In many practical situations, we do not know the exact value of the quantities characterizing the consequences of different possible actions. Instead, we often only known lower and upper bounds on these values, i.e., we only know intervals containing these values. To make decisions under such interval uncertainty, the Nobelist Leo Hurwicz proposed his optimism-pessimism criterion. It is known, however, that this criterion is not perfect: there are examples of actions which this criterion considers to be equivalent but which for which common sense indicates that one of them is preferable. These examples mean that Hurwicz criterion must be extended, to enable us to select between alternatives that this criterion classifies as equivalent. In this paper, we provide a full description of all such extensions.