# A Dynamic Contest Model of Platform Competition in Two-Sided Markets

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*J. Theor. Appl. Electron. Commer. Res.*

**2021**,

*16*(6), 2091-2109; https://doi.org/10.3390/jtaer16060117

## Abstract

**:**

## 1. Introduction

## 2. Related Literature

## 3. Model

**Stage 1:**Platform i (and symmetrically platform j) invests independently ${x}_{i,t}\ge 0$ in period $t\in \{1,2\}$. These investments can be interpreted as “contributions” that increase the asset stock ${m}_{i,t}$ of platform i in period $t\in \{1,2\}$. The contributions can be interpreted as investments in, for example, the infrastructure and technology of the platform, advertising or schooling their employees or lobbying activities. Investments generate costs, according to a weakly convex cost function ${c}_{i}\left({x}_{i,t}\right)$.

**Stage 2:**The platform that has won the contest in Stage 1 can offer its services as a monopolistic platform to two groups $(a,b)$ of agents. In Stage 2, the platform sets prices for each group to maximize its profits. We assume that network effects operate from the group-a market to the group-b market and vice versa. Following [23], we assume that the utilities of group-a agents ${u}_{i,t}^{a}$ and group-b agents ${u}_{i,t}^{b}$ on platform i in period t are as follows:

## 4. Results

#### 4.1. Optimality Conditions

#### 4.1.1. Optimal Behavior in Period 2

**Stage 2:**In Stage 2, platform manager i maximizes profits ${\pi}_{i,2}$ in Period 2 for given values of ${w}_{i,2}$ and ${x}_{i,2}$, and thus, s/he solves the following maximization problem:

**Lemma**

**1.**

**Proof.**

**Stage 1:**Using the results from Lemma 1, the expected profits of platform i in Period 2 thus amount to the following:

**Lemma**

**2.**

**Proof.**

#### 4.1.2. Optimal Behavior in Period 1

**Stage 2:**In Stage 2, platform manager i maximizes its discounted expected profits ${\pi}_{i}$ for given values of ${w}_{i,1}$ and ${x}_{i,1}$ and thus solves the following maximization problem:

**Lemma**

**3.**

**Proof.**

**Stage 1:**In Stage 1, platform $i\in \{1,2\}$ maximizes its discounted expected profit ${\pi}_{i}={\pi}_{i,1}+\beta {\pi}_{i,2}$, given by (6), with respect to ${m}_{i,1}$. The solution to this maximization problem is given in the next lemma:

**Lemma**

**4.**

**Proof.**

- In the open-loop equilibrium, managers do not take into account—by implication of the equilibrium concept—their strategic option in Period 1 to change the opponent’s incentive to invest in the platform in Period 2 by changing own platform investments in Period 1. In this case, $\partial {m}_{j,2}^{*}({m}_{j,1}^{*},{m}_{i,1})/\partial {m}_{i,1}=0$ such that ${\kappa}_{i}={\kappa}_{j}=0$ holds.
- In the closed-loop equilibrium, however, this strategic option is considered, and the terms ${\kappa}_{i}$ and ${\kappa}_{j}$ do not have to be zero. See [45], who provide a formal description of the two equilibrium concepts.

#### 4.2. Linear Costs and Heterogeneity

**Proposition**

**1.**

- (i)
- A unique equilibrium exists, and the closed-loop equilibrium coincides with the open-loop equilibrium.
- (ii)
- The optimal asset stocks of platform$i\in \{1,2\},$$i\ne j$in Period 2 are given by the following:$${m}_{i,2}^{*}=\frac{{c}_{j}(2-{\eta}_{j})}{{\left[{c}_{i}(2-{\eta}_{i})+{c}_{j}(2-{\eta}_{j})\right]}^{2}}.$$
- (iii)
- The optimal asset stocks of platform$i\in \{1,2\},$$i\ne j$in Period 1 are given by the following:$${m}_{i,1}^{*}=\frac{1}{1-\beta (1-\delta )}\frac{{c}_{j}(2-{\eta}_{j})}{{\left[{c}_{i}(2-{\eta}_{i})+{c}_{j}(2-{\eta}_{j})\right]}^{2}}.$$

**Proof.**

**Proposition**

**2.**

- (i)
- The asset stocks for both platforms are larger in Period 1 than in Period 2, i.e.,${m}_{i,1}^{*}>{m}_{i,2}^{*}$.
- (ii)
- The asset stock of platform i is larger than the asset stock of platform j in period$t\in \{1,2\}$iff$\frac{{c}_{j}}{{c}_{i}}>\frac{2-{\eta}_{i}}{2-{\eta}_{j}}$.
- (iii)
- Stronger network effects on platform i always increase the asset stock of platform i, while stronger network effects on platform j increase the asset stock of platform i iff$\phantom{\rule{4pt}{0ex}}\frac{{c}_{j}}{{c}_{i}}>\frac{2-{\eta}_{i}}{2-{\eta}_{j}}$.
- (iv)
- The winning probability of platform i increases with larger network effects on the own platform and it decreases with stronger network effects on the other platform, i.e.,$\frac{\partial {w}_{i,t}^{*}}{\partial {\eta}_{i}}>0$and$\frac{\partial {w}_{i,t}^{*}}{\partial {\eta}_{j}}<0$. Hence, larger network effects${\eta}_{i}$on platform i increase the balance of the contest if$\frac{{c}_{j}}{{c}_{i}}<\frac{2-{\eta}_{i}}{2-{\eta}_{j}}$.

**Proof.**

**Proposition**

**3.**

- (i)
- In Period 1, the manager of platform i invests the following:$${x}_{i,1}^{*}={m}_{i,1}^{*}-(1-\delta ){m}_{i,0}=\frac{1}{1-\beta (1-\delta )}\frac{{c}_{j}(2-{\eta}_{j})}{{\left[{c}_{i}(2-{\eta}_{i})+{c}_{j}(2-{\eta}_{j})\right]}^{2}}-(1-\delta ){m}_{i,0}$$
- (ii)
- In Period 2, the manager of platform i invests the following:$${x}_{i,2}^{*}={m}_{i,2}^{*}-(1-\delta ){m}_{i,1}^{*}=\frac{\delta -\beta (1-\delta )}{1-\beta (1-\delta )}\frac{{c}_{j}(2-{\eta}_{j})}{{\left[{c}_{i}(2-{\eta}_{i})+{c}_{j}(2-{\eta}_{j})\right]}^{2}}$$

**Proof.**

**Proposition**

**4.**

- (i)
- Expected profits of platform$i\in \{1,2\}$in equilibrium are given by the following:$${\pi}_{i}^{*}={\pi}_{i,1}^{*}+\beta {\pi}_{i,2}^{*}=(1+\beta )\frac{{c}_{j}^{2}{(2-{\eta}_{j})}^{2}}{(2-{\eta}_{i}){\left[{c}_{i}(2-{\eta}_{i})+{c}_{j}(2-{\eta}_{j})\right]}^{2}}+{c}_{i}(1-\delta ){m}_{i,0}$$
- (ii)
- We derive the following comparative statics:$$\begin{array}{cc}\hfill \frac{\partial {\pi}_{i}^{*}}{\partial {m}_{i,0}}& >0\phantom{\rule{4pt}{0ex}},\phantom{\rule{4.pt}{0ex}}\frac{\partial {\pi}_{i}^{*}}{\partial {c}_{i}}>0,\phantom{\rule{4.pt}{0ex}}\frac{\partial {\pi}_{i}^{*}}{\partial {c}_{j}}<0\hfill \\ \hfill \frac{\partial {\pi}_{i}^{*}}{\partial {\eta}_{i}}& >0,\phantom{\rule{4.pt}{0ex}}\frac{\partial {\pi}_{i}^{*}}{\partial {\eta}_{j}}<0\hfill \end{array}$$

**Proof.**

**Corollary**

**1.**

**Proof.**

#### 4.3. Quadratic Costs and Homogeneity

**Proposition**

**5.**

- (i)
- A lower depreciation rate δ or stronger network effects η increase asset stocks${m}_{i,t}^{*}$for platform i in period t.
- (ii)
- A higher depreciation rate δ or stronger network effects η increase the speed of convergence of asset stocks${m}_{i,t}^{*}$for platform i in period t.

**Proof.**

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

PC | Personal Computer |

VHS | Video Home System |

OTT | Over-The-Top |

CSF | Contest Success Function |

CB | Competitive Balance |

## Appendix A

#### Appendix A.1. Proof of Lemma 1

#### Appendix A.2. Proof of Lemma 4

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**Figure 1.**Asset stocks and profits for linear costs. (

**a**) Case $2-{\eta}_{i}>2-{\eta}_{j}$, (

**b**) Case $2-{\eta}_{i}<2-{\eta}_{j}$.

**Figure 2.**Asset stocks for quadratic costs. (

**a**) Platform 1: First period, (

**b**) Platform 2: First period, (

**c**) Platform 1: Second period, (

**d**) Platform 2: Second period.

**Figure 3.**Winning probabilities and competitive balance for quadratic costs. (

**a**) Platform 1: First period, (

**b**) Platform 2: First period, (

**c**) Platform 1: Second period, (

**d**) Platform 2: Second period, (

**e**) First period, (

**f**) Second period.

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## Share and Cite

**MDPI and ACS Style**

Grossmann, M.; Lang, M.; Dietl, H.M.
A Dynamic Contest Model of Platform Competition in Two-Sided Markets. *J. Theor. Appl. Electron. Commer. Res.* **2021**, *16*, 2091-2109.
https://doi.org/10.3390/jtaer16060117

**AMA Style**

Grossmann M, Lang M, Dietl HM.
A Dynamic Contest Model of Platform Competition in Two-Sided Markets. *Journal of Theoretical and Applied Electronic Commerce Research*. 2021; 16(6):2091-2109.
https://doi.org/10.3390/jtaer16060117

**Chicago/Turabian Style**

Grossmann, Martin, Markus Lang, and Helmut M. Dietl.
2021. "A Dynamic Contest Model of Platform Competition in Two-Sided Markets" *Journal of Theoretical and Applied Electronic Commerce Research* 16, no. 6: 2091-2109.
https://doi.org/10.3390/jtaer16060117