An increasing convex utility function prefers a nondegenerate lottery to its sure mean by the Jensen inequality. Its relative risk aversion coefficient is negative when its second derivative is strictly positive. This differs from risk neutrality, which corresponds to affine utility.
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Risk-seeking, also known as risk-seeking behavior, refers to a preference for engaging in actions or making decisions that involve higher levels of uncertainty and potential negative outcomes, in exchange for the possibility of greater rewards. Individuals or entities that exhibit risk-seeking behavior are willing to take on more risk than is strictly necessary, often driven by the potential for significant gains. This behavior can be observed in various contexts, including finance, investment, entrepreneurship, and personal decision-making.