I was reading some online notes on the 4 conditions that cannot all hold given the Myerson-Satterthwaite theorem and this definition of efficiency confused me:
An efficient mechanism selects the choice that maximizes the sum of agents’ utilities, disregarding monetary payments. Also known as social welfare maximization.
The part about disregarding payments confused me because shouldn't this be taken into account in order to determine the outcome that maximizes social welfare?
Mathematically, a mechanism is efficient if in equilibrium, the outcome $x$ satisfies: $$\sum_i v_i(x) \geq \sum_i v_i(x')$$ For all players $i$ and all other possible outcomes $x'$.
This definition of efficiency assumes that utility functions are quasi-linear in money. Hence, payments cancel and may be ignored.