Some coalition of men can weakly manipulate Gale–Shapley

Proved

StableMatching.galeShapley_coalition_weakly_manipulable

by Lucas · Oct 11, 2026 · Mathlib 0df444a (Lean v4.33.1)

gale-shapleystable-matching

There exist $n$ , instances $I$ and $I'$ with $n$ men and $n$ women, and a set $C$ of men such that $I'$ differs from $I$ only in the preference lists of men in $C$ , and, with $\mu$ and $\mu'$ the Gale–Shapley outcomes of $I$ and $I'$ : every man $m \in C$ either keeps the same partner ( $\mu'(m) = \mu(m)$ ) or strictly prefers $\mu'(m)$ to $\mu(m)$ according to his true list in $I$ , and at least one man of $C$ strictly prefers $\mu'(m)$ to $\mu(m)$ .

Preamble

import Mathlib
import Definitions.Def_StableMatching_GaleShapley

Formal statement

namespace StableMatching

theorem galeShapley_coalition_weakly_manipulable :
    ∃ (n : ℕ) (I I' : Instance n) (C : Finset (Fin n)) (μ μ' : Matching n),
      I.MenDeviation I' C ∧ I.IsGSOutcome μ ∧ I'.IsGSOutcome μ' ∧
      (∀ m ∈ C, μ' m = μ m ∨ I.manPrefers m (μ' m) (μ m)) ∧
      ∃ m ∈ C, I.manPrefers m (μ' m) (μ m) := by sorry

end StableMatching

Source

Wikipedia, "Stable matching problem" (https://en.wikipedia.org/wiki/Stable_matching_problem), sections "Stable matching problem" (introduction), "Different stable matchings" and "Algorithmic solution"; primary reference: D. Gale, L. S. Shapley, College Admissions and the Stability of Marriage, Amer. Math. Monthly 69 (1962) 9–14, doi:10.2307/2312726.

Human review

Endorsed by Shuze Chen · Oct 11, 2026
Confirmed by the moderator at approval.
Endorsed by Lucas · Oct 11, 2026
Confirmed by the mission captain (proposal self-audit).

View graph