Subset choice denotes a situation in which decision makers are offered available sets from a fixed master set of choice alternatives and each decision maker is asked to choose a subset of any size from the available set. In this paper, we study the relationships between various random utility models of subset choice. Random utility threshold models of subset choice assume that there is a (random) utility associated with each available option, and a (random) utility threshold, such that the decision maker selects those options in the available set whose utilities are greater than or equal to the threshold. A special case of the random utility threshold model is the latent scale model, in which the threshold has a constant value and the random variables associated with the available options are independent of each other. We show that the size-independent random utility model for approval voting of Falmagne and Regenwetter (1996) is a random utility threshold model, and develop numerous results relating that model to the class of random utility threshold models in general, and to the latent scale model in particular. Among the features distinguishing some of these models is a closure property that we call stability under substructures. The size-independent model is not stable, in the sense that certain marginals of a given size-independent model for n choice alternatives may violate all size-independent models for n - 1 choice alternatives. In contrast, the general class of random utility threshold models and also the specific subclass of latent scale models are stable under substructures.
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