  Bilateral trade is a fundamental economic scenario comprising a strategically
acting buyer and seller, each holding valuations for the item, drawn from
publicly known distributions. A mechanism is supposed to facilitate trade
between these agents, if such trade is beneficial. It was recently shown that
the only mechanisms that are simultaneously DSIC, SBB, and ex-post IR, are
fixed price mechanisms, i.e., mechanisms that are parametrised by a price p,
and trade occurs if and only if the valuation of the buyer is at least p and
the valuation of the seller is at most p. The gain from trade is the increase
in welfare that results from applying a mechanism; here we study the gain from
trade achievable by fixed price mechanisms. We explore this question for both
the bilateral trade setting, and a double auction setting where there are
multiple buyers and sellers. We first identify a fixed price mechanism that
achieves a gain from trade of at least 2/r times the optimum, where r is the
probability that the seller's valuation does not exceed the buyer's valuation.
This extends a previous result by McAfee. Subsequently, we improve this
approximation factor in an asymptotic sense, by showing that a more
sophisticated rule for setting the fixed price results in an expected gain from
trade within a factor O(log(1/r)) of the optimal gain from trade. This is
asymptotically the best approximation factor possible. Lastly, we extend our
study of fixed price mechanisms to the double auction setting defined by a set
of multiple i.i.d. unit demand buyers, and i.i.d. unit supply sellers. We
present a fixed price mechanism that achieves a gain from trade that achieves
for all epsilon > 0 a gain from trade of at least (1-epsilon) times the
expected optimal gain from trade with probability 1 - 2/e^{#T epsilon^2 /2},
where #T is the expected number of trades resulting from the double auction.
