Exploring the Tractability Frontier of Hylland-Zeckhauser Models
Ioannis Panageas, Avi Shein, Shayan Taherijam, Vijay V. Vazirani
The Hylland-Zeckhauser (HZ) mechanism provides a market-based approach to one-sided matching, making efficient equilibrium computation central to its implementation. Under bi-valued utilities, it admits a strongly polynomial-time algorithm [VY], whereas with four values per buyer, the problem is PPAD-hard under inverse polynomial approximation [CCPY]. To understand the tractability boundary, we study two extensions of the bi-valued case: a third linear utility value and separable concave utilities with two common slopes. For every fixed rational $a\in(0,1)$, we prove that computing a normalized additive $\varepsilon_{*}(a)/N$-equilibrium in an $N$-buyer HZ market is PPAD-complete for some constant $\varepsilon_{*}(a)>0$, even when all utilities belong to the same prescribed set $\{0,a,1\}$. This resolves the main question left open by [VY, CCPY]: whether three utility values already suffice for PPAD-hardness. Assuming the PCP-for-PPAD conjecture, we strengthen the above-stated result to constant additive-error hardness. For the concave extension, we compute an exact equilibrium in strongly polynomial time with two marginal utility slopes common across goods for each buyer and arbitrary rational buyer-good breakpoints. With three common marginal slopes, suitable breakpoints encode the three valued linear case, yielding PPAD-hardness. Thus, our paper identifies two thresholds separating tractable and intractable cases of HZ models with utility functions specified by utility values or marginal slopes. In both cases, the tractability threshold lies between two and three.
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