Exact price of anarchy of simultaneous first-price auctions with subadditive valuations

Determine the exact price of anarchy of simultaneous first-price auctions with budget-constrained bidders and subadditive valuations.

Background

The paper proves an upper bound of 3 for simultaneous first-price auctions with budget-constrained bidders having subadditive valuations. The paper does not establish a matching lower bound, so the exact price of anarchy for this valuation class remains unresolved. The authors identify resolving this gap, together with the generalized second-price auction question, as an open direction requiring potentially new techniques.