  We provide a mathematical definition of fragility and antifragility as
negative or positive sensitivity to a semi-measure of dispersion and volatility
(a variant of negative or positive "vega") and examine the link to nonlinear
effects. We integrate model error (and biases) into the fragile or antifragile
context. Unlike risk, which is linked to psychological notions such as
subjective preferences (hence cannot apply to a coffee cup) we offer a measure
that is universal and concerns any object that has a probability distribution
(whether such distribution is known or, critically, unknown). We propose a
detection of fragility, robustness, and antifragility using a single
"fast-and-frugal", model-free, probability free heuristic that also picks up
exposure to model error. The heuristic lends itself to immediate
implementation, and uncovers hidden risks related to company size, forecasting
problems, and bank tail exposures (it explains the forecasting biases). While
simple to implement, it outperforms stress testing and other such methods such
as Value-at-Risk.
