From the Origins to the St Petersburg Paradox
The idea of probability originates with gambling. The so-called “classical theory of probability” was developed in order to deal with the particular objective uncertainty in the games of chance.
Its history was masterfully reconstructed by Ian Hacking (1975). Mathematical theory of probability is generally taken to begin in 1654 with a correspondence between Blaise Pascal and Pierre de Fermat where some gambling problems were analysed. In 1657 Christiaan Huygens published a Libellus de Ratiociniis in Ludo Aleae (On Calculations in the Game of Dice) that “for nearly half a century... was the unique introduction to the theory of probability” (David 1962: 115). In this Libellus the notion of expectation in gambling was clearly defined in reference to the problem of the fair price for a gamble. The notion of fair price or of a fair entry fee for a game of chance was of practical importance for gamblers deciding to participate in a bet; or for when it was necessary to divide a stake because the gamble was interrupted before its conclusion; or again for when a gambler was asked by another to sell his or her place in the gamble. The mixture of money and chance was considered a natural idea, and the basic tenet of “equi-possibility”, as it was called later by Pierre-Simon de Laplace, was based on the symmetry (fairness) of the device used for a gamble. It was therefore possible to consider only games with even chance (even lay). The interest for these problems most likely originated from a robust economic incentive. The rough technology used for the construction of the gambling machine (dice, roulette wheels, and so on) allowed professional gamblers to systematically gain if they made their bets after having discovered the odds realized in a single device. The alternative reference point is the law of chance governing an ideally fair device. The notion of the expected value of a game of chance is the value of a game in a fair device. The fair price is then equated to the expected value of a fair game. This is Huygens’s postulate:That my Chance or Expectation to win any thing is worth just such a Sum, as wou’d procure me in the same Chance an Expectation at a fair Lay. As for Example, if any one shou’d put 3 Shillings in one Hand, without letting me know which, and 7 in the other, and give me Choice of either of them; I say, it is the same thing as if he shou’d give me 5 Shillings; because with 5 Shillings I can, at a fair Lay, procure the same even Chance or Expectation to win 3 or 7 Shillings. (Huygens 1657: 1)
At around the same time Pascal formulated a problem now considered as the first appearance of decision theory under uncertainty. The question “Either God is or he is not” is considered similar to the tossing of a (fair) coin which will come down head or tail; and “you must wager”:
Let us weigh up the gain and the loss involved in calling heads that God exists. Let us assess the two cases: if you win you win everything, if you lose you lose nothing. Do not hesitate then; wager that he does exist... here there is an infinity of infinitely happy life to be won. (Pascal 1670 [1985]: 154)
Pascal’s wager testified for the first time that it is possible in the calculation of the expected value of a game to exchange monetary value with the happiness derived from winning the game. This is the first conceptual step toward Daniel Bernoulli’s solution of the St Petersburg paradox.
In 1738 Daniel Bernoulli published in Latin his “Specimen Theoria Nova de Mensura Sortis” (“Exposition of a new theory on the measurement of risk”). In fact, the general problem tackled by Bernoulli is the classical one about the fair price of a gamble. Contrary to Huygens’ approach - the fair price of a game is its expected value - Bernoulli proposed to replace money with the advantage (Latin word: emolumentum) derived from money.
So, the fair price of the game became the emolumentum medium of the game. In the first English translation (1954) “emolumentum” became “utility”, and “emolumentum medium” became “moral expectation” giving a modern flavour to Bernoulli’s intuition:If the utility of each possible profit expectation is multiplied by the number of ways in which it can occur, and we then divide the sum of these products by the total number of possible cases, a mean utility [moral expectation] will be obtained, and the profit which corresponds to this utility will equal the value of the risk in question. (Bernoulli 1738 [1954]: 24)
All of the three cited authors lacked the explicit reference to the notion of probability. This is not surprising because they used the more basic idea of odds ratio. They had in mind something corresponding to the aleatory notion of possibility: in the ideal model of the game of chance it is possible to list all possible outcomes and to count the favourable outcomes. In the definition of expectation there was not probability, but the ratio of favourable to total possible outcomes. According to Hacking (1975), the idea of possible outcome was interpreted in terms of a physical possibility, or the physical propensity of each outcome to happen. The application of this kind of analysis was therefore confined to cases where it was possible to list all possible outcomes; each outcome having the same physical propensity to happen.
Laplace accomplished the treatment of these questions in terms of probability for good (Laplace 1812: 432-45). It was through Laplace’s work that the distinction between fortune physique (physical advantage) and fortune morale (moral advantage) became common in nineteenth-century probability treatises (Todhunter 1865). Since then, the notions of the expected value of a game and of expected utility may be written in its modern notation, replacing odds ratios with probability. Let xi be the monetary value - the Laplacean fortune physique - of the i-th outcome of a gamble and pi its probability; then the expected value of the gamble is E = 2,PX'l.
If we let u(xi) be the moral advantage derived - that is, the subjective evaluation in terms of utility or satisfaction - from the monetary value x,, it is possible to calculate the expected utility of the gamble as EU = 2 n= 1Piu (x).Bernoulli’s memoir is universally known for containing the St Petersburg paradox - so called because it was published in the Commentarii of the St Petersburg Academy. When the fair price of a gamble is equated to its expected value, and a particular gamble is considered, the following paradox emerges (the seminal contribution on this is Samuelson 1977): the particular gamble consists in repeatedly tossing a coin in the air; if heads comes up on the first toss, the gambler will receive 2 shillings; if it does not come up until the second toss, he will receive 4 shillings; heads only on the third toss will pay 8 shillings, and so on. The expected value for the n-th toss is 2,. 2n shillings; if the game continues indefinitely, the expected value of the game is infinite, given that E = 2 m⅛ 2n = ∞. The fair price of the game is then infinite, but simple observation tells us that “no one would be willing to purchase it at a moderately high price” (Bernoulli 1738 [1954]: 31). If, according to Bernoulli, the expected value is replaced by a bounded expected utility, the paradox vanishes. Bernoulli proposed the following model where a person possessing a certain quantity of money w, receives an additional quantity dw. Bernoulli maintained that the relative value of this increase is directly proportional to dw and inversely proportional to w, that is du = kdWW where k is a constant. From this derives the logarithmic function of the fortune morale, that is, u = a + k log w. When this formula is used, the fair price of the St Petersburg game becomes finite; a reasonable result for thought experiments with gambling agents.