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.

翻译：本文研究了在允许玩家对市场价格存在小常数误差（即近似Hylland-Zeckhauser均衡）的条件下，利用等收入竞争均衡（CEEI）为单位需求玩家寻找分数分配方案的计算难度。我们证明，假设$(\varepsilon,\delta)$-广义电路问题是PPAD难的（即“关于PPAD的PCP”猜想），则寻找近似HZ均衡同样是PPAD难的。这一结果进一步为证明“关于PPAD的PCP”猜想提供了动机，该猜想是获取市场稳健计算难度结果的重要工具。此外，我们引入了一个关于近似HZ均衡的自然限制条件：玩家的分配组合在给定价格下仍可能仅近似最优，但不得包含价格为正且玩家效用为零的物品。我们无条件证明存在常数$\varepsilon$，使得寻找受限$\varepsilon$-HZ均衡是PPAD难的。