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In robust optimization one seeks to make a decision under uncertainty, where the goal is to find the solution with the best worst-case performance. The set of possible realizations of the uncertain data is described by a so-called uncertainty set. In many scenarios, a decision maker may influence the uncertainty regime she is facing, for example, by investing in market research, or in machines which work with higher precision. Recently, this situation was addressed in the literature by introducing decision dependent uncertainty sets (endogenous uncertainty), i.e., uncertainty sets whose structure depends on (typically discrete) decision variables. In this way, one can model the trade-off between reducing the cost of robustness versus the cost of the investment necessary for influencing the uncertainty. However, there is another trade-off to be made here. With different uncertainty regimes, not only do the worst-case optimal solutions vary, but also other aspects of that solutions such as max-regret, best-case performance or predictability of the performance. A decision maker may still be interested in having a performance guarantee, but at the same time be willing to forgo superior worst-case performance if those other aspects can be enhanced by switching to a suitable uncertainty regime. We introduce the notion of uncertainty preference in order to capture such stances. We present three ways to formalize uncertainty preferences and study the resulting mathematical models. The goal is to have reformulations/approximations of these models which can be solved with standard methods. The workhorse is mixed-integer linear and conic optimization. We apply our framework to the uncertain shortest path problem and conduct numerical experiments for the resulting models. We can demonstrate that our models can be handled very well by standard mixed-integer linear solvers.