On a Robust Approach for Stochastic Equilibrium Problems

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Stochastic programming has been extensively used by operation researchers, economists and various decision makers/practitioners to model optimal decision making in economics, management, engineering, transportation networks and the environment. When a decision problem involves not only uncertainty, but also severaldecision makers who are in a competitive relationship, it becomes a stochastic game. An important approach in understanding such a game is to look at the equilibrium outcomes. These are the set of possible outcomes atthe end of competition, given that each player seeks to optimize their own payoff.A fundamental issue in stochastic programming and equilibrium concerns the representation ofuncertainty. Many of the models in the literature assume complete knowledge of the distributions of random variables (representing the uncertainty). Inmany practical cases, however, such distributions are not known precisely and have to be either estimated from historical data or constructed usingsubjective judgements. The available information is often insufficient to give confidence in the distribution identified. In the absence of full information on the underlying distribution, it may still be possible toidentify a set of possible probability distributions within which the true distribution lies. While a robust optimization approach to this problem isbased on making the decision that would be appropriate given the worst probability distribution in the set of possible distributions, robust analysis of stochastic equilibrium is to look into worst equilibrium outcomes given the incomplete information of the underlying stochastic elements and robust designrequires one to set out optimal policy/parameters which accommodate any worst equilibrium outcomes.The project is proposed to develop a mathematical framework that allows one to carry out robust anaysis of a stochastic equilibrium problem with incomplete information on the underlying uncertainty, identify optimal policy/design which accommodate the worst possible equilibrium outcomes, develop efficient numerical methods for solving the new mathematical models and apply apply them to some interesting practical problems in economics and engineering with a particular focus on energy industry.