Approximation and Irrationality in Hylland--Zeckhauser Equilibria
We study the computation of Hylland--Zeckhauser (HZ) equilibria beyond the bi-valued setting. First, we give a polynomial-time algorithm that, for general multi-valued utilities, computes an $1/e$-approximate HZ equilibrium. This yields, to our knowledge, the first polynomial-time constant-error approximation guarantee for this setting. The key technical ingredient is a utility-stratification construction that embeds a multi-valued market into a structured bi-valued instance, allowing us to apply the exact algorithm of Vazirani and Yannakakis. Second, we show that the rational structure of exact equilibria breaks down already for tri-valued utilities: there exists a $5\times5$ HZ instance with utilities in $\{0,\frac12,1\}$ such that all of whose equilibria are irrational. Taken together, these results show that while the bi-valued case can be used as a base for approximation algorithms, rational exact equilibria cannot be guaranteed even for tri-valued utilities.