This paper introduces a new class of market games featuring multiple posts per commodity, in which trading posts are privately owned. It is demonstrated via three robust counterexamples, that in this setting the law of one price fails, thus showing, contrary to longstanding belief in the literature, that price dispersion in large market games is extremely robust. Most importantly, it is established that even in economies with a continuum of small agents and infinitely many atoms (all of whom can arbitrage prices if they so wish), and an infinite number of markets per commodity, the set of equilibria—and the resulting market structure—is influenced, both by strategic behaviour, and private ownership of posts.