This paper introduces a concept for the subset of K-L-L-S increases in riskdefined by Kroll, Leshno, Levy, and Spector (1995; called K-L-L-S); a‘relatively strong increase in risk in the K-L–L-S sense’ (RSIRK). Our newnotion of K-L-L-S increases in risk extends the Rothschild-Stiglitz definitionof risk to a larger set of cumulative distribution functions, but use somewhatstronger restrictions on the structure of the decision model and the set ofdecision-makers. The decision model used in this paper consists of a utilityfunction of one scalar variable that is affected by one-dimensional choicevariable (and another random variable) to avoid problems involvingmultidimensionality. We show that, by restricting the payoff function to belinear in the random variable ( 0 = xx z ) and limiting our analysis to decisionmakerswho are prudnet ( 0 ≥ ′ u ), we are able to generate comparativestatics results for the RSIRK order.