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## Bidding Models and Repeated Games with Incomplete Information: A Survey

Using a simplified multistage bidding model with asymmetrically informed agents, De Meyer and Saley [17] demonstrated an idea of endogenous origin of the Brownian component in the evolution of prices on stock markets: random price fluctuations may be caused by strategic randomization of “insiders.” The model is reduced to a repeated game with incomplete information. This paper presents a survey of numerous researches inspired by the pioneering publication of De Meyer and Saley.

We investigate a model of one-stage bidding between two differently informed stockmarket agents for a risky asset (share). The random liquidation price of a share may take two values: the integer positive m with probability p and 0 with probability 1−p. Player 1 (insider) is informed about the price, Player 2 is not. Both players know the probability p. Player 2 knows that Player 1 is an insider. Both players propose simultaneously their bids. The player who posts the larger bid buys one share from his opponent for this price. Any integer bids are admissible. The model is reduced to a zero-sum game with lack of information on one side. We construct the solution of this game for any p and m: we find the optimal strategies of both players and describe recurrent mechanism for calculating the game value. The results are illustrated by means of computer simulation.

According to the currently prevalent theory, hippocampal formation constructs and maintains cognitive spatial maps. Most of the experimental evidence for this theory comes from the studies on navigation in laboratory rats and mice, typically male animals. While these animals exhibit a rich repertoire of behaviors associated with navigation, including locomotion, head movements, whisking, sniffing, raring and scent marking, the contribution of these behavioral patterns to hippocampal activity has not been sufficiently studied. Instead, many publications have considered animal position in space as the single variable that affects the firing of hippocampal place cells and entorhinal grid cells. Here we argue that future work should focus on a more detailed examination of different behaviors exhibited during navigation in order to interpret the cause of spatial tuning in hippocampal neurons. As a step in this direction, we have analyzed data from two datasets, shared online, containing recordings from rats navigating in square and round arenas. Our analyses revealed structured, grid-like navigation patterns, evident from the spatial maps of animal position, velocity and acceleration. Moreover, grid cells available in the datasets exhibited the same spatial periodicity as the navigation parameters. These findings cast doubt on the cognitive-map interpretation of grid cells, since they suggest that neuronal spatial patterns could be caused by behaviors associated with navigation instead of representing a hierarchically high spatial map. Additionally, we speculate that scent marks left by navigating animals could contribute to neuronal responses while rats and mice sniff their environment.

This paper is concerned with Random walk approximations of the Brownian motion on the Affine group Aff(R). We are in particular interested in the case where the innovations are discrete. In this framework, the return probabilities of the walk have fractional exponential decay in large time, as opposed to the polynomial one of the continuous object. We prove that in tegrating those return probabilities on a suitable neighborhood of the origin, the expected polynomial decay is restored. This is what we call a Quasi-local theorem.

This book presents recent research developments in social networks, economics, management, marketing and optimization applied to sports. The volume will be of interest to students, researchers, managers from sports, policy makers and as well athletes. In particular the book contains research papers and reviews addressing the following issues: social network tools for player selection, movement and pricing in team sports, methods for ranking teams and evaluating players' performance, economics and marketing issues related to sports clubs, techniques for predicting outcomes of sports competitions, optimal strategies in sports, scheduling and managing sports tournaments, optimal referee assignment techniques and the economics and marketing of sports entertainment.

Nowadays the random search became a widespread and effective tool for solving different complex optimization and adaptation problems. In this work, the problem of an average duration of a random search for one object by another is regarded, depending on various factors on a square field. The problem solution was carried out by holding total experiment with 4 factors and orthogonal plan with 54 lines. Within each line, the initial conditions and the cellular automaton transition rules were simulated and the duration of the search for one object by another was measured. As a result, the regression model of average duration of a random search for an object depending on the four factors considered, specifying the initial positions of two objects, the conditions of their movement and detection is constructed. The most significant factors among the factors considered in the work that determine the average search time are determined. An interpretation is carried out in the problem of random search for an object from the constructed model.The important result of the work is that the qualitative and quantitative influence of initial positions of objects, the size of the lattice and the transition rules on the average duration of search is revealed by means of model obtained. It is shown that the initial neighborhood of objects on the lattice does not guarantee a quick search, if each of them moves. In addition, it is quantitatively estimated how many times the average time of searching for an object can increase or decrease with increasing the speed of the searching object by 1 unit, and also with increasing the field size by 1 unit, with different initial positions of the two objects. The exponential nature of the growth in the number of steps for searching for an object with an increase in the lattice size for other fixed factors is revealed. The conditions for the greatest increase in the average search duration are found: the maximum distance of objects in combination with the immobility of one of them when the field size is changed by 1 unit. (that is, for example, with 4x4 at 5x5) can increase the average search duration in e^1,69≈5,42. The task presented in the work may be relevant from the point of view of application both in the landmark for ensuring the security of the state, and, for example, in the theory of mass service.

Repeated bidding games were introduced by De Meyer and Saley (2002) to analyze the evolution of the price system at finance markets with asymmetric information. In the paper of De Meyer and Saley arbitrary bids are allowed. It is more realistic to assume that players may assign only discrete bids proportional to a minimal currency unit. This paper represents a survey of author's results on discrete bidding games with asymmetric information.

We describe optimal contest success functions (CSF) which maximize expected revenues of an administrator who allocates under informational asymmetry a source of rent among competing bidders. It is shown that in the case of independent private values rent administrator’s optimal mechanism can always be implemented via some CSFs as posited by Tullock. Optimal endogenous CSFs have properties which are often assumed a priori as plausible features of rent-seeking contests; the paper therefore validates such assumptions for a broad class of contests. Various extensions or optimal CSFs are analyzed.

We consider multistage bidding models where two types of risky assets (shares) are traded between two agents that have different information on the liquidation prices of traded assets. These prices are random integer variables that are determined by the initial chance move according to a probability distribution p over the two-dimensional integer lattice that is known to both players. Player 1 is informed on the prices of both types of shares, but Player 2 is not. The bids may take any integer value.

The model of n-stage bidding is reduced to a zero-sum repeated game with lack of information on one side. We show that, if liquidation prices of shares have finite variances, then the sequence of values of n-step games is bounded. This makes it reasonable to consider the bidding of unlimited duration that is reduced to the infinite game G1(p). We offer the solutions for these games.

We begin with constructing solutions for these games with distributions p having two and three-point supports. Next, we build the optimal strategies of Player 1 for bidding games G1(p) with arbitrary distributions p as convex combinations of his optimal strategies for such games with distributions having two- and three-point supports. To do this we construct the symmetric representation of probability distributions with fixed integer expectation vectors as a convex combination of distributions with not more than three-point supports and with the same expectation vectors.

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.