  We propose the star-shaped acceptability indexes as generalizations of both
the approaches of Cherny and Madan (2009) and Rosazza Gianin and Sgarra (2013)
in the same vein as star-shaped risk measures generalize both the classes of
coherent and convex risk measures in Castagnoli et al. (2022). We characterize
acceptability indexes through star-shaped risk measures, star-shaped acceptance
sets, and as the minimum of a family of quasi-concave acceptability indexes.
Further, we introduce concrete examples under our approach linked to Value at
Risk, risk-adjusted reward on capital, reward-based gain-loss ratio, monotone
reward-deviation ratio, and robust acceptability indexes.
