1. Zhu Han, Dusit Niyato, Walid Saad, and Tamer Basar
Game Theory in Wireless and Communication Networks: Theory, Models, and Applications
Lecture 12 Variational Inequalities, Mathematical Programs with Equilibrium Constraints (MPEC) and Equilibrium Programs with Equilibrium Constraints (EPEC)
Mathematical Programs with Equilibrium Constraints and its Application in Solving Stackelberg Games
Zhu Han, Dusit Niyato, Walid Saad, and Tamer Basar
Thanks for Xiao Tang, Xi’an Jiaotong University

2. Overview of Lecture Notes
Introduction to Game Theory: Lecture 1, book 1
Non-cooperative Games: Lecture 1, Chapter 3, book 1
Bayesian Games: Lecture 2, Chapter 4, book 1
Differential Games: Lecture 3, Chapter 5, book 1
Evolutionary Games: Lecture 4, Chapter 6, book 1
Cooperative Games: Lecture 5, Chapter 7, book 1
Auction Theory: Lecture 6, Chapter 8, book 1
Matching Game: Lecture 7, Chapter 2, book 2
Contract Theory, Lecture 8, Chapter 3, book 2
Learning in Game, Lecture 9, Chapter 6, book 2
Stochastic Game, Lecture 10, Chapter 4, book 2
Game with Bounded Rationality, Lecture 11, Chapter 5, book 2
Equilibrium Programming with Equilibrium Constraint, Lecture 12, Chapter 7, book 2
Zero Determinant Strategy, Lecture 13, Chapter 8, book 2
Mean Field Game, Lecture 14, UCLA course, book 2
Network Economy, Lecture 15, Dr. Jianwei Huang, book 2

3. Outline Introduction Variational Inequalities MPEC EPEC
Stackelberg Game
Variational Inequalities
Definition
Examples
MPEC
EPEC

4. Stackelberg Game Definition and features
Modeling sequential decision-makings;
Solution by backward induction.
Leader’s Problem
Decision making:
Follower chooses the action by reacting to leader’s action
Backward induction:
Leader makes decisions by considering follower’s reaction
Follower’s Problem
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5. Basics on Variational Inequality

6. Basics on Variational Inequality

7. Basics on Variational Inequality

8. Basics on Variational Inequality

9. Example: IterativeWater-Filling

10. IterativeWater-Filling

11. IterativeWater-Filling

12. IterativeWater-Filling

13. IterativeWater-Filling

14. IterativeWater-Filling

15. Numerical result
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16. Xi’an Jiaotong University | University of Houston
Basics on Quasi-VI
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Xi’an Jiaotong University | University of Houston

17. Energy-Efficient Multi-Channel Power Allocation
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18. Energy-Efficient Multi-Channel Power Allocation
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19. Energy-Efficient Multi-Channel Power Allocation
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20. Energy-Efficient Multi-Channel Power Allocation
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21. Energy-Efficient Multi-Channel Power Allocation
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22. Outline Introduction Variational Inequalities MPEC EPEC
Stackelberg Game
Variational Inequalities
Definition
Power Control with Aggregated Interference Constraints
MPEC
Examples
EPEC

23. MPEC Problem Definition
A mathematical optimization with part of the constraints being in the form of complementarity conditions
complementarity condition w.r.t. y
(parameterized by x)
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24. MPEC Problem Definition
A mathematical optimization with part of the constraints being in the form of another optimization (bi-level optim.)
Optimization w.r.t. y as constraint (parametrized by x)
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25. MPEC Problem Features Bi-level (multi-level) structure;
Suit for the Stackelberg game analysis
Leader’s problem
Follower’s problem
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26. Two-Tier Power Control
System Model
Two-tier heterogeneous networks
Overlapping frequency reuse
Multi-channel transmissions
Rate maximization
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27. Two-Tier Power Control
Problem Formulation
Stackelberg game
Leader: Macro-cell users; Follower: Small-cell users
Hierarchical power control
Maximize:
w.r.t.:
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28. Two-Tier Power Control
Analysis
Follower’s power allocation
Water-filling:
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29. Two-Tier Power Control
Analysis
Leader’s power allocation
MPEC reformulation as an optimization:
MPEC formulation:
Follower’s optimal as constraints
Backward induction
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30. Two-Tier Power Control
Analysis
Leader’s power allocation
MPEC reformulation as an optimization:
Water-filling based solution
Based on follower’s optimal:
p is an implicit function of q
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31. Two-Tier Power Control
Analysis
Water-filling based part
Implicit function part:
Denote the implicit function as and apply KKT:
Leader’s power allocation as a fixed-point iteration
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32. Two-Tier Power Control
Analysis
Structural results on lower water-filling
Linear structure
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33. Two-Tier Power Control
Analysis
Solution to the MPEC (leader’s problem)
Note
The structural results of the follower’s problem is the crucial to solve the MPEC problem.
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34. Jamming-Aided Eavesdropping
System Model
Multi-channel communications
Full duplex eavesdropper
Improve eavesdropping by jamming attacks
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35. Jamming-Aided Eavesdropping
Problem Formulation
Stackelberg game
Leader’s problem:
Follower’s problem:
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36. Jamming-Aided Eavesdropping
Analysis
Follower’s problem
Convex problem and solved with Lagrange
Generalized water-level:
Implicit function of PE
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37. Jamming-Aided Eavesdropping
Analysis
Leader’s problem
backward induction: intractable optimization problem
Intractable optimization due to the lack of explicit analytical expression
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38. Jamming-Aided Eavesdropping
Analysis
MPEC problem
Changed optimization parameters
The lower optimal condition as a constraint rather than part of the objective
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39. Jamming-Aided Eavesdropping
Analysis
Equivalent optimization problem
Changed optimization parameters
A well-defined optimization:
Solved with regular method
violating the constraint qualification (CQ)
KKT condition in replace of the optimality condition
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40. Jamming-Aided Eavesdropping
Note
Replacing the optimality constraints with its KKT equivalent is very useful in solving MPEC;
The violation of CQ is a common problem which should be tackled carefully.
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41. Smoothing Method for General MPEC
General Problem
Convex problem
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42. Smoothing Method for General MPEC
Equivalent Reformulation
violating the constraint qualification (CQ)
Well-defined optimization with slack variable z
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43. Smoothing Method for General MPEC
Smoothing Function
Definition:
Properties:
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44. Smoothing Method for General MPEC
Smoothing Approximation
-Parameterized optimization
Well-defined,
smooth optimization
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45. Outline Introduction Variational Inequalities MPEC EPEC
Stackelberg Game
Variational Inequalities
Definition
Power Control with Aggregated Interference Constraints
MPEC
Examples
EPEC

46. Reduce latency and transmission costs
Introduction
Internet of Things
Big Data
Compu ting and Storage
Cloud Computi ng
Fog Computin g
Reduce latency and transmission costs
Data Service Operators (DSOs)
Data Centers
Fog Nodes (FNs)
Authorized Data Service Subscribers (ADSSs) 47

47. Motivation Large number of FNs deployed at various locations
How to manage computing resources?
How to perform large-scale optimization?
Equilibrium Problem with Equilibrium Constraints (EPEC)
Competition among multiple DSOs and ADSSs
How to deal with multiple conflicting entities?
Alternating Direction Method of Multipliers (ADMM)
Proposed method!!! 48

48. Real Time Data Service Scenario
K DSOs, N ADSSs
Computing resources from data centers or FNs – Computing Resource Blocks (CRBs)
DSOs manage CRBs for serving the ADSSs
ADSSs request for CRBs from the DSOs 49

49. Utility Functions of DSOs and ADSSs
Price set for one CRB by DSO i for ADSS j – Ɵi,j
Number of CRBs purchased from DSO i by ADSS j – xi,j
Utility function for DSO i
Utility function for ADSS j
Revenue from the CRBs provided to ADSSs
Operational and measurement costs
Revenue from workload data
Cost of service delay 50

50. Maximization of Profits for DSOs and ADSSs
Optimization problem for DSO i
Optimization problem for ADSS j
Constraint on budget to purchase CRBs
Constraint on available CRBs
Constraint on service delay
Constraint on service delay
DSOs and ADSSs aim to maximize their own profits (optimal Ɵi,j and xi,j)
DSOs provide incentives to ADSSs, to choose xi,j to obtain optimal Ɵi,j 51

51. Equilibrium Problem with Equilibrium Constraints (EPEC)
Optimization for the DSOs
Optimization for the ADSSs
Optimization of the utilities of DSOs, while considering the utilities of ADSSs (conflicting objectives)
Two level hierarchical optimization problem
Equilibria and constraints exist at both upper and lower levels 52

52. Alternating Direction Method of Multipliers (ADMM)
Objective function of this form
Constraints of this form
Method for large scale optimization
Fast convergence when objective function is convex
EPEC + ADMM
Conflicting objectives + Large network!!! 53

53. ADMM based EPEC in Fog Computing
Convergence check for outer loop
Optimization for the ADSSs
Optimization for the DSOs
The outer loop terminates when the total profit of the DSOs converges to an optimal value, i.e.,
Value at current iteration – Value at previous iteration < Error threshold 54

54. Total Profit of ADSSs vs. Number of ADSSs
Fog Computing
Cloud Computing
Data Center Service
Optimization using ADMM increases the total profit of the ADSSs in Fog Computing compared to Cloud Computing 55

55. Summary VI MPEC and EPEC
Use GNEP to tackle the common constraints in games;
VI assisted convergence analysis in distributed algorithms;
Use Quasi-NE to tackle the non-convex games.
The VI theory is a very powerful mathematical tool which has applications in many areas.
By building up the equivalence between VI and game, the VI theory provides us alternative manner in the investigation on the properties of Nash equilibrium, particularly on the uniqueness and convergence.
MPEC and EPEC
Provide a tool to facilitate the discussion on more complicated hierarchical game model;
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56. Xi’an Jiaotong University | University of Houston
References
[1]. S. Guruacharya, D. Niyato, D. I. Kim, and E. Hossain, “Hierarchical competition for downlink power allocation in OFDMA femtocell networks,” IEEE Trans. Wireless Commun., vol. 12, no. 4, pp. 1543–1553, Apr [2]. K. Zhu, E. Hossain, and A. Anpalagan, “Downlink power control in two-tier cellular ofdma networks under uncertainties: a robust Stackelberg game,” IEEE Trans. Commun., vol. 63, no. 2, pp. 520–543, Feb [3]. J. Wang, M. Peng, S. Jin, and C. Zhao, “A generalized nash equilibrium approach for robust cognitive radio networks via generalized variational inequalities,” IEEE Trans. Wireless Commun., vol. 13, no. 7, pp. 3701–3714, Jul [4]. Q. Han, B. Yang, X. Wang, K. Ma, C. Chen, and X. Guan, “Hierarchical-game-based uplink power control in femtocell networks,” IEEE Trans. Veh. Technol., vol. 63, no. 6, pp. 2819–2843, Jul [5]. Z. Luo, J. Pang, and D. Ralph, Mathematical Programs with Equilibrium Constraints. Cambridge, UK: Cambridge University Press, 1996.
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Xi’an Jiaotong University | University of Houston