The theoretical literature on asymmetric first-price auctions has focused mainly on settings with either (1) exactly two bidders or (2) an arbitrary number of bidders with types in a common support. Even though closed form solutions are typically impossible, there is enough structure in the problem to guide both empirical work and numerical solutions or simulations. However, casual observation of real-world auctions suggests that the assumptions are restrictive. Indeed, the empirical literature has ventured beyond these models. We explain the relevant complications that arise if the above conditions do not hold, emphasizing that the structure of the problem changes significantly. Critically, the dimensionality of the problem appears to increase as bidders may now tender bids over different supports (bid bifurcation). The main conceptual contribution of this paper is to establish a method by which the dimensionality of the problem can once again be reduced. This insight is used to construct the first robust solution algorithm that allows for bid bifurcation. Accurate solution methods are essential to e.g. evaluate different counterfactual policies. We provide sufficient conditions, and in some cases necessary and sufficient conditions, for bid bifurcation to occur and demonstrate its relevance through a series of analytical and numerical examples. Implications for empirical work are emphasized.
