A tractable multi-leader multi-follower peak-load-pricing model with strategic interaction

While single-level Nash equilibrium problems are quite well understood nowadays, less is known about multi-leader multi-follower games. However, these have important applications, e.g., in the analysis of electricity and gas markets, where often a limited number of firms interacts on various subsequent markets. In this paper, we consider a special class of two-level multi-leader multi-follower games that can be applied, e.g., to model strategic booking decisions in the European entry-exit gas market. For this nontrivial class of games, we develop a solution algorithm that is able to compute the complete set of Nash equilibria instead of just individual solutions or a bigger set of stationary points. Additionally, we prove that for this class of games, the solution set is finite and provide examples for instances without any Nash equilibria in pure strategies. We apply the algorithm to a case study in which we compute strategic booking and nomination decisions in a model of the European entry-exit gas market system. Finally, we use our algorithm to provide a publicly available test library for the considered class of multi-leader multi-follower games. This library contains problem instances with different economic and mathematical properties so that other researchers in the field can test and benchmark newly developed methods for this challenging class of problems.

Why Students Don't Study and Employers Don't Pay High Salaries?

The article examines the behavior of students and employers as a bimatrix game. With the tools of game theory, it is generally proven that the optimal strategy for employers is to pay low wages, and for students – not to study or to study too little. These two strategies form the Nash’s equilibrium in pure strategies. No specific numbers were used in the evidence, but only plausible assumptions about the relationships between the used parameters. This generalizes the conclusions made in the general case of higher education. Such a study of the question using game theory has not been done yet.

Game Self-organization of Hamiltonian Cycle of the Graph

This paper proposes a new application of the stochastic game model to solve the problem of self- organization of the Hamiltonian cycle of a graph. To do this, at the vertices of the undirected graph are placed game agents, whose pure strategies are options for choosing one of the incident edges. A random selection of strategies by all agents forms a set of local paths that begin at each vertex of the graph. Current player payments are defined as loss functions that depend on the strategies of neighboring players that control adjacent vertices of the graph. These functions are formed from a penalty for the choice of opposing strategies by neighboring players and a penalty for strategies that have reduced the length of the local path. Random selection of players’ pure strategies is aimed at minimizing their average loss functions. The generation of sequences of pure strategies is performed by a discrete distribution built on the basis of dynamic vectors of mixed strategies. The elements of the vectors of mixed strategies are the probabilities of choosing the appropriate pure strategies that adaptively take into account the values of current losses. The formation of vectors of mixed strategies is determined by the Markov recurrent method, for the construction of which the gradient method of stochastic approximation is used. During the game, the method increases the value of the probabilities of choosing those pure strategies that lead to a decrease in the functions of average losses. For given methods of forming current payments, the result of the stochastic game is the formation of patterns of self-organization in the form of cyclically oriented strategies of game agents. The conditions of convergence of the recurrent method to collectively optimal solutions are ensured by observance of the fundamental conditions of stochastic approximation. The game task is extended to random graphs. To do this, the vertices are assigned the probabilities of recovery failures, which cause a change in the structure of the graph at each step of the game. Realizations of a random graph are adaptively taken into account when searching for Hamiltonian cycles. Increasing the probability of failure slows down the convergence of the stochastic game. Computer simulation of the stochastic game provided patterns of self-organization of agents’ strategies in the form of several local cycles or a global Hamiltonian cycle of the graph, depending on the ways of forming the current losses of players. The reliability of experimental studies is confirmed by the repetition of implementations of self-organization patterns for different sequences of random variables. The results of the study can be used in practice for game-solving NP-complex problems, transport and communication problems, for building authentication protocols in distributed information systems, for collective decision-making in conditions of uncertainty.

Strategy for Balancing Interests of Business, Government and Society in the Field of International Trade in the Digital Economy

Global control of business by a state causes a conflict of interests and additional barriers to the development of international trade, which necessitates the formation of an appropriate strategy for balanced relations between business and society in the digital economy. Taking into account the main provisions of the agency theory, the choice of an effective strategy for interaction between state institutions and FEA participants requires the adoption of an effective toolkit for estimating such interaction. Using the main provisions of game theory, the study developed a corresponding game model, which includes three participants: state, economic operators, society. Based on the modelling of relevant processes, it was concluded that a state has 80 pure strategies, economic operators have 20 pure strategies, and society has 5 pure strategies, while results of a current game make it possible to determine the most advantageous situation of interaction for a state and economic operators.Thus, from the standpoint of regulating international trade, the issues of forming a balance of interests of all participants in this process allow to obtain positive results not only for specific business entities, but also for society as a whole.

THE CORRELATION DEPENDENCE OF PAYOFFS FOR MIXED STRATEGIES IN GAMES WITH NATURE

Цель исследования состоит в развитии и применении к задачам инвестирования методов принятия решений в играх с природой, учитывающих корреляцию случайных значений выигрышей для каждой пары чистых стратегий. При этом рассматриваются два критерия: математическое ожидание выигрыша и среднеквадратическое отклонение как оценка риска. Двухкритериальная модель принятия решений формализована путем перевода оценки риска в ограничение. Для такой обобщенной задачи квадратичного программирования получены аналитические методы решения. Приведен пример применения предложенного метода на реальных статистических данных. The aim of the research is to develop and apply to investment problems the methods of decision-making in games with nature, considering the correlation of random values of payoffs for each pair of pure strategies. In this case, two criteria are considered: the mathematical expectation of a payoff and the standard deviation as a risk assessment. The two-criteria decision-making model is formalized by translating the risk assessment into a constraint. For such a generalized quadratic programming problem, analytical solution methods are obtained. An example of applying the proposed method to real statistical data is given.

Nash blocks

A product set of pure strategies is a Nash block if it contains all best replies to the Nash equilibria of the game in which the players are restricted to the strategies in the block. This defines an intermediate block property, between curb (Basu and Weibull, Econ Lett 36(2):141–146, 10.1016/0165-1765(91)90179-O, http://www.sciencedirect.com/science/article/pii/016517659190179O, 1991) and coarse tenability (Myerson and Weibull (2015) Econometrica 83(3):943–976, 10.3982/ECTA11048). While the new concept is defined without reference to the consideration-set framework that defines tenability, the framework can be used to characterize Nash blocks in terms of potential conventions when large populations of individuals recurrently interact. Although weaker than curb, Nash blocks nevertheless maintain several robustness properties of curb sets. For example, every Nash block contains an essential component and is robust against payoff perturbations.

Every normal-form game has a Pareto-optimal nonmyopic equilibrium

It is well known that Nash equilibria may not be Pareto-optimal; worse, a unique Nash equilibrium may be Pareto-dominated, as in Prisoners’ Dilemma. By contrast, we prove a previously conjectured result: every finite normal-form game of complete information and common knowledge has at least one Pareto-optimal nonmyopic equilibrium (NME) in pure strategies, which we define and illustrate. The outcome it gives, which depends on where play starts, may or may not coincide with that given by a Nash equilibrium. We use some simple examples to illustrate properties of NMEs—for instance, that NME outcomes are usually, though not always, maximin—and seem likely to foster cooperation in many games. Other approaches for analyzing farsighted strategic behavior in games are compared with the NME analysis.

Acceptable-and-attractive Approximate Solution of a Continuous Non-Cooperative Game on a Product of Sinusoidal Strategy Functional Spaces

A problem of solving a continuous noncooperative game is considered, where the player’s pure strategies are sinusoidal functions of time. In order to reduce issues of practical computability, certainty, and realizability, a method of solving the game approximately is presented. The method is based on mapping the product of the functional spaces into a hyperparallelepiped of the players’ phase lags. The hyperparallelepiped is then substituted with a hypercubic grid due to a uniform sampling. Thus, the initial game is mapped into a finite one, in which the players’ payoff matrices are hypercubic. The approximation is an iterative procedure. The number of intervals along the player’s phase lag is gradually increased, and the respective finite games are solved until an acceptable solution of the finite game becomes sufficiently close to the same-type solutions at the preceding iterations. The sufficient closeness implies that the player’s strategies at the succeeding iterations should be not farther from each other than at the preceding iterations. In a more feasible form, it implies that the respective distance polylines are required to be decreasing on average once they are smoothed with respective polynomials of degree 2, where the parabolas must be having positive coefficients at the squared variable.

Evolution of altruistic punishments among heterogeneous conditional cooperators

It has been known that altruistic punishments solve the free rider problem in public goods games. Considering spatial structure and considering pure strategies significant advances have been made in understanding the evolution of altruistic punishments. However, these models have not considered key behavior regularities observed in experimental and field settings, where the individuals behave like conditional cooperators who are more willing to donate and are also more willing to punish free riders. Considering these behavioral regularities, without imposing a spatial structure on the population, I propose an evolutionary agent-based model in which agents behave like conditional cooperators, each agent’s donation conditional on the difference between the number of donations in the past and the threshold value and the propensity value of the agent. Altruistic punishment depends on the difference between the threshold value of the focal agent and the randomly matched another agent. The simulations show that, for certain inflicted costs of punishments, generous altruistic punishments evolve and stabilize cooperation. The results show that, unlike previous models, it is not necessary to punish all free riders equally; it is necessary to do so in the case of the selfish free riders but not in the case of negative reciprocators.

Mistakes can stabilise the dynamics of rock-paper-scissors games

A game of rock-paper-scissors is an interesting example of an interaction where none of the pure strategies strictly dominates all others, leading to a cyclic pattern. In this work, we consider an unstable version of rock-paper-scissors dynamics and allow individuals to make behavioural mistakes during the strategy execution. We show that such an assumption can break a cyclic relationship leading to a stable equilibrium emerging with only one strategy surviving. We consider two cases: completely random mistakes when individuals have no bias towards any strategy and a general form of mistakes. Then, we determine conditions for a strategy to dominate all other strategies. However, given that individuals who adopt a dominating strategy are still prone to behavioural mistakes in the observed behaviour, we may still observe extinct strategies. That is, behavioural mistakes in strategy execution stabilise evolutionary dynamics leading to an evolutionary stable and, potentially, mixed co-existence equilibrium.