Inverse-S: How define formally?

1. 1 w 0 1 p 0 insensitivity-region 1 Inverse-S: How define formally? Specify inflection point b, then require concavity to the left, convexity to the right? We prefer a different formalization. • Specify insensitivity region in the middle. 2. Specify through inequalities that marginal w is smaller there than in extreme regions. 3. Avoid comparisons between two extreme regions (by restricting domains of inequalities).

2. w (pw) (pr) (pb) brb wrb p worst-rank region best-rank region insensitivity-region best-rank overweighting worst-rank overweighting 2 Inverse-S, or likelihood insensitivity, holds on region [brb,wrb], if (i) and (ii) below hold. In insensitivity region, marginal w smaller than at extremes. p r r+p When introducing r and r+p, introduce them immediately as ranked probability; although r is the first symbol showing up, in words mention p as first. 1–p (pb) (pr) on* [0,wrb] (r+p  wrb) (ii) (pw) (pr) on* [brb,1] (r  brb). *: restricting domains to avoid comparisons between two extreme regions.