Competition between private and expressed opinions in binary choice: the $\alpha$-EPO $q$-voter model

People often express opinions that differ from their privately held views, a phenomenon known in economy as preference falsification. Expressed-private opinion (EPO) models capture this by assigning each agent two dynamical variables: a private (internal) and an expressed (external) opinion. Within the nonlinear $q$-voter model, two EPO variants have been studied so far: with and without self-anticonformity. In both formulations, agents update private and expressed binary opinions, one after another and at the same rate, which has led to two update schemes studied previously: AT (act then think), in which an agent first updates its expressed and then its private opinion, and TA (think then act), in which the order is reversed. To eliminate this ad hoc distinction and quantify the interplay between private and expressed opinions, we introduce the $\alpha$-EPO $q$-voter model with asynchronous updating -- in each elementary step, an agent updates its private opinion with probability $\alpha$ or its expressed opinion with complementary probability $1-\alpha$. We derive mean-field theory and, for the first time for EPO $q$-voter dynamics, a pair approximation, and validate them with Monte Carlo simulations on artificial and real organizational networks. Comparing the two model variants, we show that the collective outcome controlled by $\alpha$ strongly depends on self-anticonformity: with self-anticonformity the results are robust to $\alpha$, whereas without it $\alpha$ shifts the agreement-disagreement threshold and can change the type of phase transition. In the mean-field limit this change occurs only for $q=3$, but the pair approximation reveals an additional low-connectivity regime in which both $\alpha$ and the average degree $k$ control the emergence and width of hysteresis also for larger influence groups.