#### 4.1. Optimality Conditions

In this section, we solve for the subgame perfect equilibria by applying backward induction. First, we analyze the optimal behavior in Period 2, and then we switch to Period 1. In each period, we first solve the game in Stage 2 by analyzing the optimal price-setting behavior, and then we study the optimal behavior in the contest in Stage 1.

#### 4.1.1. Optimal Behavior in Period 2

In Period 2, platform managers observe the decisions made in Period 1 and take them as given. Additionally, platform manager i anticipates platform manager j’s optimal second-period decision as given.

**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:

subject to condition (1) with

${\lambda}^{a}\left({u}_{i,t}^{a}\right)={u}_{i,t}^{a}$ and

${\lambda}^{b}\left({u}_{i,t}^{b}\right)={u}_{i,t}^{b}.$ The solution to the maximization problem is derived in the next lemma.

**Lemma** **1.** In Period 2, the equilibrium in prices and quantities on platform$i\in \{1,2\}$are given by the following: Lemma 1 shows that group-a and group-b agents of platform i demand an equal quantity ${q}_{i,2}^{a\ast}={q}_{i,2}^{b\ast}$ in equilibrium, due to the symmetry of the two markets. The conditions ${n}_{i}^{a}<1$ and ${n}_{i}^{b}<1$ ensure positive quantities and prices. Furthermore, stronger combined network effects ${n}_{i}^{a}+{n}_{i}^{b}$ yield higher quantities for both group-a and group-b agents in equilibrium. This is intuitive, because stronger group-b network effects and thus increased combined network effects lead to an increase in the demand of group-a agents. A larger group-a network effect induces an increase in demand on the part of group-b agents.

In contrast to the equilibrium quantities, the equilibrium prices differ between group a and group b. The side with the stronger network effects pays a lower price in equilibrium. Note that the higher the price ${p}_{i,2}^{a\ast}$ for group a (${p}_{i,2}^{b\ast}$ for group b), the stronger the positive group-b network effects ${n}_{i}^{b}$ (group-a network effects ${n}_{i}^{a}$).

**Stage 1:** Using the results from Lemma 1, the expected profits of platform

i in Period 2 thus amount to the following:

Anticipating the optimal behavior on Stage 2, platform i chooses its assets stock ${m}_{i,2}$ in Stage 1 to maximize expected profits ${\pi}_{i,2}$ in Period 2 given by (4), which incorporates the intertemporal accumulation equation ${m}_{i,2}=(1-\delta ){m}_{i,1}+{x}_{i,2}$. We establish the following lemma.

**Lemma** **2.** In Period 2, platform i’s implicit reaction function is given by the following:such that the following condition must hold in equilibrium:with$i,j\in \{1,2\}$and$i\ne j$.

**Proof.** We use Equation (

4) to derive the first-order condition of platform

i. The first-order condition implicitly defines the reaction function. Combining the two reaction functions of platform

i and

j, we obtain the condition (5). □

Lemma 2 implicitly defines the optimal asset stocks ${m}_{i,2}^{*}({m}_{i,1},{m}_{j,1})$ of platform $i\in \{1,2\}$ in Period 2.

#### 4.1.2. Optimal Behavior in Period 1

In Period 1, platform manager $i\in \{1,2\}$ maximizes its discounted expected profits ${\pi}_{i}={\pi}_{i,1}+\beta {\pi}_{i,2}$, anticipating the optimal reactions $({m}_{i,2}^{*},{m}_{j,2}^{*})$ in Period 2 and assuming that platform manager j invests ${m}_{j,1}^{*}$ with $i,j\in \{1,2\}$ and $i\ne j$.

**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:

The solution to the maximization problem is derived in the next lemma.

**Lemma** **3.** In Period 1, the equilibrium in prices and quantities on platform i are given by the following: **Proof.** This is similar to Stage 2 of Period 2 and is, therefore, omitted. □

In Period 1, the optimal price setting on Stage 2 is similar to that in Period 2 on Stage 2 because prices have no transitional influence on Period 2 decisions. Only the choice of the asset stocks in Period 1 generates transitional effects in Period 2, as shown by Lemmas 1 and 2. Next, we analyze these transitional effects. In order to do that, we first use the results from Lemma 3 in (2). Thus, the discounted expected profits of platform

$i\in \{1,2\}$ amount to the following:

**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.** In Period 1, platform manager i’s implicit reaction function is given by the following:with${\kappa}_{i}\equiv \frac{\partial {\pi}_{i}}{\partial {m}_{j,2}}\frac{\partial {m}_{j,2}^{*}({m}_{j,1}^{*},{m}_{i,1})}{\partial {m}_{i,1}}$. Hence, the following condition must hold in equilibrium:with$i,j\in \{1,2\}$and$i\ne j$.

In the next section, we solve the whole model based on the derived results. In order to do that, we have to differentiate between two equilibrium concepts: open-loop and closed loop equilibrium.

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

In this subsection, we aim to analyze the effects of different kinds of heterogeneity. In order to obtain explicit solutions, we assume constant marginal costs with ${c}_{i}\left({x}_{i,t}\right)={c}_{i}{x}_{i,t}$, and ${c}_{i}>0$ and set the discriminatory power parameter to one, i.e., $\gamma =1$. For notational sake, we write ${\eta}_{i}\equiv {n}_{i}^{a}+{n}_{i}^{b},$, which denotes the combined network effect on platform $i\in \{1,2\}$.

First, we analyze the asset stocks and we obtain the following proposition derived from Lemma 2 and 4.

**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\},$$i\ne j$in Period 2 are given by the following: - (iii)
The optimal asset stocks of platform$i\in \{1,2\},$$i\ne j$in Period 1 are given by the following:

**Proof.** The proof for result (ii) is straightforward and therefore omitted. The critical issue to derive results (i) and (iii) is to perceive that ${\kappa}_{i}=0$ because $\partial {m}_{j,2}^{*}({m}_{j,1}^{*},{m}_{i,1})/\partial {m}_{i,1}=0$ due to result (ii). □

The winning probability of platform

$i\in \{1,2\}$ in period

$t\in \{1,2\}$ is the following:

Analyzing the results of Proposition 1, we obtain additional insights regarding the relative asset stocks and comparative statics summarized in the following proposition.

**Proposition** **2.** For$i,j\in \{1,2\}$with$i\ne j$, we derive the following results:

- (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.** The proof is straightforward and therefore omitted. □

Part (i) shows that the asset stock is always higher in Period 1 than in Period 2, i.e., ${m}_{i,1}^{*}>{m}_{i,2}^{*}$ because $1/\left(1-\beta (1-\delta )\right)>1$. That is, a dynamic competition leads to higher investments in asset stock today since part of the asset stock is also valuable in the future. The lower the depreciation rate $\delta $ and/or the higher the discount factor $\beta $, the larger the incentives to invest in the asset stock today. This result may explain why Apple, as the producer of iPads, makes large investments in R&D. Apple anticipates that today’s investments are also partly valuable in the future, when new versions are released.

Regarding Part (ii), we find that whether platform

i has a higher asset stock than platform

j in period

$t\in \{1,2\}$ depends on the relationship between relative costs and relative network effects. Formally, we derive:

Note that lower marginal costs and higher network effects increase a platform’s incentive to invest.

In the case of symmetric network effects, i.e., ${\eta}_{i}={\eta}_{j}$, the platform with the lower costs has the higher asset stock in equilibrium in both periods. On the other hand, in case of symmetric costs, i.e., ${c}_{i}={c}_{j}$, the platform with the stronger combined network effects has the higher asset stock in equilibrium in both periods. Combining both results, we see that stronger network effects can compensate for higher costs. This result suggests that a developer of a tablet PC should not just be interested in cost efficiency to prevail in a market. A competitive advantage may also arise from higher network effects. For instance, Apple’s App Store may increase these network effects by connecting the two sides (app producers and app users) more closely.

Part (iii) shows that asset stocks always increase in own network effects, i.e.,

$\frac{\partial {m}_{i,t}^{*}}{\partial {\eta}_{i}}>0$, while the impact of stronger network effects on the other platform depends on the relationship between relative costs and relative network effects. In particular, asset stocks on the own platform increase in the network effects of the other platform, i.e.,

$\frac{\partial {m}_{i,t}^{*}}{\partial {\eta}_{j}}>0$ if the own platform has a higher asset stock in equilibrium, i.e.,

${m}_{i,t}^{*}>{m}_{j,t}^{*}$ so that condition (

8) holds. This result suggests that Apple has incentives to improve their technology (e.g., through higher R&D investments) if the network effects on their platform increase. However, if the network effects on the competing platform increase, Apple will improve their technology only if they are already the dominating platform characterized through a better technology, i.e., a larger asset stock.

According to Part (iv), the probability that a platform wins the platform contest increases in own network effects, and it decreases in the network effects of the other platform. Assume that platform i has a higher level of asset stocks than platform j. It follows that stronger network effects ${\eta}_{i}$ on platform i makes the platform contest less balanced and stronger network effects ${\eta}_{j}$ on platform j makes the platform contest more balanced. This result suggests that Apple might defend its head start by trying to find ways to increase their network effects. In this case, Apple directly benefits through an increase in its asset stock. In addition, there is an indirect effect on the opponent’s behavior because Apple’s opponents have fewer incentives to improve their technology. As a consequence, the contest becomes less balanced and Apple can further increase its advantage.

Next, we derive the optimal investment path for both platforms in the following proposition.

**Proposition** **3.** - (i)
In Period 1, the manager of platform i invests the following:to build up asset stock given by${m}_{i,1}^{*}$.

- (ii)
In Period 2, the manager of platform i invests the following:to build up asset stock given by${m}_{i,2}^{*}.$

**Proof.** The proof is straightforward and therefore omitted. □

To see the interplay between costs and initial asset stock

${m}_{i,0}$, we assume that platform

j has a head start in the beginning of the contest in Period 0. Thus, platform

j has a higher initial asset stock than platform

i, i.e.,

${m}_{j,0}>{m}_{i,0}$. In addition, suppose that condition (

8) holds for

$t=1$. In this case, we know that platform

i will have a higher asset stock in Period 1 than platform

j. That is, independently of initial asset stocks, platform

i will have a higher probability to win the competition in Period 1. The reason is that platform

i will invest more than platform

j in Period 1, such that an immediate leapfrogging occurs. Note that the lower the

${m}_{i,0}$, the higher the first-period investments

${x}_{i,1}^{*}$ of platform

i. Thus, initial asset stocks in

$t=0$ are not decisive regarding future winning probabilities if costs are linear.

The effects of the head start in Period 0 on winning probabilities are immediately wiped out in Period 1. Initial asset stocks only affect the size of the first-period investments. One could argue that, even if Apple had some technological advance with its iPad, it can only sustain this advantage if the cost efficiency in combination with the network effects is superior, compared to the competitor. Samsung has adumbrated with its Galaxy Tab S7 that a catch up or even a leapfrogging can occur quite quickly.

Moreover, in the case that initial asset stocks are zero for both platforms, second-period investments ${x}_{i,2}^{*}$ are always lower than first-period investments ${x}_{i,1}^{*}$ since part of the first-period asset stocks can be used in the second period.

Finally, we derive expected profits of the two platforms and we analyze how costs and network effects influence expected profits.

**Proposition** **4.** - (i)
Expected profits of platform$i\in \{1,2\}$in equilibrium are given by the following:with$i,j\in \{1,2\}$and$i\ne j$.

- (ii)
We derive the following comparative statics:

**Proof.** The proof is straightforward and therefore omitted. □

The comparative statics analysis shows that higher initial asset stocks ${m}_{i,0}$ yield higher overall platform profits. This result is intuitive because a higher initial asset stock ${m}_{i,0}$ reduces investment costs ${c}_{i}{x}_{i,1}^{*}={c}_{i}{m}_{i,1}^{*}-{c}_{i}(1-\delta ){m}_{i,0}$ in Period 1. However, less intuitive and more interesting is the fact that higher marginal costs ${c}_{i}$ on the own platform produce higher expected profits. On the other hand, higher marginal costs ${c}_{j}$ on the other platform produce lower expected profits. Finally, we derive that expected profits increase in own network effects and decrease in the network effects of the other platform.

Next, we assume that

${m}_{i,0}=0$ for

$i\in \{1,2\}$. We derive the following:

with

$i,j\in \{1,2\}$ and

$i\ne j$ according to result (i). Basically, the same reasoning applies to a platform’s profit as above. Stronger network effects compensate for higher costs. The difference, however, is that the network-related effect is stronger than the cost-related effect. For

$1<\frac{2-{\eta}_{i}}{2-{\eta}_{j}}<\frac{{c}_{j}}{{c}_{i}}<{\left(\frac{2-{\eta}_{i}}{2-{\eta}_{j}}\right)}^{3/2}$, platform

i has larger asset stocks but lower expected profits than platform

j (see

Figure 1a). For

${\left(\frac{2-{\eta}_{i}}{2-{\eta}_{j}}\right)}^{3/2}<\frac{{c}_{j}}{{c}_{i}}<\frac{2-{\eta}_{i}}{2-{\eta}_{j}}<1$, platform

i has larger expected profits but lower asset stocks than platform

j (see

Figure 1b). Thus, we obtain the following corollary:

**Corollary** **1.** A dominance in a market characterized by a larger asset stock and therefore by a larger winning probability in the contest does not have to imply a higher (expected) profitability.

**Proof.** The proof is straightforward and therefore omitted. □

Corollary 1 provides a theoretical foundation that a firm’s size and profit are not necessarily positively correlated. Smaller platforms possibly are more profitable than larger platforms.

To sum up, this section has focused on the effect of heterogeneity with respect to costs and network effects. The assumption of linear costs is very helpful to simplify matters with respect to the dynamics. In the case of linear costs, symmetric marginal costs ${c}_{i}={c}_{j}$ and symmetric network effects ${\eta}_{i}={\eta}_{j}$, the winning probabilities and consequently competitive balance are equal in Periods 1 and 2 and do not depend on the depreciation rate $\delta $. Moreover, the dynamics regarding the other variables are trivial because convergence of the asset stocks immediately results in Period 1 (even for different initial asset stocks in Period 0).

However, a comparative advantage of a firm does not necessarily translate immediately into a competitive advantage. The transition from an underdog to a favorite might be driven by other factors, such as rigidities with respect to investments. These rigidities can be introduced into the model by assuming strictly convex costs, which we will model in the next section.

#### 4.3. Quadratic Costs and Homogeneity

To gain more insights into the dynamics of the contest, we introduce strictly convex costs given by ${c}_{i}\left({x}_{i,t}\right)=(1/2){x}_{i,t}^{2}$. For tractability, we assume that the platforms have symmetric cost functions and that the network effects are the same on both platforms, i.e., ${\eta}_{i}={\eta}_{j}=\eta $. Thus, the platforms only differ with respect to their initial asset stock. Note that we only analyze the open-loop equilibrium in this section.

We assume that platform 1 is the “large” platform in Period 0 and set ${m}_{1,0}=8$ and ${m}_{2,0}=4$. Moreover, we set $\beta =1,\phantom{\rule{4pt}{0ex}}\gamma =1$ and consider specific values for the network effect $\eta =[0,1)$ and the depreciation rate $\delta \in \{0.7,0.8,0.9\}$. We derive the following proposition:

**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.** We have to rely on numerical simulations because it is analytically not possible to derive the results with quadratic costs in closed form. We run numerous simulations and received always similar results (all simulations are available upon request). □

Result (i) is intuitive because it shows that a low depreciation rate (high technological carry-over) and/or stronger network effects increase incentives to invest into the platforms and thus increase the asset stocks. The result is illustrated in

Figure 2. Note that in the case of linear costs, a lower depreciation rate increases the asset stocks only in Period 1. However, the depreciation rate has no effect on the asset stocks in Period 2 for linear costs (see Proposition 1).

Results (ii) examines the speed of convergence: We know from the previous section that in the case of linear costs, the winning probabilities already converge in Period 1, resulting in a fully balanced competition. However, in the case of quadratic costs, convergence of the winning probabilities does not occur, neither in Period 1 nor in Period 2 (compare

Figure 3). Platform 1 benefits in both periods from the head start in Period 0, yielding an unbalanced contest with a competitive balance larger than those in Periods 1 and 2.

The simulation further shows that a higher depreciation rate $\delta $ and stronger network effects $\eta $ increase the balance of the contest in period t. This result means that heterogeneity with respect to asset stocks originating from the initial asset stocks are wiped out faster for a larger depreciation rate or stronger network effects. Therefore, we expect, according to our model, that a start-up firm catches up to the "incumbent" faster in an industry characterized by a high deprecation rate (low technological carry-over) and strong network effects. On the other hand, the combination of a low deprecation rate (high technological carry-over) and weak network effects can be responsible for the emergence of a long-lasting dominance of a specific platform.