A market impact game is concerned with a model for competing traders whose strategies can impact the prices of an asset. Knowledge of the strategies of competitors can then affect the strategic decisions of each trader. In this talk, we consider Nash equilibrium between traders for various cost functionals and several price impact models. It turns that these Nash equilibria can exhibit qualitative features that, at least at first sight, are rather surprising. For instance, under transient price impact it can happen that adding transaction costs to the model can decrease the expected costs of all participants in the market impact game. The talk includes joint work with Elias Strehle and Tao Zhang.