Hardness of Approximate Hylland-Zeckhauser Equilibria

2026-05-30

AI Summary

This study addresses the computational complexity of computing approximate Hylland–Zeckhauser (HZ) equilibria with arbitrarily small errors in Fisher markets with unit-demand buyers and equal incomes. The authors establish, without any additional assumptions, that there exists a constant ε > 0 such that finding a restricted ε-HZ equilibrium is PPAD-hard. This hardness result is achieved via a reduction from the generalized circuit problem to approximate HZ equilibrium computation, thereby forging a connection with the “PCP for PPAD” conjecture. The work thus demonstrates the inherent intractability of approximating HZ equilibria under standard complexity assumptions and provides a foundational theoretical limit on the computational feasibility of market mechanism design in this setting.

Abstract

In this paper, we investigate the computational hardness of finding fractional allocations to unit-demand players using competitive equilibria from equal incomes (CEEI), where we allow a small constant error in players' response to market prices (also known as an approximate Hylland-Zeckhauser equilibrium). We show that assuming the $\mathbf{(\varepsilon,δ)}$-Generalized Circuits problem is PPAD-hard (the "PCP for PPAD" conjecture), finding an approximate HZ equilibrium is also PPAD-hard. This result provides additional motivation for trying to prove the PCP for PPAD conjecture as a tool for obtaining robust computational hardness results about markets. Further, we introduce a natural restriction on approximate HZ equilibria, where players' bundles may still only be approximately optimal given the prices, but may not contain positive-price items for which the player has zero utility. We show unconditionally that there exists a constant $ε$ such that finding a restricted $ε$-HZ equilibrium is PPAD-hard.

Problem

• Hylland-Zeckhauser equilibrium
• computational hardness
• CEEI
• PPAD-hard
• approximate equilibrium

Innovation

• PPAD-hardness
• Hylland-Zeckhauser equilibrium
• approximate equilibrium
• Generalized Circuits
• CEEI