Frontiers of Dynamic Games

Inhaltsverzeichnis

1. Frontmatter

2. Chapter 1. Competition as a Hierarchical Multistage Game

   Petr Ageev, Svetlana Tarashnina

   Abstract

   We investigate the process of competition on the market of telecommunication services between three firms: the leader, the challenger and the follower. In this work we construct a model of competition between three players in the form of a multistage hierarchical non-zero sum game and compare it with our previous model of competition between three players in the form of a multistage non-hierarchical non-zero sum game. Compared to previous model, a hierarchical component was introduced. As solution of the game we find a subgame perfect equilibrium. We illustrate and compare the results with an example for three companies working on the Saint-Petersburg telecommunications market with the same initial conditions as in the previous work.

3. Chapter 2. Information Exchange in Price Setting Mixed Duopoly

   Sadettin Haluk Citci, Kubra Uge

   Abstract

   Consider a dynamic Bertrand competition in mixed oligopoly, where a private firm competes with a social welfare maximizing public firm. Firms produce substitute products, face stochastic demand and each firm receive noisy signals on common stochastic demand. In this mixed oligopoly, we examine incentives of public and private firms to share their private signals through an independent trade agency and we characterize equilibrium outcomes. We established two main effects of information sharing: information sharing increases production efficiency by enabling firms to predict stochastic demand shocks better. However, more precise signals increase power of private firm to capture consumer surplus and lowers social welfare. In Perfect Bayesian equilibrium of the mixed oligopoly game, private firm shares all signals it receives with the public firm, whereas public firm shares no information with the private firm. The market outcome is never optimal: it satisfy neither of informational efficiency, production efficiency and allocative efficiency.

4. Chapter 3. Reflexive and Epistemic Properties of the Tullock Rent-Seeking Game

   Denis Fedyanin

   Abstract

   This study sets out to investigate the impact of information control in the Tullock rent-seeking game. The game itself is constructed by using a normal form and making suggestions on the agents’ believes and knowledge. We found domains of parameters where monotonicity of the impact holds too. Together, these results provide valuable insights into the effects of reflexive analysis on the properties of information control.

5. Chapter 4. Solution of Differential Games with Network Structure in Marketing

   Ekaterina Gromova, Anastasiya Malakhova

   Abstract

   A marketing network model of goodwill accumulation with spillover effect is analysed in a differential game theory framework. Cooperative form of the game is considered under α-characteristic function. An approach is illustrated on a numerical example with particular values of the model parameters fixed.

6. Chapter 5. Penalty Method for Games of Constraints

   Igor Konnov

   Abstract

   We define a game problem for evaluation of composite system performance under possible external interference and in the presence of protection resources. The guaranteed joint system side decision is suggested to be found by an inexact penalty method. This enables one to essentially simplify the solution process in comparison with finding worst case strategies in the custom zero-sum game.

7. Chapter 6. Adjustment Dynamics in Network Games with Stochastic Parameters

   Alexei Korolev

   Abstract

   In this paper we introduce stochastic parameters into the network game model with production and knowledge externalities. This model was proposed by V. Matveenko and A. Korolev as a generalization of the two-period Romer model. Agents differ in their productivities which have deterministic and stochastic (Wiener) components. We study the dynamics of a single agent and the dynamics of a dyad where two agents are aggregated. We derive explicit expressions for the dynamics of a single agent and dyad dynamics in the form of Brownian random processes, and qualitatively analyze the solutions of stochastic equations and systems of stochastic equations.

8. Chapter 7. New Characteristic Function for Cooperative Games with Hypergraph Communication Structure

   David A. Kosian, Leon A. Petrosyan

   Abstract

   Cooperation in the games with hypergraph communication structure is considered. As usual in cooperative game theory, to define the allocation rule, the characteristic function is used. The communication possibilities are described by the hypergraph in which the nodes are players and hyperlinks are the communicating subgroups of players. The payoff of each player is influenced by actions of other players dependent from a distance between them on hypergraph. The new approach for constructing the characteristic function in the game is proposed. This approach does not require the use of maxmin operations which substantially simplifies the calculations. It is proved that the constructed characteristic function satisfies the convexity property. The results are shown in an example.

9. Chapter 8. Minimax Generalized Solutions of Hamilton-Jacobi Equations in Dynamic Bimatrix Games

   Nikolay A. Krasovskii, Alexander M. Tarasyev

   Abstract

   The paper deals with the dynamic bimatrix game whose dynamics describes the motion of flows corresponding to the control signals. A specific feature of the model is connected with increasing the dimension of the matrix game to 2 × 3 payoff matrices. This increase of the dimension is necessary as an initial step in solving the dynamic version of the well-known matrix game “rock-paper-scissors”, which is an algorithmically complex problem. For the considered 2 × 3 dynamic bimatrix game the sets of acceptable situations for players are constructed in the static setting. Basing on these constructions, an algorithm is developed for finding the value function for the first player in the antagonistic setting of a differential game on an infinite time interval. The stability properties of the value function are verified using the conjugate derivatives apparatus in the framework of the theory of generalized minimax (viscosity) solutions of the Hamilton-Jacobi equations.

10. Chapter 9. Analysis of Economic Behaviour in Evolutionary Model of Tax Control Under Information Diffusion

    Suriya Kumacheva, Elena Gubar, Ekaterina Zhitkova, Galina Tomilina

    Abstract

    Nowadays many economic fields are dependent on information flows. Information serves as a strategic tool in economics, business and many social processes. In particular, spreading information among taxpayers can be used as a control parameter in tax control. The probability of auditing which encourages tax payments used to be considered as a unique tool to stimulate tax collection. The current study represents a combined approach where tools of evolutionary game theory and network modeling are applied to the analysis of agents’ economic behavior. Information of possible tax auditing is disseminated across the population of taxpayers and is supposed to be a main factor influencing their decision on whether to evade or not. According to the previous research, interactions and dissemination of information or rumors among taxpayers in long-term period can be formulated as an evolutionary process. It is assumed that agents tend to spread information or rumors over their own contact network of neighbors and colleagues rather than over randomly chosen agents. Thus, the network model of social interaction is constructed. Information spreading in the network of various topology (e.g. grid, random network, etc.) is considered as an evolutionary process where agents’ behavior is described by the stochastic imitation dynamics and their interaction is described by different modifications of the bimatrix games which generate evolutionary dynamics. Scenario analysis is supported by the series of experiments. Numerous simulations help visualize the process of information spreading across different types of network, imitation protocols and players payoffs. The results show that information flow helps encourage tax payments in the population.

11. Chapter 10. Cooperation Enforcing in Multistage Multicriteria Game: New Algorithm and Its Implementation

    Denis Kuzyutin, Ivan Lipko, Yaroslavna Pankratova, Igor Tantlevskij

    Abstract

    To enforce the long-term cooperation in a multistage multicriteria game we use the imputation distribution procedure (IDP) based approach. We mainly focus on such useful properties of the IDP like “reward immediately after the move” assumption, time consistency inequality, efficiency and non-negativity constraint. To overcome the problem of negative payments along the optimal cooperative trajectory the novel refined A-incremental IDP is designed. We establish the properties of the proposed A-incremental payment schedule and provide an illustrative example to clarify how the algorithm works.

12. Chapter 11. Complementarity of Goods and Cooperation Among Firms in a Dynamic Duopoly

    Mario Alberto Garcia Meza, Cesar Gurrola Rios

    Abstract

    We construct a simple model to show how complementarities between goods yield a possibility for cooperation between rival firms. To show this, we use a simple dynamic model of Cournot oligopoly under sticky prices. While cooperation in an oligopoly model with sticky prices is not feasible, there exists a feasible cooperation when good are perfect complements and not substitutes.

13. Chapter 12. Cooperative Solutions for the Eurasian Gas Network

    Ekaterina Orlova

    Abstract

    We relate three solutions for cooperative games, the Shapley value, the nucleolus and the core. We use an empirical case study, provided in Hubert and Orlova (2018) to analyze the liberalization of network access in the European gas market. For these games the Shapley value is not in the core. To obtain a differentiated picture of the (in)stability of an allocation, we propose the n𝜖-core which is a generalization of the strong 𝜖-core, and define three stability measures. We find that the liberalization of network access increases the degree of instability of the Shapley value for all three metrics. The nucleolus is a unique point in the core, hence often used to characterize stable imputations. We show that liberalization compresses the core, but not always the nucleolus corresponds well to the shifts in the minimal and maximal values which players might receive in the core.

14. Chapter 13. Optimal Incentive Strategy in a Continuous Time Inverse Stackelberg Game

    Dmitry B. Rokhlin, Gennady A. Ougolnitsky

    Abstract

    We consider a continuous time dynamic incentive problem in the case of one leader and one follower. Follower’s ε-optimal strategy is determined via an auxiliary control problem. The main result is similar to that obtained by the authors for a stochastic discrete time model. We give an illustrative example concerning a non-renewable resource extraction problem.

15. Chapter 14. The Looking Forward Approach in a Differential Game Model of the Oil Market with Non-transferable Utility

    Ovanes Petrosian, Maria Nastych, Yin Li

    Abstract

    The paper applies the Looking Forward Approach to analyze the world oil market within the framework of a differential game model of a quantity competition oligopoly. More precisely, the Looking Forward Approach is used to take into account dynamically updating information. Under information we understand the forecast of oil demand dynamics. We use a non-cooperative game modeling for the period from December 2015 to November 2016, because over this period the countries did not cooperate officially in what concerns the amounts of oil to be produced. For the period from December 2016 to May 2017, a non-transferable utility cooperative game modeling is adapted due to the agreement to reduce oil extraction signed by the largest oil exporters at the end of November 2016. We use both solutions, which correspond to the historical cooperative solution and the sub-game consistent solution proposed developed in the field of dynamic games. In order to define the parameters of the model, open source data is used, with the results of numerical simulations and comparison between the historical data and model data for both periods also presented.

16. Chapter 15. Cooperative Decision Making in Cooperative Control Systems by Means of Game Theory

    Simon Rothfuß, Jannik Steinkamp, Michael Flad, Sören Hohmann

    Abstract

    Current state cooperative control systems assisting the human in various applications, e. g. assisted driving, lack the ability of emancipated cooperative decision making. Due to certain situations in which this a crucial skill, the authors’ objective is the automation design leading to cooperative control systems able to take part in human-machine negotiations.

    In this work, we therefore introduce two games for modeling the cooperative decision making process. First, the event-based game is introduced in a complete information setting with focus on modeling the system dynamics the decision making process is based on, e.g. a vehicle. We also provide an analytical as well as a numerical solution approach to find parameterized strategies that are in a Nash equilibrium.

    The second game is an enhanced war of attrition that models the negotiation process in an incomplete information setting. It includes a new cost structure that allows a better approximation of the costs in a realistic application of cooperative control systems. We proof that the resulting strategy leads to a Bayesian Nash equilibrium.

17. Chapter 16. Social Inefficiency of Free Entry Under the Product Diversity

    Alexander Sidorov

    Abstract

    The paper addressed to a question whether the free entry of profit-seeking large firms (oligopolies) is advantageous for consumers, or the governmental restrictions to enter may have the positive effect on consumers’ well-being. The negative welfare effect of excessive enter is well-known in case of homogeneous good, though there was hypothesis that consumers’ love for variety in case of differentiated good may offset this effect. The main result of this paper is that this almost never happened. We study a general equilibrium model with imperfect Bertrand-type price competition. Firms assumed to have non-zero impact to market statistics, in particular, to consumer’s income via distribution of non-zero profit across consumers-shareholders. It is proved that the governmental restrictions in certain bounds increases Social welfare under quite natural assumptions on utilities, which hold for most of the commonly used classes of utility functions, such as Quadratic, CARA, HARA, CES, etc.

18. Chapter 17. An Alternative Pursuit Strategy in the Game of Obstacle Tag

    Igor Shevchenko

    Abstract

    According to the generalized Isaacs’ approach, when solving a differential game, one has to fill the state space with trajectories on which the value function in some sense meets the main equation. Singular surfaces are manifolds where the value function or its derivatives are discontinuous. The obstacle tag game is a prototypical example which was used by R. Isaacs to illustrate some phenomena arising in differential games. The solution proposed by J.R. Isbell contains no singular surfaces. Afterward, several solutions with corner surfaces were described. J. Breakwell first constructed the field of optimal trajectories with a focal line and then with two switch envelops. A. Melikyan formed a field with two equivocal surfaces. In the paper, we consider the obstacle chase as an alternative pursuit game. In the part of the state space where the segment PE crosses the obstacle and alternatives are not consistent, as compared to Cases 7 and 8 of the Isbell’s solution, the generated pursuit strategy with memory allows P to switch geodesic lines a finite number of times only on boundaries of the secondary domain, and thereby prevents sliding motions. Numerical simulations for particular states show that the guaranteed results for this strategy are quite close to the value functions for fixed alternatives and to those that constructed by J. Breakwell and A. Melikyan.