Risk-loving

In economics and behavioral finance, a risk lover is a person who has a preference for risk. While most investors are considered risk averse, one could view casino goers as risk loving. If offered either €50 or a 50% chance of each of €100 and nothing, a risk seeking person would prefer the gamble even though the gamble and the sure thing both have the same expected value; in fact, the risk lover would be indifferent to accepting a less than 50% chance of €100 versus the sure €50 (how much less would depend on how risk loving the person is). The risk lover would also be indifferent to a 50% chance of each of €X and nothing versus the sure €50, where €X is some amount less than €100 (again, how much less would depend on how risk loving the person is).

The risk-loving behavior can be observed in the negative domain $$x<0$$ for Prospect theory value functions, where the functions are convex for $$x<0$$; while the functions are concave for $$x>0$$.

The risk loving utility function
Choice under uncertainty is often characterized as the maximization of expected utility. Utility is often assumed to be a function of profit or final portfolio wealth, with a positive first derivative. The utility function whose expected value is maximized is convex for a risk lover, concave for a risk averse agent, and linear for a risk neutral agent. Its convexity in the risk loving case has the effect of causing a mean-preserving spread of any probability distribution of wealth outcomes to be preferred over the unspread distribution.