  In non-truthful auctions, agents' utility for a strategy depends on the
strategies of the opponents and also the prior distribution over their private
types; the set of Bayes Nash equilibria generally has an intricate dependence
on the prior. Using the First Price Auction as our main demonstrating example,
we show that $\tilde O(n / \epsilon^2)$ samples from the prior with $n$ agents
suffice for an algorithm to learn the interim utilities for all monotone
bidding strategies. As a consequence, this number of samples suffice for
learning all approximate equilibria. We give almost matching (up to polylog
factors) lower bound on the sample complexity for learning utilities. We also
consider a setting where agents must pay a search cost to discover their own
types. Drawing on a connection between this setting and the first price
auction, discovered recently by Kleinberg et al. (2016), we show that $\tilde
O(n / \epsilon^2)$ samples suffice for utilities and equilibria to be estimated
in a near welfare-optimal descending auction in this setting. En route, we
improve the sample complexity bound, recently obtained by Guo et al. (2021),
for the Pandora's Box problem, which is a classical model for sequential
consumer search.
