# Solving a system of $3 \arg \max$ equations

Considering these 3 equations:

$$\alpha_{1} = \arg \max\limits_{\alpha \geq 0} \quad \Big \{ (\alpha - 1) \big (\frac{1/\alpha}{1/\alpha + 1/2 \alpha_{2} + 1/2 \alpha_{3}} \big)^2 \Big \} \\ \alpha_{2} = \arg \max \limits_{\alpha \geq 0} \quad \Big \{ (\alpha - 1) \big (\frac{1/2 \alpha}{1/\alpha_{1} + 1/2 \alpha + 1/2 \alpha_{3}} \big)^2 \Big \} \\ \alpha_{3} = \arg \max \limits_{\alpha \geq 0} \quad \Big \{ (\alpha - 1) \big (\frac{1/2 \alpha}{1/\alpha_{1} + 1/2 \alpha_{2} + 1/2 \alpha} \big)^2 \Big \} \\$$

What would be the most appropriate numerical method to solve the system ? Thanks.

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Solution is alpha_1 = 5,56 and alpha_{2 & 3} = 3,56. Reference : page 14 of a Columbia Business School paper link. Any idea about an appropriate numerical method ? –  juju Nov 8 '12 at 13:38