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V<math>v(C\empty) = 0 \\v(\{I\}) = 0 \\v(\{E\}) = 0 \\v(\{I,E\}) = 1^\frac{1}{2}</math> :<math>\phi_i(v)=\sum_{S \subseteq N \setminus\{i\}} \frac{|S|!\; (n-|S|-1)!}{n!}(v(S\cup\{i\})-v(S))</math> :<math>\therefore \phi_I(v)= \frac{1!0!}{2!}(1^\frac{1}{2} - 0) + \frac{0!1!}{2!}(0 - 0) = \frac{1}{2}^\frac{1}{2} \approx 0.707\,</math>
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===The players===
The players are an Entrepreneur (<math>E\,</math>) and a VCinvestor (<math>I\,</math>), both are risk neutral.
===The Value Function===
:<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} = 1^\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:
