Generalizations of risk-weighted expected utility

Buchak’s risk-weighted expected utility considers not just the probability of an outcome, but also the probability of getting a strictly better outcome, when weighting the contribution that outcome gives to the evaluation of a gamble. It uses a risk-weighting function upper R R $R$ sending probabilities in left bracket 0 comma 1 right bracket [0,1] $\left[ {0,1} \right]$ to decision weights left bracket 0 comma 1 right bracket [0,1] $\left[ {0,1} \right]$ . I adapt this to allow weights in any real interval. Finite intervals yield nothing new, but if the interval is infinite, then the resulting rule can incorporate maximin or maximax preferences (or both!) while still satisfying stochastic dominance. There are advantages to working with marginal risk-weighting, upper R R $R$ ’s derivative, r r $r$ .