  We study the high-frequency limits of strategies and costs in a Nash
equilibrium for two agents that are competing to minimize liquidation costs in
a discrete-time market impact model with exponentially decaying price impact
and quadratic transaction costs of size $\theta\ge0$. We show that, for
$\theta=0$, equilibrium strategies and costs will oscillate indefinitely
between two accumulation points. For $\theta>0$, however, strategies, costs,
and total transaction costs will converge towards limits that are independent
of $\theta$. We then show that the limiting strategies form a Nash equilibrium
for a continuous-time version of the model with $\theta$ equal to a certain
critical value $\theta^*>0$, and that the corresponding expected costs coincide
with the high-frequency limits of the discrete-time equilibrium costs. For
$\theta\neq\theta^*$, however, continuous-time Nash equilibria will typically
not exist. Our results permit us to give mathematically rigorous proofs of
numerical observations made in Schied and Zhang (2013). In particular, we
provide a range of model parameters for which the limiting expected costs of
both agents are decreasing functions of $\theta$. That is, for sufficiently
high trading speed, raising additional transaction costs can reduce the
expected costs of all agents.
