Imagine that you and I are playing a simple game of chance. We each throw $50 into a pot and start flipping a coin. Heads, you score a point; Right, I have one. The first person to reach 10 points walks away with the full $100. The game begins and the score is currently eight to six in your favor. Suddenly my phone rings: there is an emergency and I have to leave quickly. Now we have a problem. You don’t want to just give me back my $50 because you win. But I’m reluctant to give you the whole pot because I still have a chance to hit a lucky streak and stage a comeback. What is the fairest way to share the money?
Known as the “points problem” or “stakes distribution problem,” this conundrum has baffled mathematicians for more than 150 years. And this, for good reason: probability theory had not been invented when the problem was first posed. Two great names in 17th century mathematics, Blaise Pascal and Pierre de Fermatcorresponded about the problem in a famous series of letters. They not only discovered the correct way to split the pot, but also laid the foundations of modern probability theory. To date, the solution forms the basis of risk assessments of all kinds, helping us make smarter bets on everything from buying stocks to insuring a house along a coastline.
In 1494, the Italian mathematician Luca Pacioli first looked very early on the problem of points in his manualwhose title translates as Summary of arithmetic, geometry, proportions and proportionality. He proposed that players split the pot in proportion to the number of points they each had at the time of the interruption. In our running example, you have won eight of 14 flips so far. According to Pacioli’s solution, you would win eight-fourteenths of the pot, which equates to approximately $57.14. I would take the remaining six-fourteenths. The solution seems reasonable, but more than 50 years later, Niccolò Fontana “Tartaglia” noticed that it failed in cases where the points ratio between players was extreme. What if the interruption occurred after a single draw? Under Pacioli’s rule, the winner of this roll would win the entire pot, even if the game was far from decided. That would clearly be unfair – and the problem with points is finding a fair distribution.
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Tartaglia proposed an alternative method. Imagine that, in our hypothetical game, you are two rounds ahead. You have a fifth of the 10 throws needed to win. Because it’s one-fifth closer to the goal, Tartaglia estimated that you should get your entire stake back and take one-fifth of my stake: the original $50 you put in plus one-fifth of my $50, for a total of $60. This new approach seems to work more equitably, especially at the extremes. Now, if the game was interrupted after a throw, the winner of that throw would only take a tenth of their opponent’s bet instead of the whole thing. While Pacioli’s method rewards the winning player based on the size of his lead relative to the number of throws made so far, Tartaglia’s method rewards him based on the size of his lead relative to the total duration of the game. Tartaglia, however, doubted his own innovation: while writing“However the division is effected, there will be cause for dispute.” He believed that there was no perfect mathematical solution and that the problem was designed to arouse controversy. It turns out he was at least right to doubt his own solution. Imagine one player has 199 points and the other has 190 points in a game with a 200 point goal. Tartaglia would award the first player only nine two-hundredths of his opponent’s stake, or $2.25, even though his opponent would need 10 tails in a row to win. The first player’s meager payout hardly seems to reflect his high probability of winning at this stage of the game.
The debate came to nothing until the mid-17th century, when a French gambler and worldly intellectual enlisted the help of a mathematician. Blaise Pascal. Pascal immediately saw that the solution did not lie in the partition at the time of the interruption but in the future possibilities of the score, and he wrote to his friend and fellow mathematician Pierre de Fermat to help him prove it. Their correspondence resulted in two very unique approaches to the problem. Surprisingly, their disparate approaches always resulted in the same solution. This convergence sealed their confidence in their results, and mathematicians now agree that they have found the most equitable way to distribute the stakes.
Fermat’s solution was to examine all possible continuations of the game after the point where it was interrupted and count the number of these continuations that result in a victory for each player. A fair percentage of the total pot awarded to a player should be the percentage of possible futures in which that player wins the game. Consider the score of eight to six from our recent example game with a goal of 10 points; Fermat would note that the game must end within five draws. If the first player won one roll and the second won three, then they would be tied at nine to nine and the game would end on the next roll. If play stopped at this point, Fermat’s method of splitting the pot would list all the possible outcomes of those five coin flips and then count those that yielded 10 points for each player. In some of these possible futures, a player will win in fewer than five flips, but that’s okay: we can imagine that if the game ends early, players flip the coin a few more times just to make accounting easier. The figure below reveals the answer to our riddle. The first player wins in 26 of the 32 possible outcomes of the game, so he gets 26/32 = 81.25 percent of the pot, or $81.25.
Amanda Montánez
Fermat’s solution, although elegant, suffered from a major drawback: what if there were too many possible sequences to list? Even if there are only 20 flips left in our game, we would need to consider over a million imaginary futures to discover a fair distribution. Pascal came up with a brilliant answer and, in doing so, provided the first reasoning for what would become the concept of expected value, which remains a fundamental pillar of modern probability theory.
Pascal’s method begins with an uncontroversial statement: if the game is tied at the time of interruption, then both players must share the pot equally. If the score was nine to nine at the time of the interruption, each player would get $50 back. Now we work backwards from there. If the score was nine to eight in favor of the first player, Pascal’s approach would ask what would happen after one more throw. There would be a 50 percent chance that the leading player would win this toss, reach 10 points and win the entire pot. On the other hand, there would be a 50 percent chance that the other player won the flip and tied the game at nine to nine, which would mean they would have to split the pot. The winnings of the first player would amount on average to:
50 percent of $100 + 50 percent of $50 = $75
So, if the game is interrupted with a score of nine to eight, the first player should win $75. We can apply this type of reasoning recursively to determine the appropriate allocation for any situation.
The key is to consider what your fair winnings would be if another coin toss occurred and what they would be if another coin toss occurred. You then find the average of these two possibilities. With a score of nine to seven, the first player should win $87.50: an additional face would earn him $100, and an additional face would earn him $75, as this would be the case of nine to eight that we have just analyzed. With a score of nine to six, they would take $93.75. A score of eight to seven would earn $68.75, the average of their fair rating on a score of nine to seven with their fair rating on a score of eight to eight. And finally, with a score of eight to six, the first player should win: 50% of $93.75 + 50% of $68.75 = $81.25.
This is exactly the same solution as Fermat’s method. Fermat and Pascal had the same idea: equitable distribution depends on possible futures, and each possible future must be weighed according to its probability of realization. Today we recognize these equations as expected values or weighted averages of all possible future outcomes. Fermat listed these future outcomes exhaustively, considering every possible way for the next five draws. Pascal has developed a clever way of working backwards: you calculate the fair distribution when you have five coin flips remaining. based on the fair is divided with four coin tosses to be made, which you in turn calculate on the basis of three tosses to be made, and so on.
The concept of expected value was not limited to 17th-century board games. It is the mathematical engine that drives almost all modern risk assessments. When an actuary evaluates a life insurance premium, a Wall Street analyst evaluates a portfolio of stocks or a the player weighs the risks of a betthey perform exactly the same calculation. They multiply the financial effect of each possible scenario by its probability, then calculate the sum of these results to quantify the value of a decision. Uncertainty is inevitable, and we owe much of our current technological stature to our ability to rigorously address it. For millennia, mathematicians have dealt with problems of chance using unsystematic conjectures. The correspondence of Pascal and Fermat replaced these conjectures with a framework. Even if we still can’t predict the future, we at least know how to price it.