  We consider a Nash equilibrium between two high-frequency traders in a simple
market impact model with transient price impact and additional quadratic
transaction costs. Extending a result by Sch\"oneborn (2008), we prove
existence and uniqueness of the Nash equilibrium and show that for small
transaction costs the high-frequency traders engage in a "hot-potato game", in
which the same asset position is sold back and forth. We then identify a
critical value for the size of the transaction costs above which all
oscillations disappear and strategies become buy-only or sell-only. Numerical
simulations show that for both traders the expected costs can be lower with
transaction costs than without. Moreover, the costs can increase with the
trading frequency when there are no transaction costs, but decrease with the
trading frequency when transaction costs are sufficiently high. We argue that
these effects occur due to the need of protection against predatory trading in
the regime of low transaction costs.
