  In online bilateral trade, a platform posts prices to incoming pairs of
buyers and sellers that have private valuations for a certain good. If the
price is lower than the buyers' valuation and higher than the sellers'
valuation, then a trade takes place. Previous work focused on the platform
perspective, with the goal of setting prices maximizing the gain from trade
(the sum of sellers' and buyers' utilities). Gain from trade is, however,
potentially unfair to traders, as they may receive highly uneven shares of the
total utility. In this work we enforce fairness by rewarding the platform with
the fair gain from trade, defined as the minimum between sellers' and buyers'
utilities. After showing that any no-regret learning algorithm designed to
maximize the sum of the utilities may fail badly with fair gain from trade, we
present our main contribution: a complete characterization of the regret
regimes for fair gain from trade when, after each interaction, the platform
only learns whether each trader accepted the current price. Specifically, we
prove the following regret bounds: $\Theta(\ln T)$ in the deterministic
setting, $\Omega(T)$ in the stochastic setting, and $\tilde{\Theta}(T^{2/3})$
in the stochastic setting when sellers' and buyers' valuations are independent
of each other. We conclude by providing tight regret bounds when, after each
interaction, the platform is allowed to observe the true traders' valuations.
