Variational Inequality Approach to Stochastic Nash Equilibrium Problems with an Application to Cournot Oligopoly

In this note we investigate stochastic Nash equilibrium problems by means of monotone variational inequalities in probabilistic Lebesgue spaces. We apply our approach to a class of oligopolistic market equilibrium problems where the data are known through their probability distributions.Comment: 19 pages, 2 table

Generalized Nash equilibrium problems with partial differential operators: Theory, algorithms, and risk aversion

PDE-constrained (generalized) Nash equilibrium problems (GNEPs) are considered in a deterministic setting as well as under uncertainty. This includes a study of deterministic GNEPs with nonlinear and/or multivalued operator equations as forward problems and PDE-constrained GNEPs with uncertain data. The deterministic nonlinear problems are analyzed using the theory of generalized convexity for set-valued operators, and a variational approximation approach is proposed. The stochastic setting includes a detailed overview of the recently developed theory and algorithms for risk-averse PDE-constrained optimization problems. These new results open the way to a rigorous study of stochastic PDE-constrained GNEPs

Generalized Nash equilibrium problems with partial differential operators: Theory, algorithms, and risk aversion

PDE-constrained (generalized) Nash equilibrium problems (GNEPs) are considered in a deterministic setting as well as under uncertainty. This includes a study of deterministic GNEPs with nonlinear and/or multivalued operator equations as forward problems and PDE-constrained GNEPs with uncertain data. The deterministic nonlinear problems are analyzed using the theory of generalized convexity for set-valued operators, and a variational approximation approach is proposed. The stochastic setting includes a detailed overview of the recently developed theory and algorithms for risk-averse PDE-constrained optimization problems. These new results open the way to a rigorous study of stochastic PDE-constrained GNEPs

On the analysis of stochastic optimization and variational inequality problems

Uncertainty has a tremendous impact on decision making. The more connected we get, it seems, the more sources of uncertainty we unfold. For example, uncertainty in the parameters of price and cost functions in power, transportation, communication and financial systems have stemmed from the way these networked systems operate and also how they interact with one another. Uncertainty influences the design, regulation and decisions of participants in several engineered systems like the financial markets, electricity markets, commodity markets, wired and wireless networks, all of which are ubiquitous. This poses many interesting questions in areas of understanding uncertainty (modeling) and dealing with uncertainty (decision making). This dissertation focuses on answering a set of fundamental questions that pertain to dealing with uncertainty arising in three major problem classes: [(1)] Convex Nash games; [(2)] Variational inequality problems and complementarity problems; [(3)] Hierarchical risk management problems in financial networks. Accordingly, this dissertation considers the analysis of a broad class of stochastic optimization and variational inequality problems complicated by uncertainty and nonsmoothness of objective functions. Nash games and variational inequalities have assumed practical relevance in industry and business settings because they are natural models for many real-world applications. Nash games arise naturally in modeling a range of equilibrium problems in power markets, communication networks, market-based allocation of resources etc. where as variational inequality problems allow for modeling frictional contact problems, traffic equilibrium problems etc. Incorporating uncertainty into convex Nash games leads us to stochastic Nash games. Despite the relevance of stochastic generalizations of Nash games and variational inequalities, answering fundamental questions regarding existence of equilibria in stochastic regimes has proved to be a challenge. Amongst other reasons, the main challenge arises from the nonlinearity arising from the presence of the expectation operator. Despite the rich literature in deterministic settings, direct application of deterministic results to stochastic regimes is not straightforward. The first part of this dissertation explores such fundamental questions in stochastic Nash games and variational inequality problems. Instead of directly using the deterministic results, by leveraging Lebesgue convergence theorems we are able to develop a tractable framework for analyzing problems in stochastic regimes over a continuous probability space. The benefit of this approach is that the framework does not rely on evaluation of the expectation operator to provide existence guarantees, thus making it amenable to tractable use. We extend the above framework to incorporate nonsmoothness of payoff functions as well as allow for stochastic constraints in models, all of which are important in practical settings. The second part of this dissertation extends the above framework to generalizations of variational inequality problems and complementarity problems. In particular, we develop a set of almost-sure sufficiency conditions for stochastic variational inequality problems with single-valued and multi-valued mappings. We extend these statements to quasi-variational regimes as well as to stochastic complementarity problems. The applicability of these results is demonstrated in analysis of risk-averse stochastic Nash games used in Nash-Cournot production distribution models in power markets by recasting the problem as a stochastic quasi-variational inequality problem and in Nash-Cournot games with piecewise smooth price functions by modeling this problem as a stochastic complementarity problem. The third part of this dissertation pertains to hierarchical problems in financial risk management. In the financial industry, risk has been traditionally managed by the imposition of value-at-risk or VaR constraints on portfolio risk exposure. Motivated by recent events in the financial industry, we examine the role that risk-seeking traders play in the accumulation of large and possibly infinite risk. We proceed to show that when traders employ a conditional value-at-risk (CVaR) metric, much can be said by studying the interaction between value at risk (VaR) (a non-coherent risk measure) and conditional value at risk CVaR (a coherent risk measure based on VaR). Resolving this question requires characterizing the optimal value of the associated stochastic, and possibly nonconvex, optimization problem, often a challenging problem. Our study makes two sets of contributions. First, under general asset distributions on a compact support, traders accumulate finite risk with magnitude of the order of the upper bound of this support. Second, when the supports are unbounded, under relatively mild assumptions, such traders can take on an unbounded amount of risk despite abiding by this VaR threshold. In short, VaR thresholds may be inadequate in guarding against financial ruin

International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book

The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions. This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more

Regularity of Pure Strategy Equilibrium Points in a Class of Bargaining Games

For a class of n-player (n ? 2) sequential bargaining games with probabilistic recognition and general agreement rules, we characterize pure strategy Stationary Subgame Perfect (PSSP) equilibria via a finite number of equalities and inequalities. We use this characterization and the degree theory of Shannon, 1994, to show that when utility over agreements has negative definite second (contingent) derivative, there is a finite number of PSSP equilibrium points for almost all discount factors. If in addition the space of agreements is one-dimensional, the theorem applies for all SSP equilibria. And for oligarchic voting rules (which include unanimity) with agreement spaces of arbitrary finite dimension, the number of SSP equilibria is odd and the equilibrium correspondence is lower-hemicontinuous for almost all discount factors. Finally, we provide a sufficient condition for uniqueness of SSP equilibrium in oligarchic games.Local Uniqueness of Equilibrium, Regularity, Sequential Bargaining.