Dual representation of choice and aspirational preferences

We consider choice over a set of monetary acts (random variables) and study a general class of preferences. These preferences favor diversification, except perhaps on a subset of sufficiently disliked acts, over which concentration is instead preferred. This structure encompasses a number of known models in this setting. We show that such preferences can be expressed in dual form in terms of a family of measures of risk and a target function. Specifically, the choice function is equivalent to selection of a maximum index level such that the risk of beating the target function at that level is acceptable. This dual representation may help to uncover new models of choice. One that we explore in detail is the special case of a bounded target function. This case corresponds to a type of satisficing and has descriptive relevance. Moreover, the model results in optimization problems that may be efficiently solved in large-scale.