# Development of the system of sports betting

## History of sports betting. The "cash out" option in sports betting. The main determinants of the betting process. To test the hypothesis that the risk-aversion as the factor that makes individuals accept the amount to "cash out" proposed by a bookmaker.

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Table of Contents

- Chapter 1. Introduction
- 1.1 Problem statement
- 1.2 Rationale for the study
- 1.3 Research aims and objectives
- Chapter 2. Literature review
- 2.1 Introduciton
- 2.2 History of sports betting
- 2.3 How does the betting process look like
- 2.4 The “Cash out” option in sports betting
- 2.5 The main determinants of the betting process
- 2.6 Conclusion
- Chapter 3. The theoretical model
- 3.1 The main assumptions of the model
- 3.2 Extreme case analysis

3.3 Generalizing the model

- Chapter 4. Conclusion
- List of references
- Appendices

Chapter 1. Introduction

The history of betting has been evolving for centuries and in retrospect, the earliest and most recognizable forms of betting could be traced back to Greece, home of Olympic Games, and Ancient Rome. Early Romans perceived the betting outcome as a decision of a Goddess Fortuna, one of the most popular cults of those years. They considered the concept of sports betting as part of their religious inclinations and spiritual obligations. Pliny the Elder, one of the authorities of the 1st century, said: “We are so much at the mercy of chance that Chance is our god”.

The increased popularity of gambling was witnessed in the period preceding the Middle Ages, many nations tried to outlaw it, but then, the phenomenon of betting was reinvented in the Renaissance period. Since that time gambling was developing and becoming more popular. Thus, starting with wagering on “how much time one needs to pass a given distance” people ended up focusing on professional sports betting (such as football, golf, tennis, etc.).

That leaded to the creation of several bodies which were meant to organize betting process among various sporting activities. The advances that have been made in the information and communication technology have largely shaped the techniques of undertaking wagering: several innovations that have been developed improved the effectiveness and the fun involved in the sport betting and made the process easier for the bettors. For example, nowadays it is very popular to use online services (betting sites, sportsbooks). One of the latest trends in sports betting is the “Cash out” option, the tool that gives to bettors the ability to take a return before the end of the event, thus, allowing to secure their profits or minimize their losses.

**1.1 Problem statement**

**sports betting bookmaker**

Despite the advances that have been made in the sports betting industry, little emphasis has been placed on the analysis of the use and effectiveness of the “Cash out" option. Bettors continue to utilize it without necessarily considering the advantages, disadvantages and viability of that option in the betting process. The question that arises is what makes the bettors use the “Cash out”? While the apparent benefits of this option is that this could spell a boon for a player who has had a bad day in the event, it curbs natural gambling instinct to go for the big `kill' and earn large, handsome bounties for their gambling efforts. According to what rationality implies, if one accepts the bet he will not cash out until the final whistle of the referee unless any certain factors affect this decision. Moreover, the return from the bet taken before the end of the event would be quite less than the face value of the bet. Thus, using “Cash Out” robs the player of watching the nail biting, thrilling finish of the event and leaves him with mediocre sum which could have indeed been very large and substantial had he waited for the game to finally end(McGowan & Mahon, 2013). The above analysis implies that there are mixed benefits and disadvantages associated with the use of the “Cash out” option in betting, and there is dire need to undertake a research in order to understand why do the bettors utilize that option, what does affect their decision and what amount of cash has to be proposed to them in order to make them accept the offer. Intuitively, such behavior can be explained by the bettors' and bookmakers' preferences over risk. More precisely, the risk aversion of a bettor can lead to the acceptance of the “cash out” offer proposed by a risk-neutral bookmaker at some point in time during the game. What is analyzed in this paper is whether the risk-aversion of a bettor is the factor that makes him accept an amount of money proposed by a risk-neutral bookmaker before the end and not wait until the end of the event.

**1.2 Rationale for the study**

It is of vital importance to rationalize such proposition before undertaking the study and conducting a theoretical model. If the hypothesis suggested would have an irrelevant logical foundation the model would be preposterous and would not give any pertinent results.

There will be made several assumptions in the corresponding chapter which the theoretical model will be based on. The main idea of making the suggestion that risk-aversion matters is the following. Starting with different beliefs about the outcome of the event, a bettor and a bookmaker change their opinions in the same way. The closer the end of the event, the closer to each other are the beliefs of these two players. Saying that a bettor is risk-averse while a bookmaker is risk-neutral implies that the expected utilities based on their subjective beliefs become closer even more. This allows to make a proposition that there is a point in time when the subjective beliefs of a bookmaker and a bettor are so close to each other that a bookmaker can offer a certain amount to cash out which will be beneficial for a bettor according to his utility expectations of an event outcome.

**1.3 Research aims and objectives**

The following research objectives were developed in order to guide in the undertaking of the study:

How do betting companies undertake the betting processes and what differences are depicted in the diversified betting processes and options?

What are the constructs of the “cash out” option in the betting process?

Whether risk-aversion of a bettor is the factor that makes “cash out” option useful?

Chapter 2. Literature review

2.1 Introduction

Sports betting is the placement of finances normally referred to as stakes on events with uncertainty about the occurrence with the hope of a particular outcome that will make a bettor earn additional money. What is important to note about betting is that the in the eventuality that the outcome of an event does not work out as anticipated, the bet is lost hence the money (or the stake) is lost (Takahashi, 2012). Betting therefore is a game of win or loss determined by the outcome of the event in focus.

Elements of a bet that are worth noting are chance, consideration and a price. The placing of bets is done with the hope that chance favors the selected choice for one to win a price in this case the stakes involved (McMillen, 1993). However for one to be considered for a bet one must be able to place a bet and hope that it works as he expects. The outcome of the event is usually one that is determined after a short while since the events in which bets are placed are usually those that take a short while before the outcome is known and the lucky ones are rewarded.

Bets are made on numerous events especially in the sports arena. In sports people bet on football, boxing, horse racing, other animals racing, athletics, rugby and even indoor events. The bets are hinged on luck to some extend because even the clear favorites do lose in unexpected events (Gutierrez, 2012c). In betting the stakes are usually high on those considered as underdogs since most people would avoid placing bets on such people. The assumption is usually that the underdogs are bound to fail therefore there is no need to take a risk on them. This however tends to work otherwise in some cases as the bets on underdogs pay off very handsomely.

2.2 History of sports betting

Betting has been around for the long time dating back to the days when the roman emperor ruled the world. Back then they would bet on horse races in which races puling chariots would race while they watched, amused themselves and betted as well for fun. It is this racing for fun that led to the organization of a Grand National that involved horse racing at a national stage. Such a stage provided a perfect forum for people who then betted heavily on their horses in hope that they would win (Gutierrez, 2012a). Eventually with time this trend developed and was learnt by many through observation of what others were doing and how they used to place their bets.

This marked the beginning of the revolution of betting which opened the evolution of betting. In 1934 William Hill started trading in bets allowing people to place stakes on the different sports events taking place over the country. William Hill sport betting still exists today serving as a reminder of how far sport betting has come over the years. In 1961, three decades later after William Hill had started sports bookmaking, betting was officially recognized and legalized with the requisite legislation put in place to guide its implementation. More people and shops then ventured into this line of business opening shops and encouraging people to visit and make bets. However, this was not the peak of betting yet as we know it today. Revolution towards the betting today's started in 1993 when the bookmaking was taken online after internet was introduced. People were now able to make bets online without having to go to shops and they could have their money online making it easier and more convenient to bet (Seitz, 2011). Even with this there still exist bet shops to contain the crop of people who still prefer that as an option. Sports betting has evolved and gone through a lot to get where it is today although this may not be the last of it. In 2004 betting officially became a financial affair, the reason was an insertion of William Hill into the FTSE 100 that marked the realization that sport betting was not the chance people would use to earn good money.

So question is does all this development mark the maturity of sport betting? Does it still have a future? The answer is there is still more to come from sport betting with the likelihood of service diversification being the likely occurrence in the near future. This will means there is going to be the introduction of specialized bets entailing peer to peer betting, personalization of the betting and betting being a forum for socialization and interaction (Owen, 2012). Sport betting was an event born out of the need for a gathering to derive more fun from a sporting event and this is why it is unlikely to die anytime soon. It will instead be a pillar upon which friendships will be cemented and interaction also initiated and developed.

2.3 How does the betting process look like

First of all, some main definitions of the essential components of the betting process have to be mentioned.

What every bettor considers before placing a bet is a numerical expression of the likelihood that a certain outcome of the event will take place. The tool that shows that is called odds. There are several ways of expressing the odds to customers, some platforms even offer bettors the option to choose the expression that is the most convenient for them. In this paper decimal (European, digital) odds will be used for simplicity. Such expression is the clearest one.

The other thing that is worth noting is that there are many different forms of bets, and one of the most famous forms is straight betting. It entails choosing a team one feels will win in advance. In such a case the team one determines is the one he places the bet on before the game starts. At the end of the game a bettor either collect a prize that corresponds to the initial odds or looses the cash he bet (the stake).

The betting process can be constructed in several ways. Depending on type of betting proposed betting platforms can denote either bookmaking (sportsbooks) companies or betting exchanges.

Bookmaking companies (which offer the same what traditional bookies do) offer odds on the occurrence of some outcome of the event, the customers bet according that odds. Such betting process implies wagers placing the bet on the particular outcome and a bookmaker betting against that outcome (or on all the other possible outcomes). Bookmaker decides what odds to offer according to his subjective beliefs and to the amount of stakes received for each outcome.

Betting exchange companies represent a marketplace where wagers bet against each other, not against the company. The revenue of Betting exchanges consists of transaction fees, so the profit of such company does not depend on the outcome of the event. Customers of betting exchanges have more options than those of bookmaking companies: bettors here can bet both on some team's win and against it. Putting it simply, bettors can take the role of the bookmaker. They suggest the odds that they find convenient and then are matched with the other bettor who accepts the offer and bets with the same odds on the opposite outcome.

2.4 The “Cash out” option in sports betting

“Cash out” in sports betting is an option proposed to customers which allows bettors to take the money before the end of the event. Intuitively, when at some point in time during the game there is a score that a bettor bet on, and he is afraid that the final outcome may change, he can use “cash out” to take the amount of money that is proposed by a bookmaker at that particular point of time. This will give a customer some profit which will be less than the initial price, but at least he will be confident that he will earn something.

The other case when this option may seem useful is when during the game a bettor feels that the score he bet on is not likely to occur in the end of the event, he can “cash out” for another reason: he will not have the profit in this case, actually, he will loose money, but the loss will be smaller then the initial stake.

So, this option helps to secure profits of the wagers and minimize losses.

However there is a catch to this type of betting in that the odds that were initially used to bet are usually higher than what is used to pay an individual who opts to cash out of the bet.

Being a safe kind of bet it is often tricky and demanding hence requiring a lot of discipline and carefully planning and consideration. This is because the odds often change in response to the changes in the remaining uncertainty.

One tricky element is that the bookmakers also appreciate the kind of loophole they create for betters by creating such an option. As such this kind of option that is cashing out is limited to a certain number of games only. This means that when placing bets where one expects such options it is important to be careful not to make the wrong decisions and incur huge losses. Cashing out looks like an easy way to make money and truly it is although only successful when there is careful planning and strategizing on the part of the better. Emphasis on discipline is also another important dimension since such options are limited to certain games by the bookmakers because they see such games as impossible or with slimmest chance to achieve such a goal. As such set targets should be stuck to so as to avoid the open traps that have been laid for them (Gutierrez, 2012b).

Cash out options are also applicable to single bets considering the complexity were they to be used on multiply games. This means that is only possible to use cash out option on one single game rather than on a bet consisting of a series of bets in a single game. This cash out option could be convenient for tournament bets such as the world cup. In a scenario where an individual bets on a particular country say German to win the world cup. However despite the mixed performances of draws, losses and wins they manage to pull off and reach the final. However their opponents in the final looks like the team likely to beat them. What an individual does is to exercise the cash out option and not only get back the initial cash used on the bet but also with a little profit (Altman, 1985).

The use of cash out option is particularly good for the accumulator kind of bets. This entails placing a series of bets on the same team as highlighted above. The option of cash out comes in when the next opponent to the team the bet is on is likely to lose. That is when one exercises the option to cash out. It is therefore an option to mitigate from a complete loss by allowing the individual who put in the bet to salvage a little from what has been accumulated from the initial bet. Use of cash out option however is tricky and should often be thoroughly examined before being adopted (Birch, 1996).

2.5 The main determinants of the betting process

What is of the main importance while analyzing a betting process mathematically is what the betting consists of. There are two “players” who will be analyzed: a bookmaker and a bettor.

Despite the fact that they have the same goal - to “win”, they also have a lot of differences in their behavior and preferences.

-Utility function

The crucial factor that determines the behavior of an agent is his utility function. Bernoulli (1738) was the first who proposed an interpretation for utility of wealth in his theory of decision-making among risky prospects with monetary outcomes. The second, more recent, interpretation was proposed by von Neumann and Morgenstern (1944), and that is one which is used in this paper. The utility of wealth is treated as a measure of an agent's intensity of preference for some wealth with reference to outcome probabilities. (Fishburn, Bell, 2000)

-Agents' preferences

When the choice under uncertainty takes place, it is common to say that people may have different attitudes toward risk, they can be: risk-averse, risk-neutral and risk-loving. Economists typically assume that an individuals are risk-averse: they have concave utility function and they make a decision among risky alternatives by maximizing their expected utilities.

-Subjective beliefs

It is important to understand that beliefs that help agents in decision-making are subjective. Probabilities that individuals have in mind are not real, they are subjective. Savage (1954) proposed a subjective probability theory which states that decision makers behave according to their subjective estimates of the chances of uncertain events which may even be not expressible quantitatively.

-Certainty equivalent

When facing the risk (the uncertainty) the agents may prefer to “quit” the situation, exchanging the uncertain outcome by a certain one (which can be smaller in absolute terms). The certainty equivalent (in terms of wealth) is such amount of wealth utility of which is equal to the expected utility of wealth in the case of staying in uncertain situation.

2.6 Conclusion

Sport betting has come a long way and it has evolved into a worthwhile business venture a part from just being a leisurely activity for fun. There are however a lot of risks concerned with the game ranging from losses to cons out to manipulate and joyride on the naïve. Therefore it is important to make wise decisions after careful consultations especially on the online bets and bets through agents. Use of cash out option in sports betting may also be discipline that it needs. Still, the question remains open: who uses this option and why? What factors affect the bettors' decision about the usage of that option? Do the utilize it because of the risk-aversion? The model proposed in the following chapter tries to analyze that question.

Chapter 3. The theoretical model

3.1 The main assumptions of the model

I attempt to test the hypothesis that the risk-aversion is the factor that makes individuals accept the amount to “cash out” proposed by a bookmaker. This hypothesis is reasonable, such a prediction can be verified by a simple model with specific assumptions.

Suppose there is a 90-minute game (a kind of a football game) which an individual wants to place a bet on. He comes to the bookmaker and consider the odds proposed. For simplicity, assume that a bettor believes that the final result of the game will be the score “0:0” and he wants to place a bet on this result. The following assumptions have to be made:

1. Nothing “significant” happens during the game, nothing that can affect the beliefs significantly. This assumption is made in order to test whether the risk-aversion is the crucial factor that affects the decision of a bettor to accept a “cash out” amount proposed by a bookmaker

2. The wealth of a bettor before placing the bet is . The stake that a bettor makes is . A bookmaker offers odds . Depending the result of the game, the wealth of the bettor will be either or

3. Bettor is risk-averse, which means that he has a concave in wealth utility function. For simplicity assume that his utility function is

4. Bookmaker is risk-neutral

5. For simplicity, assume that time is discrete and is measured in minutes

6. Both a bettor and a bookmaker have their subjective beliefs. Each of them considers that there is a probability of a goal at each minute of the game which is a constant. Assume, a bettor believes that the probability of a goal at each minute is and a bookmaker believes that it is . It is reasonable to assume that , which means that a bookmaker's subjective probability of a goal at each minute is higher than that of a bettor. If it were not true, the bet will not occur. A bettor believes that the desired outcome is more likely to occur that it is proposed by a bookmaker and that is why a bettor accepts the offered odds a places a bet.

7. When talking about some point in time t assume that there were no goal until that time.

8. From the beginning of the game a bettor and a bookmaker have subjective beliefs about the probability of the score “0:0” being the final result. In this model it is assumed that this probability is increasing in time, but it is not linear in t. Having in mind Bernoulli distribution, it can be easily derived that the subjective probabilities of “0:0” being the final result at each point of time can be described in the following way: , where for a bettor and for a bookmaker, respectively If is the subjective probability of a goal at each minute, then is the subjective probability that there will be no goal at each minute. Such a behavior of the subjective beliefs of the bettor and bookmaker can be explained in the following way: if nothing changes during the game, the subjective probability of the desired outcome is approximately the same until the last minutes of the game. In the last minutes beliefs of the “win” increase rapidly (both of the bookmaker and the bettor). See Figure 2

9. When deciding how much to offer to a bettor to cash out, a bookmaker has to analyze the behavior of his rival. For simplicity, assume that a bookmaker knows according to what beliefs a customer acts.

There is a minimum amount of money, less of which a bettor will not accept at each point of time. It can be found with the help of the knowledge of the bettor's utility function and his expectations. That minimum amount will make a bettor achieve the certainty equivalent. In this case the certainty equivalent at each point of time can be found in the following way:

Expected utility at time t is equal to the wealth that is supposed to be in the case of a “win” multiplied by the probability of the “win” plus the wealth that is supposed to be in the case of a “loss” multiplied by the probability of the “loss”, i.e.

The equation that is used to find the certainty equivalent is the following:

Solving this equation the CE can be found See Appendix 1 for the detailed solution:

Thus, a bookmaker knows that a risk-averse bettor will not accept a “cash out” offer that is less than the amount which will make him achieve his certainty equivalent, so a bookmaker has to offer:

. is the amount of money that a bettor has independently of the outcome of the event. So, by paying the difference between the certainty equivalent of the bettor and the amount he already has, the bookmaker will make his opponent to reach the certainty equivalent.

3.2 Extreme case analysis

Let's first consider an extreme case when a bettor is so confident that his beliefs will work out that he stakes all his initial wealth, , which mean that the certainty equivalent is now the following:

,

and a bookmaker has to pay the minimum amount which equals to .

In the beginning of the event a bookmaker has to decide the policy according to which he will offer the specific amount of cash at each point of time. What he has to do is to construct an expected payout function and to minimize that. A bookmaker desires to make a bettor cash out at the point of time when the expected payout for him is the smallest one.

Assume that a bookmaker doesn't want to pay more than the minimum that a bettor desires to get. Hence, the expected payout function looks like a certainty equivalent of a bettor at each point of time t multiplied by the probability that there will be no goal until that point of time t.

The probability that there will be no goal until some point of time t is the probability of no goal at each minute in the power of t, i.e. .

Thus, a bookmaker has to minimize the following function:

,

Which is the same as

The derivative of this function is the following See Appendix 2 for the detailed solution:

Depending on the values of and , three different situations may occur It is clear that is a positive constant, and are positive as well. So, the sign of a derivative is affected by the sign of the expression in the latter brackets. .

I. Suppose that bettor's and bookmaker's subjective probabilities of a goal at each minute are such that

,

Which means that

or

In this case the derivative of the bookmaker's expected payout function is strictly positive what means that the expected payout function is strictly increasing in time. It means that the minimum payout that a bookmaker has to give to a bettor occurs at the very beginning of the game. Hence, this means that in such case the bet will not occur at all.

Intuitively, this means that the beliefs of a bookmaker and a bettor are so close to each other, that a bookmaker considers a bet senseless because he believes in a certain output approximately in the same way as a bettor does.

This case shows that when probabilities are such that is found above, the “cash out” will not be used because the bookmaker would like to offer cash out as soon as possible, and a bet will possibly not occur at all.

II. Suppose now that bettor's and bookmaker's subjective probabilities of a goal at each minute are such that

,

Which means that

or

In this situation the derivative of the expected payout amount that a bookmaker has to pay as a “cash out” is strictly negative what implies that the expected payout amount is strictly decreasing in time. This means that the expected payout that a bookmaker has to pay as “cash out” is at its minimum in the very end of the game, at the 90th minute. The bookmaker's subjective expectations of the outcome are such that it is less expensive for him to wait until the end of the game hoping that someone scores than to interrupt the game and make a bettor cash out. Hence, “cash out” option will not be utilized again as a bookmaker will not desire to offer a certainty equivalent (or more) until the end of the event and the bettor will not accept any amount less of that.

This case shows that the bet will occur but the “cash out” will not be used again. See an example in Appendix 4

III. The last case that has to be considered is the case when bettor's and bookmaker's subjective probabilities of a goal at each minute are such that:

,

Which is the same as

or

In this case the derivative of the expected payout amount that a bookmaker has to pay as a “cash out” equals to zero what implies that the expected payout amount is constant over time. Such result means that a bookmaker is indifferent whether to offer a CE as a “cash out” amount at the very beginning of the game, sometime during the game or to wait until the end. As a bettor wants to avoid the uncertainty, he will accept the cash out offer in the very beginning of the game at the price that a bookmaker will offer, and the amount proposed by a bookmaker will be fair. Again, this is the case when a “cash out” option is senseless See an example in Appendix 5.

As the analysis above has shown, when a bettor is very confident in his beliefs, the “cash out” option remains unused even though the risk-aversion of a bettor exists.

3.3 Generalizing the problem

Let's now consider the cases when a bettor is not as confident as was assumed above, thus, he stakes some part of his wealth, .

Recall the expression which represents the certainty equivalent:

A bookmaker now faces the following expected payout (say, ) function:

,

Or,

Costs minimization problem will be the following:

,

The derivative of the expected payout function is:

As is always positive, what affects the sign of the derivative is the expression in the latter brackets:

.

What have to be compared are two following expressions:

and

is a positive constant. Both and are continuous functions but if the time period of 90 minutes is considered, the quotient of those functions is almost a constant. See Appendix 8 for the details

Thus, depending on the value of the bookmaker's expected payout function can be increasing, decreasing or constant. The discussion of each case will have the same conclusion as each case of paragraph 3.2 has. Hence, the expected payout function again will not have any minimum point except either the 0th or the 90th minute. This means that a “cash out” option will not be used. See Appendix 9 for the numerical examples

Chapter 4. Conclusion

Lots of innovations have been made in the sports betting industry which were not analyzed by researchers. The “Cash out” option is one of those advances that is becoming popular in betting. Bettors use this option in the betting process on order to secure their profits. The question that arises is what makes the bettors use the “Cash out”? According to the rationality implications, if one accepts the bet he will not cash out until the final whistle of the referee unless any certain factors affect this decision. What was examined in this paper is whether the risk-aversion is the factor that affects the decision of bettors in risky situations to quit the game before its official end and take the money proposed by a bookmaker. It was assumed that there are not any other factor affecting the decision. The analysis above has shown that if the risk-aversion is the only factor that a bookmaker rely on while making a decision, a “cash out” option will not be utilized. There will not be any point in time during the game (except the 0th and the last minute) when a bookmaker and a bettor could coincide in decision to quit the game before the official end. Thus, there might be other factors that are important in such decision-making. It might be something else that happens during the game (like a strong player removal/arrival) and affects beliefs of a bettor and a bookmaker so much that they decide to quit the game and not to wait until the end. The research has to be developed by analyzing other factors that may be the reason of “cash outing”. What else could be made is providing an investigation of the more realistic cases when there are different bettors in the market and an imperfect information, which means that a bookmaker doesn't know the type of a bettor he is “playing” with. To examine the empirical evidence the real bet could be placed and the real amounts of “cash out” proposed could be analyzed.

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23. Takahashi D. (2012), “Casino Games Surpass Farm Games as the Darlings of Social Networks”, VentureBeat, April 2012. http://venturebeat.com/2012/04/27/casino-games-surpass-farm-games-as-the-darlings-of-social-networks/

24. Quandt, R.E (1986). “Betting and Equilibrium”, The Quarterly Journal of Economics Vol. 101, No. 1 (Feb., 1986) , pp. 201-208, Oxford University Press. http://www.jstor.org/stable/1884650

25. Vázquez, Y., López, J., & Pintó, J. J. (2012). Legal Regulation of Sports Betting in Spain and its History. In Sports Betting: Law and Policy (pp. 735-762). TMC Asser Press

26. Woodland B.M, Woodland L.M, (1991). “The effects of Risk-aversion on Wagering”, Journal of Political Economy Vol. 99, No. 3 (Jun., 1991) , pp. 638-653, The University of Chicago Press. http://www.jstor.org/stable/2937746

Appendix 1

(Figures)

Appendix 2

.

,

,

,

Appendix 3

Taking the first derivative of the given function with respect to t:

Taking the common multiples and out of the brakets:

.

Appendix 4

Suppose that bettor's and bookmaker's subjective probabilities of a goal at each minute are such that

,

or

Let's take some numbers to demonstrate this case properly.

Assume that initially a bettor has , he makes a stake with odds .

This means that if the outcome will be the one he placed bet on, the wealth of the bettor will be:

.

If, the outcome will be the one that was not desired by the bettor, his wealth will be:

.

Consider the case, when a bettor subjectively believes that there is a probability of a goal at each minute of the game which equals to , i.e. there is a 0,56% chance that a goal will take place at each minute of the game.

According to the constraint given in this situation, believes of the bookmaker have to be the following:

Bookmaker believes that there is a probability of a goal at each minute which is smaller than .

Figure 4.1 shows how the subjective probabilities could be located in this situation. The blue line represents bettor's subjective probability of a “0:0” score at the end of the game at each minute during the game, if no goal took place before that minute. The red lines show the same probability for a bookmaker, the smaller the probability of a goal at each minute according to his beliefs (), the higher is the red line.

It is clear from the picture that the lower , the closer the beliefs of the bookmaker and the bettor are. If their opinions almost coincide, placing a bet doesn't make sense. A bettor believes in “0:0” outcome as well as a bookmaker does, even if a bet occur, a bookmaker will offer an amount to cash out at the very beginning, that amount will be acceptable for a bettor and will be equal to the price of the bet.

To understand it clearly see the following figures:

The Figure 4.2 shows how does the certainty equivalent change with time. Remember,

,

What in this specific case equals to:

.

The Figure 4.3 shows how does the expected payout of the bookmaker change over time.

Expected payout function is the following:

,

What in this concrete case equals to:

, where

It is shown that the lower the probability of a goal at each minute (according to the bookmaker's beliefs), the faster the expected payout function increases. Intuitively it means that the closer the beliefs of a bookmaker are to those of a bettor, the more expensive it is for a bookmaker to wait to offer the cash out, what means that he wants to interrupt the bet as soon as possible.

Appendix 5

Suppose that bettor's and bookmaker's subjective probabilities of a goal at each minute are such that

,

or

Let's take some numbers to demonstrate this case properly.

Assume that initially a bettor has , he makes a stake with odds .

This means that if the outcome will be the one he placed bet on, the wealth of the bettor will be:

.

If, the outcome will be the one that was not desired by the bettor, his wealth will be:

.

Consider the case, when a bettor subjectively believes that there is a probability of a goal at each minute of the game which equals to , i.e. there is a 0,56% chance that a goal will take place at each minute of the game.

According to the constraint given in this situation, believes of the bookmaker have to be the following:

Bookmaker believes that there is a probability of a goal at each minute which is higher than .

Figure 5.1 shows how the subjective probabilities could be located in this situation. The blue line represents bettor's subjective probability of a “0:0” score as the score at the end of the game at each minute during the game if no goal took place before that minute. The red lines show the same probabilities for a bookmaker, the higher the probability of a goal at each minute according to his beliefs (), the lower is the red line.

It is clear from the picture that the higher , the more distant the beliefs of the bookmaker and the bettor are. If their opinions are so different, so a bookmaker is very confident in his beliefs that the score will not be “0:0” in the end and he will not desire to offer a cash out and to end the bet before the final whistle of a referee.

To understand it clearly see the following figures:

The Figure 5.2 shows how does the certainty equivalent change with time. Remember,

What in this specific case equals to:

.

The Figure 5.3 shows how does the expected payout of the bookmaker change over time.

Expected payout function is the following:

,

What in this concrete case equals to:

, where

It is shown that the higher the probability of a goal at each minute (according to the bookmaker's beliefs), the faster the expected payout function decreases. Intuitively it means that the more disctinct the beliefs of the bookmaker and of the bettor are, the more expensive it is for a bookmaker to offer the cash out soon, what means that he doesn't want to interrupt the bet and wants to wait as long as possible. Hence, the cash out amount that will be available for the customer will be so low, that a bettor will not accept it, and the bet will hold until the end of the event.

Appendix 6

Suppose that bettor's and bookmaker's subjective probabilities of a goal at each minute are such that

,

or

Let's take some numbers to demonstrate this case properly.

Assume that initially a bettor has , he makes a stake with odds .

This means that if the outcome will be the one he placed bet on, the wealth of the bettor will be:

.

If, the outcome will be the one that was not desired by the bettor, his wealth will be:

.

Consider the case, when a bettor subjectively believes that there is a probability of a goal at each minute of the game which equals to , i.e. there is a 0,56% chance that a goal will take place at each minute of the game.

According to the constraint given in this situation, believes of the bookmaker have to be the following:

Bookmaker believes that there is a probability of a goal at each minute which equals .

Figure 6.1 shows how the subjective probabilities could be located in this situation. The blue line represents bettor's subjective probability of a “0:0” score as the score at the end of the game at each minute during the game if no goal took place before that minute. The red line shows the same probability for a bookmaker.

The Figure 6.2 shows how does the certainty equivalent change with time. Remember,

What in this specific case equals to:

.

The Figure 6.3 shows how does the expected payout of the bookmaker change over time.

Expected payout function is the following:

,

What in this concrete case equals to:

It is shown that the expected payout is constant over time. It means that bookmaker is indifferent whether to offer cash out and interrupt the betting or to wait until the end. Hence, he will offer the acceptable for the bettor amount of money from the very beginning of the event. As bettor is risk-averse he desires to avoid uncertainty when it is possible. Thus, if the acceptable amount will be offered in the very beginning of the game, a bettor will accept it (actually, it will be the price of the bet) in order to avoid uncertainty.

It is clear that in this case the “cash out” option will be senseless as the commitment between the bettor and the bookmaker will take place at the 0th minute.

Appendix 7

,

The derivative of the expected payout function is the following:

.

,

=0 as is a constant.

.

Thus, equals to:

,

which is the same as

.

Appendix 8

The sign of the expression above has to be analyzed.

, which corresponds to

dividing both sides by :

vs ,

and this is the same as

vs

Appendix 9

The derivative of the has to be found.

Let's take some number to simplify calculations.

Assume that initial wealth is , the stake is , and the odds . A bettor subjectively believes that the probability of a goal at each minute of game equals .

The certainty equivalent is now the following:

or, simply:

Figure 9.1 shows the certainty equivalent of a bettor in this concrete case.

A bookmaker has to pay

, this amount is shown in the Figure 9.2:

The derivative of the CE function is the following:

Figure 9.2 shows that the derivative of the certainty equivalent function is also increasing.

What has to be analyzed is the ratio:

Figure 9.4 demonstrates that this ratio is almost constant and is approximately equal to 0.006:

Appendix 10

Consider the situation observed in Appendix 9.

Assume that initial wealth is , the stake is , and the odds . A bettor subjectively believes that the probability of a goal at each minute of game equals .

There was found that in such case the ratio .

For the derivative of the bookmakers' expected payout function to be positive, the following inequality must hold:

,

,

Thus, when a bookmaker believes that with probability more than 0.6% there will be a goal each minute, then the expected payout amount of “cash out” will be increasing in time and he will decide to end the bet at the very beginning, i.e. at 0th minute.

When , the opposite situation occurs, the expected payout amount of “cash out” decreases in time, the minimum amount will be achieved at the very end of the event, i.e. at 90th minute.

When , a bookmaker is indifferent as the expected payout is constant over time.

It means that bookmaker is indifferent whether to offer cash out and interrupt the betting or to wait until the end. Hence, he will offer the acceptable for the bettor amount of money from the very beginning of the event. As bettor is risk-averse he desires to avoid uncertainty when it is possible. Thus, if the acceptable amount will be offered in the very beginning of the game, a bettor will accept it (actually, it will be the price of the bet) in order to avoid uncertainty.

It is clear that in this case the “cash out” option will be senseless as the commitment between the bettor and the bookmaker will take place at the 0th minute.

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