We provide a Bayes–Nash equilibrium analysis of the simultaneous ascending auction (SAA) when local bidders interested in a single item compete against global bidders interested in aggregating many items. We first assume that each local bidder values only a specific item, e.g. the license for the region where it has monopoly power, and that global bidders' valuation functions are convex. For this environment we show that a global bidder faces an exposure problem with adverse consequences for revenue and efficiency. In the limit when the number of items grows large, the SAA is revenue and efficiency equivalent to the Vickrey–Clarke–Groves (VCG) mechanism. We extend our analysis to include the case where items are substitutes for local bidders, so that price arbitrage will occur as observed in many spectrum auctions. This environment, which combines substitutes and complements, results in an aggravated exposure problem. Consequently, the SAA is no longer efficient and may yield dramatically lower revenues than the VCG mechanism. Finally, we relax the assumption that global bidders' valuation functions are convex by considering an environment with medium-sized global bidders who demand fewer than all items. We show that global bidders divide the market at low prices when market sharing is feasible while they engage in mutually destructive bidding when there is a fitting problem. In both cases, the SAA again underperforms relative to the VCG mechanism.