Extended Stable Set in Social Choice

• Extended Stable Set is a concept that restructures cyclic dominance by grouping alternatives into equipotence classes to form an acyclic relation.
• It replaces direct strict dominance with an extended dominance relation, ensuring core-inclusive choice sets even when the traditional core is empty.
• Utilizing contraction and topological conditions, the framework applies to both finite and infinite alternative spaces in decision and game theory.

The Extended Stable Set is a general solution concept for abstract decision problems $(X,R)$ in social choice theory and game theory, designed for settings in which cyclic dominance prevents the existence of maximal alternatives. In the framework of irreflexive binary relations on a non-empty set of alternatives, it modifies the classical Von Neumann–Morgenstern stable set by replacing direct strict dominance with an extended dominance relation that collapses cyclic structures into equipotence classes and thereby yields an acyclic comparison relation (Andrikopoulos et al., 13 Aug 2025). This construction is intended to produce non-empty, cycle-respecting, core-inclusive choice sets even when the ordinary core $\mathcal{M}(X,R)$ is empty, and the 2025 characterization framework extends the concept from finite domains to infinite sets while providing topological conditions for existence (Andrikopoulos et al., 13 Aug 2025).

1. Abstract decision problems and the failure of maximality

An abstract decision problem is an ordered pair $(X,R)$, where $X$ is a non-empty set of alternatives and $R \subseteq X\times X$ is an irreflexive dominance relation. The paper uses $xRy$ for $(x,y)\in R$, $P(R)$ for the asymmetric part of $R$,

$$P(R)=\{(x,y)\in X\times X \mid xRy \text{ and not } yRx\},$$

$\mathcal{M}(X,R)$0 for the transitive closure of $\mathcal{M}(X,R)$1, and $\mathcal{M}(X,R)$2 for the transitive closure of $\mathcal{M}(X,R)$3 (Andrikopoulos et al., 13 Aug 2025).

The classical benchmark is the core, or set of maximal elements,

$\mathcal{M}(X,R)$4

For irreflexive relations this is equivalent to the absence of any strictly dominating alternative. When $\mathcal{M}(X,R)$5 is acyclic, non-empty cores can often be obtained on compact domains. When $\mathcal{M}(X,R)$6 is cyclic, however, $\mathcal{M}(X,R)$7 may be empty. The standard example is the Condorcet-type cycle $\mathcal{M}(X,R)$8, $\mathcal{M}(X,R)$9, $(X,R)$0, in which every alternative is dominated by another and no maximal alternative exists (Andrikopoulos et al., 13 Aug 2025).

This failure motivates general solution theories, which replace maximality by non-empty set-valued solution concepts. The classical Von Neumann–Morgenstern stable set is one such concept, but it can itself fail to exist on odd cycles or more complex cyclic structures. The Extended Stable Set is introduced precisely to address this defect by preserving the logic of internal and external stability while regularizing the underlying dominance relation (Andrikopoulos et al., 13 Aug 2025).

2. Extended dominance and the formal definition

The starting point is the ordinary stable set. A subset $(X,R)$1 is a Von Neumann–Morgenstern stable set of $(X,R)$2 if it satisfies internal stability,

$(X,R)$3

and external stability,

$(X,R)$4

The collection of such sets is denoted $(X,R)$5 (Andrikopoulos et al., 13 Aug 2025).

The Extended Stable Set replaces $(X,R)$6 by an extended dominance relation $(X,R)$7. Its construction begins with equipotence: for $(X,R)$8,

$(X,R)$9

if either $X$0, or $X$1 and $X$2. Thus two alternatives are equipotent exactly when they lie in the same strongly connected component of the strict-dominance reachability relation (Andrikopoulos et al., 13 Aug 2025).

Using equipotence, the extended dominance relation is defined as follows: $X$3 $X$4-dominates $X$5, written $X$6, if there exist $X$7 such that

$X$8

Conceptually, a representative of the equipotence class of $X$9 strictly dominates a representative of the equipotence class of $R \subseteq X\times X$0. The decisive structural fact is Lemma 3.2: $R \subseteq X\times X$1 is acyclic, even when the original relation $R \subseteq X\times X$2 contains cycles (Andrikopoulos et al., 13 Aug 2025).

An Extended Stable Set $R \subseteq X\times X$3 is then defined exactly analogously to a stable set, but relative to $R \subseteq X\times X$4: $R \subseteq X\times X$5 and

$R \subseteq X\times X$6

Internal stability excludes extended domination within the solution set; external stability requires every outsider to be extended-dominated by some insider (Andrikopoulos et al., 13 Aug 2025).

A plausible interpretation is that the Extended Stable Set does not eliminate cycles by arbitrary tie-breaking. Instead, it contracts cyclic strict-dominance patterns into equipotence classes and evaluates dominance only between those classes. This suggests that the concept is not a rejection of cyclic preferences but a reorganization of them into an acyclic quotient structure.

3. Contraction, strong components, and comparison with related variants

The paper studies the Extended Stable Set alongside several other stable-set variants: the generalized stable set $R \subseteq X\times X$7, the socially stable set $R \subseteq X\times X$8, the $R \subseteq X\times X$9-stable set $xRy$0, and the $xRy$1-stable set $xRy$2 (Andrikopoulos et al., 13 Aug 2025). The generalized stable set is the VNM stable set with respect to $xRy$3. Socially stable and $xRy$4-stable sets allow internal dominance chains provided they are symmetric. The $xRy$5-stable set forbids all internal $xRy$6-domination and imposes an external bidirectionality condition. A separate 2024 paper provides a topological characterization for the existence of $xRy$7-stable sets (Andrikopoulos et al., 2024).

A central organizational device is the contraction $xRy$8 of $xRy$9. Here $(x,y)\in R$0 is the set of strong components $(x,y)\in R$1 under $(x,y)\in R$2, and

$(x,y)\in R$3

holds if there exist $(x,y)\in R$4 and $(x,y)\in R$5 with $(x,y)\in R$6. This induced component relation is acyclic. Let $(x,y)\in R$7 denote the family of $(x,y)\in R$8-maximal components (Andrikopoulos et al., 13 Aug 2025).

Within this framework, the paper gives structural characterizations of several solution concepts. The Schwartz set is the union of all maximal strong components,

$(x,y)\in R$9

A generalized stable set selects one representative from each maximal component,

$P(R)$0

An $P(R)$1-stable set is the union of some maximal components. Socially stable sets must intersect every maximal component (Andrikopoulos et al., 13 Aug 2025).

For the Extended Stable Set, the main structural theorem is Theorem $P(R)$2. Under compactness, Nachbin closedness of $P(R)$3, and the $P(R)$4-order separation property with respect to $P(R)$5, if $P(R)$6 is the stable set of the contracted problem $P(R)$7, then

$P(R)$8

is an Extended Stable Set, and every Extended Stable Set has this form (Andrikopoulos et al., 13 Aug 2025). In other words, the solution is obtained by selecting exactly one representative from each component in the stable set of the contraction.

This representation clarifies the relation to the Schwartz set. The Schwartz set contains whole maximal components, whereas the Extended Stable Set selects representatives at the component level. This suggests that the Extended Stable Set is more discriminating than component-union solutions while remaining explicitly cycle-sensitive.

4. Infinite domains and topological characterization of existence

A major contribution of the 2025 framework is the extension of stable sets and their variants from finite alternative sets to infinite $P(R)$9. The relation $R$0 is still assumed irreflexive, but need not be transitive, complete, or acyclic. The extended relation $R$1 is defined in exactly the same way as in the finite case, and its acyclicity continues to hold (Andrikopoulos et al., 13 Aug 2025).

Existence is no longer obtained by finite combinatorics alone. Instead, the paper derives a topological equivalence. Theorem $R$2 states that for an abstract decision problem $R$3, the following are equivalent:

• there exists a non-empty Extended Stable Set of $R$4 in $R$5;
• there exists a compact topology $R$6 on $R$7 such that $R$8 satisfies the $R$9-order separation property with respect to $$P(R)=\{(x,y)\in X\times X \mid xRy \text{ and not } yRx\},$$0, and $$P(R)=\{(x,y)\in X\times X \mid xRy \text{ and not } yRx\},$$1 is Nachbin closed (Andrikopoulos et al., 13 Aug 2025).

The paper develops this through a more general topological machinery. It defines weak and strong $$P(R)=\{(x,y)\in X\times X \mid xRy \text{ and not } yRx\},$$2-order separation relative to a relation, Nachbin closedness via closedness of the graph of the induced preorder in the product topology, and uses domain-theoretic constructions involving the MacNeille completion, Frink ideals, precontinuity, and the Lawson topology. The Lawson topology on a complete lattice is compact and $$P(R)=\{(x,y)\in X\times X \mid xRy \text{ and not } yRx\},$$3, and under continuity assumptions the relevant order relation becomes Nachbin closed (Andrikopoulos et al., 13 Aug 2025).

Theorem $$P(R)=\{(x,y)\in X\times X \mid xRy \text{ and not } yRx\},$$4 gives the parallel result for ordinary stable sets under acyclic $$P(R)=\{(x,y)\in X\times X \mid xRy \text{ and not } yRx\},$$5: existence of a stable maximal set is equivalent to compactness, $$P(R)=\{(x,y)\in X\times X \mid xRy \text{ and not } yRx\},$$6-order separation, and Nachbin closedness of $$P(R)=\{(x,y)\in X\times X \mid xRy \text{ and not } yRx\},$$7. The Extended Stable Set theorem then arises by applying this template to the acyclic relation $$P(R)=\{(x,y)\in X\times X \mid xRy \text{ and not } yRx\},$$8 rather than to $$P(R)=\{(x,y)\in X\times X \mid xRy \text{ and not } yRx\},$$9 itself (Andrikopoulos et al., 13 Aug 2025).

This yields a precise existence principle: cyclicity in the original dominance relation is not itself the obstacle. The obstacle is whether, after passage to extended dominance, one can endow the domain with a compact order-topological structure compatible with the resulting acyclic relation.

5. Interpretation under cyclic preferences and normative role

The Extended Stable Set is motivated by the need to define a valid choice set when maximal alternatives do not exist. Under cyclic dominance, the ordinary core can be empty, and the classical stable set may also fail to exist. The extended dominance relation treats all alternatives inside a top $\mathcal{M}(X,R)$00-cycle as equipotent and evaluates dominance only across these strongly connected blocks (Andrikopoulos et al., 13 Aug 2025).

Under this interpretation, alternatives within the same cycle are not treated as successively superior and inferior. Rather, they are regarded as members of a single equivalence-like class. If any representative of one class strictly dominates a representative of another, then the entire first class extended-dominates the second. Because the resulting relation is acyclic, the Extended Stable Set can recover a notion of generalized maximality even when no alternative is maximal under the original $\mathcal{M}(X,R)$01 (Andrikopoulos et al., 13 Aug 2025).

The paper emphasizes several decision-theoretic implications. Extended Stable Sets preserve the logic of internal and external stability, are intended to be core-inclusive, and are robust to cycles because they do not require arbitrary cycle-breaking. They also treat equipotent alternatives symmetrically and can be applied to social choice rules with cyclic majority relations, bargaining settings, and game-theoretic dominance structures over infinite alternative spaces, provided the topological conditions are met (Andrikopoulos et al., 13 Aug 2025).

A plausible implication is that the Extended Stable Set occupies a middle ground between maximality-based and component-based solution theories. It is more selective than the Schwartz set, because it does not simply return whole maximal components, but less fragile than the classical stable set, because its dominance relation has already been regularized to eliminate cyclic obstruction.

6. Non-uniqueness, limitations, and terminological scope

The concept does not guarantee uniqueness. Different selections of representatives from the relevant contracted components yield different Extended Stable Sets. In the finite asymmetric setting cited by the paper, if $\mathcal{M}(X,R)$02 is the family of Extended Stable Sets on the contracted problem, then any Extended Stable Set on $\mathcal{M}(X,R)$03 is of the form

$\mathcal{M}(X,R)$04

Non-uniqueness is therefore structural rather than pathological: it reflects the freedom to choose representatives within equipotent components (Andrikopoulos et al., 13 Aug 2025).

Existence can also fail if the topological side of Theorem $\mathcal{M}(X,R)$05 fails. Although $\mathcal{M}(X,R)$06 is always acyclic, one may lack a compact topology with the required $\mathcal{M}(X,R)$07-order separation property, or $\mathcal{M}(X,R)$08 may fail to be Nachbin closed. In that case the equivalence breaks and no Extended Stable Set is guaranteed (Andrikopoulos et al., 13 Aug 2025).

The paper also identifies open directions, including comparison with admissible sets, Duggan sets, and absorbing sets in infinite contexts; weakening compactness or Nachbin closedness; and comparative statics under perturbations of the dominance relation (Andrikopoulos et al., 13 Aug 2025). These questions indicate that the current framework is a characterization theorem rather than a final comparative taxonomy.

A common terminological confusion arises because “stable set” also denotes an independent set in graph theory, and “extended stable set” may appear in the literature on stable set polytopes and extended formulations. That usage concerns graph optimization and polyhedral complexity rather than social choice solution concepts (Bousquet et al., 2017, Conforti et al., 2019). The Extended Stable Set considered here is the social-choice and game-theoretic refinement based on extended dominance, equipotence classes, and contraction of cyclic dominance relations (Andrikopoulos et al., 13 Aug 2025).

In this sense, the Extended Stable Set is best understood as a cycle-regularized stable-set solution for abstract decision problems: it replaces direct maximization by representative selection from maximal structures of an acyclic quotient relation, and on infinite domains it exists exactly when that quotient relation admits the appropriate compact topological realization (Andrikopoulos et al., 13 Aug 2025).