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We provide the methods for searching for the unique Nash equilibrium using mixed integer programming techniques. We also simulate the model parameters using this technique and derive the numerical solution to show the characteristics of a key player providing both public goods as suggested by Bergstrom et al. (1986). We present two key results. First, when the number of players decreases, the appearance rate of the unique equilibrium accompanying a key player increases. Second, when the variance of the preference or price parameters decreases, the appearance rate of the unique equilibrium accompanying a key player increases.