Теоретико-игровые модели формирования сетей с асимметричными игроками тема диссертации и автореферата по ВАК РФ 00.00.00, кандидат наук Сунь Пин

• Сунь Пин
• кандидат наук
• 2023

Оглавление диссертации кандидат наук Сунь Пин

Contents

Introduction

Chapter 1. Utility Functions for Determining Stable Networks with Ordered Group Partitioning

1.1 Model of network formation

1.1.1 Definitions and notations

1.1.2 Utility functions

1.1.3 Stable networks

1.2 Stable networks for a given partition of players

1.2.1 Benefits without decay (Cases I and II)

1.2.2 Discussion of the results for Cases I and II

1.2.3 Benefits with decay (Cases III and IV)

1.2.4 Discussion of the results for Cases III and IV

1.3 Stable network for a special class of group partitioning

1.4 Conclusion to Chapter

Chapter 2. Dynamic Network Formation with Ordered Partitioning and Incomplete Information

2.1 Dynamic network formation process

2.2 Stable networks under complete and incomplete information

2.2.1 Strategies

2.2.2 Stable equilibrium and stable network

2.3 Theoretical results

2.3.1 General partition

2.3.2 Uniform partition of players into groups

2.4 Superior information updating rule

2.5 Simulations

2.5.1 Time to stable network

2.5.2 Heterogeneity index

2.5.3 Impact of group size gaps on interconnections

2.6 Conclusion to Chapter

Chapter 3. Stochastic Model of Network Formation with Asymmetric Players

3.1 Model

3.1.1 Definitions and notations

3.1.2 Stochastic network formation process

3.1.3 Random factors in the dynamic process

3.2 Stochastic network formation games in extensive form

3.3 Main functional equations for the stochastic game of network formation

3.4 Cooperative stochastic network formation games

3.4.1 Defining characteristic function

3.4.2 The imputation set

3.4.3 The CIS-value as a cooperative solution

3.4.4 Imputation distribution procedure

3.5 Subgame consistency in stochastic network formation games

3.6 The core as a cooperative solution of stochastic network formation games

3.6.1 Regularization of the core

3.6.2 Strong subgame consistency of the core

3.7 Numerical examples

3.8 Conclusion to Chapter

Chapter 4. Network Formation with Asymmetric Players and Chance Moves

4.1 Definitions and notations

4.2 Model

4.2.1 Players of various types

4.2.2 Games with chance moves and players of various types

4.2.3 Two-stage network formation game

4.3 Two-stage game as game in an extensive form

4.3.1 Construction of a tree graph

4.3.2 Two-stage game on a tree graph

4.3.3 The expected payoff in the extensive-form game

4.4 Main results

4.5 Three-player game with a major player

4.6 Special n-player game

4.7 Conclusion to Chapter

Conclusions

Bibliography

Введение диссертации (часть автореферата) на тему «Теоретико-игровые модели формирования сетей с асимметричными игроками»

Introduction

Relevance of thesis topic

Social and economic interactions can be represented by networks where individual members - be they people, organizations, economic actors, or countries - are linked and affected by their relations within and between networks. It is becoming apparent that diverse networks make up an important role in a wide variety of social and economic interactions. It is thus fundamentally interesting to examine the nature of social communication as represented by network structures, such as the stability of network structures, as well as how they are formed and evolve through dynamics. Moreover, the theory of network games has become an attractive and actively developing branch of the game-theoretical research in the past decades.

With the support of network games, one can mathematically model and find solutions to a variety of significant and relevant game issues. For instance, one can establish a setting which may reflect certain realistic circumstances, detecting to certain extent stable configurations in the sense of which, no individual prefers to modify the communication structure among the whole community considering her own utility. Undoubtedly, the problem of such stability attracts a great concern in a lot of fields, e.g., politics, economy, human society, etc. Meanwhile, if some periods of network modification are acceptable, one may create an environment in which individuals are able to interact with each other by forming or severing connections through the dynamic process, and set a problem that combines examination of the optimal network configuration and investigation of the players' optimal behavior in the dynamic game. Moreover, with respect to the stability from the perspective of cooperation sustainability, one may also consider the cooperative behavior in network formation where players cooperate together forming a joint network project, and the issue of payoff distribution among all stages may be investigated to prevent the cooperation break-up.

The above issues quite clearly outline the area of practical application of the the-

ory of network games, while it should be noted that they are fairly extensive. As the extension, one may also involve heterogenous players, random factors or incomplete knowledge of players in the fundamental network game models mentioned above, which are even actual. Indeed, individual heterogeneity is a crucial condition that explains a number of phenomena in social networks as reality is full of mixtures of people with different characteristics - old and young, married and unmarried, different levels of education, etc. These features are often related to their interaction patterns. For instance, in the context of information networks, it is often the case that some individuals are better informed, and this allows them to obtain larger benefits. In addition, when modeling the possible scenarios of development in economic systems or political events, an analysis of random factors is broadly asked for since many real-world projects occasionally terminate because of unexpected events, such as the government policies, natural disasters, financial crisis, etc. Moreover, the actions may be failed when people behave, and it is actual to assume that it can happen with certain probabilities varying from person to person. In the big world with varieties of subgroups, it is nearly impossible for people to be accessible to issues in any subgroup. Thus, incomplete knowledge of people to other human beings is a more practical circumstance. Especially in the educational system, exploring the influence of incomplete information may largely contribute to the educational reform.

The thesis is devoted to studying network formation games involving asymmetric players who are various in terms of utility evaluation in network topology, successful probability of action-making, position in the community, etc. At the same time, we also aim to explore the impact from some realistic stochastic factors and incomplete knowledge of players on their interaction patterns.

Overview of the results in this area

Representative models of network formation with a game-theoretic approach originate from literature [3], which is one of the first papers modeling network formation explicitly as a game. The authors describe network formation from the perspective of the theory of cooperative games with communication structures (see [3]). In [54], a comprehensive course and overview of the current trends in network games are provided. In network formation games, it is natural to examine the topology of social communication networks, which are stable in some sense. A wide variety of concepts for stable networks exists in the literature regarding network games from

different points of view.

One of the most well-known concepts is called pairwise stability, which is initially proposed by Jackson and Wolinsky [43], where they account for stability when no player benefits by unilaterally severing an existing link, and no pair earns by forming a new connection. An interesting variation of pairwise stability that considers coalitions containing more than two players contrary to pairs of players is examined by Dutta and Mutuswami [24]. In the context of Jackson and Wolinsky's model, paper [85] investigates the dynamic evolution of networks and proves that the dynamic process of network formation does not always converge to the efficient network. Using a purely noncooperative approach to network formation resulting into so-called Nash network, where both one-way flow and two-way flow (or undirected) networks are examined under the assumption of homogeneous costs and benefits across players, in the paper [8] it is demonstrated that the Nash network has particular identifying structures, such as the wheel or the star. In the work of Jackson and Watts [42], a model of dynamic network formation process is introduced and the set of stable networks is mainly characterized. Integrating the one-way and two-way flow models of network formation in [8], paper [58] addresses the issue of stability by characterizing the Nash and strict Nash structures for the whole range of intermediate models. An interesting perspective for examining the stability of networks with all uncovered nodes is proposed in [45] by the definition of which, a node is covered by another node if all its neighbors are also the neighbors of another node. The work of Caffarelli [14] presents a model of noncooperative network formation and characterizes Nash networks by showing that all equilibrium networks display a center-periphery structure and may be disconnected. The recent work [39] studies the general properties of pairwise stable and pairwise Nash stable networks when players are ex-ante homogeneous. Here general link externality conditions are imposed on players' utilities and the existence of pairwise Nash stable networks of various structures is demonstrated.

The bulk of the aforementioned literature assumes homogeneity principle, i.e., the players are identical and other elements of the game, such as benefits, costs, and knowledge are homogeneous. In each chapter of the thesis, we allow heterogenous players from various aspects. In Chapter 1, for instance, we consider a society partitioned into ordered groups of players and examine stable networks by making use of different approaches to define players' utilities which rely on the group difference

(see [80]). Indeed, heterogeneity is a crucial condition that explains a number of phenomena in social networks. The idea of heterogeneity has been developed in the literature by assuming the presence of agents different in their benefits or costs of creating links. For instance, built on the work [9], the model proposed in [37] introduces heterogeneity by considering various probabilities of failure for different links. Nash networks that are connected, super-connected, or stars are given a particular consideration in this work. A one-way flow connection model, in which players are heterogeneous with respect to the values and costs of establishing links, is proposed in [26], and it is shown that heterogeneity in values and costs is fairly vital in determining the level of connectedness and the architecture of equilibrium structures. In the context of both [18] and [19], the economic model of friendship with a population formed by communities of different sizes is developed. The feature of the model is that the player's utility relies on the numbers of her friends who are of the same type, or they are of different types. Namely, each player distinguishes only between "the same" and "different" in making connections. The paper [47] extends the connection model from [43] by introducing heterogeneity in a cost structure, where the cost of creating a link is proportional to the distance between two individuals. Work [33] reports experimental research on network formation with varieties of players and provides a complete characterization of the equilibrium network set. Moreover, in [10], the concept of the strict Nash network in the framework of partner heterogeneity is defined, whereas any player's payoff obtained from a link depends on the identity of that link's partner. In particular, it is demonstrated that setting partners' heterogeneity impacts strict Nash networks. The model of [41] is established based on [43] but with separable heterogeneous costs supposing that the whole cost for each player is uniquely proportional to its degree. In [77], a model of two-stage network formation game is constructed where the player set consists of a leader and a finite number of other common players who are divided into two type subsets, the passive and the positive. Paper [46] considers a game-theoretic model of external control influence on opinion dynamics and reaching a consensus in social networks, assuming that one or several players influence the opinions of players.

In some works, combination of network and group structures is especially a basis to which the first two models of the thesis are fairly closed. As in [29], Galeotti et al. characterize the Nash networks in an insider-outsider model where the society is formed by several spatially arranged groups. In [11], the authors survey some models

of group and network formation by farsighted players as well as within dynamic contexts, comparing various procedures of network and coalition formation. The idea of finding networks which are stable in some sense relative to a given nonordered partition of players has also been investigated in [15] and [30]. Additionally, in [2], the authors investigate both a network's influence on the formation of coalitions and vice versa; i.e., how a network adapts to the current coalition structure and thus forms a social feedback loop. In [20], a model of society with two groups is introduced where players can either attempt to link only to a player of a similar type or make a costly effort to search for a partner within the whole population. Moreover, Tarbush and Teytelboym present a dynamic model of network formation in which players interact in overlapping social groups, and each player interacts with others with a probability depending on the mutual social group sizes and on the sizes of their overlaps (see [83]). In the work of Avrachenkov et al. [6], the network partitioning is examined as a cooperative game with partial cooperation presented by graph, and the community detection is analyzed through game-theoretic methods. In the framework of [53], Mauleon et al. explore group structures and propose the concept of constitutional stability to examine the group structures formed to emerge at equilibrium in many-to-many matchings. In [7], the authors suggest two cooperative game theory-based approaches for community detection in networks, both of which allow to detect clusters with various resolutions. Under the background of labor turnover in enterprises, work [51] explores the dynamic formation of the coalition structure when the utility of each player is determined by both the coalition structure and a provided communication structure among players. Paper [17] introduces a network formation game where players form one or more clusters that have positive relationships among their members but negative relationships among the members of other clusters. Work [22] analyzes the necessary and sufficient conditions for various networks being stable assuming that players are partitioned into two different communities. In [80], the stable conditions of some specific network structures are examined when the society is partitioned into ordered groups and the cost functions are affected by a given partition in two different ways. More recently, Boncinelli et al. analyze a team-formation process that generalizes matching and network formation models in [12].

In most of the literature studying how heterogeneity affects network formation, such as [26, 27, 29, 76, 86, 87], even though the theoretical frameworks vary from one

to another, complete information is still a common assumption, and the predictions in the above papers are often restricted to a few, specific, types of network topologies such as stars, wheels or core-periphery networks with "high-value" or low-cost players enjoying higher connectivity than others. In addition to heterogenous players, we also model a couple of dynamic network formation games with incomplete information where players do not know the identities (types) of other players. To date, merely a relatively small body of literature exists which has investigated the role of incomplete information on network formation. For instance, Jackson and Yariv present a model in [44] where players are uncertain over how well connected others are, and they primarily investigate incomplete information pertaining to the structure of the network. In the model proposed in [28], individuals are partially informed about the structure of a social network, and it is shown that an incomplete information setting reduces the multiplicity of Nash equilibria as obtained under complete information. Paper [30] develops a strategic network formation model where agents have heterogeneous knowledge of the network. In [50], network formation is modeled as a simultaneous game of incomplete information when decisions about linking depend on the network structure and the attributes of players. In the framework of [21], the individual's rewards and the strength of interactions are partially known by the players. In [75], a dynamic network formation game with incomplete information is examined where players do not know the players' types beforehand, but have to learn them through the dynamics.

It is also useful to take into account stochastic factors that influence the network formation process, especially while focusing on the dynamics. As an application of stochastic games, we present several stochastic network formation models where the mechanisms of random duration, random order of players' action-making, or chance moves of the Nature are designed. The starting point of stochastic game theory is a paper of Shapley [73] in which he introduces the definition of a stochastic game and proves the existence of a value of zero-sum game with infinite duration. In recent years, game-theoretical models on dynamic network formation with stochastic factors proposed as a new trend have received greater attention. Random factors can describe the uncertainty associated with the duration of the game (see [35, 78, 81] for references on cooperative multistage games with stochastic time horizon, also [63] examining a class of dynamic games implemented on an event tree with random terminal time), with the implementation of the strategies chosen by the players,

random moves of the Nature (or shocks), etc. (see [49], where multicriteria dynamic games with random moves are investigated). In [69], the authors investigate and synthesize network formation models based on random graphs and random processes, paying special attention to the influence of homophylic relationships on the structure of the network. The dynamic process of network formation is considered in [48] under the assumption of a 'blind' correction of network formation, taking into account the property of near-sightedness, and the set of stochastically stable networks is analyzed. In [74], Skyrms and Pemantle explore a dynamic network model in which players play repeated games in pairs determined by a randomly evolving social network. Work [25] analyzes a process of network formation in the framework, according to which players may form or delete links occasionally making mistakes, and primarily identifies the final structures which the formation process will converge to. In [79], the authors explore a two-stage network formation game with chance moves and players of various types where the Nature selects a type vector for players based on the given probability distribution, and each player keeps in mind only his own type and the leader's type.

In the stochastic network formation models, we also focus on the cooperative behaviors of players and in particular concern about the sustainability of cooperation along the dynamics. Cooperation in the dynamic network games is examined by many authors. For instance, a network formation issue is explored using cooperative game theory in [5], where the cooperative network formation game is solved with the Nash bargaining solution approach. In [66], the multistage network games with perfect information are considered where players can change the network structure at each stage, and a method for finding optimal behavior for players is proposed. In [55], the paired interactions in networks are analyzed using cooperative game theory methods, in particular, the characteristic function is defined as the maximal number of pairs that can be formed in some communication network. In the cooperative case, one of the most valuable methods of finding consistent solutions in dynamic network games is a construction of the time-consistent distribution procedures of the cooperative solutions (see [65]). The problem of time consistency is initially proposed in [64] for the class of cooperative differential games, and later a special mechanism of stage payments — an imputation distribution procedure — was designed to overcome the time inconsistency of cooperative solution concepts (see [65]). In particular, a two-stage network game model is constructed in [31] where

first players form the network following some rules, and then realize cooperative strategies, and both issues of time consistency and strong time consistency are studied for network games. Paper [71] studies the strong subgame consistency of the core on a class of linear-state games, and it is proved that the core is always strongly subgame consistent in the case of symmetric players in that model. Petrosyan et al. obtain a strongly subgame consistent core based on specially designed modified and limiting characteristic functions in [67]. The strongly subgame consistent core is defined for stochastic games in [60]. In [82], a model of two-stage network games is introduced, and it is proved that a cooperative subgame is convex which ensures the nonemptiness of the core.

Summarizing the above, it is essential and significant to investigate the impact of heterogeneity, incomplete information and random factors on network formation, in particular the stable network topologies, players' interaction patterns along the dynamics, cooperative behavior as well as the equilibrium structure.

Goals of the thesis

The central goal of the thesis is to study network formation games with asymmetric players by different approaches and from various perspectives, in a static or dynamic way, and from the noncooperative or cooperative point of view. To pursuit the central goal, several specific questions are addressed and answered throughout the four chapters in the thesis.

One of the goals of the thesis is to study the influence of asymmetric players on stable network structures by defining original utility functions which rely on players' types. Another goal is to develop a network formation model with incomplete information where players are heterogeneous in some observable characteristics, and examine the impact of incomplete knowledge on players' interactions over time. Moreover, the thesis also aims at the construction of game-theoretic network formation models using the theory of stochastic games and verification of the principles of stable cooperation, namely, subgame consistency and strongly subgame consistency of classical cooperative solutions. The aim of the thesis is also to investigate the players' different performances when they have different utility functions.

Main tasks

To achieve the goals, we list the following main tasks and accomplish them within this research:

1. The influence of asymmetric players on stable networks is explored, and inno-

vative utility functions are proposed depending on both the network structures and the degree of heterogeneity on players. The task is to compare the difference of conditions for specific network structures (i.e., empty network, complete network, minimal and minimally connected network, inner star and inner complete network) to be pairwise stable when various utility functions are applied.

2. For the dynamic network formation game in discrete time and with an infinite horizon, schemes incorporating asymmetric players and incomplete information are designed. We formulate the tasks of finding the stable equilibrium and investigating the properties of stable networks formed as a result of players' stable equilibrium behavior. Moreover, the most significant task is to explore the impact of the incomplete knowledge of players on the stable network structures through theoretical results and particularly designed simulation experiments.

3. One of the main tasks formulated in the work is to create a game-theoretical model of network formation taking into account stochastic factors. To do this, we define a class of stochastic network formation games with asymmetric players and random terminal time, and find the cooperative strategy profile maximizing the total expected payoff of the players. We also obtain the formula of calculating the new stage payoffs which help to guarantee the cooperation sus-tainability, namely, the subgame consistency and strong subgame consistency of the cooperative solutions.

4. One of the tasks is to define the stochastic network formation games with chance moves when players have different types. We propose the original definition of the stable partially Bayesian equilibrium which describes the corresponding behavior of any player depending on her type. One of the tasks in this work is to explore the connection between the new equilibrium concept and the Nash equilibrium in the game. Moreover, we also formulate the tasks of investigating the properties of such an equilibrium and characterizing the network structures formed when players follow such an equilibrium in games with special structures of characteristic functions.

Scientific novelty

In this thesis, innovative functions which evaluate players' utilities in any network structure are proposed. Such utility functions significantly influence players'

incentives in the formation of links and, consequently, network structure, and they are fairly applicable since they may reflect a lot of circumstances in economic and social lives. A number of novel results are obtained in the investigation of conditions for stable networks regarding a given ordered partition of players under these utility functions. Both theoretical results and numerical examples present particular characteristics of our model and exhibit fascinating phenomena. More importantly, these phenomena come in contrast to most literature dealing with stable networks with different ways of defining players' utilities. They are highlighted in this thesis.

For the dynamic network formation game with ordered partitioning and incomplete information, a novel information updating capability for each player to be faced with incomplete knowledge over other players' identities is proposed. One numerical example presented in the work helps to show an apparent comparison between the simple updating rule originated from the previous literature (see [75]) and the novel updating rule proposed in this thesis. In addition, we investigate a lot of properties of stable networks, as well as we design a series of simulation experiments to explore how the lack of information influences the process of stable network formation, supporting the theoretical results. Specially, the issues we investigate in those experiments are seldom considered in the literature before and fairly practical. Moreover, the experimental findings are consistent with the theoretical results and match common perception as expected.

The subgame consistency and strong subgame consistency of cooperative solution concepts have already been considered in the past literature in order to guarantee the sustainability of cooperation. We first apply them to the initially defined stochastic network formation game with heterogenous players where both the terminal time and the formation of the links proposed by the players are random. We obtain a recursive formula to derive the CIS-value in any cooperative subgame and investigate the subgame consistency and strong subgame consistency of the core in the game. We finally focus on the networks formed ultimately with the largest possibility in the stochastic game.

In the thesis, we also construct a novel model of network formation as a two-stage game with chance moves, players of various types and incomplete information. In the game, a novel definition of the stable partially Bayesian equilibrium which depicts the corresponding behavior of every player with each type is proposed, and its existence is verified. The relation between the new equilibrium concept and the

classical Nash equilibrium is also examined. Moreover, we in particular investigate the characteristics of the network structures formed as a result of players' stable partially Bayesian equilibrium behavior in a three-player game with a major player as well as in an n-player game with a specific characteristic function. All these results are first obtained in the works of the author.

Research methods

In the thesis, we use the methods of static game theory (investigation of conditions for pairwise stable networks), dynamic game theory (multi-stage games and the imputation distribution procedure, verification of subgame consistency and strong subgame consistency of cooperative solutions, and determination of stable partially Bayesian equilibria), graph theory (investigation for the stability of graph structures with specific topologies), cooperative game theory (derivation of characteristic functions, the core and the CIS-value), optimization theory (existence of maximal and minimal values of uni- and multivariable functions), and probability theory (chance moves, distributions of random variables and stochastic processes).

Theoretical and practical significance

The results presented in the thesis are related to the theory of network games and applications. Their significance is in the development of the theory of network games in different ways of defining players' heterogeneity, players' utility incorporating both the network structure and players' heterogeneity, players' interaction patterns through the dynamic network formation process, the stochastic elements involved in the dynamics of network structure, and the equilibrium concept which may well reflect the characteristics of the network games, as well as in the application of stable mechanisms of cooperation, and methods of construction of subgame consistent and strongly subgame consistent cooperative solutions to stochastic network formation games.

The realistic significance of the work is determined by the range of applications of network games, in particular, with heterogenous players: in mathematical modeling of economical, social matching and learning processes, in solving problems of information sharing and coordination between human beings. Therefore, the area of applications of the obtained results can be estimated by the described quite wide area of applications of network games, in which the interaction between players, sustainability of cooperation, stability of the network describing interpersonal relationships are significant and meaningful.

In Chapter 1, both the theoretical results and numerical examples very well explain the homophily principle and patterns of segregation in human society. The network formation game with incomplete information modeled in Chapter 2 reflects the actual circumstance in economic and social lives, for instance, the friendship formation process, and the information updating rule proposed in the model depicts individuals' leaning ability and knowledge updating process. In particular, the experimental findings are consistent with the theoretical results and match the common perception of people to realistic situations. Chapter 3 of the thesis is devoted to the construction of applied models using the theory of stochastic games where the involved random factors are fairly common in the coordination of joint projects, and the payment mechanism is investigated to decrease the cooperation vulnerability caused by the unexpected payoff. In the game model of Chapter 4, the circumstance that players are of various types and the existence of the role 'Nature' to some extent illustrate the realistic firm or community systems. Moreover, the novel equilibrium concept may provide people with the corresponding behavior suggestion when they act with different roles in the community.

The research conducted in the thesis is supported by the Chinese Government Scholarship (CSC) No. 202009010011 (2020-2023); the Russian Science Foundation (RSF) grant No. 22-21-00346 "Game theoretic methods of opinion dynamics control in social networks" (2022-2023).

Brief description of the thesis structure

The thesis consists of Introduction, four chapters, Conclusions and Bibliography. The content of each chapter is composed of a part of basic notations together with definitions, a detailed model description, the obtained theoretical results as well as numerical examples for better illustration and a brief summary for the chapter. The thesis is of 165 pages (in Russian version 176 pages), and it contains 18 tables and 35 figures. The bibliography cites 87 items listed in the alphabetical order.

The first chapter of the thesis is devoted to examining the stable networks formed by asymmetric players who are partitioned into ordered groups when the distance between group labels estimates their dissimilarity. Four different utility functions with varying costs and benefits are introduced, and the main purpose of this chapter is to investigate how these utilities influence a stable network structure. In Section 1.1, we introduce the basic terminology on network structures and partitioning of players into ordered groups, define the four utility functions, and introduce the

concept of stable network which is primarily studied in this chapter. The content presented in Section 1.1 is also the fundamental mathematical material needed in the subsequent chapters where the same content is omitted. In Section 1.2, we first present the comparison of conditions for some special network structures to be stable with respect to the two cost functions when there is no decay in benefits through both theoretical results and numerical examples, and then show the comparison when there is a decay in benefits. Then in Section 1.3, a special class of partitions which is prevalently employed in economics and management is considered, and particular results are obtained. Section 1.4 includes a brief summary of the first chapter.

Adopting the two utility functions employed in the static model of the first chapter, the second chapter develops a dynamic network formation model with incomplete information and asymmetric players who are partitioned into ordered groups. First, in Section 2.1, we describe the dynamic network formation process, and in particular explain the implication of incomplete knowledge of players and information updating rule along the dynamic process. In Section 2.2, we analyze the model in details based on assumption that players behave myopically, specifically, the concepts of stable equilibrium which is achieved by the myopic behavior of the players, and stable network ultimately formed are well defined and analyzed. Then Section 2.3 exhibits the theoretical results showing the apparent difference between complete information and incomplete information contexts through the dynamics and illustrates them through several numerical examples. Later in Section 2.4, we discuss an alternative information updating rule in the case of which players possess improved or superior learning capability. Section 2.5 covers all the simulation experiments and the corresponding analysis, including the examination of time to stable network, heterogeneity index, and impact of size gap of two groups on stable networks. We finally conclude Chapter 2 in Section 2.6.

A game-theoretical model of dynamic network formation with random factors is constructed and explored in the third chapter. Specifically, Section 3.1 briefly introduces some basic notations and describes the stochastic network formation process. In Section 3.2, the stochastic network formation process is presented as a game on a finite tree graph. Then the main functional equations for calculating the expected payoffs of the players are given in Section 3.3. In Section 3.4, the cooperative version of the stochastic game is constructed, and a recurrent formula for

the derivation of the CIS-value in any cooperative subgame is obtained. In Section 3.5, the problem of subgame consistency is investigated. Then the regularization of the core is carried out and the strong subgame consistency of the core is examined in Section 3.6, in particular, the sufficient conditions of the strongly subgame consistent core are obtained. In Section 3.7, two numerical examples respectively with regard to the CIS-value and the core are provided to better illustrate the theoretical results presented in this chapter. We briefly conclude in Section 3.8.

We propose a model of network formation as a two-stage game with chance moves and players of various types in the last chapter. In Section 4.1, the basic definitions and notations are briefly introduced. In Section 4.2, the model of a two-stage network formation game with nonsymmetric players and chance moves is presented. A special form of the two-stage game in an extensive form is described and investigated in Section 4.3. In Section 4.4, we propose the new definition of the stable partially Bayesian equilibrium, prove its existence, and examine the relation between the stable partially Bayesian equilibrium and the Nash equilibrium in the game. The numerical example is provided to illustrate the obtained results. Then, Section 4.5 specifically explores a three-player game with a major player, and the network structures formed when players follow the stable partially Bayesian equilibrium are separately found for various conditions. An n-player game with two projects for the leader and a specific characteristic function is considered in Section 4.6. The characteristics of the network structures formed under the stable partially Bayesian equilibrium in such a game are summarized. The conclusions of the last chapter are presented in Section 4.7.

The conclusion of the thesis contains a brief description of the results obtained in the work.

Results submitted for defense

1. The definition of the novel utility functions for the network games with ordered partitioning when the cost for a direct connection is determined by both the distance between connected players and the composition of a player's neighborhood. The stability conditions of the specific network structures (i.e., empty network, complete network, minimal and minimally connected network, inner star and inner complete network) when various utility functions are applied. The comparison of stability conditions for specific network structures corresponding to the two cost functions when there is a decay or no decay in bene-

fits.

2. Stability conditions of the specific network structures (i.e., empty network, complete network, minimal and minimally connected network, inner star and inner complete network) when different utility functions are defined for a special class of partitions when one majority group contains more than one player and all other groups represent singletons. Comparison of stability conditions for specific network structures corresponding to the two cost functions when there is a decay or no decay in benefits for the special class of partitions.

3. Formalization of a dynamic network formation game with infinite horizon and incomplete information, in which players initially do not know each other's labels, but learn them upon being connected.

4. The characterization for the features of stable network structures respectively formed in the case of complete information and incomplete information. The comparison of stable network features depending on the complete and incomplete information cases through the dynamic process.

5. The sufficient conditions guaranteeing that the set of stable networks formed under the case of complete information belongs to the one formed under incomplete information case.

6. The proposal and analysis of a novel information updating rule reflecting higher learning capability of players in dynamic network formation process, and analysis of how the two updating rules influence the stable network structures through certain theoretical results and numerical examples.

7. Simulation experiments for the dynamic network formation process and their analysis on three issues: the time it takes to reach a stable network, the heterogeneity index, and the impact of the group size gap.

8. Formalization of a stochastic network formation game with random terminal time and realization of players' actions with probabilities. The recursive formula to derive the CIS-value in the stochastic network formation game.

9. The construction of the imputation distribution procedure guaranteeing the subgame consistency of cooperative solution concepts for the stochastic network

formation game. Regularization of the core as well as the sufficient conditions of the strongly subgame consistent core in the stochastic network formation game.

10. Formalization of a two-stage network formation game with chance moves and players of various types where the Nature selects a type vector for players based on the given probability distribution and each player has only a belief over other players' payoff functions.

11. The definition of a stable partially Bayesian equilibrium in a two-stage network formation game, and the investigation of the relation between the stable partially Bayesian equilibrium and the Nash equilibrium in the two-stage network formation game.

12. The characterization of network structures formed under the stable partially Bayesian equilibrium in a three-player game with a major player as well as in an n-player game with a specific characteristic function.

Verification of results

The main results of the thesis were presented at the International Society of Dynamic Games (ISDG) - China Chapter Conference on "Dynamic Games and Game Theoretic Analysis" (Ningbo, 2017); International Conferences "Stability and control processes" (Saint Petersburg, 2019, 2020, 2021, 2022); International Conferences "Game Theory and Management" (Saint Petersburg, 2020, 2021); International Conference "Game Theory and Applications" (Saint Petersburg, 2022); "Dynamic games and applications" seminar of GERAD (Online, 2021); International Conference on "Mathematical Optimization Theory and Operations Research" (Petrozavodsk, 2022); Russia-China Scientific Seminar on "Mathematical Game Theory" (Online, 2022); International Symposium on Dynamic Games and Applications (Porto, 2022).

Publications

Based on the results of the thesis, the following works were published: [51, 76, 77, 78, 79, 80, 81, 82]. The following items [77, 78, 79, 80, 81, 82] are published in peer-reviewed journals from the list of the Higher Attestation Commission.

Acknowledgments

The author expresses her deep appreciation and thanks to Doctor of Physical and Mathematical Sciences, Professor Elena Mikhailovna Parilina (Saint Petersburg

State University) for her biggest support, valuable suggestions, long-term collaboration, constant encouragement, comprehensive assistance and for all-round consultation during the preparation of this thesis. The author is also grateful to Professor Gao Hongwei (Qingdao University, China) for his support all the time. The author thanks grandparents and parents for their love, care, understanding and encouragement in all situations.

Заключение диссертации по теме «Другие cпециальности», Сунь Пин

Заключение

Диссертация посвящена изучению сетевых игр с асимметричными игроками. В рамках данного исследования предложено несколько моделей сетевых игр, которые формализованы с помощью различных подходов: статического и динамического, а также некооперативного и кооперативного. В статических сетевых играх (глава 1) исследуются условия устойчивости нескольких специальных сетевых структур в случае, когда множество игроков разделено на упорядоченные группы. В моделях динамического формирования сети, которые изучаются с некооперативной точки зрения (глава 2 и 4), проведено сравнение влияния полной и неполной информационных структур на структуру устойчивых сетей и моделей взаимодействия игроков (см. главу 2). В главе 4 изучаются свойства новой концепции равновесия, устойчивого частично байесовского равновесия, которое определяет поведение каждого игрока в зависимости от его типа. При анализе кооперативного поведения игроков в стохастических играх формирования сети (глава 3) исследуется позиционная состоятельность и сильная позиционная состоятельность некоторых кооперативных решений.

Основными результатами работы являются следующие:

1. Для конкретных сетевых структур (пустая сеть, полная сеть, минимальная и минимально связная сеть, внутренне звездообразная и внутренне полная сети), найдены условия устойчивости при применении различных функций полезности. В частности, получены условия устойчивости для особого класса разбиений игроков на группы, когда имеется одна большая группа, состоящая из более чем одного игрока, а все остальные группы являются индивидуальными игроками. Как теоретические результаты, так и численные примеры демонстрируют два интересных явления, когда принимается новая функция затрат. Эти выводы противоположны тем, которые получены и описаны во многих исследованиях, посвященных устойчивым сетям с различными способами определения полезности игроков: (1) неминималь-

ная сеть может быть устойчивой, даже если нет затухания выигрышей по путям в сети, т. е. игроки могут иметь связи, которые не принесут им прямой прибыли, но снизят средние издержки на связь; (п) игрок может предпочесть связи с игроками из других групп с большими средними издержками на связь, отказавшись от создания ребер с игроками из той же группы. (см. [80].)

2. Сравнение влияния полных и неполных информационных структур на устойчивые сетевые структуры, формирующиеся в результате устойчивого равновесного поведения игроков. Это сравнение проведено в игре динамического формирования сети с бесконечной продолжительностью, а также проведено сравнение нового правила обновления информации, предложенного в данном исследовании, в основу которого легло простое правило обновления, предложенное в [75]. Объединяя как теоретические результаты, так и результаты численных экспериментов, которые согласуются с теоретическими результатами и соответствуют общепринятому представлению, сделано заключение, что характеристики устойчивых сетей существенно различаются для случаев полной и неполной информации, когда параметр затухания прибыли равен единице; т. е. когда затухания выигрышей нет. В случае, когда этот параметр меньше единицы, разница в результатах незначительна. Также в работе определено, что улучшение способности игроков

к обучению не обязательно способствует увеличению связности общества.

3. Определена стохастическая игра формирования сети с асимметричными игроками и случайным временем окончания. Исследуются свойства позиционной состоятельности и сильной позиционной состоятельности кооперативных решений. Для обеспечения позиционной состоятельности предлагается механизм выплат игрокам, процедура распределения дележей, заданная в определенной форме, чтобы избежать отклонений игроков от совместно оптимальной траектории. Предложена процедура регуляризации выплат игрокам в случае, если не допускаются отрицательные выплаты. Получено достаточное условие сильно позиционно состоятельного с-ядра (см. [78, 81]).

4. При изучении стохастической игры формирования сети со случайными ходами и игроками разных типов предложена концепции устойчивого частич-

но байесовского равновесия, описывающего равновесное поведение игроков разных типов. В частности, проверяется связь этого равновесия с равновесием по Нэшу в двухшаговой игре. Найдены устойчивые частично байесовские равновесия и сетевые структуры, образующиеся при реализации этих равновесий в двух специальных играх, а именно, игре трех лиц с главным игроком и игре п лиц со специальной характеристической функцией (см. [79]).

Все задачи, сформулированные в рамках данного исследования, решены, а поставленные цели выполнены в полном объеме.

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