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:<math> \frac{\partial V_2}{\partial x_1} =0 \implies x_1 = x_2 = \frac{1}{2}\,</math>
\therefore V_2 = \frac{1}{2}^\frac{1}{2} + \frac{1}{2}^\frac{1}{2} = 1^\frac{1}{2} \approx 1.41
How much should be allocated to the investor?
Using Shapley values, Nash Bargaining and infinite Rubenstein bargaining will all imply each party gets \frac{1}{2}^\frac{1}{2}\approx 0.707, assuming equal outside options of zero and equal bargaining power.
Proof using the Shapley value for a single stage of negotiation:
V(C
