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The assignment game, introduced by Shapley and Shubik [1971], is a classic model for two-sided matching markets between buyers and sellers. In the original assignment game, it is assumed that payments lead to transferable utility and that buyers have unit-demand valuations for the items being sold. There has since been substantial work studying various extensions of the assignment game. The first main area of extension is to imperfectly transferable utility, which is when frictions, taxes, or fees impede the transfer of money between agents. The second is with more complex valuation functions, in particular gross substitutes valuations, which describe substitutable goods. Multiple efficient algorithms have been proposed for computing a competitive equilibrium, the standard solution concept in assignment games, in each of these two settings. However, these lines of work have been mostly independent, with no algorithmic results combining the two. Our main result is an efficient algorithm for computing competitive equilibria in a setting encompassing both those generalizations. We assume that sellers have multiple copies of each items. A buyer $i$'s quasi-linear utility is given by their gross substitute valuation for the bundle $S$ of items they are assigned to, minus the sum of the payments $q_{ij}(p_j)$ for each item $j \in S$, where $p_j$ is the price of item $j$ and $q_{ij}$ is piecewise linear, strictly increasing. Our algorithm combines procedures for matroid intersection problems with augmenting forest techniques from matching theory. We also show that in a mild generalization of our model without quasilinear utilities, computing a competitive equilibrium is NP-hard. 

Gale and Shapley’s stable assignment problem has been extensively studied, applied, and extended. In the context of school choice, mechanisms often aim at finding an assignment that is more favorable to students. We investigate two extensions introduced in this framework—legal assignments and the efficiency adjusted deferred acceptance mechanism (EADAM) algorithm—through the lens of the classic theory of stable matchings. In any instance, the set [Formula: see text] of legal assignments is known to contain all stable assignments. We prove that [Formula: see text] is exactly the set of stable assignments in another instance. Moreover, we show that essentially all optimization problems over [Formula: see text] can be solved within the same time bound needed for solving it over the set of stable assignments. A key tool for this latter result is an algorithm that finds the student-optimal legal assignment. We then generalize our algorithm to obtain the assignment output of EADAM with any given set of consenting students without sacrificing the running time, hence largely improving in both theory and practice over known algorithms. Finally, we show that the set [Formula: see text] can be much larger than the set of stable matchings, connecting legal matchings with certain concepts and open problems in the literature.