We introduce a modified Weighted Cascade model that integrates asymmetric budgets and product scores, providing new insights into the Generalized Asymmetric Influence Maximization problem, which we establish as NP-hard. Our simulations demonstrate that players with higher budgets possess a distinct advantage in networks characterized by larger diameters, whereas players with superior product scores exhibit a significant advantage in networks with smaller diameters. Moreover, we identify a robust linear relationship between graph size and the magnitude of influenced nodes. In densely connected networks we derive bounds for the probabilities of influence that are independent of network size. Our examination of Nash equilibria in this domain underscores the absence of a guaranteed pure Nash equilibrium, suggesting that the strategic enhancement of budgets or product scores may yield more substantial benefits than the pursuit of an optimal strategy in this context.