Two-Sided Markets: Mapping Social Welfare to Gain from Trade

Though the definition of gain from trade extends the definition of social welfare from auctions to markets, from a mathematical point of view the additional dimension added by gain from trade makes it much more difficult to design a gain from trade maximizing mechanism. This paper provides a means of understanding when a market designer can choose the easier path of maximizing social welfare rather than maximizing gain from trade. We provide and prove the first formula to convert a social welfare approximation bound to a gain from trade approximation bound that maintains the original order of approximation. This makes it possible to compare algorithms that approximate gain from trade with those that approximate social welfare. We evaluate the performance of our formula by using it to convert known social welfare approximation solutions to gain from trade approximation solutions. The performance of all known two-sided markets solutions (that implement truthfulness, IR, BB, and approximate efficiency) are benchmarked by both their theoretical approximation bound and their performance in practice. Surprisingly, we found that some social welfare solutions achieve a better gain from trade than other solutions designed to approximate gain from trade.