Set-Valued Mean Fields

• Set-valued mean fields are mappings that assign a set (often convex or compact) to each bounded subset, generalizing traditional averages.
• They integrate tools from analysis, probability, and control theory to model convergence, robustness, and ambiguity in systems with nonunique limits.
• Applications span random set theory, controlled SPDEs, and mean field games, providing insights into equilibrium convergence and optimal strategy design.

Set-valued mean fields describe a class of mathematical objects and limiting procedures where both the "inputs" and "outputs" of averaging operations, propagation of chaos, or equilibrium concepts are sets (often convex or compact), rather than single values or distributions. This construct appears across analysis and probability (mean-sets and random set theory), stochastic control (mean-field SPDEs), and game theory (set-valued value functions in mean-field games), unifying the study of nonunique, robust, or ambiguous limiting behaviors.

1. Set-Valued Means: Algebraic and Measure-Theoretic Structure

A set-valued mean (or "mean-set") is a mapping

$\mathrm{MS}: \mathcal{F} \to 2^X$

where $X$ is a base space (typically $\mathbb{R}$ or $\mathbb{R}^d$) and $\mathcal{F}$ is the family of nonempty bounded subsets of $X$. The principal axiom is internality: $\mathrm{MS}(H) \subseteq [\inf H, \sup H]$ for all $H\in\mathcal{F}$. Further structural properties include strong internality (valid for infinite $H$), monotonicity under set inclusion, translation/reflection invariance, homogeneity, convexity, and finite-independence. These endow the family of mean-sets with semigroup or group structure under composition, union, or intersection. Classical means (arithmetic, geometric, etc.) are recovered as singleton-valued mean-sets.

Three prototypical examples are:

• The topological interior: $\mathrm{MS}(H) = \mathrm{Int}(H)$
• The mid-point set: $X$0
• The halving-measure set: For Lebesgue measure $X$1 and $X$2,

$X$3

These constructions show how set-valued averaging generalizes the notion of central tendency—capturing the "typical" subset rather than a unique point or number (Losonczi, 2018).

2. Set-Valued Mean Fields in Random Set Theory

In the theory of random sets, the Aumann expectation provides a canonical set-valued mean. Let $X$4 be a (measurable, integrably bounded) random compact convex set. The Aumann expectation is

$X$5

alternatively characterized via support functions, with $X$6 for each $X$7.

Given i.i.d. random sets $X$8, the sample mean

$X$9

(where "$\mathbb{R}$0" denotes Minkowski addition) converges almost surely to $\mathbb{R}$1 in the Pompeiu–Hausdorff metric (Artstein–Vitale strong law). A "set-valued law of large numbers" arises: $\mathbb{R}$2 Boundary fluctuations of $\mathbb{R}$3 about $\mathbb{R}$4 are governed by central limit theorems, either in function space (support functions) or locally at exposed points, tangent planes, or facets, inducing multivariate or univariate Gaussian fluctuations. Thus, $\mathbb{R}$5 acts as a set-valued mean field to which sample means converge, with boundary geometry dictating local behavior (Pichler, 2017).

3. Set-Valued Mean Fields in Controlled McKean–Vlasov SPDEs

For controlled path-dependent McKean–Vlasov SPDEs, the set-valued mean field framework arises naturally. Let $\mathbb{R}$6 be a separable Hilbert space, $\mathbb{R}$7 a compact metrizable action set, and $\mathbb{R}$8 the path space. The dynamics of an $\mathbb{R}$9-particle system with feedback controls $\mathbb{R}^d$0 and empirical measure $\mathbb{R}^d$1 are given by

$\mathbb{R}^d$2

The law of the particle system and of empirical measures define sets $\mathbb{R}^d$3 and $\mathbb{R}^d$4. The mean field limit consists of the set $\mathbb{R}^d$5 of all possible laws for McKean–Vlasov SPDEs with feedback, and $\mathbb{R}^d$6 as those laws on the empirical measure space supported on $\mathbb{R}^d$7.

Under compactness or spectral assumptions on the semigroup, and growth/Lipschitz/convexity conditions on $\mathbb{R}^d$8, $\mathbb{R}^d$9, propagation of chaos occurs in a set-valued sense: $\mathcal{F}$0 for the Hausdorff distance in Wasserstein topology. This provides a robust framework for limit theorems and control: value maps and optimal strategies converge uniformly, and accumulation points of controls or value laws in finite systems approximate mean-field solutions (Criens et al., 2023).

4. Set-Valued Valuation in Mean Field Games

Mean field games with potentially multiple equilibria motivate the study of the set of values across all mean field equilibria, termed the set value $\mathcal{F}$1. For a population state law $\mathcal{F}$2 at time $\mathcal{F}$3, with controls $\mathcal{F}$4, the raw set value is

$\mathcal{F}$5

and the (regularized) set value is

$\mathcal{F}$6

These sets are typically nonempty and closed, reducing to singleton values under monotonicity (i.e., uniqueness of mean field equilibrium). Dynamic programming principles extend to the set-valued case: $\mathcal{F}$7 is time-consistent, and satisfies a set-valued PDE generalizing the classical master equation.

For $\mathcal{F}$8-player games, the set of equilibrium values $\mathcal{F}$9 converges (in the Hausdorff sense) to $X$0 as $X$1, under regularity and proper alignment of control classes (closed-loop, Lipschitz). Allowing heterogeneity or relaxed controls expands the limiting set value, with strict inclusions

$X$2

stressed in the literature (Iseri et al., 2021).

5. Metric and Topological Frameworks

All set-valued mean field concepts rely on robust topologies on sets of sets:

• The Pompeiu–Hausdorff metric for compact convex sets in Euclidean space.
• The Wasserstein metric for probability measures on path spaces, and Hausdorff metric for sets of probability measures.
• Kuratowski or Hausdorff set convergence for stability of value sets in mean field games.
• Support function embeddings provide a means of linearizing and analyzing set-valied convergence in function spaces.

Compactness in these topologies is central: it guarantees existence of accumulation points to which finite-particle laws or value sets can converge, and ensures continuity of value functionals and performance measures.

6. Applications and Extensions

Set-valued mean fields find immediate application in:

• Stochastic optimal control, where robust optimal strategies and performance bounds are naturally framed in terms of convergence of set-valued cost functionals (Criens et al., 2023).
• Mean field games with nonunique equilibria or robust/adversarial controls, in which time consistent set-valued value functions underpin the limiting and stability analysis (Iseri et al., 2021).
• Random set theory, where the Aumann mean field describes population-level average geometries or resource allocations in uncertain environments (Pichler, 2017).
• Nonlinear and sublinear expectation models, e.g., set-valued propagation of chaos for G-Brownian motion under drift interaction (Criens et al., 2023).
• Abstract algebraic and measure-theoretic studies, where mean-sets illuminate generalizations of classical averaging and symmetry properties, extending to sequence spaces, measure spaces, and beyond (Losonczi, 2018).

The sensitivity of set-valued limits to definitions of admissible controls, allowable feedback regularity, and structural conditions is repeatedly emphasized. Lack of regularity or inappropriate matching of $X$3-player and mean field control classes can break time consistency and limit convergence.

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Set-valued mean fields thus offer a unifying lens for analyzing, quantifying, and controlling the limiting "shape" or value of complex, possibly nonunique or nonconvex systems, subsuming both quantitative and topological uncertainty in infinite-population or sample-size limits.