  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. In this paper, we cast the
bilateral trade problem in a regret minimization framework over $T$ rounds of
seller/buyer interactions, with no prior knowledge on their private valuations.
Our main contribution is a complete characterization of the regret regimes for
fixed-price mechanisms with different feedback models and private valuations,
using as a benchmark the best fixed-price in hindsight. More precisely, we
prove the following tight bounds on the regret:
  - $\Theta(\sqrt{T})$ for full-feedback (i.e., direct revelation mechanisms).
  - $\Theta(T^{2/3})$ for realistic feedback (i.e., posted-price mechanisms)
and independent seller/buyer valuations with bounded densities.
  - $\Theta(T)$ for realistic feedback and seller/buyer valuations with bounded
densities.
  - $\Theta(T)$ for realistic feedback and independent seller/buyer valuations.
  - $\Theta(T)$ for the adversarial setting.
