A player’s menu—the set of all allocations she might receive given other players’ reports—is a foundational concept in mechanism design. Menus capture the structure of strategyproofness: a mechanism is strategyproof if and only if each player always receives her favorite allocation from her menu. This leads to a number of natural questions: How can canonical mechanisms be described in terms of a player picking from her menu? How does one player picking from their menu affect other players’ allocations and menus? In this paper, we investigate such questions for the canonical assignment problem with unit demand and transferable utility: the Unit Demand VCG auction (UD). We study ascending and descending unit-demand auctions, a natural algorithmic model for calculating and describing auctions. We establish (i) a descending-price auction can calculate the outcome of UD while computing any particular bidder’s menu along the way, while (ii) no ascending-price auction can do the same. We then study complexity measures concerning menus in UD, which capture how one bidder’s pick from her menu influences the full allocation and another bidder’s menu. We show that (iii) the outcome-effect complexity is \(\widetilde{\varOmega }(n^2)\) , meaning that a bidder can affect the full allocation in a way that is intractable-to-characterize. Finally, we show (iv) the options-effect complexity is \(\widetilde{O}(n)\) , giving an efficient representation of how one bidder can affect another’s menu. Our results for (i) and (iv) stem from a new characterization of how removing one vertex effects the max-weight matching in a bipartite graph. Perhaps interestingly, the complexity in (ii) and (iii) arises not from one bidder’s impact on the allocation of items, but their impact on the prices other bidders must pay.