Fixed Price Approximability of the Optimal Gain from Trade

Abstract

Bilateral trade is a fundamental economic scenario comprising a strategically acting buyer and seller (holding an item), 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 dominant strategy incentive compatible, strongly budget balanced, and ex-post individually rational, 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 ǫ > 0 a gain from trade of at least (1 − ǫ) times the expected optimal gain from trade with probability 1 − 2/e #T ǫ 2 /2 , where #T is the expected number of trades resulting from the double auction. This can be interpreted as a "large market" result: Full efficiency is achieved in the limit, as the market gets thicker.

On Multi-Dimensional Gains from Trade Maximization

“…Our result is a generalization of the result by Colini-Baldeschi et al [15], where they provide the same approximation using a fixed posted price mechanism for bilateral trade. Importantly, our approximation ratio has the optimal dependence on r up to a constant factor.…”

“…In bilateral trade, simply setting the posted price to be the median of the buyer's value is sufficient to obtain 1/2 of the optimal GFT from E ij as shown by McAfee [27]. The log(1/r)-approximation by Colini-Baldeschi et al [15] essentially follows from this argument.…”

“…1. log(1/r)-Approximation (Section 3): Our starting point is similar to Colini-Baldeschi et al [15]. We first argue that the probability space of each item i can be partitioned into…”

“…Despite the additional challenges, significant progress has very recently been made in understanding singledimensional two-sided markets [1,2,5,6,7,15]. Yet, in reality, many two-sided markets involve agents with multidimensional preferences.…”

Double Auctions in Markets for Multiple Kinds of Goods

“…In contrast to our work, they approximate the social welfare rather than the gain-from-trade. Note that any mechanism that attains a fraction α of the optimal GFT also attains a fraction of at least α of the optimal social welfare [Brustle et al, 2017;Colini-Baldeschi et al, 2017a], but the opposite is not true. Moreover, their competitive ratios (1/4, 1/6 or 1/16) do not approach 1.…”