We study the problem of computing a competitive equilibrium with approximately optimal bundles in Fisher markets with separable piecewise-linear concave (SPLC) utility functions, meaning that every buyer receives a $(1-δ)$-optimal bundle, instead of a perfectly optimal one. We establish the first intractability result for the problem by showing that it is PPAD-hard for some constant $δ> 0$, assuming the PCP-for-PPAD conjecture. This hardness result holds even if all buyers have identical budgets (competitive equilibrium with equal incomes), linear capped utilities, and even if we also allow $\varepsilon$-approximate clearing instead of perfect clearing, for any constant $\varepsilon < 1/9$. Importantly, we show that the PCP-for-PPAD conjecture is in fact required to show hardness for constant $δ$: showing PPAD-hardness for finding such approximate market equilibria in a broad class of markets encompassing those generated by our hardness result would prove the conjecture. This is the first natural problem where the conjecture is provably required to establish hardness for it.

翻译：我们研究在具有可分分段线性凹（SPLC）效用函数的Fisher市场中计算带有近似最优束的竞争均衡问题，即每个买家获得的束是$(1-δ)$-最优的，而非完全最优。我们首次建立了该问题的难解性结果：假设PCP-for-PPAD猜想成立，对于某个常数$δ>0$，该问题是PPAD难解的。即便所有买家预算相同（等收入竞争均衡）、采用线性封顶效用函数，并且允许$\varepsilon$-近似结算（而非完美结算）且任意常数$\varepsilon < 1/9$，该难解性结果仍然成立。重要的是，我们证明，要显示常数$δ$下的难解性，PCP-for-PPAD猜想实际上是必需的：在包含我们难解性结果所生成市场的一类广泛市场中，证明寻找此类近似市场均衡的PPAD难解性将等价于证明该猜想。这是首个被严格证明需要该猜想才能建立其难解性的自然问题。