In this paper we study a discrete time multi-period optimal portfolio selection game problem between two risk-averse institutional investors when each investor takes into account his relative performance by comparison to his competitor. Both investors can invest freely in the risk-free asset and only one of the two correlated risky stocks is available to each investor, reflecting asset specialization. Each investor chooses a dynamic portfolio strategy to maximize his expected terminal utility of the weight sum of his wealth and the difference between his wealth and that of his competitor. We first characterize explicitly the unique Nash equilibrium portfolio strategies. Secondly, the Nash equilibrium portfolio strategy and the value function of each investor are obtained in closed forms for the case of each investor with an exponential utility function. The effects of the relative performance on the Nash equilibrium portfolio strategy and the value function are also analyzed. Furthermore, we obtain the closed-form expressions of the Nash equilibrium portfolio strategy and the value function under that the compounded return of two risky stocks are normally distributed and asset diversification. Sensitivity analysis is also provided to illustrate how the Nash equilibrium portfolio strategy and the value function change when some model parameters vary. Finally, the setting with and without asset specialization is analyzed by numerical examples. The results reveal that competition can change the institutional investor risk taking. Different investment possibilities may greatly influence the Nash equilibrium portfolio strategy and the value function of the institutional investor.