### Book

## Collective decision making

The most of solutions for games with non-transferable utilities (NTU) are NTU extensions of solution concepts defined for games with transferable utilities (TU). For example, there are three NTU versions of the Shapley value due to Aumann(1985), Kalai--Samet(1977), and Maschler--Owen(1992). The Shapley value is {\it standard} for two-person games. An NTU analog of standard solution is called the {\it symmetric proportional solution (SP)} (Kalai 1977), and the most of NTU solutions are SP solutions for two-person games. Another popular TU game solution which is not standard for two-person case is the {\it egalitarian Dutta-Ray solution (Dutta, Ray (1989), Dutta 1990). It was defined for the class of convex TU games and then extended to the class of all TU games (Branzei et al. 2006). . The DR solution for superadditive two-person TU games is the solution of constrained egalitarianism, it chooses the payoff vectors the closest to the diagonal of the space R^N. Its extension to superadditive two-person NTU games and then to n-person bargaining problems is the lexicographically maxmin solution}: for each game/bargaining problem it is the individually rational payoff vector which is maximal w.r.t. the lexmin relation. This solution if positively homogenous, but is not covariant w.r.t. shifts of individual payoffs. In the presentation this solution is extended to the class of NTU non-levelled games which are both ordinal and cardinal convex. Since convex TU games considered in NTU setting are ordinal and cardinal convex, the NTU DR solution is, in fact, an extension of the original TU version to the mentioned class of NTU games. It turns out that in this class the DR solution is single-valued and belongs to the core. A result similar to that of Dutta for TU convex games is obtained: the DR solution for the class of non-levelled ordinal and cardinal convex games is the single solution being the lexicographically maxmin solution for two-person games and consistent in (slightly modified) Peleg's definition (Peleg 1985) of the reduced games.

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.

Interval cooperative games are models of cooperative situation where only bounds for payoffs of coalitions are known with certainty. The extension of solutions of classical cooperative games to interval setting highly depends on their monotonicity properties. However. both the prenucleolus and the tau-value are not aggregate monotonic on the class of convex TU games Hokari (2000, 2001). Therefore, interval analogues of these solutions either should be defined by another manner, or perhaps they exist in some other class of interval games. Both approaches are used in the paper: the prenucleolus of a convex interval game is defined by lexicographical minimization of the lexmin relation on the set of joint excess vectors of lower and upper games. On the other hand, the tau-value is shown to satisfy extendability condition on a subclass of convex games -- on the class of totally positive convex games. The interval prenucleolus is determined , and the proof of non-emptiness of the interval \tau-value on the class of interval totally positive games is given.

Inconsistency of business processes can affect company profits and lead to the loss of regular customers and reputation in the market. Well managed business process has one key distinctive feature – a consistency. Checking the consistency of business process helps to reveal hidden bugs in the process model, but requires considerable labor costs and analytics. We compared two approaches to verifying consistency. The first approach is based on generating object life cycles for each object type used in process and supported by special tool as an extension for IBM WebSphere Business Modeler. Another one is a proposition to use DEMO methodology for verifying consistency. The results of research show that DEMO methodology enables significantly reduce labor costs and improve quality of analyze

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.

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.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.