Abstract
In fixed-odds numbers games, the prizes and the odds of winning are known at the time of placement of the wager. Both players and operators are subject to the vagaries of luck in such games. Most game operators limit their liability exposure by imposing a sales limit on the bets received for each bet type, at the risk of losing the rejected bets to the underground operators. This raises a question—how should the game operator set the appropriate sales limit? We argue that the choice of the sales limit is intimately related to the ways players select numbers to bet on in the games. There are ample empirical evidences suggesting that players do not choose all numbers with equal probability, but have a tendency to bet on (small) numbers that are closely related to events around them (e.g., birth dates, addresses, etc.). To the best of our knowledge, this is the first paper to quantify this phenomenon and examine its relation to the classical Benford’s law. We use this connection to develop a choice model, and propose a method to set the appropriate sales limit in these games.
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Notes
The BBC news article on this can be accessed at http://news.bbc.co.uk/1/hi/uk/6174648.stm.
In fact, China Mobile’s Jiangxi branch held an auction to sell a “lucky” phone number recently, and one such number-with six consecutive eights—was sold for RMB 44,000!
Interestingly, our data suggests that players in Pennsylvania have an aversion to the digit 2, but favor digits 7 and 8.
Note that this simplifying assumption may not hold in general, as some players may pad the numbers with trailing zeros, and some may simply duplicate the numbers to reach a 3-digit number. Halpern and Devereaux (1989) mentioned that triplets like 111 or 888 are very popular in the Pick-3 game in Pennsylvania. Unfortunately, it does not appear possible to incorporate such features into the model, without sacrificing the simplicity and tractability of the calibration model.
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Chou, M.C., Kong, Q., Teo, CP. et al. Benford’s Law and Number Selection in Fixed-Odds Numbers Game. J Gambl Stud 25, 503–521 (2009). https://doi.org/10.1007/s10899-009-9145-9
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DOI: https://doi.org/10.1007/s10899-009-9145-9