### Working paper

## On evaluation of the power indices with allowance of agents’ preferences in the anonimous games

We consider the problem of manipulability of social choice rules in the impartial anonymous and neutral culture model (IANC) and provide a new theoretical study of the IANC model, which allows us to analytically derive the difference between the Nitzan-Kelly index in the Impartial Culture (IC) and IANC models. We show in which cases this difference is almost zero, and in which the Nitzan-Kelly index for IANC is the same as for IC. However, in some cases this difference is large enough to cause changes in the relative manipulability of social choice rules. We provide an example of such cases.

This paper demonstrates that most existing voting schemes represent or can be rewritten as weighted games. However, axiomatics for power indices defined on simple games are not directly applied to weighted games, since related operations become ill-posed. The author shows that the majority of axiomatics can be adapted to weighted games. Finally, a series of examples are provided.

The canon of classical Greek and Latin poetry is built around big names, with Homer and Virgil at the centre, but many ancient poems survive without a firm ascription to a known author. This negative category, anonymity, ties together texts as different as, for instance, the orally derived Homeric Hymns and the learned interpolation that is the Helen episode in Aeneid 2, but they all have in common that they have been maltreated in various ways, consciously or through neglect, by generations of readers and scholars, ancient as well as modern. These accumulated layers of obliteration, which can manifest, for instance, in textual distortions or aesthetic condemnation, make it all but impossible to access anonymous poems in their pristine shape and context.
The essays collected in this volume attempt, each in its own way, to disentangle the bundles of historically accreted uncertainties and misconceptions that affect individual anonymous texts, including pseudepigrapha ascribed to Homer, Manetho, Virgil and Tibullus, literary and inscribed epigrams, and unattributed fragments.
*Poems without Poets* will be of interest to students and scholars working on any anonymous ancient texts, but also to readers seeking an introduction to classical poetry beyond the limits of the established canon.

In the general case, complexity of the algorithm to calculate the power indices grows exponentially with the number of voting agents. Yet the volume of calculations may be reduced dramatically if many coalitions have equal numbers of votes. The well-known algorithm for calculation of the Banzhaf and Shapley-Shubik indices was generalized, which enables fast calculation of the power indices where entry of the voting agent into a coalition depends on its preferences over the set of the rest of agents.

This article investigates the problem of identifying a person on the Internet by legal and technical means. The practice of identifying people in Russia and the UK was studiedand compared. Russia was selected because its legislation is well known to the authors, and the UK was selected as it has developed a mature system for the online identification of individuals and relationships and a certain legal regulation in this sphere.An analysis of two government programs was made, namely: the UK Identity Assurance Programme of the Government Digital Service and the Russian Government Decree on “The development of the Federal state information system”. In terms of technological background for person’s identification, the practice of using IPv4 and IPv6 was explored. Russia's specific problems are analysed via the protection of privacy in the case of personal identification and the processing of personal data on the Internet. The authorsdraw conclusions about the division of the concepts of identification and individualization of people on the Internet. Weintroduceourown definition of personal identification on the Internet and proposean amendment to the Russian concept of personal data: the definition of personal data should include the IP address of a person.

We offer a general approach to describing power indices that account for preferences as suggested by F. Aleskerov. We construct two axiomatizations of these indices. Our construction generalizes the Laruelle-Valenciano axioms for Banzhaf (Penrose) and Shapley-Shubik indices. We obtain new sets of axioms for these indices, in particular, sets without the anonymity axiom.

At calculation of the power indices, both well-known (Banzhaf, Shapley-Shubik and others and new (depending on the agent preferences) indices, one generally has to enumerate almost all coalitions, that is, the subsets of the set of players, which makes calculations impossible if the number of players exceeds fifty. Yet, if all players have an integer number of votes, there are players with the same number of votes, many coalitions have equal total number of votes or the sum of votes of all players is small, then the algorithms based on calculations using the generating functions become efficient. But these algorithms works only for classical power indices and some particular types of the power indices based on agents’ preferences. In this paper we consider an important specific case when all players have the same number of votes. For classical power indices in this case all players have the same power. However, it is not the case for the indices which allow preferences of agents. We introduce effective algorithms for calculation of the latter indices for most types of these indices.

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.