PPAD-hardness for constant-δ approximate competitive equilibria in SPLC Fisher markets requires the PCP-for-PPAD conjecture, which the paper proves is necessary to establish such hardness.
Thomas Chen, Xi Chen, Binghui Peng, and Mihalis Yannakakis
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Fisher Markets with Approximately Optimal Bundles and the Need for a PCP Theorem for PPAD
PPAD-hardness for constant-δ approximate competitive equilibria in SPLC Fisher markets requires the PCP-for-PPAD conjecture, which the paper proves is necessary to establish such hardness.