In the vaccination game, the spread of disease and the human decision-making to obtain pre-emptive vaccinations are coordinately united in successive seasons. This is backed both by epidemiological models, such as SIR, and by evolutionary game theory, assuming a given strategy-updating rule. Several rules have been proposed by the community and rely on either the imitation concept or the switching-action concept. The latter directly stipulates whether or not an agent commits to a course of action based on a rule, such as the aspiration concept. In contrast, the former borrowed its fundamental idea from the spatial version of a two-player, two-strategy (2 × 2) game, such as the spatial prisoner's dilemma (SPD). The pairwise Fermi (PW-Fermi) strategy has been heavily employed as the most representative idea. The present study modifies PW-Fermi, which consists of two processes: one for selecting a pairwise opponent to imitate and the other giving the probability of copying from the opponent. Instead of a random selection, our proposed model applies a stochastically skewed selection in which a neighbor who has a similar degree to the focal player is preferentially selected. This specific rule allows us to establish a quite efficient society, in which hub agents spontaneously obtain vaccination, but lower-degree agents do not. To this end, a small number of higher-degree agents, who are exposed to higher infection risk, are urged to be vaccinated, whereas many other agents enjoy free-riding. This produces a relatively small vaccination cost as a social sum and also effectively suppresses the spread of disease, resulting in a small disease cost for society as a whole. Copyright © 2021 Elsevier B.V. All rights reserved.