Distributionally Robust Markov Decision Processes and Their Connection to Risk Measures

We consider robust Markov decision processes with Borel state and action spaces, unbounded cost, and finite time horizon. Our formulation leads to a Stackelberg game against nature. Under integrability, continuity, and compactness assumptions, we derive a robust cost iteration for a fixed policy of the decision maker and a value iteration for the robust optimization problem. Moreover, we show the existence of deterministic optimal policies for both players. This is in contrast to classical zero-sum games. In case the state space is the real line, we show under some convexity assumptions that the interchange of supremum and infimum is possible with the help of Sion’s minimax theorem. Further, we consider the problem with special ambiguity sets. In particular, we are able to derive some cases where the robust optimization problem coincides with the minimization of a coherent risk measure. In the final section, we discuss two applications: a robust linear-quadratic problem and a robust problem for managing regenerative energy.

Game Theory Basics

Game theory is the science of interaction. This textbook, derived from courses taught by the author and developed over several years, is a comprehensive, straightforward introduction to the mathematics of non-cooperative games. It teaches what every game theorist should know: the important ideas and results on strategies, game trees, utility theory, imperfect information, and Nash equilibrium. The proofs of these results, in particular existence of an equilibrium via fixed points, and an elegant direct proof of the minimax theorem for zero-sum games, are presented in a self-contained, accessible way. This is complemented by chapters on combinatorial games like Go; and, it has introductions to algorithmic game theory, traffic games, and the geometry of two-player games. This detailed and lively text requires minimal mathematical background and includes many examples, exercises, and pictures. It is suitable for self-study or introductory courses in mathematics, computer science, or economics departments.

Unlocking the Mystery of the Origins of John von Neumann’s Growth Model

In his famous “A Model of General Economic Equilibrium,” von Neumann wrote that it was “obvious to what kind of theoretical models [his] assumptions correspond.” To date, however, his sources of economic insights about the functioning of the continuously growing price-economy that he modeled have remained a total mystery. Based on archival material, this mystery is solved in this account by making visible the specific influences from economics and mathematics that inspired him. I argue that von Neumann’s 1937 paper resulted from a deep engagement with economics as it was emerging at the beginning of the 1930s and that this happened as he was travelling and crossing national boundaries while bridging distinct branches of mathematics with different local perspectives in economics. His encounters with Jacob Marschak in Berlin, Nicolas Kaldor in Budapest, and Frank Graham in Princeton as well as his reading of Walras’s, Wicksell’s and Cassel’s work would be key. I also explain how he came to realize that there existed a formal analogy between systems of linear equations and inequalities with which he characterized (stationary and dynamic) economies and the minimax theorem for two-person zero-sum games that he had conceived and proved in 1928.

The Existence of Nontrivial Solutions to a Class of Quasilinear Equations

In this paper, we study the following quasilinear equation: − div ϕ ∇ u ∇ u + ϕ u u = f u in ℝ N , where ϕ ∈ C 1 ℝ + , ℝ + and Φ t = ∫ 0 t s ϕ ∣ s ∣ d s . In the Orlicz-Sobolev space, by variational methods and a minimax theorem, we prove the equation has a nontrivial solution.

A saddle point type solution for a system of operator equations

Let Ω⊂Rn n>1 and let p,q≥2. We consider the system of nonlinear Dirichlet problems equation* brace(Au)(x)=Nu′(x,u(x),v(x)),x∈Ω,r-(Bv)(x)=Nv′(x,u(x),v(x)),x∈Ω,ru(x)=0,x∈∂Ω,rv(x)=0,x∈∂Ω,endequation* where N:R×R→R is C1 and is partially convex-concave and A:W01,p(Ω)→(W01,p(Ω))* B:W01,p(Ω)→(W01,p(Ω))* are monotone and potential operators. The solvability of this system is reached via the Ky–Fan minimax theorem.