In this paper, we first introduce the concepts of Levitin-Polyak α-well-posedness by perturbations and Levitin-Polyak α-well-posedness by perturbations in the generalized sense for strong vector mixed quasivariational inequality problems of the Minty type and the Stampacchia type (for short, (MQVI) and (SQVI), respectively). Sufficient conditions for such problems to be LevitinPolyak α-well-posedness by perturbations are established. We also introduce the gap functions for (MQVI) and (SQVI) and study some their properties which are used to study the Levitin-Polyak α-well-posedness by perturbations of such problems. As applications, we investigate the oligopolistic market problem (OM) which can be formulated as a special of considered problems. Using these results, we are able to establish some sufficient conditions, which ensure the LevitinPolyak α-well-posedness by perturbations of (OM).