The competition between Island and Nasdaq at the beginning of the century offers a natural laboratory to study competition between and within trading platforms and its consequences for liquidity supply. Our empirical strategy takes advantage of the difference between the pricing grids used on Island and Nasdaq, as well as of the decline in the Nasdaq tick. Using the finer grid prevailing on their market, Island limit order traders undercut Nasdaq quotes, much more than they undercut one another. The drop in the Nasdaq tick size triggered a drop in Island spreads, despite the Island tick already being very thin before Nasdaq decimalization. We also estimate a structural model of liquidity supply and find that Island limit order traders earned rents before Nasdaq decimalization. Our results suggest that perfect competition cannot be taken for granted, even on transparent open limit order books with a very thin pricing grid.