  We study single-item single-unit Bayesian posted price auctions, where buyers
arrive sequentially and their valuations for the item being sold depend on a
random, unknown state of nature. The seller has complete knowledge of the
actual state and can send signals to the buyers so as to disclose information
about it. For instance, the state of nature may reflect the condition and/or
some particular features of the item, which are known to the seller only. The
problem faced by the seller is about how to partially disclose information
about the state so as to maximize revenue. Unlike classical signaling problems,
in this setting, the seller must also correlate the signals being sent to the
buyers with some price proposals for them. This introduces additional
challenges compared to standard settings. We consider two cases: the one where
the seller can only send signals publicly visible to all buyers, and the case
in which the seller can privately send a different signal to each buyer. As a
first step, we prove that, in both settings, the problem of maximizing the
seller's revenue does not admit an FPTAS unless P=NP, even for basic instances
with a single buyer. As a result, in the rest of the paper, we focus on
designing PTASs. In order to do so, we first introduce a unifying framework
encompassing both public and private signaling, whose core result is a
decomposition lemma that allows focusing on a finite set of possible buyers'
posteriors. This forms the basis on which our PTASs are developed. In
particular, in the public signaling setting, our PTAS employs some ad hoc
techniques based on linear programming, while our PTAS for the private setting
relies on the ellipsoid method to solve an exponentially-sized LP in polynomial
time. In the latter case, we need a custom approximate separation oracle, which
we implement with a dynamic programming approach.
