An Algorithm for the Assignment Game Beyond Additive Valuations

TL;DR

The paper addresses computing competitive equilibria in generalized assignment games where valuations are gross substitutes and utility transfer is imperfectly transferable, formalized as the $QITU$ model. It introduces a novel algorithmic framework that combines augmenting-path techniques with matroid-intersection methods, using a dynamic marginal-demand graph and maximal alternating trees to maintain partial stability while increasing allocations and prices. A central contribution is the MAT-preserving price increase procedure, which reduces to a minimum-weight common-basis problem and leverages LP duality to extract price updates, ensuring progress toward equilibrium in polynomial time. The work also proves NP-hardness for a mild non-quasilinear extension, delineating the boundary of tractability and unifying prior results on TU/ITU and GS settings, with practical implications for markets exhibiting substitution and nonlinear transfer frictions.

Abstract

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. Two important and mostly independent lines of work have studied more general settings with imperfectly transferable utility and gross substitutes valuations. Multiple efficient algorithms have been proposed for computing a competitive equilibrium, the standard solution concept in assignment games, in these two settings. Our main result is an efficient algorithm for computing competitive equilibria in a setting with both imperfectly transferable utility and gross substitutes valuations. Our algorithm combines augmenting path techniques from maximum matching and algorithms for matroid intersection. We also show that, in a mild generalization of our model, computing a competitive equilibrium is NP-hard.

Figures (3)

• Figure 1: An example of a marginal demand graph and MAT. Note that $\text{ubuy}(1) = \{1,2\}$, $\text{ubuy}(2) = \{3,4\}$, and $\text{ubuy}(3) = \{5,6\}$. Solid lines represent the current matching, and dashed lines represent unmatched edges in each (unit-)buyer's demanded set.
• Figure 2: An example of a MAT to be preserved in a MAT-preserving price increase. Solid lines represent the current matching, and dashed lines represent the other edges in the marginal demand graph given the current matching.
• Figure 3: An example of the structure of the marginal demand graph as shown in Lemma \ref{['lem:alt_cycles']}, where $k_1,k_2,k_3\in \text{ubuy}(i)$. Consider the portion of a marginal demand graph shown in Figure \ref{['fig:MDG_structure']}(a). The matching $\nu$ as shown by the solid lines is partially stable. Then, by Lemma \ref{['lem:alt_cycles']}, either the matching shown in Figure \ref{['fig:MDG_structure']}(b) also provides $i$ with optimal utility $u_i(\cdot, p)$ among bundles of size $|\nu(\text{ubuy}(i))|$, or we can find additional unmatched edges in the marginal demand graph. If part (b) of Lemma \ref{['lem:alt_cycles']} holds for the sequence $x_1 = 1, x_2=2$, then the marginal demand graph would also contain the edges in Figure \ref{['fig:MDG_structure']}(c).

Theorems & Definitions (63)

• Theorem 1
• Theorem 2
• Definition 3: Gross Substitutes kelso1982job
• Definition 4: Competitive equilibrium
• Remark 5
• Lemma 6: Single improvement property (SI) gul2000english
• Definition 7: Stability
• Lemma 8: Competitive equilibrium coincides with feasible and stable
• proof
• Definition 9: Matroid