Central results in economics guarantee the existence of efficient equilibria for various classes of markets. An underlying assumption in early work is that agents are price-takers, i.e., agents honestly report their true demand in response to prices. A line of research in economics, initiated by Hurwicz (1972), is devoted to understanding how such markets perform when agents are strategic about their demands. This is captured by the Walrasian Mechanism that proceeds by collecting reported demands, finding clearing prices in the reported market via an ascending price t\atonnement procedure, and returns the resulting allocation. Similar mechanisms are used, for example, in the daily opening of the New York Stock Exchange and the call market for copper and gold in London. In practice, it is commonly observed that agents in such markets reduce their demand leading to behaviors resembling bargaining and to inefficient outcomes. We ask how inefficient the equilibria can be. Our main result is that the welfare of every pure Nash equilibrium of the Walrasian mechanism is at least one quarter of the optimal welfare, when players have gross substitute valuations and do not overbid. Previous analysis of the Walrasian mechanism have resorted to large market assumptions to show convergence to efficiency in the limit. Our result shows that approximate efficiency is guaranteed regardless of the size of the market.
We extend our results in several directions. First, our results extend to Bayes-Nash equilibria and outcomes of no regret learning via the smooth mechanism framework. We also extend our bounds to any mechanism that maximizes welfare with respect to the declared valuations and never charges agents more than their bids. Additionally, we consider other classes of valuations and bid spaces beyond those satisfying the gross substitutes conditions. Finally, we relax the no-overbidding assumption, and present bounds that are parameterized by the extent to which agents are willing to overbid.