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Pascal and Fermat (1654)

Blaise Pascal and Pierre de Fermat’s correspondence in a series of letters over the summer of 1654 represents the starting point for mathematical probability. The two men were the pre-eminent mathematicians of their generation (Newton and Leibnitz came a little later), and they achieved lasting fame for their work in the physical sciences and number theory respectively. They initiated a long line of seventeenth and eighteenth century mathematicians and physicists of the highest order who made fundamental contributions to probability and statistics whilst dedicating most of their time to other areas of scientific research. Halley, De Moivre, Gauss and Laplace certainly fall under this category. Leibnitz might also be added, though his contributions to probability were less direct.

Pascal and Fermat’s letters discussed solutions to what was known as the ‘problem of points’. Suppose we have a game of chance between two players where the winner is the first player to reach a specified number of points and the winner receives the two stakes of the players. The two players play consecutive rounds until one player reaches the required number of points; in each round one player wins one point, and we assume the players have equal probability of winning any round. If the players agree to stop playing part-way through their game, how should they split the stakes? To take the specific example used in the letters: if the stake to play the game is 32 pistoles, the winner is first to reach three points; the first player has two points and the second player has one point, so how should the total 64 pistoles be split between the two players?

The problem of points was not new—it was discussed at length in fifteenth century Italy and then again by Cardano in the sixteenth century—but a solution had continually eluded each period’s best mathematicians. It may appear surprising that such a seemingly trivial problem caused the finest mathematical minds of the day such difficulty. After all, the first half of the seventeenth century saw complex mathematical developments being applied in fields such as astronomy with startling results. But the maths of the problem of points was not the hard part: the type of thinking required to solve it was conceptually new.

Pascal and Fermat both independently arrived at the correct solution to the problem of points, albeit in subtly different ways. Both approached the problem by defining the mathematical expectation of each player’s pay-out from the game as the fair settlement. This mathematical formulation of expectation was the crucial conceptual breakthrough required for the problem of points and it represented new territory. The Pascal and Fermat letters provide the first recorded instance of mathematical expectation being explicitly calculated. But neither of them used the term ‘expectation’ in their correspondence. It was the Dutch mathematician, Christiaan Huygens, who first introduced the term ‘expectatio’ in the Dutch translation of his 1657 paper on why it was appropriate to use mathematical expectation to price claims on uncertain cashflows (he focused mainly on fair lotteries).[1] Pascal and Fermat’s 1654 correspondence was not published until 1679, but Huygens visited Paris in 1655 and, whilst he did not meet Pascal or Fermat, he was introduced to their ideas by mutual acquaintances.

The idea of using mathematical expectations to value uncertain claims can be viewed as a more technical treatment of the already established legal doctrine of equity, and its application to the treatment of aleatory contracts.[2] The term ‘aleatory’ referred to contracts where there was a settlement of a fixed payment today in exchange for uncertain future cashflows. Insurance and annuities fell under this category, and so too did a wider array of contractual arrangements such as settlement of inheritance expectations or, more generally, risky business investments. Like all contract law, a principle of fair and equitable treatment for both parties existed. For aleatory contracts, a qualitative, heuristic notion of expectation was already part of established law in the mid-seventeenth century. This did not make explicit use of probabilities, but did recognise degree of likelihood as an important factor in determining equitable contract settlements. From here, the development of mathematical expectation can be viewed as a natural quantitative development.

Fermat’s solution to the calculation of mathematical expectation was to tabulate all the possible (and equally likely) combinations of wins/losses that could occur over the remainder of the game. He then worked out the probability of each player winning by enumerating the winning combinations and dividing by the total possible number of combinations. The equitable split of the stakes was then found as the winning probability multiplied by the total stake, i.e. the expected pay-off from the game. This basic concept of aleatory probability as a ratio of the number of equally possible favourable events to the total number of equally possible events was not new—as we saw above, it was used by Cardano and Galileo a century earlier. The breakthrough was in the conception of mathematical expectation—the probability-weighted value— and its use as a measure of fair value. But from a mathematical perspective,

Fermat’s approach is unexciting. Furthermore, the process of tabulating each possible combination would be increasingly computationally cumbersome as the number of players or the number of points required to win the game became larger.

Pascal’s proposed method was essentially mathematically identical to Fermat’s, but had a subtle difference in implementation which was arguably quite profound. Pascal’s solution used a method of backward recursion to calculate the mathematical expectation that applied at any point in the game. Todhunter’s translation5 of Pascal’s explanation of his approach is worth recounting in full:

Suppose that the first player has gained two points and the second player one point; they have now to play for a point on this condition, that if the first player gains he takes all the money which is at stake, namely 64 pistoles, and if the second player gains each player has two points, so that they are on terms of equality, and if they leave ofl^ playing each ought to take 32 pistoles. Thus, if the first player gains, 64 pistoles belong to him, and if he loses, 32 pistoles belong to him. If, then, the players do not wish to play this game, but to separate without playing it, the first player would say to the second ‘I am certain of the 32 pistoles even if I lose this game, and as for the other 32 pistoles perhaps I shall have them and perhaps you will have them; the chances are equal. Let us then divide these 32 pistoles equally and give me also the 32 pistoles of which I am certain’. Thus the first player will have 48 pistoles and the second 16 pistoles.

Next, suppose that the first player has gained two points and the second player none, and that they are about to play for a point; the condition then is that if the first player gains this point he secures the game and takes 64 pistoles, and if the second player gains this point the players will then be in the situation already examined, in which the first player is entitled to 48 pistoles, and the second to 16 pistoles. Thus if they do not wish to play, the first player would say to the second ‘If I gain the point I gain 64 pistoles; if I lose it I am entitled to 48 pistoles. Give me then the 48 pistoles of which I am certain, and divide the other 16 equally, since our chances of gaining the point are equal’. Thus the first player will have 56 pistoles and the second player 8 pistoles.

Finally, suppose that the first player has gained one point and the second player none. If they proceed to play for a point the condition is that if the first player gains it the players will be in the situation first examined, in which the first player is entitled to 56 pistoles; if the first player loses the point each player has then a point, and each is entitled to 32 pistoles. Thus if they do not wish to play, the first player would say to the second, give me the 32 pistoles of which I am certain and divide the remainder of the 56 pistoles equally, that is, divide 24

pistoles equally. Thus the first player will have the sum of 32 and 12 pistoles, that is 44 pistoles, and consequently the second will have 20 pistoles.

This is the first record of a backward recursive method being used to evaluate expectations through a path of stochastic steps. Its appeal to Pascal lay in its relative computational efficiency and elegance, but, in retrospect, it offers a great insight into the behaviour of the prices of assets with claims on uncertain cashflows: his method provides insight not only into how to price the claim at any given point in time, but also on how that price will change as uncertain events crystallise. Those with a familiarity with standard option pricing theory will immediately recognise the similarity between Pascal’s logic and the binomial tree approach to option pricing[3] (which is used both to intuitively illustrate the mathematics of option pricing to the uninitiated and to provide solutions to path-dependent option pricing problems that are too complex for analytical treatment). But the path behaviour of asset prices was not of interest to Pascal—when he realised that the arithmetical triangle could efficiently identify the binomial coefficients required by Fermat’s ‘brute force’ method, he advocated using this approach to solving the problem of points and similar combinatorial problems.

In summary, Pascal and Fermat’ s solutions to the problem of points weaved together a handful of concepts that were new or at best half-baked at the time of their writing:

  • • Mathematical expectation as the probability-weighted sum of uncertain outcomes, where the probability is calculated by defining the set of exhaustive and equiprobable outcomes and counting the relevant sub-set.
  • • The principle that an equitable claim (price) on an uncertain cashflow should be set by assessing the mathematical expectation of the cashflow. This can be viewed as an explicit probabilistic rendering of the legal concept of equity that had been an established part of the civil law relating to ‘aleatory’ contracts and which was applied using heuristic judgement.
  • • Backward recursion as an efficient way of calculating expectations (and hence prices) through stochastic paths.
  • • The arithmetic triangle as an efficient computational means of producing binomial coefficients. Pascal did not invent the arithmetic triangle (Cardano discussed it in the sixteenth century, and the Chinese and other ancient civilisations are also thought to have known of it). But Pascal was the first to identify its use in binomial expansions and how that could be applied to probability problems.

Together, these ideas and insights represent a fundamental breakthrough in thinking, both with respect to mathematical probability and to the related topic of valuing claims on uncertain cashflows.

  • [1] Huygens (1657).
  • [2] See, for example, Section 1.3, Daston (1988) for a scholarly discussion of the legal doctrine of equityand its influence on the development of mathematical expectation.
  • [3] Cox et al. (1979).
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