Материал: Development of the system of sports betting

Figure 5.3 shows how does the expected payout of the bookmaker change over time.payout function is the following:

,

What in this concrete case equals to:

, where

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

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

,

 or

’s take some numbers to demonstrate this case properly.that initially a bettor has , he makes a stake  with odds .means that if the outcome will be the one he placed bet on, the wealth of the bettor will be:

.

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

.

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.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.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.payout function is the following:

,

in this concrete case equals to: 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.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:

is the same as

.

Appendix 8

The sign of the expression above has to be analyzed.

 , which corresponds to

dividing both sides by :

 vs ,

this is the same as

 vs

Appendix 9

The derivative of the  has to be found.’s take some number to simplify calculations.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 .certainty equivalent is now the following:

, simply:

Figure 9.1 shows the certainty equivalent of a bettor in this concrete case.bookmaker has to pay

, this amount is shown in the Figure 9.2:

derivative of the CE function is the following:

Figure 9.2 shows that the derivative of the certainty equivalent function is also increasing.has to be analyzed is the ratio:

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

Appendix 10

the situation observed in Appendix 9.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 .was found that in such case the ratio .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., 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., a bookmaker is indifferent as the expected payout is constant over time.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.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.