Changes

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. ==Thoughts== *We shouldn't include effort from the entrep. - we want a model that has no contract theory, just bargaining.*I added effort to be able to calculate a Shapley Value. Otherwise, you can't divide the pie between the two sides, as you don't know the contribution of the other side. The effort is assumed to be binary (0 or 1), so the solution will be easy.

The players are an Entrepreneur (<math>E\,</math>) and a VCinvestor (<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).

I think the best idea for a One possible stopping constraint is to have the exit occur when :

Actually===Bargaining=== 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. ===Simple First Steps=== Address the question: How does the optimal policy compare to the current way of calculating shares and values? Assume a fixed number of rounds: <math>t={1, given how we built 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> and  Again <math>V(0)=0 \,</math>.  :<math> \therefore V_2 = x_1^\frac{1}{2} + x_2^\frac{1}{2}\,</math> Recalling that <math> x_2 = 1 - x_1 \, </math>  :<math>V\frac{\partial V_2}{\partial x_1} =0 \implies x_1 = x_2 = \frac{1}{2}\,</math> is concave :<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\, so it </math> How much should have a natural maximum (where be allocated to the marginal increase in value investor?  Using Shapley values, Nash Bargaining and infinite Rubenstein bargaining will be 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 marginal cost which is 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>k  :<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\, so I don't think we need to go that far with the exit value.</math>

:<math>V_1 = x_1^\sum_t (x_t) frac{1}{2} = \ge frac{1}{2}^\overlinefrac{x1}{2} \approx 0.707\,</math>

or we could just fix <math>tv(\{\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, but it would be nice to have it endogenousthe 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\, otherwise </math>  And we would need to justify discrete rounds seperately (have the same result as we did yesterday evening with the state-tree perhaps)single negotiation version of the game.

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.