Maxmin under risk

Let $\succsim $ be a continuous and convex weak order on the set of lotteries defined over a set Z of outcomes. Necessary and sufficient conditions are given to guarantee the existence of a set $\mathcal{U}$ of utility functions defined on Z such that, for any lotteries p and q, <p>\[ p\succsim q \Leftrightarrow \min_{u\in{\mathcal U}}{\Bbb E} _p\left[ u\right] \geq \min_{u\in{\mathcal U}}{\Bbb E} _q\left[ u\right] . \] <p>The interpretation is simple: a conservative decision maker has an unclear evaluation of the different outcomes when facing lotteries. She then acts as if she were considering many expected utility evaluations and taking the worst one.