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:<math> \frac{\partial V_2}{\partial x_1} =0 \implies x_1 = x_2 = \frac{1}{2}\,</math>
:<math>\therefore V_2 = \frac{1}{2}^\frac{1}{2} + \frac{1}{2}^\frac{1}{2} = 2\cdot\frac{1}{2}^\frac{1}{2} \approx 1.41\,</math>
How much should be allocated to the investor?
Using Shapley values, Nash Bargaining and infinite Rubenstein bargaining will all imply each party gets :<math>\frac{1}{2}^\frac{1}{2}\approx 0.707\,</math>, assuming equal outside options of zero and equal bargaining power.
Proof using the Shapley value for a single stage of negotiation:
<math>v(\{\empty\}) = 0 , \\;v(\{I\}) = 0 , \\; v(\{E\}) = 0 , \\; v(\{I,E\}) = 2\cdot\frac{1}{2}^\frac{1}{2}\,</math>
:<math>\therefore \phi_I(v)= \frac{1!0!}{2!}(2\cdot\frac{1}{2}^\frac{1}{2} - 0) + \frac{0!1!}{2!}(0 - 0) = \frac{1}{2}^\frac{1}{2} \approx 0.707\,</math> The entrepreneur gets the same (the efficient outcome is realised and the profits are fully distributed, so you know he must without doing the math). For two stages of negotiation, the intermediate value of the firm is :<math>V_1 = x_1^\frac{1}{2} = \frac{1}{2}^\frac{1}{2} \approx 0.707\,</math> and the characteristic function is: <math>v(\{\empty\}) = 0, \;v(\{I\}) = 0, \; v(\{E\}) = 0, \; v(\{I,E\}) = \frac{1}{2}^\frac{1}{2}\,</math> This gives: :<math>\phi_{I(1)}(v)= \frac{1!0!}{2!}(\frac{1}{2}^\frac{1}{2} - 0) + \frac{0!1!}{2!}(0 - 0) = \frac{1}{4}^\frac{1}{2} \approx 0.354\,</math> For the second stage, the characteristic function is: :<math>v(\{\empty\}) = 0, \;v(\{I\}) = \frac{1}{2}^\frac{1}{2}, \; v(\{E\}) = \frac{1}{2}^\frac{1}{2}, \; v(\{I,E\}) = 2\cdot\frac{1}{2}^\frac{1}{2}\,</math> This gives: :<math>\phi_{I(2)}(v)= \frac{1!0!}{2!}(2\cdot\frac{1}{2}^\frac{1}{2}-\frac{1}{2}^\frac{1}{2}) + \frac{0!1!}{2!}(\frac{1}{2}^\frac{1}{2} - 0) = \frac{1}{2}^\frac{1}{2} \approx 0.707\,</math> And we have the same result as the single negotiation version of the game. Note that this assumes that value isn't created or destroyed by the presence of the investor or the entrepreneur alone after the first stage - the first stage value just sits there, waiting to be built upon by the combination of the investor and the entrepreneur together. This is a model where the outside options of both players are zero. If the entrepreneur doesn't turn up for both rounds the firm is worth zero, and likewise for the entrepreneur. Also, but differently, the bargaining strength is equal. To express different bargaining strengths we would use a weighted Shapley value. Note that this could still be used with zero outside options.
