VC Bargaining

This 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.

A Basic Model

The players

The players are an Entrepreneur and a VC, both are risk neutral.

The Value Function

[math]V_t=V_{t-1} + f(x_t) - k \,[/math]

with

[math]V_0=0, f(0)=0, f'\gt 0, f''\lt 0, k\gt 0 \,[/math]

should do us just fine. Having [math]k\gt 0\,[/math] will force 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 stopping constraint is to have the exit occur when

[math]V_t \ge \overline{V}\,[/math]

with

[math]\overline{V} \sim F(V)\,[/math]

where the distribution is known to both parties.

Actually, given how we built [math]f[/math] and [math]V[/math], [math]V[/math] is concave, so it should have a natural maximum (where the marginal increase in value will be equal the marginal cost which is [math]k[/math]), so I don't think we need to go that far with the exit value.

However, I wanted to start much simpler - assume there is a fixed number of rounds and investments, how does the optimal policy compare to the current way of calculating shares and values?

Old Ideas

There are some other methods that come to mind:

we could force an exit once

[math]\sum_t (x_t) \ge \overline{x}\,[/math]

or we could try to induce an optimum value

[math]f'(0) \gt 0, f''\lt 0, \exist z^* s.t. \forall z \gt z^* f'(z)\lt 0\,[/math]

though now that I look at this I realize it isn't going to work using just investment...

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).

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.