We analyze optimal execution strategies when multiple traders are simultaneously in­volved in optimal execution. In this case, we obtain new trading strategies that follow from a direct extension of the mean variance approach of Grinold and Kahn, and Almgren and Chriss. However, as we show below, the proposed strategies can be quite different from the standard ones obtained in Grinold and Kahn, and Almgren and Chriss. This is because each trader (assu­med to be rational) is trying to minimize her trading cost or ‘implementation shortfall’ and therefore takes into account the price impacts caused by herself and all other traders. We also obtain a close form characterization for the dynamic Nash equilibrium in terms of the system of second-order ODEs, which can be solved explicitly. The resulting equilibrium strategies describe different types of predatory and defensive behavior, though aggregate order flow profile have some properties of standard Almgren, Chriss strategies, e.g. is monotoneous and convex. We show that the traders with smaller holdings are involved in predatory strategies, while traders with larger holdings tend to defend themselves against potential predators by following the de­layed trading strategies. We also show that depending on liquidity and volatility parameters, predatory traders may be frontrunners or contrarian traders.