  In this study, we investigate auctions in two-sided markets with budget
constraints on buyers. Our goal is to propose an efficient mechanism that
satisfies dominant strategy incentive compatibility (DSIC), individual
rationality (IR), and weak budget balance (WBB). To avoid several known
impossibility theorems for each of two-sided markets and budget constraints, we
assume prior information on sellers' valuations and investigate the efficiency
of mechanisms by means of liquid welfare (LW), an efficiency measure used for
budget-constrained auctions, in addition to social welfare (SW).
  Our first result is to improve the efficiency guarantees of the Polyhedral
Clinching Auction by Hirai and Sato (2022), proposed for two-sided markets in
which each seller is assumed to be truthful. We show that even under budget
constraints and polymatroid constraints, their mechanism achieves an LW of more
than 1/2 of the optimal LW, and an SW of more than the optimal LW. This
significantly strengthens the efficiency guarantees of the existing studies on
clinching auctions.
  Our second result is the extension of the Polyhedral Clinching Auction to a
single-sample mechanism. Under the assumption that a single sample is provided
from each seller's distribution on her valuation, we utilize the first result
and propose an efficient mechanism that satisfies DSIC, IR, and WBB. Our
mechanism achieves an LW of more than 1/4 of the optimal LW, and an SW of more
than 1/2 of the optimal LW, in expectation. This result can be viewed as a
budget extension of D"{u}tting et al. (2021).
