Models of simultaneous discrete choice may be incomplete, delivering multiple values of outcomes at certain values of the latent variables and covariates, and incoherent, delivering no values. Alternative approaches to accommodating incompleteness and incoherence are considered in a unifying framework afforded by the Generalized Instrumental Variable models introduced in Chesher and Rosen (2017). Sharp identification regions for parameters and functions of interest defined by systems of conditional moment equalities and inequalities are provided. Almost all empirical analysis of simultaneous discrete choice uses models that include parametric specifications of the distribution of unobserved variables. The paper provides characterizations of identified sets and outer regions for structural functions and parameters allowing for any distribution of unobservables independent of exogenous variables. The methods are applied to the models and data of Mazzeo (2002) and Kline and Tamer (2016) in order to study the sensitivity of empirical results to restrictions on equilibrium selection and the distribution of unobservable payoff shifters, respectively. Confidence intervals for individual parameter components are provided using a recently developed inference approach from Belloni, Bugni, and Chernozhukov (2018). The relaxation of equilibrium selection and distributional restrictions in these applications is found to greatly increase the width of resulting confidence intervals, but nonetheless the models continue to sign strategic interaction parameters.