Table of Content

30 March 2024, Volume 12 Issue 1

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Preface: The Special Issue on Dynamic and Networking Games

Hong-Wei Gao, Vladimir Mazalov

2024, 12(1):  1-3.  doi:10.1007/s40305-023-00536-7

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A Dynamic Network Game of the Fintech Industry

David W. K. Yeung, Leon A. Petrosyan, Ying-Xuan Zhang

2024, 12(1):  5-33.  doi:10.1007/s40305-022-00434-4

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Economies of scale, economies of scope, and technology spillover are decisive economic elements that are crucial to the development in the Fintech industry. These positive externalities are often realized through network links. In this paper, we present a dynamic network of financial firms which exhibits these decisive elements. The network game equilibria are characterized. A Pareto efficient solution involving collaboration of all firms is provided. To obtain a fair-share distribution of cooperative gains, the Shapley value is adopted as the sharing mechanism. Payoff distribution mechanisms which guarantee the fulfilment of the Shapley value distribution in each stage of the cooperation duration are derived.

A Note on Transition Kernels for the Most Unfavourable Mixed Strategies of the Market

Sergey N. Smirnov

2024, 12(1):  35-50.  doi:10.1007/s40305-023-00490-4

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We consider a deterministic model of market evolution with trading constraints and apply a game-theoretic approach to the superhedging problem. We obtain sufficient conditions for the game equilibrium and prove under these conditions the existence of a Borel-measurable transition kernel describing dependence on price prehistory of the most unfavourable mixed strategy of the market.

Differential Game Model of Resource Extraction with Continuous and Dynamic Updating

Ovanes Petrosian, Tihomirov Denis, Jiang-Jing Zhou, Hong-Wei Gao

2024, 12(1):  51-75.  doi:10.1007/s40305-023-00484-2

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This paper is devoted to a new class of differential games with continuous and dynamic updating. The direct application of resource extraction in a case of dynamic and continuous updating is considered. It is proved that the optimal control (cooperative strategies) and feedback Nash equilibrium strategies uniformly converge to the corresponding strategies in the game model with continuous updating as the number of updating instants converges to infinity. Similar results are presented for an optimal trajectory (cooperative trajectory), equilibrium trajectory and corresponding payoffs.

Equilibrium Arrivals to Preemptive Queueing System with Fixed and Random Population Size

Julia Chirkova, Vladimir Mazalov

2024, 12(1):  77-92.  doi:10.1007/s40305-023-00461-9

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A single-server queueing system with preemptive access is considered. Each customer has one attempt to enter the system at its working interval [0, T]. As soon as the customer request enters the system, the server immediately starts the service. But when the next request arrives in the system, the previous one leaves the system even he has not finished his service yet. We study a non-cooperative game in which the customers wish to maximize their probability of obtaining service within a certain period of time. We characterize the Nash equilibrium and the price of anarchy, which is defined as the ratio between the optimal and equilibrium social utility. Two models are considered. In the first model the number of players is fixed, while in the second it is random and obeys the Poisson distribution. We demonstrate that there exists a unique symmetric equilibrium for both models. Finally, we calculate the price of anarchy for both models and show that the price of anarchy is not monotone with respect to the number of customers.

Essential Players in Cooperative Games with Graph Communication Structure

Guang Zhang, Jing-Yi Ge

2024, 12(1):  93-108.  doi:10.1007/s40305-023-00463-7

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A class of cooperative games with graph communication structure is studied in this paper by considering some important players, namely essential players. Under the assumption that only connected coalitions containing essential players are able to cooperate and obtain their worths, the class of graph games with essential players is proposed as well as an allocation rule. The proposed value follows the spirit of the Myerson value defined by applying the Shapley value on a modified game. Three properties, feasible component efficiency, the inessential component property, and fairness, are provided to fully characterize this value, where feasible component efficiency and fairness follows the same ideas of component efficiency and fairness for classical graph games, and the inessential component property says that the total payoffs of the players in a non-feasible component is zero. Moreover, some computational aspects of the proposed value and comparisons with disjunctive permission value for games with permission structure are also studied, respectively.

Opinion Dynamics in Two-Layer Networks with Hypocrisy

Chi Zhao, Elena Parilina

2024, 12(1):  109-132.  doi:10.1007/s40305-023-00503-2

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We propose a general concealed voter model (GCVM), in which individuals interact in two layers and can exchange their opinions in the internal layer. This interaction is not allowed in a CVM. By exchanging opinions in the internal layer, we mean that individuals share their real or internal opinions with their close friends. The process of opinion formation in GCVM is presented in the paper. We make the series of numerical simulations of GCVM with different network structures (both external and internal) and get some counterintuitive conclusions. For instance, we find out that sometimes with a relatively simple network structure of an external layer the consensus within the individuals’ opinions cannot be reached, and if individuals in the network are not good at expressing their opinions publicly (in an external layer), exchanging opinions with their close friends (in an internal layer) is almost useless.

Competitive Resource Allocation Among Urban Congestion Areas in a Modern Big City

Alexander Krylatov, Anastasiya Raevskaya

2024, 12(1):  133-153.  doi:10.1007/s40305-023-00530-z

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The continuing growth of modern big cities leads to their spatial expansion and the emergence of new road connections and urban areas. Areas where large transportation flows of pedestrians, passengers, and drivers come together create demand points, which attract business companies that strive to allocate their resources in the most sought-after places. However, the law of supply and demand restrains companies from allocating all their resources solely in the most popular congestion areas since the more valuable an urban area, the higher the cost to be paid for a resource unit allocation there. As a result, companies act in a non-cooperative manner and try to minimize their own overall costs when allocating resources across available commercial areas in a big city. Non-cooperative behavior of companies leads to the problem of Nash equilibrium search in the game of competing entrepreneurs. In this paper, we study the corresponding resource allocation game under affine cost functions and obtain Nash equilibrium strategies in explicit form. These findings allow us to develop a simple procedure for computing Nash equilibria in the game of companies allocating their resources among urban congestion areas. The computational study demonstrates the dependence of the average price for resource allocation on the number of players and their resource volumes. The outcome of the paper contributes to flow theory and seems to be fresh and useful for managers.

The Generalized Stackelberg Equilibrium of the Two-Person Stopping Game

Marek Skarupski, Krzysztof J. Szajowski

2024, 12(1):  155-168.  doi:10.1007/s40305-023-00460-w

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In modeling the bilateral selection of states of the process, Dynkin (Dokl Akad Nauk USSR 185:241–288, 1969) proposed a two-person game in which players use stopping moments as strategies. The purpose of this work is to present a model of the game in which the players have different information about the process itself, as well as various laws to stop the process and accept its state. The game model uses the stochastic process apparatus, in particular, the ability to create different filters for the same process. The sets of stopping moments based on different filters are not identical, which allows us to model different sets of strategies for players. We show that the follower, by observing the behavior of a rational leader, can recover information that is lost due to the lack of complete observation of the state of the process. In the competition of two opponents for the maximum of the i.i.d. sequence, one of whom has access to full information and the other only knows their relative ranks, we found the generalized Stackelberg equilibrium. If the priority of a player observing the relative ranks is less than 50%, then that player modifies his strategy based on the behavior of the second player. For a player with full information, information about the behavior of the player observing the relative ranks is useless.

Zero-Sum Continuous-Time Markov Games with One-Side Stopping

Yurii Averboukh

2024, 12(1):  169-187.  doi:10.1007/s40305-023-00502-3

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The paper is concerned with a variant of the continuous-time finite state Markov game of control and stopping where both players can affect transition rates, while only one player can choose a stopping time. The dynamic programming principle reduces this problem to a system of ODEs with unilateral constraints. This system plays the role of the Bellman equation. We show that its solution provides the optimal strategies of the players. Additionally, the existence and uniqueness theorem for the deduced system of ODEs with unilateral constraints is derived.

Strong Subgame Consistency of the Core in Stochastic Network Formation Games

Ping Sun, Elena Parilina

2024, 12(1):  189-213.  doi:10.1007/s40305-022-00442-4

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We consider a model of network formation as a stochastic game with random duration proposed initially in Sun and Parilina (Autom Remote Control 82(6):1065–1082, 2021). In the model, the leader first suggests a joint project to other players, i.e., the network connecting them. Second, the players are allowed to form fresh links with each other updating the initially proposed network. The stage payoff of any player is defined depending on the network structure. There are two types of randomness in the network formation process: (i) links may fail to be formed with different probabilities although players intend to establish them, (ii) the game process may terminate at any stage or transit to the next stage with a certain probability distribution. Finally, a network is formed as a result of players’ decisions and realization of random variables. The cooperative version of the stochastic game is investigated. In particular, we examine the properties of subgame consistency as well as strong subgame consistency of the core. We provide a payment mechanism or regularization of the core elements to sustain its subgame consistency and avoid the player’s deviations from the cooperative trajectory. In addition, the distribution procedure of the core elements is regularized in case there are negative payments to achieve only nonnegative payments to the players at any stage. The sufficient condition of a strongly subgame consistent core is also obtained. We illustrate our theoretical results with a numerical example.

Structural Stability of the Financial Market Model: Continuity of Superhedging Price and Model Approximation

Sergey N. Smirnov

2024, 12(1):  215-241.  doi:10.1007/s40305-023-00524-x

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The present paper continues the topic of our recent paper in the same journal, aiming to show the role of structural stability in financial modeling. In the context of financial market modeling, structural stability means that a specific “no-arbitrage” property is unaffected by small (with respect to the Pompeiu–Hausdorff metric) perturbations of the model’s dynamics. We formulate, based on our economic interpretation, a new requirement concerning “no arbitrage” properties, which we call the “uncertainty principle”. This principle in the case of no-trading constraints is equivalent to structural stability. We demonstrate that structural stability is essential for a correct model approximation (which is used in our numerical method for superhedging price computation). We also show that structural stability is important for the continuity of superhedging prices and discuss the sufficient conditions for this continuity.

Irrational-Behavior-Proof Conditions for Stochastic Games over Event Trees

Lei Wang, Cui Liu, Hong-Wei Gao, Chong Lin

2024, 12(1):  243-263.  doi:10.1007/s40305-022-00446-0

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In this paper, the irrational-behavior-proof conditions in a class of stochastic dynamic games over event trees are presented. Four kinds of irrational-behavior-proof conditions are proposed by the imputation distribution procedure, and their relationships are discussed. More specific properties for the general transformation of characteristic functions are developed, based on which, the irrational-behavior-proof conditions are proved to be true in a transformed cooperative game.