Risk Preferences and their Robust Representation: Beyond Random Variables, a Setup Independent Approach

Due to the plurality of interpretations of risk, we concentrate on context invariant features related to this notion: diversification and monotonicity. We introduce and study the general properties of three key concepts in an abstract framework: risk order, risk measure and risk acceptance family. Our main result is a uniquely characterized robust representation of lower semi-continuous risk orders.
We will also give some new results when monotonicity actually ensures the lower semicontinuity of risk orders.
This general approach to risk perception leaves room for different interpretations of risk perception within one concept depending on the setting and one's perspective.
We illustrate this with several settings.
In the setup of random variables, where risk perception can be interpreted as a model risk, we give a robust representation for numerous risk measures: various certainty equivalents, or a general version of Aumann and Serrano's economic index of riskiness among others.
In the setup of lotteries where risk perception can be seen as a distributional risk, we show that the Value at Risk is a risk measure on this level (not for random variables) and provide a robust representation for it. In the setup of consumption patterns where risk perception is related to a discounting risk, we provide a robust representation of intertemporal risk measures \`a la Hindy, Huang and Kreps or Epstein and Zin.
We finally discuss in the setting of state dependent lotteries à la Anscombe and Aumann, the interplay between model risk and distributional risk.