  Bilateral trade, a fundamental topic in economics, models the problem of
intermediating between two strategic agents, a seller and a buyer, willing to
trade a good for which they hold private valuations. Despite the simplicity of
this problem, a classical result by Myerson and Satterthwaite (1983) affirms
the impossibility of designing a mechanism which is simultaneously efficient,
incentive compatible, individually rational, and budget balanced. This
impossibility result fostered an intense investigation of meaningful trade-offs
between these desired properties. Much work has focused on approximately
efficient fixed-price mechanisms, i.e., Blumrosen and Dobzinski (2014; 2016),
Colini-Baldeschi et al. (2016), which have been shown to fully characterize
strong budget balanced and ex-post individually rational direct revelation
mechanisms. All these results, however, either assume some knowledge on the
priors of the seller/buyer valuations, or a black box access to some samples of
the distributions, as in D{\"u}tting et al. (2021). In this paper, we cast for
the first time the bilateral trade problem in a regret minimization framework
over rounds of seller/buyer interactions, with no prior knowledge on the
private seller/buyer valuations. Our main contribution is a complete
characterization of the regret regimes for fixed-price mechanisms with
different models of feedback and private valuations, using as benchmark the
best fixed price in hindsight. More precisely, we prove the following bounds on
the regret:
  $\bullet$ $\widetilde{\Theta}(\sqrt{T})$ for full-feedback (i.e., direct
revelation mechanisms);
  $\bullet$ $\widetilde{\Theta}(T^{2/3})$ for realistic feedback (i.e.,
posted-price mechanisms) and independent seller/buyer valuations with bounded
densities;
  $\bullet$ $\Theta(T)$ for realistic feedback and seller/buyer valuations with
bounded densities;
  $\bullet$ $\Theta(T)$ for realistic feedback and independent seller/buyer
valuations;
  $\bullet$ $\Theta(T)$ for the adversarial setting.
