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*We shouldn't include effort from the entrep. - we want a model that has no contract theory, just bargaining.===The players===
==A Basic Model==The players are an Entrepreneur (<math>E\,</math>) and a VC investor (<math>I\,</math>), both are risk neutral.
should do us just fine. Having <math>k>0\,</math> will force forces a finite number of rounds as the optimal solution providing there is a stopping constraint on <math>V_t\,</math> (so players don't invest forever). One possible stopping constraint is: :<math>V_t \ge \overline{V}\,</math> with :<math>\overline{V} \sim F(V)\,</math> where the distribution is known to both parties. ===Bargaining===
There are some methods that come to mind: In each period there is Rubenstein finite bargaining, with potentially different patience, and one player designated as last. This will give a single period equilibrium outcome with the parties having different bargaining strength.
we could force an exit once ===Simple First Steps===
or onceAssume a fixed number of rounds: <math>t={1,2}\,</math>Assume a fixed total investment: <math>\sum_t x_t = 1\,</math>Assume a functional form for <math>f(x_t): f(x_t) = x_t^\frac{1}{2}\,</math>
:Again <math>V_t \ge \overline{V_t}V(0)=0 \,</math>.
or we could try to induce an optimum value
though now Recalling that I look at this I realize it isn't going to work using just investment...<math> x_2 = 1 - x_1 \,</math>
or we could just fix <math>t\,</math>, but it would be nice to have it endogenous, otherwise we would need to justify discrete rounds seperately (as we did yesterday evening with the state-tree perhaps).
In each period 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 is Rubenstein finite bargaining, with potentially different patience, waiting to be built upon by the combination of the investor and one player designated as lastthe entrepreneur together. This will give is a single period equilibrium outcome with 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 parties having bargaining strength is equal. To express different bargaining strengthstrengths we would use a weighted Shapley value. Note that this could still be used with zero outside options.
→Simple First Steps
This page (and the discussion page) is for Ed and Ron to share their thoughts on VC Bargaining. Access is restricted to those with "Trusted" access.
==ThoughtsA Basic Model==
===The Value Function===
:<math>V_0=0, f(0)=0, f'>0, f''<0, k>0 \,</math>
Address the question:<math>\sum_t (x_t) \ge \overline{x}\,</math>How does the optimal policy compare to the current way of calculating shares and values?
:<math>f'(0) >0, f''<0, \exist ztherefore V_2 = x_1^* s.t. \forall z > zfrac{1}{2} + x_2^* f'(z)<0\frac{1}{2}\,</math>
:<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 Bargainingand 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>\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!}(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.
