Identities for maximum, minimum, and maxmin random utility models

- We derive an identity relating worst choice probabilities to best choice probabilities.
- A similar identity is relating expected minimum utilities to expected maximum utilities.
- Expected minimum utility and best choice probabilities are related by a Roy's type identity.
- We build a new discrete choice model: the maxmin logit model.

We generalize Roy's identity for discrete choice models, focusing on the worst choices. To do so, we derive a relation between the expected minimum utility and the worst choice probabilities for additive random utility models. We extend this relationship to maxmin random utility models, applying this framework to model ambiguity in a discrete choice setting.