We reconsider complementary pivoting algorithms on the base of a modified form of Scarf’s primitive sets and Tuy’s abstract pivotal method. Here, the pivots are to be controled by a new type of labeling. The analysis of the limiting stage provides a new variety of fixed point results, where the usual convexity assumptions are relaxed drastically. The results mentioned also connect separation and fixed point theory in a natural way and are even applicable outside the domain of linear spaces. Moreover, they focus on the differences between integer and vector labeling in a very accurate manner.