This paper considers sequencing situations with non-linear cost functions under optimal order consistency. Specifically, we study sequencing situations with discounting cost functions and logarithmic cost functions of the completion time. In both settings, we show that the neighbor switching gains are non-negative and non-decreasing for every misplaced pair of players. We derive new conditions on the time-dependent neighbor switching gains in a sequencing situation under optimal order consistency to guarantee convexity of the associated sequencing game. Furthermore, we define two types of gain splitting rules for the class of sequencing situations under optimal order consistency. Each one of them is based on a procedure that specifies a path from the initial order to an optimal order, dividing the neighbor switching gains in every step among the two involved players. We prove that these allocations are stable under the same conditions that are required for convexity. These requirements are fulfilled for discounting and logarithmic sequencing situations, as well as in other settings, such as in sequencing situations with exponential cost functions.