  We consider a single security market based on a limit order book and two
investors, with different speeds of trade execution. If the fast investor can
front-run the slower investor, we show that this allows the fast trader to
obtain risk free profits, but that these profits cannot be scaled. We derive
the fast trader's optimal behaviour when she has only distributional knowledge
of the slow trader's actions, with few restrictions on the possible prior
distributions. We also consider the slower trader's response to the presence of
a fast trader in a market, and the effects of the introduction of a `Tobin tax'
on financial transactions. We show that such a tax can lead to the elimination
of profits from front-running strategies. Consequently, a Tobin tax can both
increase market efficiency and attract traders to a market.
