New characterizations of increasing risk

I present alternative constructions of gambles with greater risk. Rothschild and Stiglitz (1970) demonstrate that gamble Y has greater risk than X when Y is equal in distribution to X+Z, where Z is noise. Gambles called positive-upper-conditional-mean errors are introduced, and I show that Y has greater risk than X when Z is a PUCME and is not noise. Simple examples demonstrate that the set of PUCMEs is strictly greater than the set of gambles that are noise.