Bingo pricing: a game simulation and evaluation using the derivatives approach

The Bingo game is well known and played all over the world. Its main feature is the sequential drawing without repetition of a set of numbers. Each of these numbers is compared to the numbers contained in the boxes printed on the different rows (and columns) of the score-cards owned by the Bingo participants. The winner will be the participant that firstly is able to check all the boxes (numbers) into a row (Line) or into the entire score-card (Bingo). Assuming that the score-card has a predetermined purchase price and that the jackpot is divided into two shares, respectively for the Bingo and the Line winner, it is evident that all the score-cards show the same starting value (initial price). After each drawing, every score-card will have different values (current price(s)) according with its probability to gain the Line and/or the Bingo. This probability depends from the number of checked boxes in the rows of the score-card and from the number of checked boxes in the rows of all the other playing score-cards. The first aim of this paper is to provide the base data structure of the problem and to formalize the needed algorithms for the initial price and current price calculation. The procedure will evaluate the single score-card and/or the whole set of playing score-cards according to the results of the subsequent drawings. In fact, during the game development and after each drawing, it will be possible to know the value of each score-card in order to choose if maintain it or sell it out. The evaluation will work in accordance to the traditional Galilee's method of "the interrupted game jackpot repartition". This approach has been also mentioned by Blaise Pascal and Pierre de Fermat in their mail exchange about the "jackpot problem". More advanced objective of the paper would be the application of the stock exchange techniques for the calculation of the future price of the score-card (and/or of a set of score-cards) that will have some checked numbers after a certain number of future drawings. In the same way will be calculated the value of the right to purchase or sell a score-card (and/or of a set of score-cards) at a pre-determined price (option price). Especially during the prototyping phase, the modelling and the development of these kind of problems need the use of computational environments able to manage structured data and with high calculation skills. The software that meet these requirements are APL, J and Matlab , as for their capability to use nested arrays and for the endogenous parallelism features of the programming environments. In this paper we will show the above mentioned issues through the use of Apl2Win/IBM . The formalisation of the game structure has been made in a general way, in order to foresee particular cases that act differently from the Bingo. In this way it is possible to simulate the traditional game with 90 numbers in the basket, 3 rows per 10 columns score-cards, 15 number for the Bingo and 5 numbers for the Line but already, for example, the Roulette with 37 (or 38) numbers, score-cards with 1 (or more) row and 1 column and Line with just 1 number.