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Bilateral trade is a fundamental economic scenario comprising a strategicallyacting buyer and seller, each holding valuations for the item, drawn frompublicly known distributions. A mechanism is supposed to facilitate tradebetween these agents, if such trade is beneficial. It was recently shown thatthe only mechanisms that are simultaneously DSIC, SBB, and ex-post IR, arefixed 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 andthe valuation of the seller is at most p. The gain from trade is the increasein welfare that results from applying a mechanism; here we study the gain fromtrade achievable by fixed price mechanisms. We explore this question for boththe bilateral trade setting, and a double auction setting where there aremultiple buyers and sellers. We first identify a fixed price mechanism thatachieves a gain from trade of at least 2/r times the optimum, where r is theprobability that the seller's valuation does not exceed the buyer's valuation.This extends a previous result by McAfee. Subsequently, we improve thisapproximation factor in an asymptotic sense, by showing that a moresophisticated rule for setting the fixed price results in an expected gain fromtrade within a factor O(log(1/r)) of the optimal gain from trade. This isasymptotically the best approximation factor possible. Lastly, we extend ourstudy of fixed price mechanisms to the double auction setting defined by a setof multiple i.i.d. unit demand buyers, and i.i.d. unit supply sellers. Wepresent a fixed price mechanism that achieves a gain from trade that achievesfor all epsilon > 0 a gain from trade of at least (1-epsilon) times theexpected 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.

Fixed Price Approximability of the Optimal Gain From Trade