We show that if limit orders are required to vary smoothly, then strategic (Nash) equilibria of the double auction mechanism yield competitive (Walras) allocations. It is not necessary to have competitors on any side of any market: smooth trading is a substitute for price wars. In particular, Nash equilibria are Walrasian even in a bilateral monopoly.
Dubey, Pradeep and Sondermann, Dieter, "Perfect Competition in a Bilateral Monopoly (In Honor of Martin Shubik)" (2005). Cowles Foundation Discussion Papers. 1821.