Socially Stable Set in Decision Theory

• Socially stable set is defined as a subset that satisfies symmetric internal reachability under the transitive closure and direct external dominance.
• It refines classical stable sets, distinguishing itself from m-stable, w-stable, and extended stable sets in cyclic and complex dominance structures.
• Topological conditions like compactness, T1 separation, and Nachbin closedness are crucial to proving the existence of socially stable sets in infinite domains.

A socially stable set is a stable-set variant for an abstract decision problem $(X,R)$, where $X$ is a non-empty set of alternatives and $R\subseteq X\times X$ is an irreflexive binary dominance relation. In the formulation developed for possibly infinite sets of alternatives, it combines an internal condition stated through reachability under the transitive closure $\overline{P(R)}$ restricted to the candidate set, with an external condition requiring direct strict domination of every excluded alternative by some included alternative. Within the broader stable-set literature, it is treated as a refinement of classical stable sets and as a distinct alternative to the $m$-stable, $w$-stable, and extended stable set constructions; its existence is characterized topologically via a compact topology, a $T_1$-order separation property, and Nachbin closedness of a trap relation (Andrikopoulos et al., 13 Aug 2025).

1. Abstract decision-theoretic framework

The ambient object is an abstract decision problem $(X,R)$, where $X\neq\varnothing$ and $R$ is irreflexive, meaning

$X$0

The asymmetric part of $X$1 is

$X$2

and the transitive closure of $X$3 is denoted $X$4. The framework also uses the restriction $X$5 of $X$6 to a subset $X$7, as well as background notions such as $X$8-maximal elements

$X$9

top cycles, and the Schwartz set $R\subseteq X\times X$0 (Andrikopoulos et al., 13 Aug 2025).

This setting is designed for environments in which ordinary maximality may fail, especially under cyclic dominance. The stable-set family then provides non-singleton solution concepts that remain meaningful even when $R\subseteq X\times X$1. The socially stable set belongs to this family, but it uses a mixed criterion: transitive-closure reachability internally and direct strict dominance externally.

2. Formal definition of a socially stable set

A subset $R\subseteq X\times X$2 is a socially stable set of $R\subseteq X\times X$3 if it satisfies two clauses.

First, its internal stability is: $R\subseteq X\times X$4

Second, its external stability is: $R\subseteq X\times X$5

The family of all socially stable sets is denoted

$R\subseteq X\times X$6

The internal clause says that within $R\subseteq X\times X$7, any dominance chain from $R\subseteq X\times X$8 to $R\subseteq X\times X$9 that stays inside $\overline{P(R)}$0 must be matched by reverse reachability. The external clause is classical in form: every outsider must be directly strictly dominated by some insider. The resulting solution concept is therefore not simply a stable set with $\overline{P(R)}$1 substituted everywhere. Its internal and external components are intentionally asymmetric: internal robustness is evaluated through restricted transitive closure, while external coverage is evaluated through direct asymmetric dominance $\overline{P(R)}$2 (Andrikopoulos et al., 13 Aug 2025).

The paper motivating this notion states that socially acceptable outcomes should be internally robust against dominance chains and should cover all outsiders by direct dominance. This suggests a solution concept aimed at cyclic environments in which direct maximality is too weak, but full closure-based exclusion of outsiders would be too strong.

3. Position within the stable-set family

The socially stable set is defined alongside several other stable-set variants. The core distinctions are concentrated in the internal and external clauses.

Concept	Internal condition	External condition / reference relation
Stable set	$\overline{P(R)}$3	$\overline{P(R)}$4
Generalized stable set	Stable relative to $\overline{P(R)}$5	Stable relative to $\overline{P(R)}$6
Socially stable set	$\overline{P(R)}$7	$\overline{P(R)}$8
$\overline{P(R)}$9-stable set	$m$0	$m$1
$m$2-stable set	$m$3	$m$4
Extended stable set	Stable relative to $m$5	Stable relative to $m$6

Relative to the classical stable set, the socially stable set strengthens only the internal clause. Classical internal stability excludes direct strict domination within $m$7, whereas social internal stability requires symmetric reachability for any transitive dominance chain that remains inside $m$8. Relative to $m$9-stable and $w$0-stable sets, it is neither identical nor reducible to them: those concepts alter both internal and external conditions and work more directly with $w$1 rather than with $w$2 plus direct $w$3-externality (Andrikopoulos et al., 13 Aug 2025).

The contrast with $w$4-stability is especially sharp. A $w$5-stable set requires

$w$6

and

$w$7

which the corresponding paper presents as a robust mutual non-dominance criterion. It also proves that under compactness and upper $w$8-semicontinuity, $w$9-stable sets are exactly those obtained by selecting exactly one alternative from each maximal strong component of the contraction (Andrikopoulos et al., 2024). By contrast, the socially stable set uses a different internal symmetry condition and a direct-dominance external condition.

4. Structural relations: contraction, maximal components, Duggan set, and trap relation

The contraction framework decomposes $T_1$0 into strong components. If $T_1$1 is the collection of ground sets of the strong components and

$T_1$2

then $T_1$3 is acyclic, and its maximal elements are denoted

$T_1$4

Within this contraction language, the socially stable set satisfies a component-intersection property: $T_1$5 Thus a socially stable set must intersect every maximal strong component of the contraction (Andrikopoulos et al., 2024, Andrikopoulos et al., 13 Aug 2025).

The same framework connects socially stable sets to the Duggan set and to a special auxiliary relation. The trap relation is defined by

$T_1$6

The paper establishes that the Duggan set is exactly $T_1$7, and the topological existence theorem for socially stable sets is formulated directly in terms of $T_1$8. It also shows that under the relevant topological conditions, the Duggan set $T_1$9 is itself socially stable (Andrikopoulos et al., 13 Aug 2025).

These results make the socially stable set a component-sensitive solution concept. It is not merely a dominance-free subset: it is constrained by the strong-component structure of the decision problem and by the acyclic behavior captured by the trap relation.

5. Topological characterization of existence

The central existence theorem states that for an abstract decision problem $(X,R)$0, the following are equivalent: $(X,R)$1 and

$(X,R)$2

1. $(X,R)$3 satisfies the $(X,R)$4-order separation property with respect to $(X,R)$5;
2. $(X,R)$6 is Nachbin closed.

For the relation under consideration, $(X,R)$7-order separation is given in generic form as follows: for $(X,R)$8 with $(X,R)$9, there exists an open neighborhood $X\neq\varnothing$0 of $X\neq\varnothing$1 such that $X\neq\varnothing$2, and for every net $X\neq\varnothing$3, eventually $X\neq\varnothing$4. The weak version requires only the neighborhood clause. Nachbin closedness means that the graph of the order relation is closed in the product topology: $X\neq\varnothing$5 is closed in $X\neq\varnothing$6. For socially stable sets, the relevant relation is $X\neq\varnothing$7, not $X\neq\varnothing$8 itself (Andrikopoulos et al., 13 Aug 2025).

The proof uses the excluded set topology generated by a subset $X\neq\varnothing$9,

$R$0

together with compactness, acyclicity of $R$1, and the nonemptiness of the Duggan set. This replaces finite combinatorial existence arguments by an order-topological criterion suited to infinite domains.

A plausible implication is that socially stable sets are intended not merely as combinatorial refinements of stable sets, but as solution concepts robust under topological extension from finite to infinite choice spaces.

6. Terminological scope and common confusions

The term “socially stable” is used in several distinct literatures, and the socially stable set should not be conflated with the socially stable matching of two-sided markets. In the matching formulation, a bipartite social graph $R$2 or $R$3 specifies which blocking pairs are socially connected. A matching is socially stable if it is individually rational and has no social blocking pairs, meaning no blocking pair connected by an edge of the social graph. This model concerns blocking deviations in matching markets, not stable-set solution concepts for abstract dominance relations [(Askalidis et al., 2013); (Askalidis et al., 2013)].

It is also distinct from self-stability in set-rationalizable social choice. There, a set $R$4 is $R$5-stable in $R$6 if

$R$7

equivalently,

$R$8

That framework studies fixed points of a choice function over feasible sets, not stable subsets of an abstract dominance relation $R$9 (0910.3580).

A further nearby notion appears in stable sets of contracts. There, a set of contracts $X$00 is stable if

$X$01

and for every contract $X$02,

$X$03

This is the contract-market analogue of Gale–Shapley stability and operates with agent choice functions rather than a binary dominance relation on alternatives (Danilov et al., 2021).

Finally, internally stable and internally closed sets of matchings are again different. There, a set of matchings is internally stable if no matching in the set blocks another, and internally closed if it is inclusionwise maximal with that property; a von Neumann–Morgenstern stable set is internally closed plus externally stable. This is a cooperative-game style theory of families of matchings, not the socially stable set of $X$04 (Faenza et al., 2022).

The shared vocabulary of “stable,” “social,” and “set” therefore masks materially different primitives. The socially stable set in the strict sense is a solution concept for irreflexive binary relations on alternatives, with internal stability based on symmetric reachability under $X$05, external stability based on direct strict dominance, and existence characterized through compact topology and the trap relation (Andrikopoulos et al., 13 Aug 2025).