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 . 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, assigns the equal-probability lottery or expected utility , above the utility of its sure mean. Where and at positive wealth, the relative risk aversion coefficient is negative. This local curvature statement should be distinguished from strict preference when a utility has affine segments.
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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.