Implementation in advised strategies: Welfare guarantees from posted-price mechanisms when demand queries are NP-hard

State-of-the-art posted-price mechanisms for submodular bidders with m items achieve approximation guarantees of O((log log m)³) [1]. Their truthfulness, however, requires bidders to compute an NP-hard demand-query. Some computational complexity of this form is unavoidable, as it is NP-hard for truthful mechanisms to guarantee even an m¹/2−^ε-approximation for any ε > 0 [21]. Together, these establish a stark distinction between computationally-efficient and communication-efficient truthful mechanisms. We show that this distinction disappears with a mild relaxation of truthfulness, which we term implementation in advised strategies. Specifically, advice maps a tentative strategy either to that same strategy itself, or one that dominates it. We say that a player follows advice as long as they never play actions which are dominated by advice. A poly-time mechanism guarantees an α-approximation in implementation in advised strategies if there exists advice (which runs in poly-time) for each player such that an α-approximation is achieved whenever all players follow advice. Using an appropriate bicriterion notion of approximate demand queries (which can be computed in poly-time), we establish that (a slight modification of) the [1] mechanism achieves the same O((log log m)³)-approximation in implementation in advised strategies.