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91 bytes added ,  01:22, 26 May 2011
:<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?
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>
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>
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