In order to reveal information about buyer valuations, price competition leads to sorting of
buyers and sellers. In a decentralized Walrasian market with private values, we show that there is
perfect sorting of high valuation buyers with high cost sellers. Such negative assortative matching
arises generally in the absence of complementarity. Positive assortative matching obtains only if
there is a high enough degree of complementarity. The relevant condition is root-supermodularity;
i.e., the square root of the match value function is supermodular. This condition is weaker than
log-supermodularity, a condition required for positive assortative matching in markets with random
search.
Keywords: Decentralized Walrasian markets. Price Competition. Two-Sided Matching. Root-
Supermodularity.