1st Edition

# Grey Game Theory and Its Applications in Economic Decision-Making

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To make the best decisions, you need the best information. However, because most issues in game theory are grey, nearly all recent research has been carried out using a simplified method that considers grey systems as white ones. This often results in a forecasting function that is far from satisfactory when applied to many real situations. **Grey Game Theory and Its Applications in Economic Decision Making** introduces classic game theory into the realm of grey system theory with limited knowledge. The book resolves three theoretical issues:

- A
*game equilibrium of grey game* - A reasonable explanation for the
*equilibrium of a grey matrix*of static nonmatrix game issues based on incomplete information - The Centipede Game paradox, which has puzzled theory circles for a long time and greatly enriched and developed the core methods of subgame Nash perfect equilibrium analysis as a result

The book establishes a grey matrix game model based on pure and mixed strategies. The author proposes the concepts of grey saddle points, grey mixed strategy solutions, and their corresponding structures and also puts forward the models and methods of risk measurement and evaluation of optimal grey strategies. He raises and solves the problems of grey matrix games. The book includes definitions of the test rules of information distortion experienced during calculation, the design of tokens based on new interval grey numbers, and new arithmetic laws to manipulate grey numbers. These features combine to provide a practical and efficient tool for forecasting real-life economic problems.

**1. Introduction**1.1 Background, Meaning, and Purpose of the Research

1.1.1 Background

1.1.2 Significance of the Topic

1.1.3 Research Target

1.2 Status of Research and Development

1.3 Research and Technology Roadmap

1.3.1 Main Content

1.3.2 Technical Route

1.4 Main Innovative Points and Characteristics

1.4.1 Main Innovative Points

1.4.2 Main Characteristics

**2. Study of the Grey Matrix Game Model Based on Pure Strategy**

2.1 Study of the Simple Grey Matrix Game Model Based on Pure Strategy

2.1.1 Construction of a Simple (Standard) Grey Matrix Game Model Based on Pure Strategy

2.1.1.1 Analysis of a Grey Game Profit and Loss Value Matrix

2.1.1.2 Size Comparison of Interval Grey Numbers

2.1.1.3 Modeling of Standard Grey Matrix Game

2.1.2 Solution of a Simple (Standard) Grey Matrix Game Model Based on Pure Strategy

2.1.2.1 Concept of Pure Strategy Solution of a Standard Grey Matrix Game

2.1.2.2 The Sufficient and Necessary Term and the Property of Pure Strategy Solution of Standard Grey Matrix Game

2.1.2.3 Relationship between Pure Strategies of Standard and Rigorous Standard Grey Matrix Games

2.2 Study of a Pure Strategy Solution of a Grey Matrix Game Based on Unilateral Strategy Analysis

2.2.1 Analysis of Grey Game Revenue Matrix

2.2.2 Model Construction

2.2.3 Case Study

2.3 Example Analysis of the Grey Matrix Game Model in Stock Speculation for Immediate Price-Margin Based on Pure Strategies

**3. Pure Strategy Solution of a Grey Matrix Game Based on an Interval Grey Number Not to Be Determined Directly**3.1 Study of a Pure Strategy Solution and Risk of a Grey Matrix Game Based on Interval Grey Number Not to Be Determined Directly

3.1.1 Background

3.1.2 Judgment on the Relationship of the Superior, Inferior, and Equipollence Position Degrees That Include Mixed Ranges

3.1.3 The Position Optimum Pure Strategy Solutions and the Answers

3.1.4 Case Study

3.1.5 Summary

3.2 Study on Strategy Dominance and Its Optimum Solution of Grey Matrix Game Based on Interval Grey Numbers Not to Be Determined Directly

3.2.1 The Dominance Analysis of Position Optimum Pure Strategy

3.2.2 Case Study

3.2.3 Summary

3.3 Study of Risk of Position Optimum Pure Strategy Solution Based on a Grey Interval Number Matrix Game

3.3.1 Identity and Definition of Overrated and Underestimated Risks of Position Optimum Pure Strategy Solution

3.3.2 Judgment of Position Optimum Pure Strategy Solution Risk

3.3.3 Summary

**4.1 Grey Mixed Strategy and Grey Mixed Situation**

4. Grey Matrix Game Model Based on Grey Mixed Strategy

4. Grey Matrix Game Model Based on Grey Mixed Strategy

4.1.1 Background

4.1.2 Relation and Operation of Grey Events

4.1.3 Basic Concepts and Propertiesof Grey Interval Probability

4.1.4 Grey Mixed Strategy and Related Theorems

4.1.5 Summary

4.2 Characterization of an Interval Grey Number and Improvement of Its Operation

4.2.1 Background

4.2.2 Standard Interval Grey Number and Its Operation

4.2.3 The First and Second Standard Grey Numbers

4.2.4 Judgment of Quantitative Relations of Standard Grey Numbers

4.2.5 Case Study

4.2.6 Summary

4.3 The Maximum-Minimum Grey Game Value and the Grey Saddle Point of Grey Mixed Strategy

4.3.1 Theorem of the Maximum-Minimum Grey Game Value

4.3.2 Grey Saddle Point of Grey Mixed Strategy

4.3.3 Summary

4.4 Properties of a Grey Mixed Strategy and Its Grey Linear Program Model

4.4.1 Properties of a Grey Mixed Strategy

4.4.2 Grey Linear Program Model of Grey Matrix Game

4.4.3 The Concept of a Grey Linear Programming Model Solution of a Grey Matrix Game

4.4.4 Summary

4.5 Seeking Solutions of Grey Linear Programming Model of a Grey Matrix Game

4.5.1 Grey Basis Feasible Solution Corresponds to the Vertex of a Grey Feasible Domain

4.5.2 The Optimum Grey Game Value Corresponds to the Vertex of Grey Linear Programming Feasible Domain

4.5.3 Grey Linear Programming Solution Seeking of Optimum Grey Game Value

4.5.4 Summary

**5. Study of Elementary Transformations of the Grey Matrix and the Invertible Grey Matrix**

5.1 Grey Vector Groups and Grey Linear Correlations

5.1.1 Basic Concept of Grey Vectors and Grey Linear Combinations

5.1.2 Grey Linear Correlation of Grey Vectors

5.1.3 Theorems about Grey Vectors Grey Linear Correlation

5.1.4 Summary

5.2 Maximum Grey Vector Groups and the Rank of Grey Matrix

5.2.1 Basic Theorems about Grey Vector Group Grey Linear Correlations

5.2.2 Grey Vector Groups and Grey Rank of a Grey Matrix

5.2.3 Summary

5.3 The Elementary Transformation of the Grey Matrix and Its Grey Invertible Matrix

5.3.1 The Elementary Transformation of the Grey Matrix

5.3.2 Grey Invertible Matrix

5.3.3 Summary

**6.1 Matrix Solution Method of a Grey Matrix Game Based on a Grey Full-Rank Expanding Square Matrix**

6. Matrix Solution Method of a Grey Matrix Game

6. Matrix Solution Method of a Grey Matrix Game

6.1.1 Concept of an Expanding Square Matrix and Its Grey Inverse Matrix

6.1.2 Optimum Grey Game Strategy and Grey Game Value of Player 1

6.1.3 Optimum Grey Game Strategy and Grey Game Value of Player 2

6.1.4 Optimum Grey Game Strategy and Grey Game Value of Players Based on a Combined Grey Full-Rank Expanding Square Matrix

6.1.5 Summary

6.2 Construction of an Analogous Grey Full-Rank Expanding Square Matrix and Its Necessary and Sufficient Conditions

6.2.1 Construction of an Analogous Grey Full-Rank Expanding Square Matrix

6.2.2 Necessary and Sufficient Conditions of Full Rank for an Analogous Grey Expanding Square Matrix

6.2.3 Full-Rank Treatment of an Analogous Grey Rank-Decreased Expanding Square Matrix

6.2.4 Summary

6.3 Compound Standard Grey Number and

*G*(⊗)’s Infeasible Solution, Feasible Solution, and Optimal Solution Matrix

6.3.1 Concept of Compound Standard Grey Numbers and Their Determination

6.3.2 Concepts of Grey Optimal Solutions, Feasible Solutions, and Infeasible Solution Matrix

6.3.3 Sufficient and Necessary Condition of the Existence of a Grey Optimal Solution Square Matrix

6.3.4 Summary

6.4

*G*(⊗)’s Redundant Constraint, Zero Strategy Variables, and Grey Matrix Solving Method

6.4.1 Zero Strategy Variables in an Infeasible Solution Square Matrix

6.4.2 Redundant Constraint Equation and Zero Strategy Variables in a Nonsquare Matrix

6.4.3 Optimal Grey Game Solution in a Nonsquare Matrix

6.4.4

*A*(⊗)m × n’s Inferior Strategy and Its Redundant Constraint and Zero Strategy Variable

6.4.5

*G*(⊗)’s Grey Matrix Method Solving Steps

6.4.6 Summary

**7.1 Study of the Venture Problem of a Potential Optimal Pure Strategy Solution for the Grey Interval Number Matrix Game**

7. Potential Optimal Strategy Solution’s Venture and Control Problems Based on the Grey Interval Number Matrix Game

7. Potential Optimal Strategy Solution’s Venture and Control Problems Based on the Grey Interval Number Matrix Game

7.1.1 Optimal Potential Pure Strategy Solution of a Grey Interval Number Matrix Game

7.1.2 Measurement of the Optimal Potential Pure Strategy Solution

7.1.3 Conclusions

7.2 Venture and Control Problems of an Optimal Mixed Strategy for a Grey Interval Number Matrix Game

7.2.1 Recognition and Definition of the Overrated and Underrated Risks of the Potential Optimum Mixed Strategy Solution

7.2.2 Measurement for Maximal Overrated and Underrated Risks of the Potential Pure Strategy Solution

7.2.3 Information Venture Control Ability and Venture Control of Game

*G*(⊗)

7.2.4 Summary

**8.1 Duopoly Strategic Output-Making Model Based on the Experienced Ideal Output and the Best Strategy Decision- Making Coefficient**

8. Concession and Damping Equilibriums of Duopolistic Strategic Output-Making Based on Limited Rationality and Knowledge

8. Concession and Damping Equilibriums of Duopolistic Strategic Output-Making Based on Limited Rationality and Knowledge

8.2 Concession Equilibrium of the Later Decision-Maker under Nonstrategic Extended Damping Conditions: Elimination from the Market

8.3 Damping Equilibrium of the Advanced Decision-Maker under Strategic Extended Damping Conditions: Giving Up Some Market Share

8.4 Damping Loss and the Total Damping Cost When the First Decision-Making Oligopoly Has Occupied the Market Completely

8.5 Summary

**9.1 Nash Equilibrium of Grey Potential Pure Strategy Analysis of**

9. Nash Equilibrium Analysis Model of Static and Dynamic Games of Grey Pair-Wise Matrices

9. Nash Equilibrium Analysis Model of Static and Dynamic Games of Grey Pair-Wise Matrices

*N*-Person Static Games of Symmetrical Information Loss

9.1.1 Nash Equilibrium of Grey Potential Pure Strategy

9.1.2 Analyzing Methods of Absolute Grey Superior and Inferior Potential Relationships of Game

*G*(⊗)

9.1.3 Analysis of Relative Grey Superior and Inferior Relationships in Game

*G*(⊗)

9.1.4 Summary

9.2 Solving the Paradox of the Centipede Game: A New Model of Grey Structured Algorithm of Forward Induction

9.2.1 Backward Grey Structured Algorithm of a Dynamic Multistage Game’s Profit Value

9.2.2 The Termination and Guide Nash Equilibrium Analysis of Grey Structured Algorithm of Forward Induction in a Multistage Dynamic Game

9.2.3 Summary

**10. Chain Structure Model of Evolutionary Games Based on Limited Rationality and Knowledge**

10.1. Chain Structure Model of Evolutionary Games Based on a Symmetric Case

10.1.1 Establishing a Chain Structure Model for Evolutionary Games

10.1.2 Imitation of Dynamic Process of Duplication and ESS

10.1.3 Initial State and Analysis of Replication Dynamics

10.1.4 Summary

10.2. Chain Structure Models of Evolutionary Games Based on an Asymmetric Case

10.2.1 Analysis of the Chain Structure Model of Evolutionary Games

10.2.2 Establishing Chain Structure Model for Evolutionary Game

10.2.3 Imitation of Dynamic Process of Duplication and ESS

10.2.4 Initial State and Analysis of Replication Dynamics

10.2.5 Summary

10.3. Chain Structure Models of Grey Evolution of Industry Agglomeration and Stability Analysis

10.3.1 Research Background

10.3.2 Establishing a Chain Structure Model for the Evolutionry Game of Industry Agglomeration Development

10.3.2.1 Company Learning Mechanisms

10.3.2.2 Model Construction

10.3.3 Duplication Dynamic Simulation of the Development of Industry Agglomeration

10.3.4 Stability Analysis of the Industry Agglomeration Formation and Development

10.3.5 Summary

**11.1. Optimal Grey Quotation Model Based on an Evaluation of Accurate Value and Experiential Ideal Quotation**

11. Bounded Rationality Grey Game Model of First-Price Sealed-Bid Auction

11. Bounded Rationality Grey Game Model of First-Price Sealed-Bid Auction

11.1.1 Conditions of Optimal Grey Quotation Model

11.1.2 Design of Grey Quotation and Grey Expected Utility Model

11.1.3 Simulation and Analysis of Optimal Grey Quotation and Grey Expected Utility Model

11.1.4 Summary

11.2. Optimal Grey Quotation Model Based on Evaluation of Grey Value and Grey Experiential Ideal Quotation

11.2.1 Conditions of Optimal Grey Quotation Model

11.2.2 Design of Grey Quotation and Grey Expected Utility Model

11.2.3 Simulation and Analysis of Optimal Grey Quotation and Grey Expected Utility Model

11.2.4 Summary

11.3. Choice of Stock Agent: A First-Price Sealed-Bid Auction Game Model Based on Bounded Rationality

11.3.1 Construction of Optimal Quoted Price Model of Stock Dealers

11.3.2 Model Imitation of Optimization Quoted Price and Expectation Utility and Analysis of the Main Conclusions

11.3.2.1 Model Imitation

11.3.2.2 Analysis of the Imitation Conclusions

11.3.3 The Strategy in the Bid Process of Stock Dealership

11.3.3.1 Strategy 1: To Enhance Bidder’s Approval Degree to Enterprise Stock Dealership

11.3.3.2 Strategy 2: To Construct Favorable Environment for Auction of Stock Dealership

11.3.4 Summary

**12.1 Summary**

12. Summary and Prospect

12. Summary and Prospect

12.2 Research Prospects

12.2.1 Research on Grey Matrix Game’s Optimal Mix Strategy Solution Algorithm

12.2.2 Venture Control Problem of the Optimal Grey Matrix Game Solution

12.2.3 Solution of a Subsidiary Game’s Grey Potential

References

Index

References

Index

### Biography

**Zhigeng Fang** is professor and doctoral tutor at the College of Economics and management, Nanjing University of Aeronautics and Astronautics (NUAA), China. From 1996 to 1999, he studied at Xi’an University of Technology, earning a master’s degree in management science and engineering management. From 2003 to 2007, he attended NUAA, earning a doctoral degree in systems engineering. His academic part-time positions include under-secretary-general of IEEE International Professional Committee of Grey Systems; governor of the Chinese Society for Optimization, Overall Planning and Economic Mathematics; managing director of Complex Systems Research Committee; secretary-general of Grey Systems Society of China; under-secretary-general of the Systems Engineering Society of Jiangsu Province, China; deputy director of the Post-Evaluation Research Center of Jiangsu Province; deputy director of the Grey Systems Research Institute of NUAA. He is a member of Services Science Global and the Services and Logistics Technical Committee of IEEE Intelligent Transportation Systems Society. His main research directions lie in the fields of theory and methods of uncertain prediction and decision, Grey game theory, applied economic, and energy economics.

"... an excellent introductory graduate and undergraduate level text on grey game theory and its real applications in economics, management and system engineering. … This book deserves the highest rating possible because a big percentage of the actual economic problems are based on a wrong decision. ... a valuable and timely addition to the current literature on game theory. It is well-written and clearly organized with excellent figures. The style and the presentation of the text provide the reader with a broad overview of recent progress in the lively and challenging field of game theory and its applications in economic problems. This book is highly recommendable to anyone, a graduate student, a researcher or a professor who studies game theory or a director of a society or an investment man."

—Zeraoulia Elhadj, University of Tébessa, Algeria"The various means of dealing with uncertainty have been the subject of much recent online discussion, with Chinese complexity studies, particularly those under the heading of grey systems, receiving particular attention. This book presents impressive mathematical theory, with references to applications… . All the topics are treated in a fresh and stimulating way and readers are assured of their effectiveness in practice."

—Alex M. Andrew, Reading University, inKybernetes,Vol. 39 No.7, 2010