The probability of the correct majority made decision

Abstract

Aim: the probability of correctness of collective decision is studied in this paper, whereby the decision is made by majority vote of some team (group), consisting of an odd number of members, provided that the probability of correctness of individual decision of each member of the group is known.Мethods: the Bernoulli scheme, asymptotic representation, estimation by virtue of geometric progression, exponential series expansion, Wallis' product, a power scale of averages, Kolmogorov mean.Result: it was found, that if for each member of the group the probability of the correct decision is more than ½, then with an unlimited increase of the number of members of the group, the probability of the right collective decision tends towards one. The asymptotic representation and a number of two-sided assessments which characterise the speed of this tending were obtained. For a non-homogeneous group (i.e. the group the members of which make the right individual decision with different probability) the notion of a collective average was introduced here as an averaged characteristic that can be used to replace individual probability of each group member saving the probability of the right collective decision. The existence and uniqueness of a collective average was proved.A collective inequality was identified which shows that a collective average of some set of numbers is no less than the geometric mean of the same numbers, and the equality is present if and only if all members are equal at that. A collective inequality serves as analogue and addition to the known set of inequalities establishing connection between two different average values (for instance, the AM-GM inequality).Conclusion: thus, the results of the study fully meet the aim of determining the probability of correct decision made by a majority of votes under the assumptions taken. As a result, asymptotic representation and bilateral estimates characterising the speed of tending to the correct decision was obtained. For a non-homogeneous group, the existence and uniqueness of the concept of collective average as an averaged characteristic were introduced and firmly proved, which can be used to replace an individual probability of each group member, whereby preserving the probability of correctness of the collective decision. It was found that the collective average is no less than the geometric mean. Potential applications of the results obtained can be the quantitative evaluation of election procedures and the solution of problems associated with improving the reliability of recognition of weak signals of control sensors in various transport systems, including high-speed transport systems on magnetic suspension. 1 Санкт-Петербургский государственный университет 2 Петербургский государственный университет путей сообщения Императора Александра I (Санкт-Петербург, Россия) ВЕРОЯТНОСТЬ ПРАВИЛЬНОСТИ РЕШЕНИЯ, ПРИНИМАЕМОГО БОЛЬШИНСТВОМ ГОЛОСОВ Цель. Исследуется вероятность правильности коллегиального решения, которое принимается большинством голосов некоторого колле...