Bilateral Trade: A Regret Minimization Perspective

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:

• $\mathrm{\Theta }(\sqrt{T})$ for full-feedback (i.e., direct revelation mechanisms).
• $\mathrm{\Theta }(T^{2\mathrm{/}3})$ for realistic feedback (i.e., posted-price mechanisms) and independent seller/buyer valuations with bounded densities.
• $\mathrm{\Theta }(T)$ for realistic feedback and seller/buyer valuations with bounded densities.
• $\mathrm{\Theta }(T)$ for realistic feedback and independent seller/buyer valuations.
• $\mathrm{\Theta }(T)$ for the adversarial setting.