  In this paper, we study asymmetric information economies consisting of both
non-negligible and negligible agents and having ordered Banach spaces as their
commodity spaces. In answering a question of Herv\'{e}s-Beloso and
Moreno-Garc\'{i}a, we establish a characterization of Walrasian expectations
allocations by the veto power of the grand coalition. It is also shown that
when an economy contains only negligible agents a Vind's type theorem on the
private core with the exact feasibility can be restored. This solves a problem
of Pesce.
