Preserving the Rothschild-Stiglitz type of increasing risk with background risk

Background risk refers to a risk that is exogenous and is not subject to transformations by a decision-maker. In this paper, we extend the definition of the Rothschild-Stiglitz type of increasing risk to a background risk framework. We theoretically investigate a more general definition of increase in risk in the presence of background risk. The results suggest that an extended concept of expectation dependence plays a vital role.