Subjective Probability and Mercantile Speculations

Despite Laplace’s advancement, the applicability of expectations (mathematical or moral) continued to be limited to gambling or to situations where it was possible to define probability as in games of chance.

Antoine-Augustin Cournot (1843) in the Exposition de la Theorie des Chances et des Probabilites developed a pluralist view of probabilities. He maintained the notion of “mathematical probability”, traditionally associated with gambling, defined as the ratio of favourable outcomes to total outcomes, and put it side by side to a strictly subjec­tive interpretation of probability, depending on the status of imperfect knowledge of an individual (Cournot 1843: 438). This kind of probability governs the rules regarding the stakes in a “marche aleatoire” (aleatory market). He explicitly introduced, perhaps for the first time, the modern notion of lottery as the general device to treat problems of choice under uncertainty (Cournot 1843: 89-90). He suggested that in a lottery where the prize is a generic good, every ticket may be considered as an “eventual right” over the good. Every ticket obviously may be sold in a market and its fair price (“valeur venale”) is its mathematical expectation, where probability is defined in reference to the subjective status of knowledge of agents.

In the same time period in England Augustus de Morgan explicitly used mathemati­cal expectation as a device to model “mercantile speculations” and “every species of affair in which no absolute certainty exists” (De Morgan 1838: 98). For de Morgan the nature of probability was also very different from the one proposed in classical theory: “Probability is the feeling of the mind, not the inherent property of a set of circum­stances” (De Morgan 1838: 7), and this probability is different for different persons depending on the status of their knowledge and “impressions”. This enlargement of the notion of probability paved the way to its rigorous application to all “questions involving loss and gain” (De Morgan 1838: 103): that is, not only to gambling but also to matters of “commercial speculation” (De Morgan 1847: 404) and principally to prob­lems regarding insurance offices. The passage from the notion of mathematical expecta­tion to that of “moral expectation” involves, according to De Morgan, a supplementary problem concerning “the temperament of the individual”, that is, in modern jargon, the question of risk propensity and aversion:

different persons will look forward in the same circumstances with different degrees of hope. One man will consider himself better off than before when he has bartered one pound certain for an even chance of two; a second will contemplate loss more strongly than gain, and will consider himself damnified by the exchange. (De Morgan 1847: 409)

Indeed, according to De Morgan the main problem in practical applications arises not from the side of utility, but from the side of probability: when the probability of an outcome is very small, and benefits depend upon this vanishing probability, then “the mathematical expectation is not a sufficient approximation to the actual phenomenon of the mind, even when the fortune of the player forms no part of the consideration” (De Morgan 1847: 409). In the St Petersburg paradox, for example, neither mathemati­cal expectation nor moral expectation may be considered as the fair price of the game, because both depend on very small probabilities. This intuition is similar to the notion of weight of probability in prospect theory (see below).