greater than one-to-one, and when more one bet at a time. The Kelly criterion is also known to economists and financial theorists under such names as "strategy of maximizing the geometric mean of the portfolio", maximizing logarithmic utility, optimal growth strategy, capital growth criterion.

2 Description of the "Kelly test" method and its properties

This chapter discusses the properties of the Kelly criterion. For simplicity, we will illustrate it with the simplest case - coin toss, but the concept and conclusions are easily generalized.

Let's say we are playing with an infinitely wealthy opponent who will place repeated bets on independent events - coin tosses. Further, suppose that for each throw, our probability of winning is p> 1/2, and the probability of losing is q = 1 - p. Our starting capital is XO. Suppose our goal is to maximize the expected value E (Xn) in n attempts. How much will we bet, Bk, on the k-th attempt? Let Tk = 1, if the kth attempt is winning and Tk = -1, if it is lost, then Xk = Xk-1 + Tk Bk for k = 1,2,3 .., and Xn = XO +? Nk = 1TkBk. Then

Since the game has a positive expectation, that is, p-q> 0, in this situation of equal payoffs, in order to maximize E (Xn), we would have to maximize E (Bk) for each attempt. Thus, in order to maximize the expected growth we must put all our resources in every attempt. So B1 = X0, and if we win the first bet, B2 = 2X0, etc. However, the probability of collapse in this case will be 1 - pN and for p< 1, lim n?? = 1 , так что крах почти неизбежен. Таким образом, "смелый" критерий ставок для максимизации ожидаемого роста обычно нежелателен.

Likewise, if our strategy is to minimize the likelihood of a possible crash (and a "crash" occurs if XK = 0 on the k-th attempt), we should place a minimum bet on each attempt, but this, unfortunately, also minimizes expected growth. Thus, a "timid" betting system is also unattractive.

This suggests the existence of an intermediate strategy that lies somewhere between maximizing E (Xn) (and definitely crashing) and decreasing the likelihood of crashing (and decreasing E (Xn)). An asymptotically optimal strategy was first proposed by John Kelly in 1956.

Since the probabilities and payoffs for each bet in the coin-toss game described are the same, it seems plausible that an "optimal" strategy would require you to always bet on the same fraction f of your capital. To make it possible to do this, we further assume that capital can be infinitely fragmented.

A strategy in which bets are made according to Bi = f Xi-1, where 0? f? 1 is sometimes referred to as a "fixed stake" strategy. Let S and F be the number of successes and failures in n attempts, respectively, then our capital after n attempts is

Xo (1+ f) S (1-f) F,

where S + F = n. For f in the interval 0< f < 1, Рr (Хn = 0) = 0. Таким образом, "краха", понимаемом в техническом смысле как разорение игрока, произойти не может. "Крах" будет означать, что для произвольно маленького положительного ?, limn ?? [Pr (Xn? ?)] = 1... In this sense, as we will see, a crash can still happen under certain circumstances.

Note that since

The magnitude

measures the exponential rate of growth per attempt. Kelly maximized the expected value of the growth rate coefficient, g (f),

It turns out that g (f) = (1 / n) E- (1 / n) logX0, therefore, for a fixed n, maximizing g (f) is the same as maximizing E. Calculate the derivative:

when f = f * = p - q.

then g "(f) decreases strictly monotonically on,

since g "(0) = p-q> 0 and lim f? 1 - g" (f) = -?. Due to the continuity of g "(f), g (f) has a unique maximum at the point f = f *, where g (f *) = p log p + q log q + log 2> 0. Moreover, since g (0) = 0 and lim f? 1 - g (f) = -?, Then there is a unique fC> 0 such that 0< f* < fC < 1 и g(fC) = 0.

Let's build a graph of the function g (f) from f (Figure 3.1).

Figure 3.1. Graph of the function g (f)

Based on the maximization of the function g (f), the following properties were formulated by John Kelly:

-If g (f)> 0, then it is almost certain that limn ?? Xn =?, That is, for each M, Pr = 1. This property shows that, if it were not for finite time, the welfare of the player XN would exceed any set limit of M when f is chosen in the interval (0, fс).

-If g (f)< 0, тогда почти достоверно, что limn?? Хn = 0, то есть для каждого ?>0,Pr = 1 , it turns out that the collapse is inevitable.

-If g (f) = 0, then it is almost certain that lim n ?? sup Xn =? and lim n ?? inf Xn = 0. This statement demonstrates that if g (f) = 0, then it is almost certain that lim n ?? sup Xn =? and lim n ?? inf Xn = 0.

-For a given strategy Ф *, which maximizes E and any other "substantially different" strategy Ф (not necessarily a strategy of fixed fractional rates), it is almost certain that limn ?? Xn (Ф *) / Xn (Ф) =?.

-The expected time required for the current capital Xn to reach a predetermined value C will be, asymptotically, the smallest under a strategy that maximizes E.

-If we assume that the return on one bet on the ith attempt is a binomial random variable Ui, then we assume that the probability of success is pi, where 1/2< pi < 1. Тогда E максимизируется выбором значением для ставки при каждой попытке доли f *i = pi - qi которая максимизирует E. Эта часть устанавливает справедливость использования метода Kelly выбора fi* при каждой попытке (даже если от одной попытки к следующей меняется вероятность) для максимизации E.

3 An example of using the properties of the "Kelly criterion". Generalizing Kelly's formula

Let's look at an example:

Player A is playing against an infinitely wealthy opponent. The player wins the same amount in successive independent coin tosses with probability p = 0.53 (independent events). Player A has an initial capital of X0, and the capital can be infinitely divisible. If we apply the sixth property, then we get * = p - q = 0.53 - 0.47 = 0.06, Thus, in each game he must bet 6% of the current capital so that Xn grows at the maximum rate and with zero probability collapse. If Player A constantly bets less than 6%, Xn will also grow indefinitely, but more slowly.

If Player A constantly bets more than 6% but less than fc, the same happens. Solving the equation g (f) = 0.53log (l + f) + 0.47log (l - f) = 0 numerically on a computer, we get fc = 0.11973Ї. So, if the rate is more than about 12%, then even though Player A may temporarily enjoy a fast growth rate, possible downward fluctuations will certainly bring the value of Xn to zero. The calculation gives the growth coefficient g (f *) = f (0.06) = 0.001801 so that after n consecutive bets the logarithm of the average capital of Player A will tend to a value of 0.001801 * n times the starting capital. Equating 0.001801n = log 2, we get the expected time required to double the capital, which is approximately equal to n = 385.

Games with equal payouts were considered above. But, the Callie Criterion can easily be extended to games with unequal payouts. Suppose Player A wins b units for every unit of bet. Further, suppose that on each attempt the probability of winning is p> 0 and pb - q> 0, so that the game is beneficial for Player A. Methods similar to those discussed above can be used to maximize:

Calculations give f * = (bp - q) / (b - 1), this formula is a generalized Kelly formula showing how much of the current amount of money must be allocated for each individual bet in order to maximize the growth rate g (f). If we adapt this formula for sports betting, then it takes the following form:

С = (K * V) - 1) / (K - 1),

С - coefficient of the size of the next bet, - coefficient of the bookmaker, - estimate of the probability of the player passing the event.

Chapter IV. Software implementation

1 An example of using the program for calculating betting "surebets"

It is necessary to develop a program that, according to the given odds and the amount of the bank, will find the size of the bets on each outcome of the event in order to have a guaranteed profit regardless of the outcome of the match. She also has to calculate the profit margin.

There is a BANK (amount of money) and RATES K1 and K2 (for two initiating events) and RATES K1, K2, KX (for three initiating events). It is necessary to enter values in the BANK field and in the RATIO field press the CALCULATE button. The program will calculate the margin between these odds for us (if it is positive, then there is an arbitrage situation, if it is negative or equal to zero, then there is no surebet) the amount of the bet on each outcome, in which we win the same amount regardless of the event result and the final profit.

Program dialog boxes for calculating two and three initiating events are shown in Figure 4.1. and Figure 4.2. respectively.

Figure 4.1. Program dialog box for two source events

Figure 4.2. Program dialog box for three source events

All values must be presented in numerical form; if this rule is not followed, the program will display an error message.

Examples of such situations are shown in Figure 3.3. and Figure 3.4.

Figure 4.3. Data Entry Error dialog box for the RATIO field.

Figure 4.4. Dialog box "Data entry error" for the BANK field.

2 Listing of the program for calculating betting "surebets"

Form module text

UnitMain ;, Messages, SysUtils, Variants, Classes, Graphics, Controls, Forms, Menus, StdCtrls; = class (TForm): TMainMenu ;: TMenuItem ;: TMenuItem ;: TLabel ;: TEdit ;: TLabel ;: TLabel ;: TLabel; : TLabel ;: TLabel ;: TLabel ;: TLabel ;: TEdit ;: TEdit ;: TEdit ;: TEdit ;: TEdit ;: TEdit ;: TEdit ;: TEdit ;: TEdit ;: TLabel ;: TLabel ;: TLabel; TrippleIssueClick ( Sender: TObject); DoubleIssueClick (Sender: TObject); FormActivate (Sender: TObject); LabelCalcClick (Sender: TObject);

(Private declarations)

(Public declarations) ;: TMainForm; Math ;: boolean; // indicates the selected number of outcomes in the calculation

($ R * .dfm) TMainForm.TrippleIssueClick (Sender: TObject);: = False;

// change the visibility of the fields. Visible: = True; .Visible: = True; .Visible: = True; .Visible: = True;

// nullify the fields.Text: = ""; .Text: = ""; .Text: = ""; .Text: = ""; .Text: = ""; .Text: = ""; .Caption: = "0%" ;; TMainForm.DoubleIssueClick (Sender: TObject);: = True;

// change the visibility of the fields. Visible: = False; .Visible: = False; .Visible: = False; .Visible: = False;

// nullify the fields.Text: = ""; .Text: = ""; .Text: = ""; .Text: = ""; .Text: = ""; .Text: = ""; .Caption: = "0%" ;; TMainForm.FormActivate (Sender: TObject); (nil);

// default values. Text: = "100";. Text: = "1.00";. Text: = "1.00";. Text: = "1.00" ;; TMainForm.LabelCalcClick (Sender: TObject) ;: double ;: double ;: double ;: double ;: double ;: double ;: double ;: double ;: double;

// check the input of valid data in the bank fields and coefficients: = StrToFloat (EditBank.Text); (Handle, "Invalid data entered in the" Bank "field!", "Data entry error", MB_OK) ;;;: = StrToFloat ( EditCoef1.Text);: = StrToFloat (EditCoef2.Text); (not IsDoubleForks) then: = StrToFloat (EditCoefX.Text) ;; (Handle, "Incorrect data entered in the" Factor "field!", "Data entry error", MB_OK) ;;;

// calculation for a double outcome (IsDoubleForks) then: = Round ((bank * coef2 / (coef1 + coef2)) * 100) /100;.Text:= FloatToStr (bet1);: = Round ((bank * coef1 / ( coef1 + coef2)) * 100) /100;.Text:= FloatToStr (bet2);: = bet1 + bet2;: = coef1 * bet1 - betsum; .Text: = FloatToStr (Round ((profit1) * 100) / 100 ) ;. Text: = FloatToStr (Round ((coef2 * bet2 - betsum) * 100) / 100) ;. Caption: = FloatToStr (Round ((100 * profit1 / bank) * 100) / 100) + "%"; // calculation for triple outcome: = Round ((bank * coef2 * coefX / (coef1 * coef2 + coef1 * coefX + coef2 * coefX)) * 100) /100;.Text:= FloatToStr (bet1);: = Round ( (bank * coef1 * coefX / (coef1 * coef2 + coef1 * coefX + coef2 * coefX)) * 100) /100;.Text:= FloatToStr (bet2);: = Round ((bank * coef1 * coef2 / (coef1 * coef2 + coef1 * coefX + coef2 * coefX)) * 100) /100;.Text:= FloatToStr (betX);: = bet1 + bet2 + betX;: = coef1 * bet1 - betsum; .Text: = FloatToStr (Round ( (profit1) * 100) / 100) ;. Text: = FloatToStr (Round ((coef2 * bet2 - betsum) * 100) / 100) ;. Text: = FloatToStr (Round ((coefX * betX - betsum) * 100) /100 );.Caption:= FloatToStr (Round ((100 * profit1 / bank) * 100) / 100) + "%" ;;;.

4.3 An example of using the program to calculate the bet amount using the "Kelly criterion" method

It is necessary to develop a program that, according to the given criteria (the bookmaker's coefficient, the amount of the bank and the assessment of the probability of passing the event), will calculate the amount that must be delivered to increase its own profit.

The essence of the program is as follows:

There is a bank (amount of money) and a bookmaker's coefficient for a sporting event, as well as a player's personal probabilistic assessment of this event, expressed as a percentage.

You must enter a value in the fields YOUR PROBABILITY ASSESSMENT, BOOKMEKER'S RATIO AND BANK and press the CALCULATE button. The program will calculate the bet amount and display the result in the PLAYER'S BET field.

The dialog window of the program for calculating the bet amount using the "Kelly criterion" method is shown in Figure 4.5.

Figure 4.5. Program dialog box upon pressing the CALCULATE button

The value entered in the PROBABILITY ASSESSMENT field must be in the range from 1 to 100. If this criterion is not met, the program displays an error message. An example of such a situation is shown in Figure 4.6.

Figure 4.6. Data Entry Error Dialog Box for PROBABILITY ASSESSMENT

If the fields BOOKMEKER RATIO and BANK are not filled in, the program displays input errors shown in Figures 4.7 and 4.8, respectively.

Figure 4.7. Data Entry Error Dialog Box for BOOKMEKER RATIO field

Figure 4.8. Dialog box "Data entry error" for the BANK field

If, as a result of the calculation, the sum of the bet turns out to be a negative number, then we can conclude that such a bet is not profitable for the player. An example of such a calculation is shown in Figure 4.9.

Figure 4.9. Negative program dialog box

The result of pressing the RESET button, which resets the values of all fields, is shown in Figure 4.10.

Figure 4.9. Program dialog box when you click the RESET button

4 Listing of the program for calculating the bet amount using the "Kelly criterion" method

Unit1 ;, Messages, SysUtils, Variants, Classes, Graphics, Controls, Forms, StdCtrls; = class (TForm): TLabel ;: TEdit ;: TLabel ;: TEdit ;: TLabel ;: TEdit ;: TButton ;: TLabel ;: TEdit ;: TButton; Edit1KeyPress (Sender: TObject; var Key: Char); Button1Click (Sender: TObject); Button2Click (Sender: TObject);

(Private declarations)

(Public declarations) ;: TForm1;

($ R * .dfm) TForm1.Edit1KeyPress (Sender: TObject; var Key: Char); not (key in ["0" .. "9", # 8, "."]) Then key: = # 0; ; TForm1.Button1Click (Sender: TObject); k, c, b: real;, v: integer; (Edit1.Text, v, r); r<>0 then ("Incorrect entry of the probability") ;;; not (v in) then ("The probability must be within the limits") ;;; (Edit2.Text, k, r); r<>0 then ("Incorrect entry of the bookmaker coefficient") ;;; (Edit3.Text, b, r); r<>0 then ("Incorrect entry of the player's bank") ;;;: = (k * (v / 100) -1) / (k-1) ;. Text: = FloatToStrF (c * b, ffFixed, 12,4); ; TForm1.Button2Click (Sender: TObject) ;. Text: = "" ;. Text: = "" ;. Text: = "" ;. Text: = "" ;;.

Conclusion

In this paper, arbitration situations were analyzed in detail, they were proved and described using simple linear inequalities. The essence of the bookmaker's offices, the principles of their work were told. Was analyzed and mathematically justified betting strategy using the Kelly criterion. A program was developed in the Delphi environment, which calculates the profit of a surebet, and another program in the Delphi environment has been developed, which calculates the amount of a player's bet using the Kelly criterion.

List of used literature

1.A. Izodin, I. Miklin, Bookmaking for Beginners, M .: Airis-press, 2012.

.O. Maryin, Arbitration situations in bookmakers, Moscow: Ayris-press, 2011.

.Edward O. Thorpe, The Kelly Criterion in Sports Betting, 2010.

.V.E. Gmurman, Probability Theory and Mathematical Statistics, Moscow: Higher Education, 2009.

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