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Adjusted optimal choices for BargainingPowerModel and UnobservableModel.
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manuelbieri committed Nov 4, 2021
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13 changes: 2 additions & 11 deletions README.md
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This package implements the models of [Shelegia and Motta (2021)](shelegia_motta_2021.pdf).
This package implements the models of [Shelegia and Motta (2021)](resources/shelegia_motta_2021.pdf).

![GitHub](https://img.shields.io/github/license/manuelbieri/shelegia_motta_2021)
![PyPI - Python Version](https://img.shields.io/pypi/pyversions/Shelegia-Motta-2021)
Expand Down Expand Up @@ -94,14 +94,5 @@ Generate api-documentation with the following command:
pdoc -o ./docs Shelegia_Motta_2021 --docformat "numpy" --math
```

#### Dependencies

| Package  | Version   | Annotation   |
|:-----------|:---------|:------------------------------------------------|
| matplotlib | 3.4.3 | Always needed (includes numpy) |
| jupyter | 1.0.0 | Just for the demonstration in demo.ipynb |
| pdoc | 8.0.1 | Only to generate the documentation from scratch |
<br>

#### Additional Notes
For further information about the coordinates used in the code, see dev_notes.md.
For further information about the coordinates used in the code, see resources/dev_notes.md.
12 changes: 10 additions & 2 deletions Shelegia_Motta_2021/IModel.py
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Expand Up @@ -159,14 +159,18 @@ def get_optimal_choice(self, A: float, F: float) -> Dict[str, str]:
pass

@abc.abstractmethod
def plot_incumbent_best_answers(self, axis: matplotlib.axes.Axes = None) -> matplotlib.axes.Axes:
def plot_incumbent_best_answers(self, axis: matplotlib.axes.Axes = None, **kwargs) -> matplotlib.axes.Axes:
"""
Plots the best answers of the incumbent to all possible actions of the entrant.
Parameters
----------
axis : matplotlib.axes.Axes
Axis to draw the plot on. (optional)
**kwargs
Optional key word arguments for the best answers plot.<br>
- title: title on top of the plot, instead of the default title.<br>
- options_legend: If true, an additional legend, explaining the options of the entrant and the incumbent, will be added to the plot.<br>
Returns
-------
Expand All @@ -176,14 +180,18 @@ def plot_incumbent_best_answers(self, axis: matplotlib.axes.Axes = None) -> matp
pass

@abc.abstractmethod
def plot_equilibrium(self, axis: matplotlib.axes.Axes = None) -> matplotlib.axes.Axes:
def plot_equilibrium(self, axis: matplotlib.axes.Axes = None, **kwargs) -> matplotlib.axes.Axes:
"""
Plots the equilibrium path based on the choices of the entrant and incumbent.
Parameters
----------
axis : matplotlib.axes.Axes
Axis to draw the plot on. (optional)
**kwargs
Optional key word arguments for the equilibrium plot.<br>
- title: title on top of the plot, instead of the default title.<br>
- options_legend: If true, an additional legend, explaining the options of the entrant and the incumbent, will be added to the plot.<br>
Returns
-------
Expand Down
65 changes: 43 additions & 22 deletions Shelegia_Motta_2021/Models.py
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@@ -1,4 +1,4 @@
from typing import Dict, List, Tuple, Literal
from typing import Dict, List, Tuple, Literal, Final

import matplotlib.axes
import matplotlib.pyplot as plt
Expand All @@ -15,7 +15,7 @@ class BaseModel(Shelegia_Motta_2021.IModel):
There are two players in our base model: The Incumbent, which sells the primary product, denoted by Ip, and a start-up, called the Entrant, which sells a product Ec complementary to Ip. (One may think of Ip as a platform, and Ec as a service or product which can be accessed through the platform.) We are interested in studying the choice of E between developing a substitute to Ip, denoted by Ep, or another complement to Ip, denoted by Ẽc and the choice of I between copying E’s original complementary product Ec by creating a perfect substitute Ic, or not. Since E may not have enough assets to cover the development cost of its second product, copying its current product will affect the entrant’s ability to obtain funding for the development. We shall show that the incumbent has a strategic incentive to copy when the entrant plans to compete, and to abstain from copying when it plans to create another complement.
"""

tolerance: float = 10 ** (-8)
TOLERANCE: Final[float] = 10 ** (-8)
"""Tolerance for the comparison of two floating numbers."""

def __init__(self, u: float = 1, B: float = 0.5, small_delta: float = 0.5, delta: float = 0.51,
Expand Down Expand Up @@ -235,26 +235,27 @@ def _plot(self, coordinates: List[List[Tuple[float, float]]], labels: List[str],
Axis containing the plot.
"""
if axis is None:
fig, axis = plt.subplots()
figure, axis = plt.subplots()
self._draw_thresholds(axis)

for i, coordinates in enumerate(coordinates):
poly = plt.Polygon(coordinates, ec="k", color=self._get_color(i), label=labels[i])
axis.add_patch(poly)

axis.legend(bbox_to_anchor=(1.3, 1), loc="upper left")
if kwargs.get('options_legend', False):
axis.text(-0.05, -0.4, self._create_options_legend(), verticalalignment='top')
BaseModel._set_axis_labels(axis, title=kwargs.get('title', ''),
x_label=kwargs.get('xlabel', 'Assets of the entrant'),
y_label=kwargs.get('ylabel', 'Fixed costs of copying for the incumbent'))
BaseModel._set_axis(axis)
plt.show()
return axis

def plot_incumbent_best_answers(self, axis: matplotlib.axes.Axes = None) -> matplotlib.axes.Axes:
def plot_incumbent_best_answers(self, axis: matplotlib.axes.Axes = None, **kwargs) -> matplotlib.axes.Axes:
poly_coordinates: List[List[Tuple[float, float]]] = self._get_incumbent_best_answer_coordinates()
poly_labels: List[str] = self._get_incumbent_best_answer_labels()
axis: matplotlib.axes.Axes = self._plot(title="Best Answers of the incumbent to the choices of the entrant",
coordinates=poly_coordinates, labels=poly_labels, axis=axis)
axis: matplotlib.axes.Axes = self._plot(title=kwargs.get("title", "Best Answers of the incumbent to the choices of the entrant"),
coordinates=poly_coordinates, labels=poly_labels, axis=axis, options_legend=kwargs.get('options_legend', False))
return axis

def _create_choice_answer_label(self, entrant: Literal["complement", "substitute", "indifferent"],
Expand All @@ -267,7 +268,7 @@ def _get_incumbent_best_answer_labels(self) -> List[str]:
"""
Returns a list containing the labels for the squares in the plot of the best answers of the incumbent to the choice of the entrant.
For the order of the labels refer to the file devnotes.md.
For the order of the labels refer to the file resources/dev_notes.md.
Returns
-------
Expand Down Expand Up @@ -298,7 +299,7 @@ def _get_incumbent_best_answer_coordinates(self) -> List[List[Tuple[float, float
"""
Returns a list containing the coordinates for the squares in the plot of the best answers of the incumbent to the choice of the entrant.
For the order of the squares refer to the file devnotes.md.
For the order of the squares refer to the file resources/dev_notes.md.
Returns
-------
Expand Down Expand Up @@ -330,18 +331,18 @@ def _get_incumbent_best_answer_coordinates(self) -> List[List[Tuple[float, float
(x_max, y_max), (0, y_max),
(0, self._copying_fixed_costs['F(YN)s']), (self._assets['A-s'], self._copying_fixed_costs['F(YN)s'])]]

def plot_equilibrium(self, axis: matplotlib.axes.Axes = None) -> matplotlib.axes.Axes:
def plot_equilibrium(self, axis: matplotlib.axes.Axes = None, **kwargs) -> matplotlib.axes.Axes:
poly_coordinates: List[List[Tuple[float, float]]] = self._get_equilibrium_coordinates()
poly_labels: List[str] = self._get_equilibrium_labels()
axis: matplotlib.axes.Axes = self._plot(title='Equilibrium Path in the base Model',
coordinates=poly_coordinates, labels=poly_labels, axis=axis)
axis: matplotlib.axes.Axes = self._plot(title=kwargs.get("title", "Equilibrium Path"),
coordinates=poly_coordinates, labels=poly_labels, axis=axis, options_legend=kwargs.get('options_legend', False))
return axis

def _get_equilibrium_labels(self) -> List[str]:
"""
Returns a list containing the labels for the squares in the plot of the equilibrium path.
For the order of the squares refer to the file devnotes.md.
For the order of the squares refer to the file resources/dev_notes.md.
Returns
-------
Expand All @@ -362,7 +363,7 @@ def _get_equilibrium_coordinates(self) -> List[List[Tuple[float, float]]]:
"""
Returns a list containing the coordinates for the squares in the plot of the equilibrium path.
For the order of the squares refer to the file devnotes.md.
For the order of the squares refer to the file resources/dev_notes.md.
Returns
-------
Expand All @@ -388,7 +389,7 @@ def _get_equilibrium_coordinates(self) -> List[List[Tuple[float, float]]]:

def plot_payoffs(self, axis: matplotlib.axes.Axes = None) -> matplotlib.axes.Axes:
if axis is None:
fig, axis = plt.subplots()
figure, axis = plt.subplots()
index = arange(0, len(self._payoffs) * 2, 2)
bar_width = 0.35
opacity = 0.8
Expand Down Expand Up @@ -517,7 +518,7 @@ def _draw_thresholds(self, axis: matplotlib.axes.Axes) -> None:
self._draw_horizontal_line_with_label(axis, y=self._copying_fixed_costs['F(YN)s'], label="$F^{YN}_S$")
self._draw_horizontal_line_with_label(axis, y=self._copying_fixed_costs['F(YY)c'], label="$F^{YY}_C$")

if abs(self._copying_fixed_costs['F(YY)s'] - self._copying_fixed_costs['F(YN)c']) < BaseModel.tolerance:
if abs(self._copying_fixed_costs['F(YY)s'] - self._copying_fixed_costs['F(YN)c']) < BaseModel.TOLERANCE:
self._draw_horizontal_line_with_label(axis, y=self._copying_fixed_costs['F(YY)s'],
label="$F^{YY}_S=F^{YN}_C$")
else:
Expand Down Expand Up @@ -564,6 +565,19 @@ def _draw_vertical_line_with_label(self, axis: matplotlib.axes.Axes, x: float, l
if label is not None:
axis.text(x, label_y, label)

def _create_options_legend(self, latex: bool = True) -> str:
space: str = "$\quad$" if latex else "\t"
return "Options of the entrant:\n" + \
space + self.ENTRANT_CHOICES['complement'] + ": Tries to develop an additional complementary product to a primary product.\n" + \
space + self.ENTRANT_CHOICES['substitute'] + ": Tries to develop an substitute to the primary product of the incumbent.\n" + \
space + self.ENTRANT_CHOICES['indifferent'] + " : Indifferent between the options mentioned above.\n" + \
"Options of the incumbent:\n" + \
space + self.INCUMBENT_CHOICES['copy'] + " : Tries to develop an additional complementary product to a primary product.\n" + \
space + self.INCUMBENT_CHOICES['refrain'] + " : Tries to develop an additional complementary product to a primary product.\n" + \
"Outcomes of the development:\n" + \
space + self.DEVELOPMENT_OUTCOME['success'] + " : The additional product can be developed, since the entrant has sufficient assets.\n" + \
space + self.DEVELOPMENT_OUTCOME['failure'] + " : The additional product can not be developed, since the entrant has not enough assets."

@staticmethod
def _get_color(i: int) -> str:
"""
Expand Down Expand Up @@ -793,8 +807,8 @@ def __init__(self, u: float = 1, B: float = 0.5, small_delta: float = 0.5, delta
"""
super(UnobservableModel, self).__init__(u=u, B=B, small_delta=small_delta, delta=delta, K=K, beta=beta)

def plot_incumbent_best_answers(self, axis: matplotlib.axes.Axes = None) -> matplotlib.axes.Axes:
return self.plot_equilibrium(axis=axis)
def plot_incumbent_best_answers(self, axis: matplotlib.axes.Axes = None, **kwargs) -> matplotlib.axes.Axes:
return self.plot_equilibrium(axis=axis, kwargs=kwargs)

def _create_choice_answer_label(self, entrant: Literal["complement", "substitute", "indifferent"],
incumbent: Literal["copy", "refrain"],
Expand All @@ -806,7 +820,7 @@ def _get_equilibrium_labels(self) -> List[str]:
"""
Returns a list containing the labels for the squares in the plot of the equilibrium path.
For the order of the squares refer to the file devnotes.md.
For the order of the squares refer to the file resources/dev_notes.md.
Returns
-------
Expand All @@ -823,6 +837,13 @@ def _get_equilibrium_labels(self) -> List[str]:
self._create_choice_answer_label(entrant="substitute", incumbent="refrain", development="success")
]

def get_optimal_choice(self, A: float, F: float) -> Dict[str, str]:
result: Dict = super().get_optimal_choice(A, F)
# adjust the different choices in area three -> since the kill zone does not exist in this model.
if result["entrant"] == self.ENTRANT_CHOICES["complement"]:
result = {"entrant": self.ENTRANT_CHOICES["substitute"], "incumbent": self.INCUMBENT_CHOICES["copy"], "development": self.DEVELOPMENT_OUTCOME["failure"]}
return result


class AcquisitionModel(BargainingPowerModel):
"""
Expand Down Expand Up @@ -864,6 +885,6 @@ def get_copying_fixed_costs_values(self) -> Dict[str, float]:


if __name__ == '__main__':
base_model = UnobservableModel(beta=0.6)
base_model.plot_equilibrium()
print(base_model)
bargaining_power_model = Shelegia_Motta_2021.BargainingPowerModel(beta=0.6)
bargaining_power_model.plot_equilibrium()
plt.show()
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