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Subjective beliefsis 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 equivalentfacing 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. .6 Conclusion

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 .1 The main assumptions of the model

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.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:

. Nothing “significant” happens during the game, nothing that can affect the beliefs significantly.

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 eitheror

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

. Bookmaker is risk-neutral

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

. 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 isand a bookmaker believes that it is .

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

. 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: , wherefor a bettor and for a bookmaker, respectively.

. 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:

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.

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

this equation the CE can be found:

, 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:

.

.2 Extreme case analysis

’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:

,

a bookmaker has to pay the minimum amount which equals to .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.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.probability that there will be no goal until some point of time t is the

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