A wealth-requirement axiomatization of riskiness

How are risks evaluated? Here risk is meant in the simplest sense: facing certain gains or losses, with given probabilities. The “subjective” approach considers each individual decision-maker separately, and proceeds according to that decision-maker’s preference and utility. But can risks be evaluated in an “objective” manner—depending only on the risks themselves and not on the specific decision-maker’s attitude? While at first sight this may appear to be a tall order, remember that objective measures do exist, as, for example, the return of the gamble (its expectation) and the spread of the gamble (its standard deviation). While the standard deviation is at times used also to measure riskiness, in general it is not a good measure of it. This is so, in particular, since the standard deviation is not monotonic: one may increase the gains and decrease the losses—which clearly lowers the risks—in such a way that the standard deviation actually increases. New objective measures of riskiness have recently been developed by Aumann and Serrano (2008) and Foster and Hart (2009). While the approach of Aumann and Serrano is axiomatic (it is based mainly on their “duality” axiom), the approach of Foster