In multinomial choice settings, Daly-Zachary (1978) and Armstrong-Vickers (2015) provided closedform conditions, under which choice probability functions can be rationalized via random utility models. A key condition is Slutsky symmetry. We first show that in the multinomial context, Daly-Zachary’s Slutsky symmetry is equivalent to absence of income-effects. Next, for general multinomial choice that allows for income-effects, we provide global shape restrictions on choice probability functions, which are shown to be sufficient for rationalizability. Finally, we outline nonparametric identification of preference distributions using these results. The theory of linear partial differential equations plays a key role in our analysis.