Critical-value payment ($p_i(v)=v_i{x}_i(v)-{\int }_{{\underline{v}}_i}^{v_i}{x}_i(t,v_{-i})dt$)

A monotone binary allocation is made truthful by charging its winning threshold, truncated below at the lowest allowed value, and charging zero to losers. The displayed integral formula sets the lowest type's utility to zero. To check it, hold the other reports fixed: a type above the threshold benefits from winning at that price, while a type below it benefits from losing. At a threshold, either deterministic tie choice is compatible with indifference.