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It is proved that the Nash equilibrium in the vaccination game is not unique if population heterogeneity is considered. Moreover, herd immunity is not achieved if individuals are solely driven by self-interests. There is no consensus on the best definition for acute-on-chronic liver failure (ACLF). In this study, we compared the prevalence and 30-day all-cause and transplant-free mortality of patients with ACLF identified by European Association for the Study of the Liver-Chronic Liver Failure Consortium (EASL-CLIF) and North American Con