Probabilistic Infinite Secret Sharing

The study of probabilistic secret sharing schemes using arbitrary probability spaces and possibly infinite number of participants lets us investigate abstract properties of such schemes. It highlights important properties, explains why certain definitions work better than others, connects this topic to other branches of mathematics, and might yield new design paradigms. A probabilistic secret sharing scheme is a joint probability distribution of the shares and the secret together with a collection of secret recovery functions for qualified subsets. The scheme is measurable if the recovery functions are measurable. Depending on how much information an unqualified subset might have, we define four scheme types: perfect, almost perfect, ramp, and almost ramp. Our main results characterize the access structures which can be realized by schemes of these types. We show that every access structure can be realized by a non-measurable perfect probabilistic scheme. The construction is based on a paradoxical pair of independent random variables which determine each other. For measurable schemes we have the following complete characterization. An access structure can be realized by a (measurable) perfect, or almost perfect scheme if and only if the access structure, as a subset of the Sierpi\'nski space $\{0,1\}^P$, is open, if and only if it can be realized by a span program. The access structure can be realized by a (measurable) ramp or almost ramp scheme if and only if the access structure is a $G_\delta$ set (intersection of countably many open sets) in the Sierpi\'nski topology, if and only if it can be realized by a Hilbert-space program

Non-parametric probability distributions embedded inside of a linear space provided with a quadratic metric

There exist uncertain situations in which a random event is not a measurable set, but it is a point of a linear space inside of which it is possible to study different random quantities characterized by non-parametric probability distributions. We show that if an event is not a measurable set then it is contained in a closed structure which is not a σ-algebra but it is a linear space over R. We think of probability as being a mass. It is really a mass with respect to problems of statistical sampling. It is a mass with respect to problems of social sciences. In particular, it is a mass with regard to economic situations studied by means of the subjective notion of utility. We are able to decompose a random quantity meant as a geometric entity inside of a metric space. It is also possible to decompose its prevision and variance inside of it. We show a quadratic metric in order to obtain the variance of a random quantity. The origin of the notion of variability is not standardized within this context. It always depends on the state of information and knowledge of an individual. We study different intrinsic properties of non-parametric probability distributions as well as of probabilistic indices summarizing them. We define the notion of α-distance between two non-parametric probability distributio

Игра со случайным вторым игроком и еe приложение к задаче о выборе цены проезда

The choice of the optimal strategy for a significant number of applied problems can be formalized as a game theory problem, even in conditions of incomplete information. The article deals with a hierarchical game with a random second player, in which the first player chooses a deterministic solution, and the second player is represented by a set of decision makers. The strategies of the players that ensure the Stackelberg equilibrium are studied. The strategy of the second player is formalized as a probabilistic solution to an optimization problem with an objective function depending on a continuously distributed random parameter. In many cases, the choice of optimal strategies takes place in conditions when there are many decision makers, and each of them chooses a decision based on his (her) criterion. The mathematical formalization of such problems leads to the study of probabilistic solutions to problems with an objective function depending on a random parameter. In particular, probabilistic solutions are used for mathematical describing the passenger’s choice of a mode of transport. The problem of optimal fare choice for a new route based on a probabilistic model of passenger preferences is considered. In this formalization, the carrier that sets the fare is treated as the first player; the set of passengers is treated as the second player. The second player’s strategy is formalized as a probabilistic solution to an optimization problem with a random objective function. A model example is considered. © 2021 Udmurt State University. All rights reserved.Funding. The study was funded by federal budget of the Russian Federation within the framework of the state order, the project «Optimization of the transport and logistics system based on modeling the development of transport infrastructure and models of consumer preference»