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There is a paradox in the model of social dynamics determined by voting in a stochastic environment (the ViSE model) called “pit of losses.” It consists in the fact that a series of democratic decisions may systematically lead the society to the states unacceptable for all the voters. The paper examines how this paradox can be neutralized by the presence in society of a group that votes for its benefit and can regulate the threshold of its claims. We obtain and analyze analytical results characterizing the welfare of the whole society, the group, and the other participants as functions of the said claims threshold.

We present a survey of results on models of consensus in asynchronous multiagent systems with discrete and continuous time. We consider mathematical methods developed over recent years, which are used in the analysis of stability, stabilization, and consensus problems for linear multiagent systems with discrete time. These methods are based on the idea of using the notion of joint/generalized spectral radius of a set of matrices to analyze the rate of convergence of matrix products with factors drawn from certain sets of matrices with special properties.

We describe certain classes of linear asynchronous multi-agent systems in discrete time for which the stability problem allows for a constructive solution. We also present a general analytic approach to constructing numerical characteristics similar to the generalized spectral radius in stability theory, which would provide an opportunity to analyze the stabilizability of controlled multi-agent systems. This work completes our survey “Consensus in Asynchronous Multi-Agent Systems,” whose first two parts have been published in [1, 2].

The paper studies the problem of achieving consensus in multi-agent systems in the case where the dependency digraph Γ has no spanning in-tree. We consider the regularization protocol that amounts to the addition of a dummy agent (hub) uniformly connected to the agents. The presence of such a hub guarantees the achievement of an asymptotic consensus. For the “evaporation” of the dummy agent, the strength of its influences on the other agents vanishes, which leads to the concept of latent consensus. We obtain a closed-form expression for the consensus when the connections of the hub are symmetric; in this case, the impact of the hub upon the consensus remains fixed. On the other hand, if the hub is essentially influenced by the agents, whereas its influence on them tends to zero, then the consensus is expressed by the scalar product of the vector of column means of the Laplacian eigenprojection of Γ and the initial state vector of the system. Another protocol, which assumes the presence of vanishingly weak uniform background links between the agents, leads to the same latent consensus.

We describe mathematical methods for analyzing the stability, stabilizability and consensus of linear multiagent systems with discrete time. These methods are based on the idea of using the notion of joint/generalized spectral radius of matrix sets to analyze the rate of convergence of matrix products with factors from the sets of matrices with special properties. This is a continuation of the survey by the same authors named “Consensus in Asynchronous Multiagent Systems”; the first part was published in [1].

In this paper, we study the efficiency of egoistic and altruistic strategies within the model of social dynamics determined by voting in a stochastic environment (the ViSE model) using two criteria: maximizing the average capital increment and minimizing the number of bankrupt participants. The proposals are generated stochastically; three families of the corresponding distributions are considered: normal distributions, symmetrized Pareto distributions, and Student’s t-distributions. It is found that the “pit of losses” paradox described earlier does not occur in the case of heavy-tailed distributions. The egoistic strategy better protects agents from extinction in aggressive environments than the altruistic ones, however, the efficiency of altruism is higher in more favorable environments. A comparison of altruistic strategies with each other shows that in aggressive environments, everyone should be supported to minimize extinction, while under more favorable conditions, it is more efficient to support the weakest participants. Studying the dynamics of participants’ capitals we identify situations where the two considered criteria contradict each other. At the next stage of the study, combined voting strategies and societies involving participants with selfish and altruistic strategies will be explored.

Within the model of social dynamics determined by collective decisions in a stochastic environment (ViSE model), we consider the case of a homogeneous society consisting of classically rational economic agents (or homines economici, or egoists). We present expressions for the optimal majority threshold and the maximum expected capital increment as functions of the parameters of the environment. An estimate of the rate of change of the optimal threshold at zero is given, which is an absolute constant: (2/π−π/2)/2 .

The model of voting originated social dynamics in a stochastic environment (the ViSE model) is considered. Within this model, the influence of the heaviness of distribution tails on the effectiveness of egoistic and altruistic strategies in terms of maximizing two criteria, the average capital increment and the number of non-bankrupt participants, is investigated. Homogeneous societies and four types of distributions used to generate proposals (Gaussian, logistic, Student’s with 3 degrees of freedom, and symmetrized Pareto distributions) are studied. To assess the effect of tail heaviness, all distributions are unified by quartile using scatter. Such an approach can be used to compare the heavy-tailed distributions that are commensurable by density with other distributions under consideration on an interval containing 90% of observations.

Constructing and studying distributed control systems requires the analysis of the Laplacian spectra and the forest structure of directed graphs. In this paper, we present some basic results of this analysis. We also discuss the application of these results published earlier to decentralized control and touch upon some problems of spectral graph theory.

There is a paradox in the model of social dynamics determined by voting in a stochastic environment (the ViSE model) called \"pit of losses.\" It consists in the fact that a series of democratic decisions may systematically lead the society to the states unacceptable for all the voters. The paper examines how this paradox can be neutralized by the presence in society of a group that votes for its benefit and can regulate the threshold of its claims. We obtain and analyze analytical results characterizing the welfare of the whole society, the group, and the other participants as functions of the said claims threshold.