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One of the most important contributions to the relation between game theory and

probability at the end of the 17
th
century is probably that of Bernoulli, particularly in

his topic
Ars Conjectandi
(published in 1713). Bernoulli suggested as early as 1680

a mathematical theory of probabilities and their combinations. He uses game theory

to calculate probabilities, but does not remove from probabilities their epistemolog-

ical aspect and their role in the judgment of individuals. Thus, the first formalisms

that allowed numerical probabilities to be used focused on the study of subjective

probabilities! For the most part, Bernoulli's theory focuses on probabilities that are

subjective, and that are means of measuring knowledge. They are calculated based on

the concept of “arguments”, and their properties depend on the nature of these argu-

ments. In particular, they are not always additive. The combination rules suggested by

Bernoulli, which are more complete than Hopper's, take on different forms, depend-

ing on the arguments, and not all of these forms correspond to the usual probabilistic

rules. In the last part of his work, Bernoulli established his famous law of large num-

bers. This law allows a prior unknown probability to be estimated afterwards based

on the observation of occurrence frequencies. Therefore, this theorem has more to do

with random probabilities than epistemological probabilities (according to the distinc-

tion later introduced by Lambert), since a chance that can only be known
a posteriori

is not initially a characteristic of our knowledge.

The successors of Bernoulli simplified his theory and unknowingly reduced its

scope. First of all, they were not convinced by Bernoulli whose theory, which often

seemed complicated to them, was not as well established as game theory. Addition-

ally, they mostly held on to the law of large numbers and identified probability with

the chance of appearing, thus leading to an essentially frequentist approach. We can

mention, among the successors of Bernoulli, Montmort (
Essai d'analyse sur les jeux

de hasard
, 1708), who attempts to apply game theory to other fields, and Moivre (
De

Mensura Sortis
, 1711,
Doctrine of Chances
, 1718) where we find the first explicit

additivity rule and a representation of probabilities between 0 and 1. His definition

became the classical definition. These two authors mentioned probabilities but their

theory often deals with chance. They defined probabilities as the ratio of the number

of favorable cases to the number of possible cases, but the authors were faced with the

problem of counting the number of cases, which is not possible in every field. Note

that Moivre had already discovered the Gaussian distribution.

In the 18
th
century, only Lambert pursued Bernoulli's work and distinguished ran-

dom probabilities and epistemological probabilities. The former are those that can be

known
a priori
, as in game theory, or
a posteriori
, provided by experience. The latter

are assigned to events by inferences based on effects or circumstances and are more

subjective in nature. In
Photometrica
(1760), he focused on error theory and suggested

a method known today as maximum likelihood. In
Neues Organon
(1764), he general-

ized Bernoulli's argument theory, corrected and generalized his combination laws and

discussed the cases of games, syllogisms and of different types of testimonies. His

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