>, the card game

Article describes new card game for 4 players, that requires from its participants decisions, made in presence of information asymmetry arising from secret component of players' every move. Significance for outcome of information asymmetry is demonstrated, and overview of potential lines of research in game analysis is given.

On some general scheme of constructing iterative methods for searching the Nash equilibrium in concave games

The subject of the paper is finite-dimensional concave games id est noncooperative $n$-person games with objective functionals concave with respect to `their own' variables. For such games we investigate the problem of designing iterative algorithms for searching the Nash equilibrium with convergence guaranteed without requirements concerning objective functionals such as smoothness and as convexity in `strange' variables or another similar hypotheses (in the sense of weak convexity, quasiconvexity and so on). In fact, we prove some assertion analogous to the theorem on convergence of $M$-Fej\'{er iterative process for the case when an operator acts in a finite-dimensional compact and nearness to an objective set is measured with the help of arbitrary continuous function. Then, on the base of this assertion we generalize and develop the approach suggested by the author formerly to searching the Nash equilibrium in concave games. The last one can be regarded as "a cross between" the relaxation algorithm and the Hooke-Jeeves method of configurations (but taking into account a specific character of the the residual function being minimized). Moreover, we present results of numerical experiments with their discussion.

Equilibria in a dynamic model of two firms coordination with non-fixed prices

Model of 2 companies' interaction is considered. Companies exchange two kinds of goods with each other. These goods also can be selled on the market. Prices of goods change over time. Interation is modelled by positional game which is similar to Rosental's Centipede game. Conditions for a contract beetween companies are found. For these conditions, companies can't violate a contract because it is unprofitable for them. Such conditions are modelled by Nash equilibriua in a positional game. We found that sufficient conditions for contract: it renews each 2 points of time; each company pays rather big fine if it refused to transfer good to another company.

Hierarchical games with feedback on the assumption of a lower-level player's benevolence

A new optimality principle is proposed that generalizes the Stackelberg equilibrium principle. Its connection with the classical definition is investigated. The technique of working with the new definition is discussed. As an example, solutions are found in two hierarchical games with feedback.

Sufficient condition for nondominated maximin strategy of arbitrary player with terminal payoff function in two-step positional game of n persons with strategies-syntheses and finite sets of controlling actions of players

Two-step positional game of $n$ persons with strategies-syntheses, $n\geqslant 2$, and finite sets of controlling actions of players is investigated. Sufficient condition for nondominated maximin strategy of arbitrary player, whose payoff function is terminal, has been obtained.

Game-theoretic models of battle action

The main types of combined arms combat operations are offensive and defense. Using the function of victory in battle, which is an extension of the function of conflict by G. Tullock, the following game-theoretic problems have been solved. First, the extended Gross-Germeier ''attack-defense'' model, which is a special case of a more general ''offensive-defense'' model, and describing the solution by the parties of the nearest tactical tasks, is investigated. Secondly, it has been proved that in the problem of breaking through points of defense (the closest tactical task), the criteria ``breaking through the weakest point'' and ``breaking through at least one point'' are equivalent. Thirdly, in the model of resource distribution of attackers and defenders between tactical tasks (echelons), the use of two criteria: 1) the product of the probabilities of solving the nearest and subsequent tactical tasks, 2) the minimum value of the named probabilities, - gives two fundamentally different solutions. Fourthly, the results of decisions were checked for compliance with the principles of military art and the practice of battles, battles and operations.

Machine load balancing game with linear externalities

The Machine Load Balancing Game with linear externalities is considered. A set of jobs is to be assigned to a set of machines with different latencies depending on their own loads and also loads on other machines. Jobs choose machines to minimize their own latencies. The social cost of a schedule is the maximum delay among all machines, i.e. {\it makespan. For the case of two machines in this model an Nash equilibrium existence is proven and of the expression for the Price of Anarchy is obtained.

Two-armed bandit problem and batch version of the mirror descent algorithm

We consider the minimax setup for the two-armed bandit problem as applied to data processing if there are two alternative processing methods with different a priori unknown efficiencies. One should determine the most efficient method and provide its predominant application. To this end, we use the mirror descent algorithm (MDA). It is well-known that corresponding minimax risk has the order of $N^{1/2$ with $N$ being the number of processed data and this bound is unimprovable in order. We propose a batch version of the MDA which allows processing data by packets that is especially important if parallel data processing can be provided. In this case, the processing time is determined by the number of batches rather than by the total number of data. Unexpectedly, it turned out that the batch version behaves unlike the ordinary one even if the number of packets is large. Moreover, the batch version provides significantly smaller value of the minimax risk, i.e., it considerably improves a control performance. We explain this result by considering another batch modification of the MDA which behavior is close to behavior of the ordinary version and minimax risk is close as well. Our estimates use invariant descriptions of the algorithms based on Gaussian approximations of incomes in batches of data in the domain of ``close'' distributions and are obtained by Monte-Carlo simulations.

To the individual stability of Pareto equilibrium of objections and counterobjections in a coalition differential positional 3-person game without side payments

The notion <<individual stability>> of Pareto equilibrium of objections and counter objections in one differential linear-quadratic 3-person game without side payments is used. The explicit form corresponding equilibrium is found.

Dynamic SPICE-model of the regional development: a comparative analysis of the administrative and economic control mechanisms (on the example of South Federal District)

A game theoretic formalization of the mechanisms of control over the regions as parts of a macroregion with consideration of the requirements of sustainable development is proposed. A modified Solow model is used for the description of the regional state dynamics. The model is identified on real data for the South Russian Federal District. A qualitative comparative analysis of efficiency of the administrative and economic control mechanisms is made.