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The origin of the various manifestations of the mixed modifier effect in silicate glasses remains poorly understood. Here, based on molecular dynamics simulations, we investigate the origin of a negative deviation from linearity in the hardness of a series of mixed alkaline earth aluminosilicate glasses. The minimum of hardness is shown to arise from a maximum propensity for shear flow deformations in mixed compositions. We demonstrate that this anomalous behavior originates from the existence of local structural instabilities in mixed compositions arising from a mismatch between the modifiers and the rest of the silicate network. Overall, we suggest that the mixed modifier effect manifests itself as a competition between the thermodynamic driving force for structural relaxation and the kinetics thereof.