  We study the problem of social welfare maximization in bilateral trade, where
two agents, a buyer and a seller, trade an indivisible item. We consider
arguably the simplest form of mechanisms -- the fixed-price mechanisms, where
the designer offers trade at a fixed price to the seller and buyer. Besides the
simple form, fixed-price mechanisms are also the only DSIC and budget balanced
mechanisms in bilateral trade.
  We obtain improved approximation ratios of fixed-price mechanisms in
different settings. In the full prior information setting where the designer
has access to the value distributions of both the seller and buyer, we show
that the optimal fixed-price mechanism can achieve at least $0.72$ of the
optimal welfare, and no fixed-price mechanism can achieve more than $0.7381$ of
the optimal welfare. Prior to our result the state of the art approximation
ratio was $1 - 1/e + 0.0001 \approx 0.632$. Interestingly, we further show that
the optimal approximation ratio achievable with full prior information is
identical to the optimal approximation ratio obtainable with only one-sided
prior information. We further consider two limited information settings. In the
first one, the designer is only given the mean of the buyer's (or the seller's)
value. We show that with such minimal information, one can already design a
fixed-price mechanism that achieves $2/3$ of the optimal social welfare, which
surpasses the previous state of the art ratio even when the designer has access
to the full prior information. Furthermore, $2/3$ is the optimal attainable
ratio in this setting. In the second one, we assume that the designer has
sample access to the value distributions. We propose a new family mechanisms
called order statistic mechanisms and provide a complete characterization of
their approximation ratios for any fixed number of samples.
