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## Bachet's game with lottery moves

Bachet’s game is a variant of the game of Nim. There are 𝑛 objects in one pile. Two players take turns to remove any positive number of objects not exceeding some fixed number 𝑚. The player who takes the last object loses. We consider a variant of Bachet’s game in which each move is a lottery over set {1, 2, . . . ,𝑚}. The outcome of a lottery is the number of objects that player takes from the pile. We show that under some nondegenericity assumptions on the set of available lotteries the probability that the first player wins in subgame perfect Nash equilibrium converges to 1/2 as 𝑛 tends to infinity.

This paper explores the intertwining of uncertainty and values. We consider an important but underexplored field of fundamental uncertainty and values in decision-making. Some proposed methodologies to deal with fundamental uncertainty have included *potential surprise theory*, *scenario planning* and *hypothetical retrospection*. We focus on the principle of uncertainty transduction in hypothetical retrospection as an illustrative case of how values interact with fundamental uncertainty. We show that while uncertainty transduction appears intuitive in decision contexts it nevertheless fails in important ranges of strategic game-theoretic cases. The methodological reasons behind the failure are then examined.

In this paper, we study the Maximum Happy Vertices and the Maximum Happy Edges problems (MHV and MHE for short). Very recently, the problems attracted a lot of attention and were studied in Agrawal ’17, Aravind et al. ’16, Choudhari and Reddy ’18, Misra and Reddy ’17. Main focus of our work is lower bounds on the computational complexity of these problems. Established lower bounds can be divided into the following groups: NP-hardness of the above guarantee parameterization, kernelization lower bounds (answering questions of Misra and Reddy ’17), exponential lower bounds under the Set Cover Conjecture and the Exponential Time Hypothesis, and inapproximability results. Moreover, we present an O∗(ℓk)O∗(ℓk) randomized algorithm for MHV and an O∗(2k)O∗(2k) algorithm for MHE, where ℓℓ is the number of colors used and *k* is the number of required happy vertices or edges. These algorithms cannot be improved to subexponential taking proved lower bounds into account.

This book is devoted to game theory and its applications to environmental problems, economics, and management. It collects contributions originating from the 12th International Conference on “Game Theory and Management” 2018 (GTM2018) held at Saint Petersburg State University, Russia, from 27 to 29 June 2018.

This contributed volume presents the state-of-the-art of games and dynamic games, featuring several chapters based on plenary sessions at the ISDG-China Chapter Conference on Dynamic Games and Game Theoretic Analysis, which was held from August 3-5, 2017 at the Ningbo campus of the University of Nottingham, China. The chapters in this volume will provide readers with paths to further research, serving as a testimony to the vitality of the field. Experts cover a range of theory and applications related to games and dynamic games.

This paper discusses the scientific and practical perspectives of using general game playing in business-to-business price negotiations as a part of Procurement 4.0 revolution. The status quo of digital price negotiations software, which emerged from intuitive solutions to business goals and refereed to as electronic auctions in industry, is summarized in a scientific context. Description of such aspects as auctioneers’ interventions, asymmetry among players and time- depended features reveals the nature of nowadays electronic auctions to be rather termed as price games. This paper strongly suggests general game playing as the crucial technology for automation of human rule setting in those games. Game theory, genetic programming, experimental economics, and AI human player simulation are also discussed as satellite topics. SIDL-type game descriptions languages and their formal game-theoretic foundations are presented.

The majority of social and economic interactions take place between people of different social status. Age, position, income and other factors affect the way people evaluate their position in the society. We investigate how self-estimation of the social status is formed when an individual participates in an economic experimental game. In our experiment subjects are set in pairs and play consequently the dictator game, the trust game and the labor market (contract) game. After each game we measure their subjective socioeconomic status using two different scales. We show that participation in the dictator game affects the perception of one’s social status to the greatest extent: the status of dictators is higher than the status of recipients. Prescription of roles in other games does not have such an effect. Active behavior, gender, income, etc. also affect the subjective status.

We use the vertical differentiation framework to explore the quality - price competition in the insurance market.

On the basis of A. Przeworski’s game-theoretic approach the qualitative transformation of “National Movement” – an authoritarian-dominant party in Franco’s Spain – during the global economic crisis period (1957-1958) is analyzed. The strategies of political factions (the coalition of technocrats-monarchists and postfalangists) were identified as the basic elements of the analysis.

Game theory has recently become a useful tool for modeling and studying various networks. The past decade has witnessed a huge explosion of interest in issues that intersect networks and game theory. With the rapid growth of data traffic, from any kind of devices and networks, game theory is requiring more intelligent transformation. Game theory is called to play a key role in the design of new generation networks that are distributed, self-organizing, cooperative, and intelligent. This book consists of invited and technical papers of GAMENETS 2018, and contributed chapters on game theoretic applications such as networks, social networks, and smart grid.

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.

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.