Under item response theory, the characteristic curve methods (Haebara and Stocking-Lord methods) are used to link two ability scales from separate calibrations. The linking methods use their respective criterion functions that can be defined differently according to the symmetry- and distribution-related schemes. The symmetry-related scheme relates to which scale, targeted or transformed, should be used for the definition. The distribution-related scheme refers to a way of incorporating underlying ability distributions into the definition. Through simulations, this study examined if a certain optimal combination of the two schemes exists regardless of differences in proficiency distribution between samples involved in scale linking. Concurrent calibration was considered for comparative purposes and, across all nine combinations of proficiency distributions to be linked, its performance was better in linking accuracy than the linking methods. There was no optimal combination of the symmetry- and distribution-related schemes that led to minimal linking error across the nine combinations. (Contains 1 figure and 4 tables.)