The theory of the “kill zone” has become an increasingly prominent cause for concern among economists in recent times, especially with the rise of digital companies that have become monopolists in their sectors internationally. Companies like Facebook are continuously acquiring start-ups that may have a chance of competing with them in some way in the future. The sheer size and dominance of these companies combined with their aggression regarding the acquisition of competitors makes entering these markets as a direct competitor very unattractive at first glance. However, this issue is not as one - sided as that. This paper aims to rationalize the well-known “kill zone” argument by providing a simple model that explores how and when an incumbent firm may induce an entrant to choose a “non-aggressive” innovation path and enter with the goal of being acquired.
The different models revealed that platform-owning incumbents react in diametrically opposing fashion to an entrant’s plans to develop a substitute to their platform and a complement. When a larger firm is dominating a certain sector and new firms are trying to enter this market, these firms may feel a hesitation to produce a direct competitor to the products of the incumbent, and they will veer more towards producing a compliment as the prospect of the incumbent copying or acquiring the entrant looms. This is the reason a “kill zone” may emerge. Interestingly, the possibility of an acquisition by the incumbent does not worsen the “kill zone” effect. In fact, it may even induce the entrant to develop a product that rivals the incumbent in the hope of being acquired as this can generate massive profits for the smaller entrant. Meanwhile, a two – sided market will not alter the “kill zone” significantly compared to the basic model. Only simultaneous choices of both parties will avoid the existence of a “kill zone” since the choice of the entrant cannot prevent the incumbent to copy the entrant.
The API - Documentation of the package provides the following information:
The paper, which constitutes the foundation of this program, can be found here. On GitHub you can find the source code for this project.
# setup environment
# dependencies
%matplotlib inline
import matplotlib.pyplot as plt
%config InlineBackend.figure_format ='svg'
import ipywidgets as widgets
from IPython.display import display
# install and import models
#%pip install Shelegia_Motta_2021
#%pip uninstall Shelegia_Motta_2021
import Shelegia_Motta_2021
# opens the latest version of the documentation
# Shelegia_Motta_2021.docs()
The following products are currently or potentially sold by the entrant/incumbent:
| Product | Description |
|---|---|
| $I_P$ | Primary product sold by the incumbent. |
| $I_C$ | Complementary product to $I_P$ potentially sold by the incumbent, which is an exact copy of $E_C$. |
| $E_P$ | Perfect substitute to $I_P$ potentially sold by the entrant. |
| $E_C$ | Complementary product to $I_P$ (or $E_P$) currently sold by the entrant |
| $\tilde{E}_C$ | Complementary product to $I_P$ potentially sold by the entrant. |
The following table shows the flow of the game played by the entrant ($E$) and the incumbent ($I$)
| Time | Action |
|---|---|
| 0 | The entrant chooses to either develop a substitute or an additional complement. |
| 1(i) | The incumbent decides on a strategy that is, it can either take no action or create an identical copy of the entrant’s complementary product. We assume that the copy will be immediately available. |
| 1(ii) | The first period market interaction. At this stage only the primary product of the incumbent, the complementary product of the entrant, and possibly the copy of the complementary product by the incumbent can be sold in the market. |
| 2(i) | The entrant decides whether to develop or not the second product, chosen at t = 0, at the investment cost K. |
| 2(ii) | If the entrant has obtained finance and tried to develop the product. The product will become available. |
| 2(iii) | The second period market interaction: active firms sell in the market (all market configurations are possible), payoffs are realized, and contracts are honored. |
| Parameter | Description |
|---|---|
| $u$ | Payoff gained from the primary product. |
| $B$ | Even if the development of the additional product fails, the entrant gains a private benefit, called $B$. |
| $\delta$ | Additional utility gained from any complementary product combined with a primary product. |
| $\Delta$ | Additional utility gained from the substitute compared to the primary product of the incumbent. |
| $K$ | Investment costs to develop a second product for the entrant. |
Additional parameter added in the BargainingPowerModel:
| Parameter | Description |
|---|---|
| $\beta$ | Bargaining power of the incumbent relative to the entrant. |
See "Effects of the parameters on the Kill Zone" for the impact of the parameters on the Kill Zone.
Payoffs with the parameter $\beta$ (set $\beta = 0.5$ for the payoffs in the BaseModel):
| Market Configuration | Profit Incumbent | Profit Entrant | Consumer Surplus | Total Welfare |
|---|---|---|---|---|
| $I_P$;$E_C$ | $u+\delta\beta$ | $\delta(1-\beta)$ | 0 | $u+\delta$ |
| $I_P+I_C$;$E_C$ | $u+\delta$ | 0 | 0 | $u+\delta$ |
| $I_P$;$E_P+E_C$ | 0 | $\Delta+\delta$ | $u$ | $u+\Delta+\delta$ |
| $I_P+I_C$;$E_P+E_C$ | 0 | $\Delta$ | $u+\delta$ | $u+\Delta+\delta$ |
| $I_P$;$E_C+\tilde{E}_C$ | $u+2\delta\beta$ | $2\delta(1-\beta)$ | 0 | $u+2\delta$ |
| $I_P+I_C$;$E_C+\tilde{E}_C$ | $u+\delta(1+\beta)$ | $\delta(1-\beta)$ | 0 | $u+2\delta$ |
Thresholds for the assets of the entrant (set $\beta=0.5$ for the thresholds in the BaseModel):
| Threshold | Name | Formula of the Threshold | Threshold Description |
|---|---|---|---|
| $\underline{A}_S$ | A_s | $B+K-\Delta-\delta(2-\beta)$ | Minimum level of initial assets that ensure that the project of a perfect substitute gets funded if the incumbent does not copy. |
| $\underline{A}_C$ | A_c | $B+K-3\delta(1-\beta)$ | Minimum level of initial assets that ensure that the project of another complement gets funded if the incumbent does not copy. |
| $\overline{A}_S$ | A-s | $B+K-\Delta$ | Minimum level of initial assets that ensure the project of a perfect substitute gets funded if the incumbent copies. |
| $\overline{A}_C$ | A-c | $B+K-\delta(1-\beta)$ | Minimum level of initial assets that ensure that the project of another complement gets funded if the incumbent copies. |
Thresholds for the fixed costs of copying of the incumbent (set $\beta=0.5$ for the thresholds in the BaseModel):
| Threshold | Name | Formula of the Threshold | Threshold Description |
|---|---|---|---|
| $F^{YY}_S$ | F(YY)s | $\delta(1-\beta)$ | Maximum level of copying costs that ensure that the incumbent copies the entrant if the entrant is guaranteed to invest in a perfect substitute. |
| $F^{YN}_S$ | F(YN)s | $u+\delta(2-\beta)$ | Maximum level of copying costs that ensure that the incumbent copies the entrant if the copying prevents the entrant from developing a perfect substitute. |
| $F^{YY}_C$ | F(YY)c | $2\delta(1-\beta)$ | Maximum level of copying costs that ensure that the incumbent copies the entrant if the entrant is guaranteed to invest in another complement. |
| $F^{YN}_C$ | F(YN)c | $\delta(2-3\beta)$ | Maximum level of copying costs that ensure that the incumbent copies the entrant if the copying prevents the entrant from developing another complement. |
| $F^{ACQ}_S$ | F(ACQ)s | $\frac{(u+\Delta-K)}{2}+\delta(2-\beta)$ | Maximum level of fixed costs that ensure that the incumbent acquires the entrant if the entrant develops a substitute. |
| $F^{ACQ}_C$ | F(ACQ)c | $\delta(\frac{5}{2}-3\beta)-\frac{K}{2}$ | Maximum level of fixed costs that ensure that the incumbent acquires the entrant if the entrant develops a second complement. |
The name column, denotes the name of the thresholds in dictionaries used in the program.
The base model of the project consists of two players: The incumbent, which sells the primary product, and a start-up otherwise known as the entrant which sells a complementary product to the incumbent. One way to visualize a real-world application of this model would be to think of the entrant as a product or service that can be accessed through the platform of the incumbent, like a plug in that can be accessed through Google or a game on Facebook. The aim of this model is to monitor the choice that the entrant has between developing a substitute to or another compliment to the incumbent. The second aim is to observe the choice of the incumbent of whether to copy the original complementary product of the entrant by creating a perfect substitute or not. Seeing as the entrant may not have enough assets to fund a second product, the incumbent copying its first product would inhibit the entrant’s ability to fund its projects. This report will illustrate how the incumbent has a strategic incentive to copy the entrant if it is planning to compete and that it would refrain from copying if the entrant plans to develop a compliment. The subsequent models included in this report will introduce additional factors but will all be based on the basic model.
The equilibrium path arguably supports the “kill zone” argument: due to the risk of an exclusionary strategy by the incumbent, a potential entrant may prefer to avoid a market trajectory which would lead it to compete with the core product of a dominant incumbent and would choose to develop another complementary product instead.
base_model: Shelegia_Motta_2021.IModel = Shelegia_Motta_2021.BaseModel()
base_model.plot_payoffs(opacity=0.4, products_legend=True)
<AxesSubplot:title={'center':'Payoffs for different Market Configurations'}, xlabel='Market Configurations'>
Interestingly, the welfare and the consumer surplus are maximized, whenever the entrant develops a substitute. Meanwhile, exactly for this situation the incumbent will earn its lowest profits. This fact indicates, that the incumbent does not want to compete with the entrant, so he will try to prevent this situation through copying the entrant.
fig_base, (axis_best, axis_eq) = plt.subplots(ncols=2, figsize=(12, 8))
base_model.plot_incumbent_best_answers(axis=axis_best, title="BaseModel Best Answers", x_max=0.7, y_max=2, costs_legend=True, asset_legend=True, legend_width=70)
base_model.plot_equilibrium(axis=axis_eq, title="BaseModel Equilibrium", x_max=0.7, y_max=2, options_legend=True)
<AxesSubplot:title={'left':'BaseModel Equilibrium'}, xlabel='Assets of the entrant', ylabel='Fixed costs of copying for the incumbent'>
Besides the parameters used in the paper (and in the BaseModel), this class will introduce the parameter $\beta$ in the models, called the bargaining power of the incumbent. $\beta$ describes how much of the profits from the complementary product of the entrant will go to the incumbent In the paper the default value $\beta=0.5$ is used to derive the results, which indicate an equal share of the profits.
bargaining_power_model: Shelegia_Motta_2021.IModel = Shelegia_Motta_2021.BargainingPowerModel(beta=0.6)
bargaining_power_model.plot_payoffs(opacity=0.4, products_legend=True)
<AxesSubplot:title={'center':'Payoffs for different Market Configurations'}, xlabel='Market Configurations'>
If the values of $\beta$ are not too far from the default value, the changes in the payoffs are hardly observable.
fig_bargaining, (axis_best, axis_eq) = plt.subplots(ncols=2, figsize=(12, 5))
bargaining_power_model.plot_incumbent_best_answers(axis=axis_best, title="BargainingPowerModel Best Answers", x_max=0.7, y_max=2)
bargaining_power_model.plot_equilibrium(axis=axis_eq, title="BargainingPowerModel Equilibrium", x_max=0.7, y_max=2)
<AxesSubplot:title={'left':'BargainingPowerModel Equilibrium'}, xlabel='Assets of the entrant', ylabel='Fixed costs of copying for the incumbent'>
Due to the variable parameter $\beta$, a new area between $F^{YY}_S$ and $F^{YN}_C$ emerges in the best answers of the incumbent. Since the entrant does not develop a complementary product in the equilibrium, the equilibrium is not affected from this change.
This model indicates that if the incumbent were not able to observe the entrant at the moment of choosing, the “kill zone” effect whereby the entrant stays away from the substitute in order to avoid being copied would not take place. Intuitively, in the game as we studied it so far, the only reason why the entrant is choosing a trajectory leading to another complement is that it anticipates that if it chose one leading to a substitute, the incumbent would copy, making it an inefficient strategy for entering the market. However, if the incumbent cannot observe the entrant’s choice of strategy, the entrant could not hope to strategically affect the decision of the incumbent. This would lead to the entrant having a host of new opportunities when entering the market makes the entrant competing with a large company much more attractive.
Although there may be situations where the entrant could commit to some actions (product design or marketing choices) which signals that it will not become a rival, and it would have all the incentive to commit to do so, then the game would be like the sequential moves game analyzed in the basic model. Otherwise, the entrant will never choose a complement just to avoid copying, and it will enter the “kill zone”.
unobservable_model: Shelegia_Motta_2021.IModel = Shelegia_Motta_2021.UnobservableModel(beta=0.6)
fig_unobservable, axis_eq = plt.subplots(figsize=(6, 5))
unobservable_model.plot_equilibrium(axis=axis_eq, title="UnobservableModel Equilibrium", x_max=0.7, y_max=2)
<AxesSubplot:title={'left':'UnobservableModel Equilibrium'}, xlabel='Assets of the entrant', ylabel='Fixed costs of copying for the incumbent'>
In order to explore how acquisitions may modify the entrant’s and the incumbent’s strategic choices, we extend the base model in order to allow an acquisition to take place after the incumbent commits to copying the entrant’s original complementary product (between t=1 and t=2, see demo.ipynb "Timing of the game"). We assume that the incumbent and the entrant share the gains (if any) attained from the acquisition equally.
The “kill zone” still appears as a possible equilibrium outcome, however for a more reduced region of the parameter space. The prospect of getting some acquisition gains does tend to increase the profits gained from developing a substitute to the primary product, and this explains why part of the “kill zone” region where a complement was chosen without the acquisition, the entrant will now choose a substitute instead.
acquisition_model: Shelegia_Motta_2021.IModel = Shelegia_Motta_2021.AcquisitionModel(beta=0.6)
fig_acquisition, (axis_best, axis_eq) = plt.subplots(ncols=2, figsize=(12, 10))
acquisition_model.plot_incumbent_best_answers(axis=axis_best, title="AcquisitionModel Best Answers", x_max=0.7, y_max=2, costs_legend=True)
acquisition_model.plot_equilibrium(axis=axis_eq, title="AcquisitionModel Equilibrium", x_max=0.7, y_max=2, options_legend=True)
<AxesSubplot:title={'left':'AcquisitionModel Equilibrium'}, xlabel='Assets of the entrant', ylabel='Fixed costs of copying for the incumbent'>
An alternative formulation could be introduced in order to allow acquisitions to take place before the copying decision by the incumbent. The results are qualitatively like the results of the model which had the acquisition after the copying decision. In this alternative formulation of the game, copying would never occur along the equilibrium path. Indeed, there would be an additional source of gains from acquisition consisting of avoiding the fixed cost of copying.
# setup for the interactive plots
%matplotlib inline
import matplotlib.pyplot as plt
base_model: Shelegia_Motta_2021.IModel = Shelegia_Motta_2021.BaseModel()
layout_slider = widgets.Layout(width="300px")
layout_widget = widgets.Layout(width="280px", height="25px", margin_left="50")
u_slider = widgets.FloatSlider(value = 1.0, min = 0.9, max = 1.1, step = 0.01, description = '$u$', layout=layout_slider)
B_slider = widgets.FloatSlider(value = 0.5, min = 0.4, max = 0.6, step = 0.01, description = '$B$', layout=layout_slider)
small_delta_slider = widgets.FloatSlider(value = 0.5, min = 0.4, max = 0.6, step = 0.01, description = '$\delta$', layout=layout_slider)
delta_slider = widgets.FloatSlider(value = 0.51, min = 0.4, max = 0.6, step = 0.01, description = '$\Delta$', layout=layout_slider)
K_slider = widgets.FloatSlider(value = 0.2, min = 0.1, max = 0.3, step = 0.01, description = '$K$', layout=layout_slider)
beta_slider = widgets.FloatSlider(value = 0.5, min = 0.4, max = 0.6, step = 0.01, description = '$\\beta$', layout=layout_slider)
models = widgets.Dropdown(options=['BargainingModel', 'UnobservableModel', 'AcquisitionModel'], value='BargainingModel', layout=layout_widget)
def on_button_clicked(_):
u_slider.value = 1
B_slider.value = 0.5
small_delta_slider.value = 0.5
delta_slider.value = 0.51
K_slider.value = 0.2
beta_slider.value = 0.5
button = widgets.Button(description="Reset", layout=layout_widget)
button.on_click(on_button_clicked)
def plot(model:str, u: float, B: float, small_delta: float, delta: float, K: float, beta: float):
figure, (axis_base, axis_bargaining) = plt.subplots(ncols=2, figsize=(12, 4))
base_model.plot_equilibrium(axis=axis_base, title="BaseModel (default parameters)", x_max=0.7, y_max=2)
try:
if model == 'BargainingModel':
Shelegia_Motta_2021.BargainingPowerModel(u=u, B=B, delta=delta, small_delta=small_delta,K=K, beta=beta).plot_equilibrium(axis=axis_bargaining, title=model, x_max=0.7, y_max=2)
elif model == 'UnobservableModel':
Shelegia_Motta_2021.UnobservableModel(u=u, B=B, delta=delta, small_delta=small_delta,K=K, beta=beta).plot_equilibrium(axis=axis_bargaining, title=model, x_max=0.7, y_max=2)
else:
Shelegia_Motta_2021.AcquisitionModel(u=u, B=B, delta=delta, small_delta=small_delta,K=K, beta=beta).plot_equilibrium(axis=axis_bargaining, title=model, x_max=0.7, y_max=2)
except AssertionError as e:
# prints the reason for a failed assertion
print(e)
# run the cell above, to ensure the functionality of the interactive plots
display(widgets.GridBox([u_slider, B_slider, small_delta_slider, delta_slider, K_slider, beta_slider, models, button], titles=["$u$"], layout=widgets.Layout(grid_template_columns="repeat(4, 300px)"), align_content="stretch"))
widgets.interactive_output(plot, {'model': models, 'u': u_slider, 'B':B_slider, 'small_delta': small_delta_slider, 'delta':delta_slider, 'K': K_slider, 'beta': beta_slider})
GridBox(children=(FloatSlider(value=1.0, description='$u$', layout=Layout(width='300px'), max=1.1, min=0.9, st…
Output()
In the following sections we will discuss effects of the parameters on the kill zone and the effects of copying on consumer surplus and welfare.
These effects are valid for the BaseModel and the BargainingPowerModel, but only in a limited way for the UnobservableModel and the AcquisitionModel, since this models use different thresholds or result in different equilibrium paths.
The effects described in the table below, are meant to be ceteris paribus.
| Parameter Trend | Effect | Affected Threshold(s) (Fixed costs and assets) | Explanation |
|---|---|---|---|
| $u\uparrow$ | $\uparrow$ | $F^{YN}_S\uparrow$ | As $u$ increases, the incumbent makes a higher profit from his primary product and therefore will be willing to copy the entrant in more cases, if it will prevent the entrant from developing a substitute. |
| $u\downarrow$ | $\downarrow$ | $F^{YN}_S\downarrow$ | As $u$ decreases, the incumbent makes a smaller profit from his primary product and therefore will be willing to copy the entrant in less cases, if it will prevent the entrant from developing a substitute. |
| $B\uparrow$ | $\uparrow$ | $\overline{A}_S\uparrow$ | As $B$ increases, the higher private benefit of the entrant leads to a higher burden to find potential investors for additional assets. Therefore, the entrant will rather develop a complementary product, to prevent the incumbent form copying its product. |
| $B\downarrow$ | $\downarrow$ | $\overline{A}_S\downarrow$ | As $B$ decreases, the smaller private benefit of the entrant leads to a lower burden to find potential investors for additional assets. Therefore, the entrant will more likely develop a substitute. |
| $\delta\uparrow$ | $\uparrow$ | $\Delta F^{YN}_S>\Delta F^{YN}_C$ | As $\delta$ increases, the incumbent will gain more from the complementary product produced by the entrant (this effect is stronger than the additional profit for the entrant). Therefore, the incentive for the incumbent to not copy the entrant increases, if the entrant develops a complementary product. |
| $\delta\downarrow$ | $\downarrow$ | $\Delta F^{YN}_S>\Delta F^{YN}_C$ | As $\delta$ decreases, the incumbent will gain less from the complementary product produced by the entrant (this effect is stronger than the loss of profit for the entrant). Therefore, the incentive for the incumbent to copy the entrant increases, if the entrant develops a complementary product. |
| $\Delta\uparrow$ | $\downarrow$ | $\overline{A}_S\uparrow$ | As $\Delta$ increases, the payoff for the entrant for a substitute increases. Therefore, the entrant will rather try to develop a substitute than a complementary product. |
| $\Delta\downarrow$ | $\uparrow$ | $\overline{A}_S\downarrow$ | As $\Delta$ decreases, the payoff for the entrant for a substitute decreases. Therefore, the entrant will rather try to develop a complementary product than a substitute. |
| $K\uparrow$ | $\uparrow$ | $\overline{A}_S\uparrow$ | As $K$ increases, the development costs for an additional product increase. Therefore, the entrant needs to obtain more additional assets for the development, if the incumbent copies the complementary product. |
| $K\downarrow$ | $\downarrow$ | $\overline{A}_S\downarrow$ | As $K$ decreases, the development costs for an additional product decrease. Therefore, the entrant needs to obtain less additional assets for the development, if the incumbent copies the complementary product. |
| $\beta\uparrow$ | $\uparrow$ | $\Delta F^{YN}_C>\Delta F^{YN}_S$ | As $\beta$ increases, the profits of the incumbent from the complementary product of the entrant increase. Therefore, the incumbent will less likely copy the entrant, if the entrant develops a complementary product. |
| $\beta\downarrow$ | $\downarrow$ | $\Delta F^{YN}_C>\Delta F^{YN}_S$ | As $\beta$ decreases, the profits of the incumbent from the complementary product of the entrant decrease. Therefore, the incumbent will more likely copy the entrant, if the entrant develops a complementary product. |
This section will analyze the effects of copying on welfare. Firstly, it leads to an increase in costs as the total consumer surplus does not increase when a copy of the original complement is produced. However, a wasteful fixed cost emerges. Secondly, copying may induce the entrant into not developing a new project which would be financially detrimental because we assume that whenever the entrant introduces either a substitute or complement, society gains. Finally, copying may lead E to developing a complement rather than a substitute, which may have a positive or negative effect on welfare.
| Entrant choice | Development | Conditions regarding the thresholds and parameters | Effect of copying on the welfare |
|---|---|---|---|
| $\sigma_E = C$ | $Y$ | $F_C^{YN}\le F\le F_S^{YN}, A<{\overline{A}}_S, \delta \le \Delta$ | (weakly) detrimental |
| $\sigma_E = C$ | $Y$ | $F_C^{YN}\le F\le F_S^{YN}, A<{\overline{A}}_S, \delta > \Delta$ | beneficial |
| $\sigma_E = C/S$ | $N$ | $F\le F_C^{YN}, A<{\overline{A}}_S$ | detrimental |
| $\sigma_E = S$ | $Y$ | $F\le F_S^{YY}, \overline{A}_S \le A$ | detrimental |
| $\sigma_E = S$ | $Y$ | All other cases | Unaffected as the incumbent does not copy. |
$\sigma_{E}$ denotes the choice of the entrant.
We are also interested in looking at the effects of copying on consumer surplus. A necessary condition for consumers to obtain positive surplus is that a substitute to the primary good is developed; otherwise, they have zero surplus. Given that there is a substitute, they will receive an additional surplus whenever copying arises due to competition between the entrant and the incumbent.
| Entrant choice | Development | Conditions regarding the thresholds and parameters | Effect of copying on the consumer surplus |
|---|---|---|---|
| $\sigma_E = C$ | $Y$ | $F_C^{YN}\le F\le F_S^{YN}, A<{\overline{A}}_S$ | detrimental |
| $\sigma_E = C/S$ | $N$ | $F\le F_C^{YN}, A<{\overline{A}}_S$ | detrimental |
| $\sigma_E = S$ | $Y$ | $F\le F_S^{YY}, {\overline{A}}_S\le A$ | beneficial |
| $\sigma_E = S$ | $Y$ | All other cases | Unaffected as the incumbent does not copy. |
$\sigma_{E}$ denotes the choice of the entrant.