Stochastic Graphon Games: II. The Linear-Quadratic Case

Aurell, Alexander; Carmona, René; Laurière, Mathieu

doi:10.1007/s00245-022-09839-2

Cited by 15 publications

(6 citation statements)

References 38 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This paper adds to the literature on Graphon Mean Field Games which read as Mean Field Games with networked agents, see for example Huang (2018, 2021); Lacker and Soret (2022); Delarue (2017); Parise and Ozdaglar (2019); Carmona et al (2019); Aurell et al (2022). These Graphon Mean Field Games are generalizations of Mean Field Games (see for example Caines et al (2017); Caines (2020)), and can be seen as Graphon Mean Field Games with agents on large undirected graphs.…”

Section: Introductionmentioning

confidence: 86%

Stationary Cost Nodes in Infinite Horizon LQG-GMFGs

Tchuendom¹,

Gao²,

Caines³

2022

Preprint

View full text Add to dashboard Cite

An analysis of infinite horizon linear quadratic Gaussian (LQG) Mean Field Games is given within the general framework of Graphon Mean Field Games (GMFG) on dense infinite graphs (or networks) introduced in Caines and Huang (2018). For a class of LQG-GMFGs, analytical expressions are derived for the infinite horizon Nash values at the nodes of the infinite graph. Furthermore, under specific conditions on the network and the initial population means, it is shown that the nodes with strict local maximal infinite network degree are also nodes with strict local minimal cost at equilibrium.

show abstract

Section: Introductionmentioning

confidence: 86%

Stationary Cost Nodes in Infinite Horizon LQG-GMFGs

Tchuendom¹,

Gao²,

Caines³

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Outside of the highly symmetric mean field case (1.3), it is not as clear how to construct nearoptimal distributed controls for the control problem (1.2). There have been some recent proposals to extend the mean field framework to accommodate certain models with heterogeneous interactions which possess certain asymptotic structure, by taking advantage of the theory of graphons [1,3,25,33]. For instance, suppose the cost function G (and similarly F ) takes the form…”

Section: Full-information Controlsmentioning

confidence: 99%

“…The two main assumptions behind our main bound (1.7) are (1) the boundedness of the second (but not the first) derivatives of F , G, and the Hamiltonians H i (x, p) := sup a∈R d − a•p − L i (x, a) , and (2) the convexity of F , G, and L i for each i. In fact, in the precise Theorem 4.5 below, a sharper bound is given in which the L ∞ norms in (1.7) are replaced by certain L 2 norms.…”

Section: Full-information Controlsmentioning

confidence: 99%

Approximately optimal distributed stochastic controls beyond the mean field setting

Jackson¹,

Lacker²

2023

Preprint

View full text Add to dashboard Cite

We study high-dimensional stochastic optimal control problems in which many agents cooperate to minimize a convex cost functional. We consider both the full-information problem, in which each agent observes the states of all other agents, and the distributed problem, in which each agent observes only its own state. Our main results are sharp non-asymptotic bounds on the gap between these two problems, measured both in terms of their value functions and optimal states. Along the way, we develop theory for distributed optimal stochastic control in parallel with the classical setting, by characterizing optimizers in terms of an associated stochastic maximum principle and a Hamilton-Jacobi-type equation. By specializing these results to the setting of mean field control, in which costs are (symmetric) functions of the empirical distribution of states, we derive the optimal rate for the convergence problem in the displacement convex regime. Contents 1. Introduction 2. Notation and preliminaries 3. Problem formulations and value functions 4. Near-optimality of distributed controls 5. Application to mean field control 6. Application to heterogeneous doubly stochastic interactions 7. Tradeoffs between convexity and smallness Appendix A. Well-posedness of maximum principle FBSDE References

show abstract

“…The emergence of heterogeneity in biological resource dynamics mathematically involves another dimension that is often infinite-dimensional, which increases the model complexity, although this aspect can be resolved analytically if the problem at hand admits a tractable structure. Examples include, but are not limited to, the graphon linear-quadratic game whose resolution is reduced to the Riccati equation (Aurell et al, 2022), and the moment approximation closure (Lunz et al, 2023). However, most problems that are encountered in applications do not admit such a useful mathematical structure.…”

Section: Introductionmentioning

confidence: 99%

Optimal harvesting policy for biological resources with uncertain heterogeneity for application in fisheries management

Yoshioka

2024

Natural Resource Modeling

View full text Add to dashboard Cite

Conventional harvesting problems for natural resources often assume physiological homogeneity of the body length/weight among individuals. However, such assumptions generally are not valid in real‐world problems, where heterogeneity plays an essential role in the planning of biological resource harvesting. Furthermore, it is difficult to observe heterogeneity directly from the available data. This paper presents a novel optimal control framework for the cost‐efficient harvesting of biological resources for application in fisheries management. The heterogeneity is incorporated into the resource dynamics, which is the population dynamics in this case, through a probability density that can be distorted from reality. Subsequently, the distortion, which is the model uncertainty, is penalized through a divergence, leading to a nonstandard dynamic differential game wherein the Hamilton–Jacobi–Bellman–Isaacs (HJBI) equation has a unique nonlinear partial differential term. Here, the existence and uniqueness results of the HJBI equation are presented along with an explicit monotone finite difference method. Finally, the proposed optimal control is applied to a harvesting problem with recreationally, economically, and ecologically important fish species using collected field data.

show abstract

Stochastic Graphon Games: II. The Linear-Quadratic Case

Cited by 15 publications

References 38 publications

Stationary Cost Nodes in Infinite Horizon LQG-GMFGs

Stationary Cost Nodes in Infinite Horizon LQG-GMFGs

Approximately optimal distributed stochastic controls beyond the mean field setting

Optimal harvesting policy for biological resources with uncertain heterogeneity for application in fisheries management

Contact Info

Product

Resources

About