Frontiers of Dynamic Games

Inhaltsverzeichnis

1. Frontmatter

2. Chapter 1. An Example of Reflexive Analysis of a Game in Normal Form

   Denis Fedyanin

   Abstract

   In this paper we considered a normal form game with a parameter A and suggested that this is an uncertain parameter for agents and they have to make some suggestion about it. We took a normal form game and weakened a suggestion on common knowledge. We introduced two fundamental alternatives based on dynamic epistemic logic and found equilibria for this modified game. We used the special property of these alternatives which let us calculate an equilibria by solving several normal form games with perfect information. We found direct expressions for equilibria for nonmodified game and for modified games with alternative suggestions on beliefs.

3. Chapter 2. Differential-Game-Based Driver Assistance System for Fuel-Optimal Driving

   Michael Flad

   Abstract

   Increasing the fuel-efficiency is a current and essential question for all major car manufacturers. Supporting these efforts, the paper presents a shared control driver assistance system that may help the driver to apply a fuel-efficient driving strategy. For the proposed system, both driver and assistance system can apply forces to the acceleration pedal enabling a close cooperation between the two partners. The interaction between driver and such kind of assistance system can be described by means of a differential game. By solving this differential game, the assistance system calculates optimal control outputs. For realization, the assistance system is required to solve different game theoretic problems that are presented in this paper. The assistance system was implemented on a real time system, integrated in a driving simulator and validated in a driving study. The results indicate that the proposed system is able to save in average about 10% fuel in a highway scenario.

4. Chapter 3. On the Selection of the Nash Equilibria in a Linear-Quadratic Differential Game of Pollution Control

   Ekaterina Gromova, Yulia Lakhina

   Abstract

   The work is devoted to the problem of the selection of Nash equilibrium in non-cooperative differential games with an n-dimensional state variable. We consider the problem of the control harmful emissions. When solving the problem in the class of closed-loop strategies it turns out that Hamilton–Jacobi–Bellman equation may have multiple solutions. The application of an economic criterion and a classical method used in the theory of linear-quadratic regulators (LQR) to the selection of the admissible solutions from the set of obtained solutions was shown in the considered model.

5. Chapter 4. Endogenous Formation of Cooperation Structure in TU Games

   Anna Khmelnitskaya, Elena Parilina, Artem Sedakov

   Abstract

   Our main goal is to provide comparative analysis of several procedures for endogenous dynamic formation of the cooperation structure in TU games. In the paper we consider two approaches to endogenous graph formation based on sequential link announcement and revision. For the evaluation of the pros and cons when adding of a new link is in question, along with the Myerson value we consider also the average tree solution and the centrality rewarding Shapley and Myerson values, recently introduced.

6. Chapter 5. Analysis of Competitive and Cooperative Solutions in Dynamic Auction Games

   Nikolay A. Krasovskii, Alexander M. Tarasyev

   Abstract

   The paper deals with an approach for dynamic games of searching maximum Pareto points which possess special decomposition characteristics, that inherits the properties of the competitive Nash equilibrium. Analysis of properties is implemented for the competitive Nash equilibrium. The set of Pareto points is constructed for cooperative actions of players. In the Pareto set a point of market equilibrium is defined, and the problem is posed to shift the system from the competitive Nash equilibrium to the point of market equilibrium. The shift algorithm is proposed and is based on analytical calculation of the players’ best replies to auction prices. The algorithm is implemented as a system of differential equations, right sides of which describe the mechanism of formation of auction prices and players’ best replies to these prices. Trajectories of the dynamic system describe the shift of players from the Nash equilibrium to Pareto maximum and show stable convergence of the algorithm. The results of the algorithm are demonstrated by the model of the auction game of fast growing economies.

7. Chapter 6. A-Subgame Concept and the Solutions Properties for Multistage Games with Vector Payoffs

   Denis Kuzyutin, Yaroslavna Pankratova, Roman Svetlov

   Abstract

   We deal with multistage multicriteria games in extensive form and employ so-called “A-subgame” concept to examine dynamical properties of some non-cooperative and cooperative solutions. It is proved that if we take into account only the active players at each A-subgame the set of all strong Pareto equilibria is time consistent but does not satisfy dynamical compatibility.

   We construct an optimal cooperative trajectory and vector-valued characteristic function using the refined leximin algorithm. To ensure the sustainability of a cooperative agreement we design the A-incremental imputation distribution procedure for the Shapley value which provides a better incentive for cooperation than classical incremental allocation procedure. This specific payment schedule corresponds to the A-subgame concept satisfies time consistency and efficiency condition and implies non-zero current payment to the active player immediately after her move.

8. Chapter 7. A Dynamic Model of Bertrand Competition for an Oligopolistic Market

   Zeng Lian, Jie Zheng

   Abstract

   We study an infinitely-repeated Bertrand competition game among a fixed number of firms in a market of both stochastic entry and stochastic demand. A firm’s entry into market in the next period is possible by making a positioning investment with stochastic success rate. The market demand in the next period is also stochastic and will not be realized until the firm enters the market. A successful investment allows a firm to participate in the Bertrand competition and an unsuccessful investment prevents a firm from entering the market, for the next period. We characterize the symmetric Markov perfect Nash Equilibrium (SMPNE) of such a dynamic game, where a firm’s strategy consists of two components: positioning strategy and pricing strategy. In examples with 1, 2, and 3 firms, we show the stage game market outcome, present the dynamic process of market structure, solve for the steady state of the dynamic system, and discuss about the speed of convergence to the steady state. Our work contributes to the dynamic oligopoly literature by allowing for two dimensions of stochastic uncertainty in firms’ decision-making.

9. Chapter 8. Pure and Mixed Stationary Nash Equilibria for Average Stochastic Positional Games

   Dmitrii Lozovanu

   Abstract

   In this paper we study the problem of the existence of Nash equilibria in pure and mixed stationary strategies for average stochastic positional games. We prove the existence of Nash equilibria in pure stationary strategies for an arbitrary two-player zero-sum average stochastic positional game and for an m-player average stochastic positional game in the case when the probability transition matrices induced by any profile of stationary strategies of the players are unichain. For the general case of an m-player average stochastic positional game we show that a Nash equilibrium in pure stationary strategies may not exist, however a Nash equilibrium always exists in mixed stationary strategies. Based on the proof of these results we present conditions for determining the optimal strategies of the players in the considered average stochastic positional games.

10. Chapter 9. Variational Inequalities, Nash Equilibrium Problems and Applications: Unification Dynamics in Networks

    Vladimir Matveenko, Maria Garmash, Alexei Korolev

    Abstract

    We study game equilibria in a model of production and externalities in network with two types of agents who possess different productivities. Each agent may invest a part of her endowment (for instance, time or money) on the first stage; consumption on the second period depends on her own investment and productivity as well as on the investments of her neighbors in the network. Three ways of agent’s behavior are possible: passive (no investment), active (a part of endowment is invested) and hyperactive (the whole endowment is invested). We introduce adjustment dynamics and study consequences of junction of two regular networks with different productivities of agents. We use the projection-based method for solving variational inequalities for the description of adjustment dynamics in networks.

11. Chapter 10. About the Looking Forward Approach in Cooperative Differential Games with Transferable Utility

    Ovanes Petrosian, Ildus Kuchkarov

    Abstract

    This paper presents a complete description and the results of the Looking Forward Approach for cooperative differential games with transferable utility. The approach is used for constructing game theoretical models and defining solutions for conflict-controlled processes where information about the process updates dynamically or for differential games with dynamic updating. It is supposed that players lack certain information about the dynamical system and payoff function over the whole time interval on which the game is played. At each instant, information about the game structure updates, players receive new updated information about the dynamical system and payoff functions. A resource extraction game serves as an illustration in order to compare a cooperative trajectory, imputations, and the imputation distribution procedure in a game with the Looking Forward Approach and in the original game with a prescribed duration.

12. Chapter 11. Dynamically Consistent Bi-level Cooperation of a Dynamic Game with Coalitional Blocs

    Leon A. Petrosyan, David W. K. Yeung

    Abstract

    In many real-life scenarios, groups or nations with common interest form coalition blocs by agreement for mutual support and joint actions. This paper considers two levels of cooperation: cooperation among members within a coalition bloc and cooperation between the coalition blocs. Coalition blocs are formed by players with common interests to enhance their gains through cooperation. To increase their gains coalition blocs would negotiate to form a grand coalition. A grand coalition cooperation of the coalitional blocs is studied. The gains of each coalition are defined as components of the Shapley value. Dynamically consistent payoff distributions between coalitions and among players are derived for this double-level cooperation scheme. For definition of players’ gains inside each coalition the proportional solution is used.

13. Chapter 12. Optimal Incentive Strategy in a Markov Game with Multiple Followers

    Dmitry B. Rokhlin, Gennady A. Ougolnitsky

    Abstract

    We consider a dynamic stochastic incentive problem for the case of several followers who play a Markov game. The leader ε-optimal strategy is determined via a stochastic control problem. This result generalizes a similar authors’ result for the case of one follower and the known results from the static theory of control in organizational systems. An illustrative example is given.

14. Chapter 13. How Oligopolies May Improve Consumers’ Welfare? R&D Is No Longer Required!

    Alexander Sidorov

    Abstract

    The paper studies how the industry concentration affects the Social welfare, which is measured as consumer’s indirect utility. Schumpeterian hypothesis tells that the harmful effect of oligopolization may be offset by positive externalities of concentration, such as innovations in technologies, R&D, etc. This contradicts to traditional neoliberal paradigm, which insists that concentration is always harmful for the end consumers. We study a general equilibrium model with two types of firms and imperfect price competition. Firms of the first type are monopolistic competitors with negligible impact to market statistics, subjected to typical assumptions, e.g., free entry until zero-profit cut-off. Unlike this, the firms of second type assumed to have non-zero impact to market statistics, in particular, to consumer’s income via distribution of non-zero profit across consumers-shareholders. Moreover, these large firms (oligopolies) allow for dependence of profits on their strategic choice, generating so called Ford effect. The first result we present is that in case of CES utility the concentration effect is generically harmful for consumers’ well-being. However, the result may be different for preferences, generating the demand with Variable Elasticity of Substitution (VES). We find the natural assumption on VES utilities, which hold for most of the commonly used classes of utility functions, such as Quadratic, CARA, HARA, etc., which allows to obtain the positive welfare effect, i.e., to justify Schumpeter hypothesis.

15. Chapter 14. Guaranteed Deterministic Approach to Superhedging: Lipschitz Properties of Solutions of the Bellman–Isaacs Equations

    Sergey N. Smirnov

    Abstract

    For the discrete-time superreplication problem, a guaranteed deterministic formulation is proposed: the problem is to ensure the cheapest coverage of the contingent claim on an American option under all admissible scenarios. These scenarios are set by a priori defined compacts depending on the price history; the price increment at each moment of time must lie in the corresponding compact. The market is considered without trading constraints and transaction costs. The problem statement is game-theoretic in nature and leads directly to the Bellman–Isaacs equations of a special form under the assumption of no trading constraints. In the present study, we estimate the modulus of continuity of uniformly continuous solutions, including the Lipschitz case.

16. Chapter 15. Evaluation of Portfolio Decision Improvements by Markov Modulated Diffusion Processes: A Shapley Value Approach

    Benjamin Vallejo-Jimenez, Mario A. Garcia-Meza

    Abstract

    We use the Shapley value to evaluate the introduction of a second stochastic process to an optimization problem. We find that introducing a Time-Inhomogeneous Markov Modulated Diffusion process to an asset portfolio decision problem yields higher returns to the rational decision maker. These increases in returns can be diminished by a high volatility in the state change or a high discount factor.

17. Chapter 16. Conditionally Coordinating Contracts in Supply Chains

    Nikolay A. Zenkevich, Irina Berezinets, Natalia Nikolchenko, Alina Rucheva

    Abstract

    The chapter revisits the supplier-retailer supply chain game and investigates target sales rebate and buyback contracts, which motivate participants for both individual rational and Pareto optimal behavior. The research considers the Stackelberg model in supply chain under condition of fixed retail price and stochastic demand. Authors proposed the algorithm of the conditionally coordinating problem solving for both types of contracts. A general framework is introduced, the condition for achieving the coordinating equilibrium is characterized, and a special case for uniformly distributed demand is analyzed. The models with uniformly distributed demand demonstrate that the conditional coordination can be achieved, and examples show that the problem of establishing contract parameters is a problem of sharing of supply chain expected profit under a compromise between supplier and retailer.