The Lattice Property

Let $P$ denote the set of stable payoffs. Define $(w',\pi')\succceq(w'',\pi'')$ if for all $l$ $w_l'\geq w_l''$ and for all $f$ $\pi_f'\leq\pi_f''$, each in the usual vector order.

Theorem. $(P,\succceq)$ is a complete lattice.

Consequence: There is a unique least wage payoff $(w',\pi')$ and a unique greatest wage payoff $(w'',\pi'')$ among the set of all equilibria. They are, respectively,the best for all firms and the worst for all workers, and the worst for all firms and the best for all workers.