Remarks on Nash equilibria for games with additively coupled payoffs (revision, previous title: An unusual Nash equilibrium result and its application to games with allocations in infinite dimensions)

If the payoffs of a game are affine, then they are additively coupled. In this situation both the Weierstrass theorem and the Bauer maximum principle can be used to produce existence results for a Nash equilibrium, since each player is faced with an individual, independent optimization problem. We consider two instances in the literature where these simple observations immediately lead to substantial generalizations

More on maximum likelihood equilibria for games with random payoffs and participation

Maximum likelihood Nash equilibria were introduced by Borm et al. (1995) for games with nitely many players and random payos. In Voorneveld (1999) random participation was added to the model. These existence results were extended by Balder (2000c) to continuum games with random payos and participation. However, in that paper the complicated measurability issue for the central equilibrium likelihood notion was bypassed by using inner probabilities for the central equilibrium likelihood notion. Here those measurability questions are shown to have a quite satisfactory resolution; this makes the maximum likelihood equilibrium notion more natural. Our main results are not only more general than those found in Borm et al. (1995) and Voorneveld (1999), but also improve upon them

On characterizing optimality and existence of optimal solutions for Lyapunov type optimization problems

Necessary and sucient conditions for optimality, in the form of a duality result of Fritz-John type, are given for an abstract optimization problem of Lyapunov type. The introduction of a so-called integrand constraint qualication allows the duality result to take the form of a Kuhn-Tucker type result. Special applications include necessary and sucient conditions for the existence of optimal controls for certain optimal control problems

An existence result for optimal economic growth problems

AbstractAn existence result for optimal control problems of Lagrange type with unbounded time domain is derived very directly from a corresponding result for problems with bounded time domain. This subsumes the main existence result of R. F. Baum ¦J. Optim. Theory Appl. 19 (1976), 89–116¦ and has the existence results for optimal economic growth problems of S.-I. Takekuma ¦J. Math. Econom. 7 (1980), 193–208¦ and M. J. P. Magill ¦Econometrica 49 (1981), 679–711; J. Math. Anal. Appl. 82 (1981), 66–74¦ as simple corollaries. In addition, a new notion of uniform integrability is used, which coincides with the classical notion if the time domain is bounded

On an optimal consumption problem for p-integrable consumption plans

A generalization is presented of the existence results for an optimal consumption problem of Aumann and Perles [4] and Cox and Huang [10]. In addition, we present avery general optimality principle

Nonsmooth analysis of doubly nonlinear evolution equations

In this paper we analyze a broad class of abstract doubly nonlinear evolution equations in Banach spaces, driven by nonsmooth and nonconvex energies. We provide some general sufficient conditions, on the dissipation potential and the energy functional,for existence of solutions to the related Cauchy problem. We prove our main existence result by passing to the limit in a time-discretization scheme with variational techniques. Finally, we discuss an application to a material model in finite-strain elasticity.Comment: 45 page

Value Functions and Transversality Conditions for Infinite-Horizon Optimal Control Problems

This paper investigates the relationship between the maximum principle with an infinite horizon and dynamic programming and sheds new light upon the role of the transversality condition at infinity as necessary and sufficient conditions for optimality with or without convexity assumptions. We first derive the nonsmooth maximum principle and the adjoint inclusion for the value function as necessary conditions for optimality that exhibit the relationship between the maximum principle and dynamic programming. We then present sufficiency theorems that are consistent with the strengthened maximum principle, employing the adjoint inequalities for the Hamiltonian and the value function. Synthesizing these results, necessary and sufficient conditions for optimality are provided for the convex case. In particular, the role of the transversality conditions at infinity is clarified

An extension of the usual model in statistical decision theory with applications to stochastic optimization problems

By employing fundamental results from “geometric” functional analysis and the theory of multifunctions we formulate a general model for (nonsequential) statistical decision theory, which extends Wald's classical model. From central results that hold for the model we derive a general theorem on the existence of admissible nonrandomized Bayes rules. The generality of our model makes it also possible to apply these results to some stochastic optimization problems. In an appendix we deal with the question of sufficiency reduction

Existence results without convexity conditions for general problems of optimal control with singular components

This note presents a new, quick approach to existence results without convexity conditions for optimal control problems with singular components in the sense of [11.], 438–485). Starting from the resolvent kernel representation of the solutions of a linear integral equation, a version of Fatou's lemma in several dimensions is shown to lead directly to a compactness result for the attainable set and an existence result for a Mayer problem. These results subsume those of [12.], 110–117), [13.], 74–101), [10.], [7.], 319–331) and [1.], 63–79)

Existence results without convexity conditions for general problems of optimal control with singular components

This note presents a new, quick approach to existence results without convexity conditions for optimal control problems with singular components in the sense of [11.], 438–485). Starting from the resolvent kernel representation of the solutions of a linear integral equation, a version of Fatou's lemma in several dimensions is shown to lead directly to a compactness result for the attainable set and an existence result for a Mayer problem. These results subsume those of [12.], 110–117), [13.], 74–101), [10.], [7.], 319–331) and [1.], 63–79)