On a generalization of Arrow's impossibility theorem
A complete classification of symmetric sets of choice functions with the Arrow property is obtained.
A set of related majority rule-based social choice correspondences are considered: the union of minimal Р-dominating sets MPD (Duggan 2011, Subochev 2016) the union of weakly stable sets MWS (Aleskerov & Kurbanov 1999), the union of minimal P-externally stable sets MPES (Wuffl et al. 1989, Subochev 2008) and the union of minimal R-externally stable sets MRES (Aleskerov & Subochev 2009, 2013). These tournament solutions have not attracted much attention so far. However, the analysis of their properties suggests that MPES and MRES can be useful as instruments of choice, for instance when it is necessary to aggregate rankings. Their implementation is also possible under certain conditions.
The results presented are the following.
1) In a general case of a topological space of alternatives, a sufficient and necessary condition has been provided for an alternative to belong to a minimal P-dominating set. This characteristic condition is related to some version of the covering relation. It has been established that the union of minimal P-dominating sets and the uncovered set are logically nested neither in a general case, nor in finite tournaments. The characterization obtained provides a sufficient condition of nonemptiness of MPES and MRES in a general case of a topological space of alternatives.
2) It has been found that MPES and MRES both satisfy the following axioms:
a) monotonicity with respect to changes in social preferences (P-monotonicity),
b) the generalized Nash independence of irrelevant alternatives,
c) the idempotence,
d) the Aizerman-Aleskerov property,
e) the independence of social preferences for irrelevant alternatives (the independence of losers),
but they do not satisfy the extension axiom (Sen’s property g). It has also been demonstrated that MPD satisfies neither of these axioms, and MWS satisfies P-monotonicity only.
3) It has been found that MPES and MRES both satisfy Sanver monotonicity (a.k.a. cover monotonicity). Thus, despite they are not Maskin monotonic, these social choice correspondences can be implemented in a nonstandard setting, where actors have (extended) preferences for sets of alternatives. It has also been demonstrated that MPD and MWS do not satisfy Sanver monotonicity.
One of the main tasks of the theory of collective choice is formulated in the language of functional Galois correspondences. A convenient characterization of symmetric classes of decision rules without the Arrow property is proposed.
We give an effective description of symmetric closed classes of discrete functions preserving any unary predicate.
The paper considers some new applications of the clone method in Computational Social Choice
In work we consider algorithms of sequential aggregation. We prove a classification theorem that generalizes the classification of aggregation rules without the Arrow property, previously proposed by the authors. Next, we consider non-local aggregation schemes that simulate the sequential choice.
A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.
This article is talking about state management and cultural policy, their nature and content in term of the new tendency - development of postindustrial society. It mentioned here, that at the moment cultural policy is the base of regional political activity and that regions can get strong competitive advantage if they are able to implement cultural policy successfully. All these trends can produce elements of new economic development.