We investigate how the existence of optimal portfolios can be established for investors with behavioural preferences. As the latter involve probability distortion functions and fail to be convex, the usual compactness substitutes based on convex combinations cannot be used. One has to work with weak convergence instead. However, this leads to the appearance of randomized strategies which, indeed, enhance performance. We survey available results in the context of markets with proportional transaction costs and discuss open problems. Based on joint work with Ngoc Huy Chau (Manchester University).