Riskiness for sets of gambles

Aumann–Serrano (J Polit Econ 116:810–836, 2008) and Foster–Hart (J Polit Econ 117:785–814, 2009) suggest two new riskiness measures, each of which enables one to elicit a complete and objective ranking of gambles according to their riskiness. These riskiness measures were created with a risky world in mind, but not an uncertain one. We apply similar arguments to models of decision under uncertainty and develop complete and objective rankings of sets of gambles, which arise naturally in many such models. Clearly, these results extend the previous riskiness measures, and they have a natural interpretation in terms of those measures even when uncertainty does play a significant role.