Equilibrium

A payoff is a vector $(w_l,\pi_f)_{l,f\in\ll\cup\ff}\geq0$.

An allocation is a matching-payoff pair $(x,w,\pi)$ such that:

• if $x_{lf}=1$ then $w_l+\pi_f=v_{lf}$,
• if $x_{lf}=0$ for all $f$, then $w_l=0$,
• if $x_{lf}=0$ for all $l$, then $\pi_f=0$.

An allocation $(x,w,\pi)$ is stable if no currently unmatched worker-firm pair can increase their total surplus by matching to each other:

if $x_{lf}=0$ then $w_l+\pi_f\geq v_{lf}$.