image

The author is a CS professor at SUNY, Stony Brook. This book recounts his experience of building a mathematical system to bet on the play outcomes of what is considered the fastest ball game in the world, “Jai alai”. In the English vernacular this is sometimes spelled as it sounds,that is, “hi-li”.  The book recounts the history of the game and how it made to US from Spain and France. However the focus of the book is on using mathematical modeling and computers to analyze the game and design a betting system. The game itself is designed in such a way that it is a textbook case for analyzing the game mathematically. The players enter the competition based on FIFO queue and the player who gets to score 7 points is the winner. It takes hardly a few minutes to understand the game from this wiki.

With the help of some of his grad students, the author works on the following questions :

  • Given a player starts in a specific position, what is probability that he ends up in a Win/Place/Show ?
  • What are the best combination of numbers that have the highest probability of winning a Trifecta ?
  • How does one build a statistical model to evaluate the relative skills of the players ?
  • Given that two players A and B have probabilities of winning as pb and pb, How does one construct a model that evaluates the probability of A winning over B ?
  • How does one create a payoff model for the various bets that are allowed in the game ?
  • How do you deal with missing  / corrupt data ?
  • Given the 1) payoffs of various bets, 2) the probabilities of a player winning from a specific position, and 3) the relative skillsets, how does one combine all of these elements to create a betting strategy ?

I have just outlined a few of the questions from the entire book. There are numerous side discussions that makes the book a very interesting read. Here is one of the many examples from the book that I found interesting :

Almost every person who learns to do simulation comes across Linear congruential generator(LCG), one of the basic number theory technique to generate pseudo random numbers. It has the following recursion form :

By choosing appropriate values for a, c and n, one can generate pseudo random numbers.

The book connects the above recursive form to a roulette wheel :

Why do casinos and their patrons trust that roulette wheels generate random numbers? Why can’t the fellow in charge of rolling the ball learn to throw it so it always lands in the double-zero slot? The reason is that the ball always travels a very long path around the edge of the wheel before falling, but the final slot depends upon the exact length of the entire path. Even a very slight difference in initial ball speed means the ball will land in a completely different slot.

So how can we exploit this idea to generate pseudorandom numbers?A big number (corresponding to the circumference of the wheel) times a big number(the number of trips made around the wheel before the ball comes to rest) yields a very big number (the total distance that the ball travels). Adding this distance to the starting point (the release point of the ball) determines exactly where the ball will end up. Taking the remainder of this total with respect to the wheel circumference determines the final position of the ball by subtracting all the loops made around the wheel by the ball.

The above analogy makes the appearance of mod operator in LCG equation obvious.

One does not need to know much about Jai-alai to appreciate the modeling aspects of the game and statistical techniques mentioned in the book. In fact this book is a classic story of how one goes about modeling a real life scenario and profiting from it.

Advertisements