VC Bargaining - Revision history

imported>Ed: /* Simple First Steps */

2011-05-26T05:22:09Z

Simple First Steps

2011-05-26T05:20:27Z

Simple First Steps

2011-05-26T03:17:17Z

2011-05-26T02:19:14Z

2011-05-26T02:18:36Z

Simple First Steps

2011-05-26T02:17:51Z

Simple First Steps

2011-05-26T02:04:42Z

Simple First Steps

2011-05-26T02:04:03Z

Simple First Steps

2011-05-26T02:03:35Z

2011-05-26T01:54:06Z

Simple First Steps

@@ Line 59: / Line 59: @@
 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>v(\{\empty\}) = 0, \;v(\{I\}) = 0, \; v(\{E\}) = 0, \; v(\{I,E\}) = 2\cdot\frac{1}{2}^\frac{1}{2}\,</math>
@@ Line 66: / Line 66: @@
-:<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>
+:<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>
@@ Line 90: / Line 90: @@
 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\}) = 1^\frac{1}{2}\,</math>
+:<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!}(1^\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>
+:<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>

@@ Line 49: / Line 49: @@
 :<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>
+:<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?

@@ Line 59: / Line 59: @@
 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>v(\{\empty\}) = 0, \;v(\{I\}) = 0, \; v(\{E\}) = 0, \; v(\{I,E\}) = 1^\frac{1}{2}\,</math>
@@ Line 80: / Line 80: @@
 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>
+<math>v(\{\empty\}) = 0, \;v(\{I\}) = 0, \; v(\{E\}) = 0, \; v(\{I,E\}) = \frac{1}{2}^\frac{1}{2}\,</math>
@@ Line 90: / Line 90: @@
 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\}) = 1^\frac{1}{2}\,</math>
+:<math>v(\{\empty\}) = 0, \;v(\{I\}) = \frac{1}{2}^\frac{1}{2}, \; v(\{E\}) = \frac{1}{2}^\frac{1}{2}, \; v(\{I,E\}) = 1^\frac{1}{2}\,</math>

← Older revision		Revision as of 02:19, 26 May 2011
Line 85:		Line 85:
	This gives:		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>	+	:<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>

@@ Line 85: / Line 85: @@
 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>
+:<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>
@@ Line 95: / Line 95: @@
 This gives:
-:<math>\phi_I_2(v)= \frac{1!0!}{2!}(1^\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>
+:<math>\phi_{I(2)}(v)= \frac{1!0!}{2!}(1^\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>

← Older revision		Revision as of 02:04, 26 May 2011
Line 57:		Line 57:
	Proof using the Shapley value for a single stage of negotiation:		Proof using the Shapley value for a single stage of negotiation:

−	<math>	+	<math>v(\empty) = 0, \;v(\{I\}) = 0, \; v(\{E\}) = 0, \; v(\{I,E\}) = 1^\frac{1}{2}\,</math>
−	v(\empty) = 0
−	v(\{I\}) = 0
−	v(\{E\}) = 0
−	v(\{I,E\}) = 1^\frac{1}{2}
−	</math>

@@ Line 5: / Line 5: @@
 ===The players===
-The players are an Entrepreneur and a VC, both are risk neutral.
+The players are an Entrepreneur (<math>E\,</math>) and a VC investor (<math>I\,</math>), both are risk neutral.
 ===The Value Function===
@@ Line 49: / Line 49: @@
 :<math> \frac{\partial V_2}{\partial x_1} =0 \implies x_1 = x_2 = \frac{1}{2}\,</math>
-\therefore V_2 = \frac{1}{2}^\frac{1}{2} + \frac{1}{2}^\frac{1}{2} = 1^\frac{1}{2} \approx 1.41
+:<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 \frac{1}{2}^\frac{1}{2}\approx 0.707, assuming equal outside options of zero and equal bargaining power.
+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:
-V(C
+<math>

← Older revision		Revision as of 01:54, 26 May 2011
Line 48:		Line 48:

	:<math> \frac{\partial V_2}{\partial x_1} =0 \implies x_1 = x_2 = \frac{1}{2}\,</math>		:<math> \frac{\partial V_2}{\partial x_1} =0 \implies x_1 = x_2 = \frac{1}{2}\,</math>
		+
		+	\therefore V_2 = \frac{1}{2}^\frac{1}{2} + \frac{1}{2}^\frac{1}{2} = 1^\frac{1}{2} \approx 1.41
		+
		+	How much should be allocated to the investor?
		+
		+	Using Shapley values, Nash Bargaining and infinite Rubenstein bargaining will all imply each party gets \frac{1}{2}^\frac{1}{2}\approx 0.707, assuming equal outside options of zero and equal bargaining power.
		+
		+	Proof using the Shapley value for a single stage of negotiation:
		+
		+	V(C