Markets have an exposure problem when getting to the optimal allocation requires a sequence of transactions which if started but not completed leaves at least one trader with losses. We use laboratory experiments to evaluate the effect of the exposure problem on alternative market mechanisms. The continuous double auction performs poorly: efficiency is only 20% when exposure is high and 55% when it is low. A package market effectively eliminates the exposure problem: in low and high exposure treatments efficiency is 82% and 89% respectively. Building on stability notions from matching theory we introduce the concept of mechanism stability. A model of trade that combines mechanism stability with noisy best responses and imperfect foresight explains the difference in market performance. Finally, decentralized bargaining with contingent contracts performs well with perfect information and communication but not in the more realistic case when traders’ preferences are privately known.