My Problem with the Problem of Points
In his lecture for Google’s series, “Talks at Google”, Keith Devlin, author of The Unfinished Game, begins with a brief history of the formulation of the concept of numbers throughout humanity. Perhaps not surprisingly, Devlin explains that numbers were first developed as a more convenient way to represent wealth or quantity of possessions. It is almost impossible now to imagine a world without the concept of numbers, or to imagine that it took as long as it did to come up with such a system, given the fact that it’s one of the first concepts at which modern day toddlers can master the basics. Nevertheless, though the revolution that accompanied the invention of numbers as a concept was undoubtedly one of the most important breakthroughs of our species, Devlin’s focus is on a subsequent revolution brought upon by the solution to “The Unfinished Game”, or the Problem of Points.
The situation is as follows: two competitors are in the middle of a game of chance when their game is interrupted. Though there are several iterations of how exactly this game is played or what exactly the scores were at the point of interruption, what is agreed upon is that one player is ahead of the other, and that the result of each point is generated randomly and independently of any previous points. The problem is that the two competitors had placed a wager at the beginning of the game and are unsure of how to divvy it up, since they were unable to finish their game. Do they just split it 50/50, or is that unfair to the player who was ahead? Does the actual score matter, or how close the lead is?
For the solution, we look to Fermat and Pascal, two 17th-century mathematicians. Though earlier mathematicians had attempted solutions, none had given a fully satisfying answer, and some had even deemed it impossible to solve. The supposed impossibility rested on the assumption that we cannot assign a numerical value to an event in the future. Numbers, up until this point in history, had never been used in such a way, and to many of the brightest minds of their time, this application was completely unfathomable.
Nowadays, within the psychological comfort of hindsight, the solution may seem simple, and perhaps even obvious to some. Say the game was flipping a coin, and the first player to ten wins. If Fermat leads Pascal 8–7 at the time their game was interrupted, they determined that the maximum number of rounds until the winner would definitively be decided is 4, calculated by adding the number of remaining points each player needs to win (2 for Fermat, 3 for Pascal) and subtracting 1. The complete list of possible outcomes are as follows, using “h” to denote “heads”, and “t” to denote “tails”, and wins for Fermat (playing heads) are indicated with an asterisk:
Next, the number of outcomes that result in a win for Fermat are tallied and divided by the total number of possible outcomes, so 11/16, and that ratio is used to divvy up the pot of money.
Though I’m sure many people would have no problem accepting this explanation, I initially struggled with the fact that not all four remaining rounds are guaranteed to happen. For instance, if the coin lands on heads the first two times, Fermat has won and the game is over. For the first row of outcomes on the table above, the last two rounds’ flips are irrelevant to determining the winner, and therefore my intuition was that the outcomes that begin with two “h”s shouldn’t be counted as distinct outcomes.
I was mistaken, of course, in thinking that this table considers conditional probability of a player winning given the previous coin flips. The conditional probability of Fermat winning any of the games in the first row after the first two rounds is 1, since it is a guaranteed event. One might think (as I did) that because the odds of a player winning at any point given the score at that point change from round to round and game scenario to game scenario, that that would effect the number, or probabilistic effect, of the overall outcomes. In reality, the two calculations are separate from each other, and the distinct possible outcomes, though not necessarily distinct wins, do not take into account a winner, or when the win occurs in the sequence.
My own mathematical hang-ups aside, this method of calculation essentially introduced probability theory to the world. Though now we take it for granted, the idea of assigning a numerical value to the likelihood of an event occurring in the future is truly revolutionary.
