Abstract
Decision makers can be indifferent between alternatives in a binary-choice setting, but they must somehow pick a single item. This final selection is particularly important in models with limited perception because indifference between indistinguishable objects is central to the theory. We allow for general stochastic final picking rules in a setting with limited perception, and find the revealed preference axioms that are enable identification of the preferences and picking rules from binary choice data. The model generalizes several useful special cases such as a modified logit model that allows zero shares, selection from filtered sets, and randomly picking from indistinguishable alternatives.