The Game Theory Approach for Solving the Hierarchical and Decentralized Bi-Level Problem

This paper presents a game theory approach to solve two types op problem, hierarchical and decentralized bi-level multi-objective problem with multiple objective functions at the upper level and multiple players at the lower level. The sensitivity based approach is applied for numerical approach. A sensitivity based algorithm is proposed to the Hierarchical and decentralized bi-level multi-objective problem. Two scenarios are studied in this paper for modeling the decentralized bi-level multi-objective problem. The first scenario considers the cooperative game as interaction between the players at upper level and the lower level individually. The interaction between the upper and lower level is considered as Stackelberg. On the second scenario, the interaction in the lower level is modeled by Nash game. The sensitivity based method has been used to provide an approximation to the rational reaction set (RRS) for each player. An alternate approach for generating the RRS based on design of experiments (DOE) combined with response surface methodology (RSM) is also explored. Two numerical examples are given to demonstrate the proposed algorithm for the both scenario. It is seen that the proposed sensitivity based approach is able to approximate non linear RRS. For the hierarchical approach, one numerical example is studied to show the application of the algorithm.

A Sensitivity Based Approach to Determining Non-Cooperative Solutions in Nash and Stackelberg Games

A sensitivity based approach is presented to determine Nash solution(s) in multiobjective problems modeled as a non-cooperative game. The proposed approach provides an approximation to the rational reaction set (RRS) for each player. An intersection of these sets yields the Nash solution for the game. An alternate approach for generating the RRS based on design of experiments (DOE) combined with response surface methodology (RSM) is also explored. The two approaches for generating RRS are compared on three example problems to find Nash and Stackelberg solutions. It is seen that the proposed sensitivity based approach (i) requires less computational effort than a RSM-DOE approach, (ii) is less prone to numerical errors than the RSM-DOE approach, (iii) is able to find all Nash solutions when the Nash solution is not a singleton, (iv) is able to approximate non linear RRS, and (v) is able to find better a Nash solution on an example problem than the one reported in the literature.

An Approach to Improved Decentralized Design: The Modified Approximation-Based Decentralized Design (MADD) Framework

The design of large scale complex engineering systems requires interaction and communication between multiple disciplines and decentralized subsystems. One common fundamental assumption in decentralized design is that the individual subsystems only exchange design variable information and do not share objective functions or gradients. This is because the decentralized subsystems can either not share this information due to geographical constraints or choose not to share it due to corporate secrecy issues. Game theory has been used to model the interactions between distributed design subsystems and predict convergence and equilibrium solutions. These game theoretic models assume that designers make perfectly rational decisions by selecting solutions from their Rational Reaction Set (RRS), resulting in a Nash Equilibrium solution. However, empirical studies reject the claim that decision makers always make rational choices and the concept of Bounded Rationality is used to explain such behavior. In this paper, a framework is proposed that uses the idea of bounded rationality in conjunction with set-based design, metamodeling and multiobjective optimization techniques to improve solutions for convergent decentralized design problems. Through the use of this framework, entitled Modified Approximation-based Decentralized Design (MADD) framework, convergent decentralized design problems converge to solutions that are superior to the Nash equilibrium. A two subsystem mathematical problem is used as case study and simulation techniques are used to study the impact of the framework parameters on the final solution. The discipline specific objective functions within the case study problem are unconstrained and continuous — however, the implementation of the MADD framework is not restricted to such problems.