The problem of the unfinished game is famous in the history of mathematics. It deals with the question of how to fairly divide the "kitty" or "pot" among gamblers when a series of games of chance must be quit before it's finished. The game of chance could be dice rolls, coin tosses, pulling cards from a deck—anything having an outcome completely under the influence of the laws of probability.
In addition, there must be a series of games, such as "best two-out-of-three" or "best of seven." The problem arises when one contestant must abandon the game before its end. Of course, if the gamblers agree to simply return their bets, then there is nothing to discuss. Is there a mathematically correct and fair way to divide the winnings even if the game is incomplete?
Keith Devlin is highly qualified to write on this topic and the great mathematicians who worked on it. Devlin is (among other things) Executive Director of the Stanford University Center for the Study of Language and Information, a consulting professor in Stanford's mathematics department, and he is the "Math Guy" for National Public Radio programming. He writes pretty well too.
The Beginnings of Probability Research
Games of chance have been played since ancient times. There were some tabulations of all possible outcomes from dice roles as early as the tenth century. Yet, no one had been able to use (or apparently even considered using) such information to calculate probabilities of future outcomes in such situations.
Chance was considered unapproachable by scientific means. Things just happened; be it by the hand of God, simple "luck," or under the control of other mysterious and unfathomable laws. There was no way to predict future events.
In 1494, Italian mathematician Luca Pacioli published Everything About Arithmetic, Geometry, and Proportions. It was in this work that Friar Luca posed "the problem of the points" which we now know as the problem of the unfinished game. It would be two centuries before anyone took a serious shot at it.
Fermat and Pascal Take Up the Problem
The story begins in 1654 with a letter from Blaise Pascal to Pierre de Fermat. These two Frenchmen are towering figures in the history of mathematics. Actually, the whole book is about this correspondence and its implications. Pascal wrote to Fermat, because he thought he could pierce the veil, and he wanted Fermat's opinion. Pascal had an approach that he thought would lead to a calculable solution to the "problem of the points."
Bit by bit, Devlin reveals Pascal's letter and Fermat's response. Without the author's help, the correspondence would not necessarily make an interesting read. However, Devlin takes us through it one piece at time, interpreting it and bringing in additional information, allowing the reader to appreciate both the historical and modern importance of the work.
The problem of the points is indeed solved, but not in the way one might think. It seems like an easy problem. There is more to it that merely dividing the pot according to the proportion of wins and losses at the time the game is suspended.
The back and forth between these two mathematicians, who had tremendous respect for each other, really is how mathematics is done. Truly important and interesting things take place behind the scenes and out of sight. The Unfinished Game is an accessible and intimate look into this seldom seen world. Besides, you never now when it might come in handy to know how to properly resolve an unfinished game.
References
The Unfinished Game: Pascal, Fermat, and the 17th-Century Letter that Made the World Modern, 2008, Keith Devlin, Basic Books, NY, ISBN 0465009107
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