Let $\Omega$ be a finite set of players. For a selector $\alpha:(2^{\Omega}-\{\emptyset\})\rightarrow\Omega$, we define a marginal value operator as a linear operator $m^{\alpha}$ $$m^{\alpha}:\mathbb{R}^{2^{\Omega}-\{\emptyset\}}\rightarrow \mathbb{R}^{\Omega}:\;\left(m^{\alpha}(u_{S})\right)_{\alpha(S)}=1,\;\left(m^{\alpha}(u_{S})\right)_{i\neq\alpha(S)}=0$$ ($u_{S}$ is a unanimity basis, i.e. a cooperative game $u_{S}(T)=1$ iff $S\subseteq T $, $u_{S}(T)=0$ otherwise). A special case is a selector defined by permutation $\pi$, $\pi(S)=max_{\pi}(S)$
It is known that $$ Conv\{m_{\pi}:\:\pi\:\textrm{is permutation}\}\subsetneqq Conv\{m_{\alpha}:\:\alpha\:\textrm{is selector}\} $$ except a small $\Omega$. But it is true that $$\dim(Conv\{m_{\pi}:\:\pi\:\textrm{is permutation}\})=\dim(Conv\{m_{\alpha}:\:\alpha\:\textrm{is selector}\}) $$ ?