Об устойчивых решениях в ординальной задаче выбора
A concept of k-stable alternatives is introduced. Relationship of classes of k-stable alternatives with dominant, uncovered and weakly stable sets is established.
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.
This article describesseveral impossibility results in social choice theory and demonstrates their importance for democratic theory. Since 1950s social scientists paid a great attention to the investigation of collective decision-making. This interest led to the formation of a new field of study within economics and political science, social choice theory. The main resultsof this strand of research are various impossibilitytheorems which illustrateinconsistencies indifferentvoting rules. Arrow`s impossibility theorem is usually considered to bethe most important result of this kind: however, many other impossibility theorems were proved during the last fifty years, among them the Gibbard-Satterthwaite theorem, Amartya Sen's liberal paradox and discursive dilemma. These paradoxical findingsreveal serious inner defects of democratic decision-making and therefore challenge the democratic idea itself, which is presumably the central project of modern political thought. Therefore, they are of great interest for democratic theorists.
Originally published in 1951, Social Choice and Individual Valuesintroduced “Arrow’s Impossibility Theorem” and founded the field of social choice theory in economics and political science. This new edition, including a new foreword by Nobel laureate Eric Maskin, reintroduces Arrow’s seminal book to a new generation of students and researchers.
"Far beyond a classic, this small book unleashed the ongoing explosion of interest in social choice and voting theory. A half-century later, the book remains full of profound insight: its central message, ‘Arrow’s Theorem,’ has changed the way we think.”—Donald G. Saari, author of Decisions and Elections: Explaining the Unexpected
Kenneth J. Arrow is professor of economics emeritus, Stanford University, and a Nobel laureate. Eric S. Maskin is Albert O. Hirschman Professor, School of Social Science, Institute of Advanced Study, Princeton, NJ, and a Nobel laureate.
When a society needs to take a collective decision one could apply some aggregation method, particularly, voting. One of the main problems with voting is manipulation. We say a voting rule is vulnerable to manipulation if there exists at least one voter who can achieve a better voting result by misrepresenting his or her preferences. The popular approach to comparing manipulability of voting rules is defining complexity class of the corresponding manipulation problem. This paper provides a survey into manipulation complexity literature considering variety of problems with different assumptions and restrictions.
Aleskerov et al.  and  estimated the degree of manipulability for the case of multi-valued choice (without using any tie-breaking rule) and for Impartial Culture (IC). In our paper, we address the similar question for the multi-valued choice and for Impartial Anonymous Culture (IAC). We use Nitzan-Kelly's (NK) index to estimate the degree of manipulability, which is calculated as the share of all manipulable voting situations, and calculate indices for 3 alternatives and up to 10000 voters. We have found that for the case of 3 alternatives Nanson's procedure shows the best results. Hare's procedure shows close, but a bit higher results. The worst aggregation procedure in terms of manipulability is Plurality rule. Additionally, it turned out that NK indices for IAC are smaller than NK indices for IC.
We consider certain spaces of functions on the circle, which naturally appear in harmonic analysis, and superposition operators on these spaces. We study the following question: which functions have the property that each their superposition with a homeomorphism of the circle belongs to a given space? We also study the multidimensional case.
We consider the spaces of functions on the m-dimensional torus, whose Fourier transform is p -summable. We obtain estimates for the norms of the exponential functions deformed by a C1 -smooth phase. The results generalize to the multidimensional case the one-dimensional results obtained by the author earlier in “Quantitative estimates in the Beurling—Helson theorem”, Sbornik: Mathematics, 201:12 (2010), 1811 – 1836.
We consider the spaces of function on the circle whose Fourier transform is p-summable. We obtain estimates for the norms of exponential functions deformed by a C1 -smooth phase.