VC Bargaining

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A Basic Model

The players

The players are an Entrepreneur ([math]E\,[/math]) and a VC investor ([math]I\,[/math]), 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]

Having [math]k\gt 0\,[/math] 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

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={1,2}\,[/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(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] \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:

[math] v(\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]