Frontiers of Dynamic Games

Inhaltsverzeichnis

1. Frontmatter

2. Chapter 1. Countervailing Power with Large and Small Retailers

   George Geronikolaou, Konstantinos G. Papadopoulos

   Abstract

   When concentration in the retail market increases, retailers gain more market power towards the suppliers and they hence can achieve better wholesale prices. In the 1950s, Galbraith introduced the concept of countervailing power claiming that lower wholesale prices will pass on to consumer as lower retail prices. Consequently higher concentration may turn out to be beneficial for consumers. In this model where a monopolistic supplier sells an intermediate good to M large retailers who are Cournot competitors and a competitive fringe consisting of N retailers, we show that higher concentration does not decrease retail prices and results solely to a reallocation of profits between the supplier and large retailers, thus invalidating Galbraith’s conjecture. The same result carries on when the exogenously given level of bargaining power of large retailers increases.

3. Chapter 2. Dynamic Voluntary Provision of Public Goods: The Recursive Nash Bargaining Solution

   Simon Hoof

   Abstract

   Grim trigger strategies can support any set of control paths as a cooperative equilibrium, if they yield at least the value of the noncooperative Nash equilibrium. We introduce the recursive Nash bargaining solution as an equilibrium selection device and study its properties by means of an analytically tractable n-person differential game. The idea is that the agents bargain over a tuple of stationary Markovian strategies, before the game has started. It is shown that under symmetry the bargaining solution yields efficient controls.

4. Chapter 3. Altruistic, Aggressive and Paradoxical Types of Behavior in a Differential Two-Person Game

   Anatolii Kleimenov

   Abstract

   A non-antagonistic positional (feedback) differential two-person game is considered in which each of the two players, in addition to the usual normal (nor) type of behavior oriented toward maximizing own functional, can use other types of behavior. In particular, it is altruistic (alt), aggressive (agg) and paradoxical (par) types. It is assumed that in the course of the game players can switch their behavior from one type to another. In this game, each player simultaneously with the choice of positional strategy selects the indicator function defined over the whole time interval of the game and taking values in the set {nor, alt, agg, par}. Player’s indicator function shows the dynamics for changing the type of behavior that this player adheres to. The concept of BT-solution for such game is introduced. The use by players of types of behaviors other than normal can lead to outcomes more preferable for them than in a game with only normal behavior. An example of a game with the dynamics of simple motion on a plane and phase constraints illustrates the procedure for constructing BT-solutions.

5. Chapter 4. Learning in a Game of Strategic Experimentation with Three-Armed Exponential Bandits

   Nicolas Klein

   Abstract

   The present article provides some additional results for the two-player game of strategic experimentation with three-armed exponential bandits analyzed in Klein (Games Econ Behav 82:636–657, 2013). Players play replica bandits, with one safe arm and two risky arms, which are known to be of opposite types. It is initially unknown, however, which risky arm is good and which is bad. A good risky arm yields lump sums at exponentially distributed times when pulled. A bad risky arm never yields any payoff. In this article, I give a necessary and sufficient condition for the state of the world eventually to be found out with probability 1 in any Markov perfect equilibrium in which at least one player’s value function is continuously differentiable. Furthermore, I provide closed-form expressions for the players’ value function in a symmetric Markov perfect equilibrium for low and intermediate stakes.

6. Chapter 5. Solution for a System of Hamilton–Jacobi Equations of Special Type and a Link with Nash Equilibrium

   Ekaterina A. Kolpakova

   Abstract

   The paper is concerned with systems of Hamilton–Jacobi PDEs of the special type. This type of systems of Hamilton–Jacobi PDEs is closely related with a bilevel optimal control problem. The paper aims to construct equilibria in this bilevel optimal control problem using the generalized solution for the system of the Hamilton–Jacobi PDEs. We introduce the definition of the solution for the system of the Hamilton–Jacobi PDEs in a class of multivalued functions. The notion of the generalized solution is based on the notions of minimax solution and M-solution to Hamilton–Jacobi equations proposed by Subbotin. We prove the existence theorem for the solution of the system of the Hamilton–Jacobi PDEs.

7. Chapter 6. The Impact of Discounted Indices on Equilibrium Strategies of Players in Dynamical Bimatrix Games

   Nikolay Krasovskii, Alexander Tarasyev

   Abstract

   The paper deals with construction of solutions in dynamical bimatrix games. It is assumed that integral payoffs are discounted on the infinite time interval. The dynamics of the game is subject to the system of differential equations describing the behavior of players. The problem of construction of equilibrium trajectories is analyzed in the framework of the minimax approach proposed by N. N. Krasovskii and A. I. Subbotin in the differential games theory. The concept of dynamical Nash equilibrium developed by A. F. Kleimenov is applied to design the structure of the game solution. For obtaining constructive control strategies of players, the maximum principle of L. S. Pontryagin is used in conjunction with the generalized method of characteristics for Hamilton–Jacobi equations. The impact of the discount index is indicated for equilibrium strategies of the game.

8. Chapter 7. On Control Reconstruction Problems for Dynamic Systems Linear in Controls

   Evgeniy Krupennikov

   Abstract

   In differential games the a posteriori analysis of motions, namely, trajectories of the dynamics and the analysis of the players’ controls generating these trajectories are very important. This paper is devoted to solving problems of reconstruction of trajectories and controls in differential games using known history of inaccurate measurements of a realized trajectory. A new method for solving reconstruction problems is suggested and justified for a class of differential games with dynamics, linear in controls and non-linear in state coordinates. This method relies on necessary optimality conditions in auxiliary variational problems. An illustrating example is exposed.

9. Chapter 8. Evolution of Risk-Statuses in One Model of Tax Control

   Suriya Kumacheva, Elena Gubar, Ekaterina Zhitkova, Galina Tomilina

   Abstract

   Nowadays information is an important part of social life and economic environment. One of the principal elements of economics is the system of taxation and therefore tax audit. However total audit is expensive, hence fiscal system should choose new instruments to force the tax collections. In current study we consider an impact of information spreading about future tax audits in a population of taxpayers. It is supposed that all taxpayers pay taxes in accordance with their income and individual risk-status. Moreover we assume that each taxpayer selects the best method of behavior, which depends on the behavior of her social neighbors. Thus if any agent receives information from her contacts that the probability of audit is high, then she might react according to her risk-status and true income. Such behavior forms a group of informed agents which propagate information further then the structure of population is changed. We formulate an evolutionary model with network structure which describes the changes in the population of taxpayers under the impact of information about future tax audit. The series of numerical simulation shows the initial and final preferences of taxpayers depends on the received information.

10. Chapter 9. Stationary Nash Equilibria for Average Stochastic Positional Games

    Dmitrii Lozovanu

    Abstract

    An average stochastic positional game is a stochastic game with average payoffs in which the set of states is divided into several disjoint subsets such that each subset represents the position set for one of the player and each player controls the Markov process only in his position set. In such a game each player chooses actions in his position set in order to maximize his average reward per transition. We show that an arbitrary average stochastic positional game possesses a stationary Nash equilibrium. Based on this result we propose an approach for determining the optimal stationary strategies of the players.

11. Chapter 10. Game Equilibria and Transition Dynamics in Networks with Heterogeneous Agents

    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) at the first stage; consumption at 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. In particular, we study how the behavior of nonadopters (passive agents) changes when they connect to adopters (active or hyperactive) agents.

12. Chapter 11. Non-cooperative Differential Game Model of Oil Market with Looking Forward Approach

    Ovanes Petrosian, Maria Nastych, Dmitrii Volf

    Abstract

    The paper applies Looking Forward Approach to analyze the world oil market with the framework of a differential game model of quantity competition oligopoly. Namely Looking Forward Approach is used to take into account dynamically updating information. Under the information we understand the forecast of the oil demand dynamics. We focus on the period from December 2015 to November 2016 and suppose that during this time interval countries did not cooperate officially on the amounts of oil to be produced. Therefore, their behavior can be modeled using the non-cooperative game model. As a solution concept for this conflict-controlled process we use feedback Nash equilibrium. In order to define the parameters of model open source data is used, results of numerical simulations and comparison with the historical data are presented.

13. Chapter 12. S-strongly Time-Consistency in Differential Games

    Leon A. Petrosyan, Ekaterina V. Gromova

    Abstract

    In the paper the definition of S-strongly time-consistency in differential games is introduced. The approach of the construction of S-strong time-consistent subcore of the classical core on the base of characteristic function obtained by normalization of classical characteristic function is formulated. Its relation to another characteristic function obtained by an integral extension of the original characteristic function is studied.

14. Chapter 13. Characteristic Functions in a Linear Oligopoly TU Game

    Artem Sedakov

    Abstract

    We consider a linear oligopoly TU game without transferable technologies in which the characteristic function is determined from different perspectives. In so-called γ-, δ-, and ζ-games, we study the properties of characteristic functions such as monotonicity, superadditivity, and supermodularity. We also show that these games have nonempty cores of a nested structure when the δ-characteristic function is supermodular.

15. Chapter 14. The Position Value and the Myerson Value for Hypergraph Communication Situations

    Erfang Shan, Guang Zhang

    Abstract

    We characterize the position value and the Myerson value for uniform hypergraph communication situations by employing the “incidence graph game” and the “link-hypergraph game” which are induced by the original hypergraph communication situations. The incidence graph game and link-hypergraph game are defined on the “incidence graph” and the “link-hypergraph”, respectively, obtained from the original hypergraph. Using the above tools, we represent the position value by the Shapley value of the incidence graph game and the Myerson value of the link-hypergraph game for uniform hypergraph communication situations, respectively. Also, we represent the Myerson value by the Owen value or the two-step Shapley value of the incidence graph game with a coalition structure for hypergraph communication situations.

16. Chapter 15. Bertrand Meets Ford: Benefits and Losses

    Alexander Sidorov, Mathieu Parenti, Jacques-Francois Thisse

    Abstract

    The paper carries out the detailed comparison of two types of imperfect competition in a general equilibrium model. The price-taking Bertrand competition assumes the myopic income-taking behavior of firms, another type of behavior, price competition under a Ford effect, implies that the firms’ strategic choice takes into account their impact to consumers’ income. Our findings suggest that firms under the Ford effect gather more market power (measured by Lerner index), than “myopic” firms, which is agreed with the folk wisdom “Knowledge is power.” Another folk wisdom implies that increasing of the firms’ market power leads to diminishing in consumers’ well-being (measured by indirect utility.) We show that in general this is not true. We also obtain the sufficient conditions on the representative consumer preference providing the “intuitive” behavior of the indirect utility and show that this condition satisfy the classes of utility functions, which are commonly used as examples (e.g., CES, CARA and HARA.)

17. Chapter 16. On Multilateral Hierarchical Dynamic Decisions

    Krzysztof Szajowski

    Abstract

    Many decision problems in economics, information technology and industry can be transformed to an optimal stopping of adapted random vectors with some utility function over the set of Markov times with respect to filtration build by the decision maker’s knowledge. The optimal stopping problem formulation is to find a stopping time which maximizes the expected value of the accepted (stopped) random vector’s utility.

    There are natural extensions of optimal stopping problem to stopping games-the problem of stopping random vectors by two or more decision makers. Various approaches dependent on the information scheme and the aims of the agents in a considered model. This report unifies a group of non-cooperative stopping game models with forced cooperation by the role of the agents, their aims and aspirations (v. Assaf and Samuel-Cahn (1998), Szajowski and Yasuda (1995)) or extensions of the strategy sets (v. Ramsey and Szajowski (2008)).