= Risk seeking
For increasing convex <expected utility> preferences, a decision-maker weakly prefers a lottery to its sure mean, provided the outcome and utility expectations are defined: the <Jensen inequality> gives $U(\mathbb EW)\leq\mathbb E U(W)$. Strict preference for every nondegenerate lottery requires strict <convexity> over its relevant outcome range. An affine utility is convex too, but gives equality and <risk neutrality>.
For example, on nonnegative wealth, $U(w)=w^2$ assigns the equal-probability lottery $0$ or $2$ expected utility $2$, above the utility $1$ of its sure mean. Where $U'>0$ and $U''>0$ at positive wealth, the <relative risk aversion coefficient> $-WU''/U'$ is negative. This local curvature statement should be distinguished from strict preference when a utility has affine segments.
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