# we-solve research prompt: Econ-20, Axiomatic containment of Mertens-stable sets without genericity

Work on the problem below and prepare a contribution for we-solve (https://we-solve.github.io/problems/axiomatic-containment-of-mertens-stable-sets-without-genericity/).

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## Problem statement (quoted material)
<<<
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\usepackage{amsmath,amssymb,amsthm,mathtools,mathrsfs}
\usepackage{enumitem,booktabs}
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\hypersetup{pdftitle={Axiomatic containment of Mertens-stable sets: improved We-Solve problem statement and variants},pdfauthor={LLM-assisted editorial research draft}}
\newcommand{\NE}{\operatorname{NE}}
\newcommand{\QP}{\operatorname{QP}}
\newcommand{\Stab}{\operatorname{Stab}_0}
\newcommand{\CC}{\mathcal C}
\newcommand{\A}{\mathcal A}
\newcommand{\Q}{\mathbb Q}
\newcommand{\R}{\mathbb R}
\newcommand{\U}{\mathcal U_0}
\DeclareMathOperator{\relint}{relint}
\newtheorem{proposition}{Proposition}
\title{Axiomatic containment of Mertens-stable sets without genericity\\
\large Improved problem statement, exact quantifiers, and nearby research variants}
\author{LLM-assisted editorial research draft}
\date{18 October 2026}
\begin{document}
\maketitle
\section*{Problem identity and editorial status}
\noindent\textbf{Permanent problem identifier:} \texttt{WS-107793}.\\
\textbf{Status:} Proposed editorial generalization of Govindan--Wilson
\cite{gw}; no proof or disproof of the universal statement in the
accompanying contributions. The question's status in the broader literature
and its novelty are not certified by these documents.\\
\textbf{Classification suggestion:} JEL C72 (Noncooperative Games), with C73
as a secondary classification.

\section*{Recommended replacement problem statement}
\subsection*{1. Definitions and domain}
Let $\mathfrak G$ be the domain of all finite normal-form games (with
any finite number of players, finite nonempty action sets, and real payoffs).
For $G=(I,(A_i)_{i\in I},(u_i)_{i\in I})\in\mathfrak G$, write
\[
 \Sigma_G=\prod_{i\in I}\Delta(A_i),\qquad
 \NE(G)\subseteq\Sigma_G,
\]
and
\[
 \CC(G)=\{K\subseteq\NE(G):K\ne\varnothing,\ 
                        K\text{ is closed and connected}\}.
\]
Because $\NE(G)$ is compact, every $K\in\CC(G)$ is compact.
A \emph{solution concept} is a correspondence $\Phi$ satisfying
\[
 \varnothing\ne\Phi(G)\subseteq\CC(G)
 \qquad\text{for every }G\in\mathfrak G.
 \tag{1}\label{eq:correspondence}
\]
No hypothesis says that the \emph{family} $\Phi(G)$ is itself closed
in the Hausdorff hyperspace.

Use \emph{exactly} the following fixed reference: Govindan--Wilson
\cite[Definition 2.4; Axioms 3.1, 4.7, 5.4]{gw}, arXiv:2504.16908v2.
The stability solution family $\Stab(G)$ is Mertens's
characteristic-zero stability, i.e.\ rational \v Cech-cohomological
essentiality together with the stated connexity requirement.
The superscript $0$ concerns the characteristic of the coefficient
field and is unrelated to an equilibrium's fixed-point index.

For clarity, the source's constrained-strategy perturbations use
\[
 P_\varepsilon(G)=\{\delta\tau:0\le\delta\le\varepsilon,
                                 \ \tau\in\Sigma_G\},
 \qquad
 \Sigma_i(\eta)=\{x_i\in\Delta(A_i):x_i(a)\ge\eta_i(a)
                                         \ \forall a\}.
 \tag{2}\label{eq:perturb}
\]
Writing $G(\eta)$ for the constrained game, put
\[
 E_G=\{(\eta,x):\eta\in P_1(G),\ x\in\NE(G(\eta))\},
 \qquad \pi_G(\eta,x)=\eta.
\]
If $X\subseteq E_G$, let
\[
 X_\varepsilon=X\cap\pi_G^{-1}(P_\varepsilon(G)),\quad
 \partial X_\varepsilon=X\cap\pi_G^{-1}(\partial P_\varepsilon(G)),
 \quad X_0=X\cap\pi_G^{-1}(0).
\]
The boundary $\partial P_\varepsilon(G)$ is relative to the affine
(or, equivalently, linear) span of the perturbation polytope. Its
dimension is $d=1+\sum_i(|A_i|-1)$. Conditions involving
$X_0$, connexity and the induced rational cohomology maps must be
read literally as in the cited Definition 2.4; the summary in (2) is
not a replacement definition of stability.

\subsection*{2. Exact axioms}
Require $\Phi$ to satisfy all three axioms \emph{globally}, not just
at the game used to test the conclusion:
\begin{enumerate}[label=(\roman*),leftmargin=2.5em]
\item \textbf{Backward induction $B$.} For \emph{every} perfect-recall
extensive-form representation $\Gamma$ whose normal form is $G$,
every $K\in\Phi(G)$ contains a mixed profile equivalent to a
quasi-perfect equilibrium of $\Gamma$, in the precise sense of
\cite[Definition 2.1 and Axiom 3.1]{gw}.
\item \textbf{Strongly irrelevant dominated strategies $D^*$.}
For \emph{every} strategy extension allowed by
\cite[Definition 4.4]{gw}, the two solution families coincide
under the specified face identification, as demanded in
\cite[Axiom 4.7]{gw}. Weak dominance by itself is \emph{not}
sufficient for a comparison to be admissible.
\item \textbf{Strong-embedding invariance $I^*$.} If a game $H$
strongly embeds $G$ through $f$ satisfying
\cite[Definition 5.2]{gw}, then
\[
  \Phi(G)=\{f(L):L\in\Phi(H)\},
  \tag{3}\label{eq:embedding}
\]
using the source's mapping of equilibrium profiles and their
connected sets. This is equality \emph{of families}, containing both
an image condition and an existence-of-lifts condition.
\end{enumerate}
No additional axioms (admissibility, genericity, strict equilibrium,
closedness of $\Phi(G)$, weak embedding, arbitrary payoff limits)
are silently imposed.

\subsection*{3. Principal question S1: nongeneric containment}
\noindent\textbf{Prove or disprove the following implication:}
\[
 \boxed{\begin{gathered}
 \forall\Phi:\mathfrak G\rightrightarrows\CC\quad
 \bigl[\Phi\text{ satisfies }(1),B,D^*,I^*\bigr]\\[-2pt]
 \Longrightarrow\quad
 \forall G\in\mathfrak G\ \forall K\in\Phi(G)\quad
 \exists S\in\Stab(G):\ S\subseteq K.
 \end{gathered}}
 \tag{S1}\label{eq:S1}
\]
There is no genericity, regularity, isolation, or finite-equilibrium-set
restriction on the target game. A \emph{positive} solution must treat
non-isolated equilibrium sets and the actual rational-cohomological
stability witness; an argument using only topological robustness or
existence of equilibria of perturbed games does not suffice.
A \emph{negative} solution must give a \emph{globally} defined
solution concept satisfying (1), $B,D^*,I^*$ in all relevant comparison
games, and a particular $G,K$ with $K\in\Phi(G)$ but without any
$S\in\Stab(G)$ contained in $K$. A counterexample set in one isolated
game, without constructing a compatible global rule, is insufficient.

\subsection*{4. Known theorem and the exact missing step}
Govindan--Wilson's Theorem 6.1 proves the containment conclusion for
\emph{generic} perfect-recall extensive-form games, for any finite
number of players. Their Section 10 observes that nongeneric
perturbation graphs can have higher dimension and that the fixed-point
(coincidence) property used in the generic proof does not automatically
imply the required rational-cohomological essentiality.

The original target is \emph{containment}, not either stronger demand
\[
 \Phi(G)\subseteq\Stab(G),\qquad
 \Phi(G)=\Stab(G).
\]
Nor does the assertion refer only to components with nonzero
fixed-point index. Replacing \v Cech essentiality by homotopy
essentiality would define a different problem.

\section*{Close variations and reductions}
The following variants are proposed research questions and reductions,
\emph{not} asserted to be unresolved by a comprehensive literature review.
They all keep the global axioms unchanged unless they explicitly say
otherwise.

\subsection*{V1. Two-player nongeneric containment}
Restrict the \emph{tested game} in (S1), but not the domain or axioms
of $\Phi$, to games with two players:
\[
 \forall\Phi\ [B,D^*,I^*]\Longrightarrow
 \forall G\in\mathfrak G\ (|I(G)|=2)\ \forall K\in\Phi(G)
 \ \exists S\in\Stab(G):S\subseteq K.
 \tag{V1}
\]
The known generic two-player case is not the whole of (V1).
This is a natural first setting in which to search for a nongeneric
counterexample or a stronger 2-player argument.

\subsection*{V2. Bounded players and actions}
For integers $n\ge2$, $m\ge2$, ask the same conclusion for every
tested game with at most $n$ players and at most $m$ actions per player:
\[
  \forall\Phi\ [B,D^*,I^*]\Longrightarrow
  \forall G:\ \bigl(|I(G)|\le n,\,\max_i|A_i|\le m\bigr)
  \quad\forall K\in\Phi(G)\quad
  \exists S\in\Stab(G): S\subseteq K.
  \tag{V2$_{n,m}$}
\]
Crucially, $\Phi$ is still defined on \emph{all} finite games because
an allowed strong embedding may introduce additional players or actions.
Establish the smallest $(n,m)$ for which a counterexample exists, if
(S1) fails, or prove a nontrivial bounded-size positive theorem.

\subsection*{V3. A boundary-degenerate target subclass}
Define $\mathfrak G_{\partial}$ to be games $G$ satisfying
\[
  \NE(G)\cap\relint\Sigma_G=\varnothing,
  \qquad
  \operatorname{StrictNE}(G)=\varnothing.
\]
Ask whether (S1) holds whenever the \emph{target} $G$ belongs to
$\mathfrak G_{\partial}$, with the axioms still imposed on all games.
The restrictions deliberately remove two straightforward sufficient
cases: a completely mixed equilibrium in $K$, and a strict equilibrium
in $K$. They do \emph{not} assume that no other stability witness exists.
This is a focused candidate hard class, not a claim that all genuinely
hard cases can be reduced to it.

\subsection*{V4. Global pruning: an equivalent finite-stage elimination target}
One can define the \emph{largest globally admissible rule} without
assuming (S1). Write $Q_\Gamma$ for the mixed profiles equivalent to
quasi-perfect equilibria of a representation $\Gamma$, and let
\[
 \A_0(G)=\{K\in\CC(G):K\cap Q_\Gamma\ne\varnothing
       \text{ for every representation }\Gamma\}.
\]
For each exact allowed comparison $e:H\to G$, denote the induced
continuous map on compact connected equilibrium sets by $e_*$.
Define the downward compact recursion
\[
 \A_{r+1}(G)=\A_r(G)
 \cap\bigcap_{e:G\to F}e_*^{-1}(\A_r(F))
 \cap\bigcap_{e:H\to G}e_*(\A_r(H)),
 \qquad
 \A_\infty(G)=\bigcap_{r\ge0}\A_r(G).
 \tag{4}\label{eq:prune}
\]
The accompanying partial-contribution manuscript claims a proof, using
the source's established all-game stability axioms, that $\A_\infty$
is a nonempty globally axiom-compatible rule and every other such
rule is included in it. These manuscript claims have not been independently
verified. Under those claimed results, (S1) would be equivalent to
\[
 \boxed{\forall G\quad \forall K\in\CC(G)\quad
 \bigl[K\text{ contains no }0\text{-stable subset}\bigr]
 \Longrightarrow\exists r<\infty: K\notin\A_r(G).}
 \tag{V4}\label{eq:V4}
\]
This formulation does \emph{not} assert a computable depth bound.
Each stage still ranges over generally infinitely many permitted
representations and comparisons. Prove the displayed elimination
or find a survivor that violates it.

\subsection*{V5. Is a continuous equilibrium section necessary for stable singletons?}
For a fixed finite game $G$ and $x\in\NE(G)$, define
\[
  \operatorname{Sec}(G,x)
  \iff
  \exists\rho\in(0,1)\ \exists\text{ continuous }s:P_\rho(G)\to\Sigma_G
  :s(0)=x,\ s(\eta)\in\NE(G(\eta))\ \forall\eta.
\]
The accompanying contribution claims a proof of
$\operatorname{Sec}(G,x)\Rightarrow\{x\}\in\Stab(G)$;
that proof has not been independently verified.
Determine whether the converse holds in any specified class of games,
or supply a finite-game counterexample:
\[
 \{x\}\in\Stab(G)\ \stackrel{?}{\Longrightarrow}\
 \operatorname{Sec}(G,x).
 \tag{V5}
\]
This is an independent strengthening of the \emph{witness form}
for singleton stability, not equivalent to (S1). The existence of a
cohomologically essential subset need not yield a section for arbitrary
topological maps, so a direct argument from essentiality alone would
be invalid without further game structure.

\subsection*{V6. Metastability and a minimal strengthening of invariance}
Govindan--Wilson's Section 10 discusses metastability and suggests
examining stronger combined invariance principles, potentially adding
both players and dominated actions in one comparison. A \emph{precise}
version must first explicitly define the new class $\mathscr E_{\rm comp}$
of allowed composite embeddings, including its off-path preference
requirements. Then ask:
\[
 B+D^*+I^*+I_{\mathscr E_{\rm comp}}
 \quad\stackrel{?}{\Longrightarrow}\quad\text{(S1)}.
 \tag{V6}
\]
The condition that composite embeddings have the required properties
must not be replaced by arbitrary embeddings. Until
$\mathscr E_{\rm comp}$ is specified, (V6) is a \emph{research programme},
not a single theorem-or-counterexample problem. An alternative route is
to fix the original literature's metastability rule and determine the
exact extent of its strong-embedding invariance.

\subsection*{V7. Exact stable-set selection and minimality axioms}
A strictly stronger conclusion than (S1) would require every
selected set to be itself $0$-stable:
\[
 \forall G\in\mathfrak G\ \forall K\in\Phi(G):\quad
 K\in\Stab(G).
 \tag{V7a}
\]
One may alternatively seek an \emph{explicit}, strategically motivated
minimality axiom $M$ such that $B+D^*+I^*+M$ implies (V7a), or even
$\Phi(G)=\Stab(G)$ for every finite $G$. Merely stating ``choose the
minimal stable sets'' is not an axiomatic characterization: it includes
the desired answer in the assumption. Thus a well-posed version of the
programme must specify $M$ independently and verify that $\Stab$
satisfies it. The new axiom could be tested first on nongeneric games
whose Nash set has multiple components or positive-dimensional faces.

\section*{Special-case boundaries described in the accompanying manuscript}
The accompanying manuscript claims proofs of the following \emph{sufficient}
conditions from the definition, without resolving (S1). These claims
remain subject to independent review:
\begin{enumerate}[leftmargin=2.0em]
\item Every strict Nash equilibrium and every completely mixed Nash
equilibrium has a $0$-stable singleton, using an explicit continuous
constrained-equilibrium section.
\item If $x,y\in\NE(G)$, $y$ is completely mixed, and
$[x,y]\subseteq\NE(G)$, then $\{x\}\in\Stab(G)$.
\item For binary-action zero-sum polymatrix games with
$u_i(a)=a_i\sum_jA_{ij}a_j$, $A^\mathsf T=-A$, $A\mathbf 1=0$,
every equilibrium singleton is $0$-stable; equilibrium continua can be
non-isolated and contain no strict equilibria.
\end{enumerate}
The accompanying manuscript offers two-player payoff-limit examples
intended to show that stability of a singleton need not pass to a limit of
\emph{changing games}, and a Hopf-map example intended to show that a
coincidence property alone need not force rational-cohomological essentiality.
These arguments remain subject to independent review; neither is presented
as a game-theoretic counterexample to (S1).

\section*{Acceptance criteria and possible output}
\noindent
\textbf{For a full proof:} specify a $0$-stable subset of every permitted
selected set, with a witness satisfying both parts of Definition 2.4
(or invoke an applicable theorem implying them). All cross-game
quantifiers must be respected.\\
\textbf{For a full disproof:} give an explicit rule on all finite games,
verify nonemptiness, $B,D^*,I^*$ for every permitted comparison, then
give one violating $G,K$.\\
\textbf{For a partial contribution:} prove a nontrivial nongeneric
subclass, an exact reduction, a barrier or a rigorously specified
counterexample to an intermediate assertion. Keep the missing global
step visible.

\section*{Provenance and references}
This is an editorial, LLM-assisted reformulation of the posted We-Solve
question and close variants. The source paper is prior work by
Govindan and Wilson; other names cited therein retain their credit.
The permanent public problem identifier is \texttt{WS-107793}.
The problem is published at
\href{https://we-solve.github.io/problems/axiomatic-containment-of-mertens-stable-sets-without-genericity/}{published problem page}.

\paragraph{Origin of the question.}
The open-problem suggestions motivating this entry were credited in the
preceding We-Solve statement to Rida Laraki's Facebook
comment~\cite{laraki-facebook}.

\begin{thebibliography}{9}
\raggedright
\bibitem{gw} Srihari Govindan and Robert Wilson,
\emph{Axiomatic Equilibrium Selection in Generic Extensive-Form Games},
arXiv:2504.16908v2 (26 August 2026), Definition 2.4, Axioms 3.1,
4.7 and 5.4, Theorem 6.1, Section 10, Appendices A--B.
\url{https://arxiv.org/html/2504.16908v2}.
\bibitem{ws} We-Solve, permanent problem \texttt{WS-107793}.
\href{https://we-solve.github.io/api/v1/problems/WS-107793.json}{Permanent problem lookup}.
\href{https://github.com/We-Solve/research/blob/main/problems/34/prb_34ecf5fa262f4a85a0fbe4af9837c46b/statement.tex}{Public statement source}.
\bibitem{attach} User-supplied research draft,
\emph{Local witnesses for nongeneric stability and two obstructions to
extending the generic theorem}, file \texttt{WS-107793-attempt.tex},
18 October 2026. Working material concerning \texttt{WS-107793};
not an independently refereed source.
\bibitem{laraki-facebook}
Rida Laraki.
\emph{Facebook comment suggesting open problems in game theory}.
Attribution preserved from the preceding We-Solve problem statement.
\end{thebibliography}
\end{document}
>>>

## Earlier contributions (quoted titles and claims; read them before repeating work)
- Econ-20-A2 [Partial result] Two-player partial results for axiomatic containment of Mertens-stable sets without genericity (https://we-solve.github.io/contributions/axiomatic-containment-of-mertens-stable-sets-without-genericity/attempt-2/r1/)
- Econ-20-A1 [Partial result] Axiomatic containment of Mertens-stable sets without genericity: Merged partial contribution and proof audit (https://we-solve.github.io/contributions/axiomatic-containment-of-mertens-stable-sets-without-genericity/attempt-1/r1/)

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