Strategic product compatibility in network industries

Abstract

This article considers a product’s compatibility as a strategic variable in a Cournot duopoly with network consumption externalities. It develops a non-cooperative compatibility decision game (CDG) in which firms choose whether to let products be (in)compatible. With costless compatibility, the unique (Pareto-efficient) sub-game perfect Nash equilibrium (SPNE) of the CDG is universal compatibility. With quasi-fixed compatibility costs, the SPNE depends on whether product compatibility is an endogenous (i.e., a profit-maximising) or exogenous (i.e., given by technical constraints) variable. In the case of endogenous compatibility, the SPNE can vary from one unique to multiple regimes of (in)compatibility, such as the anti-prisoner’s dilemma (deadlock), prisoner’s dilemma, and coordination game. In the case of exogenous compatibility, the SPNE can become an anti-coordination game as well. The article also discusses the welfare outcomes corresponding to the SPNE.

1 Introduction

Markets with network consumption externalities often question the compatibility of products (Katz and Shapiro 1985). The present article tackles this issue and contributes to the literature by applying a game-theoretic approach to a network Cournot duopoly to study whether the production of in(compatible) products emerges as a sub-game perfect Nash equilibrium (SPNE) in a multi-stage non-cooperative compatibility decision game (CDG) with complete information.¹

Consumption externality arises when the utility of a consumer from the consumption of goods depends on the number of users belonging to the same network. Katz and Shapiro (1985, p. 424) offer a clear, early example regarding hardware and software markets of the importance of the compatibility degree of network goods: “If two brands of hardware can work with the same software (e.g., an operating system), then the two brands are said to be compatible. Therefore, depending on how and whether software produced for working on one brand of hardware may also work on another brand of hardware, for the consumer of one brand of hardware the relevant network is the set of users who buy other brands of hardware which are totally (or partially) compatible with her brand”.

The compatibility decision must then be properly analysed, and the crucial question is made based on “whether firms will have proper [strategic] incentives to produce compatible goods or services” (Katz and Shapiro 1985, p. 425).

Despite the importance of network industries and the related issue of product compatibility,² there is still no strategic characterisation of the argument in a standard Cournot duopoly. However, researchers have devoted effort to a wide range of topics, especially (1) the timing at which product compatibility can be introduced in the market and the possibility of time inconsistencies (Regibeau and Rockett 1996), (2) the compatibility as a signalling device to evaluate the quality of a newly introduced technology for which users do not have information in a Hotelling strategic setting (Kim 2002), (3) market dominance and the relationship between the installed base and product compatibility in markets with network effects (Chen et al. 2009), (4) product compatibility and product differentiation in a Cournot duopoly (Chen and Chen 2011; Toshimitsu 2018), (5) mix-and-match compatibility in system markets (Hahn and Kim 2012), (6) the role of consumer expectations (strong, weak, and mixed) in a model à la Hotelling with network effects and product compatibility, in which firms compete on prices (Suleymanova and Wey 2012); expectations differ depending on the logical timing in which consumers make decisions relative to the firm’ pricing decisions, and (7) a dynamic network oligopoly with segmented market demand and the effects of product compatibility as an exclusionary device (Do 2023).

The studies most closely related to ours in terms of modelling approach and methodology are Chen and Chen (2011), Suleymanova and Wey (2012), and Toshimitsu (2018).

This article aims to fill the gap discussed thus far by developing a multi-stage, quantity setting CDG to appropriately model the choice to produce compatible ($K$) or non-compatible ($NK$) goods. The study also investigates the occurrence of the SPNE endogenously emerging in the market, the corresponding efficiency properties, and the social welfare outcomes. It does this by considering endogenous compatibility (the extent of product compatibility is a profit-maximising variable) and exogenous compatibility (the extent of product compatibility is determined by technological constraints).³

Firms producing network goods sustain production costs and compatibility costs. The present article considers both, and the costs to achieve compatibility are assumed to be fixed like Katz and Shapiro (1985). Examples of fixed costs of compatibility include the “costs of developing and designing a compatible product, the costs of negotiating to select a standard, and the costs of introducing a new, compatible product” (Katz and Shapiro 1985, p. 427).

The strategic choice to produce compatible or incompatible products emerges from the interaction between two counterbalancing effects on profits in the CDG. On one hand, the degree of compatibility and the intensity of the network effect (both affecting output) exert a positive effect. On the other hand, the fixed cost of compatibility (which does not affect output) has a negative effect.

If the degree of compatibility is a profit-maximising variable (endogenous on the firm side), the CDG shows a rich spectrum of SPNE depending on the network size and the extent of the fixed cost of compatibility (there exists a one-to-one relationship between these two variables in determining the relevant SPNE). This happens even though profits are always increasing in their degree of compatibility. The SPNE can vary from having one unique to multiple regimes of (in)compatibility: anti-prisoner’s dilemma (deadlock), prisoner’s dilemma, and coordination game. The article also pinpoints some relevant social welfare outcomes corresponding to the SPNE, presenting the possibility of a win–win result for society, i.e., firms and consumers are better off by producing incompatible products.

If the degree of compatibility is bounded by technological constraints (exogenous on the firm side), the CDG confirms the discussion on SPNE so far, with one relevant exception: the emergence of an anti-coordination game in which only one firm produces compatible goods. If the network effect is sufficiently low (resp. high) and, a fortiori, the degree of compatibility is sufficiently high (resp. low), the SPNE allows for a compatibility regime, which is Pareto efficient (resp. inefficient) for both firms. This is because the positive profit effect of the network and the compatibility overweighs (resp. does not overweigh) the negative profit effect of the fixed cost of compatibility. For the intermediate values of the network size and the degree of compatibility, the quasi-balance between the positive and the negative effect on profits induces the occurrence of an equilibrium in which only one firm chooses to produce compatible goods. Finally, when the degree of compatibility is high (e.g., close to the full compatibility regime), but the network effect is sufficiently low, multiple symmetric Nash equilibria emerge. This is because the strong positive effect of the high compatibility becomes sufficient to induce both firms to unilaterally produce compatible goods, but due to the low intensity of the network effect, the positive effect is not magnified enough to prevent both firms from deviating unilaterally under the compatibility regime. Then, in this case, both compatibility and incompatibility are SPNE, although for both firms, the common choice to be non-compatible would lead them to be better off.

The compatibility between two products can run in one direction (for example, because some components of one firm can be used together with those of the rival, but not vice versa).⁴ Recently, the paradigmatic illustrative case of the compatibility between the operating systems of Apple and Microsoft emerged: Windows OS can be used on Mac computers, which allows for dual booting, but Mac OS cannot be used with a PC (Kim and Choi 2015, p. 115). So far, one explanation for this type of puzzling situation has been ascribed to the fact that two firms may have various levels of network externalities or are asymmetrically located. For instance, (1) larger firms are more likely to prefer incompatibility than smaller firms (Katz and Shapiro 1985), (2) Asian computer firms are more likely to prefer intra-technology than inter-technology competition (Ferguson and Morris 1993), and (3) firms with asymmetric strategic positions with respect to quality or costs may have different preferences regarding compatibility (Einhorn 1992). The present article adds another explanation to one-way compatibility: it can emerge due to the existence of a fixed cost for compatibility.

This result provides a novel, theoretically robust, explanation emerging in an appropriate game-theoretic setting for the stylised fact of one-way compatibility, in which the firms’ history also plays a crucial role. Through the unilateral choice of compatibility, one firm can increase its market share and then use the degree of compatibility as a strategic device. The compatibility works like other well-known devices in the industrial organisation (IO) literature, such as managerial delegation or corporate social responsibility, by letting firms obtain a Pareto-efficient SPNE.

The rest of the article proceeds as follows. Section 2 presents the model and the CDG. Section 3 (resp. 4) studies the endogenous market configuration emerging in the CDG when the degree of compatibility is endogenous (resp. exogenous). Both sections also consider the social welfare outcomes corresponding to the SPNE. Section 5 outlines the main conclusions and some possible related policy issues. The Online Appendix provides modelling and mathematical details of the study presented in the main text; it also presents the CDG with horizontal product differentiation à la Singh and Vives (1984) augmenting Chen and Chen (2011) and Toshimitsu (2018), pinpointing the joint role of the extent of product differentiation and the strength of the network effect in determining the SPNE of the CDG. In this regard, (1) a reduction (resp. an increase) in product substitutability strengthens product incompatibility (resp. product compatibility) when the network consumption externality is positive (the less [resp. more] products are substitutes the weaker [resp. stronger] the bandwagon effect), and (2) an increase (resp. a reduction) in product complementarity strengthens product compatibility (resp. product incompatibility) when the network consumption externality is negative (the less [resp. more] products are complements the weaker [resp. stronger] the snob effect).

2 The model

The model directly departs from Katz and Shapiro (1985), later followed by two other relevant articles (Katz and Shapiro 1986a, b) on the topic of product compatibility in oligopolistic industries. In these contributions, the degree of compatibility was exogenously given. The IO literature has also concentrated on the case of endogenous compatibility by letting firms choose the extent of the degree of compatibility with the aim to maximise profits (e.g., Economides 1989; Kim and Choi 2015; Toshimitsu 2018; Stadler et al. 2022). The present article deals with both approaches (exogenous and endogenous compatibility) and to the literature adds the compatibility decision stage to a Cournot duopolistic industry.

The degree of product compatibility as exogenous or endogenous on the firm side depends on the main properties and the state of the art of the existing technology. If the technology does (not) allow choice of product compatibility, its extent is endogenous, and it is chosen to maximise profits (is exogenous, and it is taken as given).

If the degree of compatibility is endogenous, the CDG is a three-stage game. In the first (decision) stage, each firm chooses to let products be compatible or incompatible. In the second (intermediate) stage, each firm chooses the extent of product compatibility to maximise profits. In the third (market) stage, each firm competes à la Cournot in the product market.

If the degree of compatibility is exogenous, a two-stage game arises that includes the first decision stage and the second market stage.

2.1 Consumers and firms

Consider a Cournot duopoly in which firms produce homogeneous network goods (Katz and Shapiro 1985). The network effect (consumption externality) can be positive (e.g., mobile communications, software, internet-related activities, online social networks, fashion) or negative (e.g., traffic congestion or network congestion over limited bandwidth). Under a positive (resp. negative) externality, an increasing number of users increases (resp. reduces) the individual utility and thus the value of the goods for each consumer, thus causing the so-called bandwagon (resp. snob) effect.⁵ To tackle the issue of strategic product compatibility, we follow the main narrative of Katz and Shapiro (1985) and assume that firms are unable to commit themselves to a given output level and consumers form their expectations on total sales, which are fulfilled at equilibrium according to the standard rational expectations hypothesis (Katz and Shapiro 1985; Hoernig 2012; Suleymanova and Wey 2012).

The linear inverse market demand of firm $i$ is⁶:

$$p_{i} = 1 - q_{i} - q_{j} + n\left( {y_{i} + ky_{j} } \right),$$

(1)

where $p_{i}$ is the marginal willingness to pay toward products of network $i$ ($i,j = \left\{ {1,2} \right\}$, $i \ne j$) $q_{i}$ denotes the quantity of the goods produced by firm $i$ and $n \in \left( { - 1,1} \right)$ is the strength of the network effect (the higher the absolute value of $n$, the stronger the effect of network goods). Positive (resp. negative) values of $n$ refer to positive (resp. negative) consumption externalities generating a bandwagon (resp. snob) effect. The parameter $k \in \left[ {0,1} \right]$ (common standardisation) measures the degree of compatibility of the networks of the products of the two firms. In addition, $y_{i}$ represents an external effect denoting the consumers’ expectations about firm $i$’s equilibrium total sales, and the term $y_{i} + ky_{j}$ is the expected effective network size of firm $i$’s consumers. Consumers choose to buy the product of network $i$ or the product of network $j$ before the actual network sizes are known to them. This implies that consumers first form the expectations about the size of the networks and then the duopolistic firm $i$ and firm $j$ play a non-cooperative CDG based on Cournot rivalry by considering the hypothesis that they do not commit to an announced output level. This is done by considering consumers’ expectations as given. Consumers are rational, and their expectations are realised in equilibrium. In this sense, the model follows the basic idea of Katz and Shapiro (1985) of SPNE in which expectations are self-fulfilling.

The generic firm $i$’s profit function is given by:

$${\Pi }_{i} = \left( {p_{i} - w} \right)q_{i} - Z,$$

(2)

where $0 \le w < 1$ is the constant average and marginal cost, which is set to zero henceforth without loss of generality, and $Z > 0$ represents a quasi-fixed cost of compatibility. This assumption directly follows the original article by Katz and Shapiro (1985) and is in line with a more recent contribution by Planer-Friedrich and Sahm (2021), who assume the quasi-fixed costs of CSR with a similar narrative.

3 Endogenous product compatibility

3.1 The symmetric sub-game $K/K$

Given the expression in (2), the equilibrium output in the third stage of the game when both firms produce compatible products ($K/K$) must satisfy the first-order condition:

$$\frac{{\partial {\Pi }_{i} }}{{\partial q_{i} }} = 0 \Leftrightarrow 1 - 2q_{i} - q_{j} + n\left( {y_{i} + ky_{j} } \right) = 0.$$

(3)

Equation (3) allows us to obtain firm $i$’s reaction function, which is given by:

$$q_{i} \left( {q_{j} ,y_{i} ,y_{j} } \right) = \frac{{1 - q_{j} + n\left( {y_{i} + ky_{j} } \right)}}{2}.$$

(4)

which is negatively sloped (i.e., products are strategic substitutes). By imposing the usual “rational expectation condition” such that $y_{i} = q_{i}$ and $y_{j} = q_{j}$ (e.g., Hoernig 2012) and solving the system of output reaction functions composed by (4) and its counterpart for firm $j$, the output and profits as a function of $k$ are respectively as follows:

$$q_{i}^{K/K} = \frac{1}{{3 - n\left( {1 + k} \right)}},$$

(5)

and

$${\Pi }_{i}^{K/K} = \frac{1}{{\left[ {3 - n\left( {1 + k} \right)} \right]^{2} }} - Z,$$

(6)

where $Z < \frac{1}{{\left[ {3 - n\left( {1 + k} \right)} \right]^{2} }}: = Z_{TH}^{K/K} \left( {n,k} \right)$ is the condition that must be fulfilled to guarantee that profits are positive.

In the second (intermediate) stage of the game, firm $i$ chooses the degree of product compatibility to maximise profits. Therefore,

$$\frac{{\partial {\Pi }_{i}^{K/K} }}{\partial k} = \frac{2n}{{\left[ {3 - n\left( {1 + k} \right)} \right]^{3} }}.$$

(7)

Given the expression in (7) and knowing that ${\Pi }_{i}^{K/K}$ is a convex function of $k$, when $n > 0$ (resp. $n < 0$), the sign of $\frac{{\partial {\Pi }_{i}^{K/K} }}{\partial k}$ is positive (resp. negative) for any $k \in \left[ {0,1} \right]$. Therefore, if $n > 0$ (resp. $n < 0$), the profit-maximising degree of product compatibility under $K/K$ is a corner solution given by $k = 1$ (resp. $k = 0$).

If $n > 0$, the Nash equilibrium values of output and profits under $K/K$ are respectively given by the following expressions:

$$q_{i}^{*K/K} \left( n \right) = \frac{1}{3 - 2n},$$

(8)

and

$${\Pi }_{i}^{*K/K} \left( n \right) = \frac{1}{{\left( {3 - 2n} \right)^{2} }} - Z,$$

(9)

where $Z < \frac{1}{{\left( {3 - 2n} \right)^{2} }}: = Z_{TH}^{K/K} \left( n \right)$ is the threshold that must be satisfied to have positive profits in equilibrium.

The equilibrium consumer surplus and social welfare in this sub-game are respectively as follows:

$$CS^{*K/K} \left( n \right) = \frac{{2\left( {1 - n} \right)}}{{\left( {3 - 2n} \right)^{2} }},$$

(10)

and

$$W^{*K/K} \left( n \right) = CS^{*K/K} \left( n \right) + 2{\Pi }_{i}^{*K/K} \left( n \right) = \frac{{2\left( {2 - n} \right)}}{{\left( {3 - 2n} \right)^{2} }} - 2Z.$$

(11)

From (11), the social welfare is positive for any $Z < Z_{TH}^{K/K} \left( n \right)$.

If $n < 0$, then the Nash equilibrium values of output and profits in the sub-game $K/K$ coincide with those of the sub-game $NK/NK$.

3.2 The symmetric sub-game $NK/NK$

If both firms produce incompatible products ($NK/NK$), then $k = 0$ and $Z = 0$, so that profits of firm $i$ become ${\Pi }_{i} = p_{i} q_{i}$, where $p_{i} = 1 - q_{i} - q_{j} + ny_{i}$. Given these expressions, the equilibrium output at the third stage of the game must satisfy the first-order condition:

$$\frac{{\partial {\Pi }_{i} }}{{\partial q_{i} }} = 0 \Leftrightarrow 1 - 2q_{i} - q_{j} + ny_{i} = 0.$$

(12)

From (12), the reaction function of firm $i$ is:

$$q_{i} \left( {q_{j} ,y_{i} } \right) = \frac{{1 - q_{j} + ny_{i} }}{2}.$$

(13)

Therefore, the Nash equilibrium values of output and profits under $NK/NK$ are respectively as follows:

$$q_{i}^{*NK/NK} \left( n \right) = \frac{1}{3 - n},$$

(14)

and

$${\Pi }_{i}^{*NK/NK} \left( n \right) = \frac{1}{{\left( {3 - n} \right)^{2} }}.$$

(15)

The equilibrium consumer surplus and social welfare in this sub-game are respectively as follows:

$$CS^{*NK/NK} \left( n \right) = \frac{2 - n}{{\left( {3 - n} \right)^{2} }},$$

(16)

and

$$W^{*NK/NK} \left( n \right) = CS^{*NK/NK} \left( n \right) + 2{\Pi }_{i}^{*NK/NK} \left( n \right) = \frac{4 - n}{{\left( {3 - n} \right)^{2} }}.$$

(17)

3.3 The asymmetric sub-game $K/NK$

Assume now that firm $i$ produces compatible goods and that the rival, firm $j$, produces incompatible goods ($K/NK$). Then, the profit functions of firm $i$ and firm $j$ can respectively be written as:

$${\Pi }_{i} = p_{i} q_{i} - Z,$$

(18)

and

$${\Pi }_{j} = p_{j} q_{j} ,$$

(19)

where $p_{i} = 1 - q_{i} - q_{j} + n\left( {y_{i} + ky_{j} } \right)$ and $p_{j} = 1 - q_{j} - q_{i} + ny_{j}$.

The maximisation of the expressions in (18) and (19) for $q_{i}$ and $q_{j}$, respectively, in the third stage of the game provides the reaction functions of firm $i$ and firm $j$ under $K/NK$, which are given by Eqs. (4) and (13). By assuming $y_{i} = q_{i}$ and $y_{j} = q_{j}$ and solving the system of asymmetric output reaction functions, one gets:

$$q_{i}^{K/NK} = \frac{{1 - n\left( {1 - k} \right)}}{{\left( {3 - n} \right)\left( {1 - n} \right) + nk}},$$

(20)

$$q_{j}^{K/NK} = \frac{1 - n}{{\left( {3 - n} \right)\left( {1 - n} \right) + nk}},$$

(21)

where $q_{i}^{K/NK} > q_{j}^{K/NK}$ and

$${\Pi }_{i}^{K/NK} = \frac{{\left[ {1 - n\left( {1 - k} \right)} \right]^{2} }}{{\left[ {\left( {3 - n} \right)\left( {1 - n} \right) + nk} \right]^{2} }} - Z,$$

(22)

$${\Pi }_{j}^{K/NK} = \frac{{\left( {1 - n} \right)^{2} }}{{\left[ {\left( {3 - n} \right)\left( {1 - n} \right) + nk} \right]^{2} }}.$$

(23)

From (22), the feasibility condition that guarantees the positivity of profits of $K$-firm $i$ is $Z < \frac{{\left[ {1 - n\left( {1 - k} \right)} \right]^{2} }}{{\left[ {\left( {3 - n} \right)\left( {1 - n} \right) + nk} \right]^{2} }}: = Z_{TH}^{K/NK} \left( {n,k} \right)$.

In the second (intermediate) stage of the game, the $K$-firm chooses the degree of product compatibility to maximise profits. Therefore,

$$\frac{{\partial {\Pi }_{i}^{K/NK} }}{\partial k} = \frac{{2n\left( {1 - n} \right)\left( {2 - n} \right)\left[ {1 - n\left( {1 - k} \right)} \right]}}{{\left[ {\left( {3 - n} \right)\left( {1 - n} \right) + nk} \right]^{3} }}.$$

(24)

Given the expression in (24), when $n > 0$ (resp. $n < 0$) the sign of $\frac{{\partial {\Pi }_{i}^{K/NK} }}{\partial k}$ is positive (resp. negative) for any $k \in \left[ {0,1} \right]$. Therefore, if $n > 0$ (resp. $n < 0$), the profit-maximising degree of product compatibility chosen by the $K$-firm under $K/NK$ is a corner solution given by $k = 1$ (resp. $k = 0$).

If $n > 0$, the Nash equilibrium values of output and profits under $K/NK$ are respectively given by the following expressions:

$$q_{i}^{*K/NK} \left( n \right) = \frac{1}{{3\left( {1 - n} \right) + n^{2} }},$$

(25)

$$q_{j}^{*K/NK} \left( n \right) = \frac{1 - n}{{3\left( {1 - n} \right) + n^{2} }},$$

(26)

and

$${\Pi }_{i}^{*K/NK} \left( n \right) = \frac{1}{{\left[ {3\left( {1 - n} \right) + n^{2} } \right]^{2} }} - Z,$$

(27)

$${\Pi }_{j}^{*K/NK} \left( n \right) = \frac{{\left( {1 - n} \right)^{2} }}{{\left[ {3\left( {1 - n} \right) + n^{2} } \right]^{2} }},$$

(28)

where the threshold that must be satisfied to get positive profits in equilibrium for the $K$-firm becomes $Z < \frac{1}{{\left[ {3\left( {1 - n} \right) + n^{2} } \right]^{2} }}: = Z_{TH}^{K/NK} \left( n \right)$.

The equilibrium consumer surplus and social welfare are respectively as follows:

$$CS^{*K/NK} \left( n \right) = \frac{{\left( {1 - n} \right)\left[ {3\left( {1 - n} \right) + 1 + n^{2} } \right]}}{{2\left[ {3\left( {1 - n} \right) + n^{2} } \right]^{2} }},$$

(29)

and

$$W^{*K/NK} \left( n \right) = CS^{*K/NK} \left( n \right) + {\Pi }_{i}^{*K/NK} \left( n \right) + {\Pi }_{j}^{*K/NK} \left( n \right) = \frac{2 - n}{{3\left( {1 - n} \right) + n^{2} }} - Z.$$

(30)

From (30), the social welfare is positive for any $Z < Z_{TH}^{K/NK} \left( n \right)$.

If $n < 0$, the Nash equilibrium values of output and profits in the sub-game $K/NK$ coincide with those of the sub-game $NK/NK$.

3.4 Endogenous product compatibility and endogenous market outcomes

This section examines the first stage of the CDG, in which each firm chooses to be a $K$-firm or an $NK$-firm. The main variables of the problem are summarised in Table 1 (optimal values of $k$) and Table 2 (payoff matrix), in which the equilibrium profit functions are given by the expressions in (9), (15), (27), and (28).

Table 1 Optimal (profit maximising) values of $k$

Table 2 The compatibility decision game (payoff matrix) when $k$ is endogenous

The technical conditions that must be satisfied to have well-defined equilibria in pure strategies for every strategic profile (one for each player) are $Z < Z_{TH}^{K/K} \left( n \right)$ and $Z < Z_{TH}^{K/NK} \left( n \right)$, where $Z_{TH}^{K/NK} \left( n \right) > Z_{TH}^{K/K} \left( n \right)$ for any $0 < n < 1$. Therefore, the unique feasibility condition that must hold to guarantee meaningful Nash equilibria is $Z < Z_{TH}^{K/K} \left( n \right)$. Then, to derive all possible SPNE of the CDG, one must study the sign of the profit differentials for $i = \left\{ {1,2} \right\}, i \ne j$, that is⁷:

$${{\Delta \Pi }}_{A} \left( n \right): = {\Pi }_{i}^{*K/NK} \left( n \right) - {\Pi }_{i}^{*NK/NK} \left( n \right),$$

(31)

$${{\Delta \Pi }}_{B} \left( n \right): = {\Pi }_{i}^{*NK/K} \left( n \right) - {\Pi }_{i}^{*K/K} \left( n \right),$$

(32)

and

$${{\Delta \Pi }}_{C} \left( n \right): = {\Pi }_{i}^{*NK/NK} \left( n \right) - {\Pi }_{i}^{*K/K} \left( n \right).$$

(33)

The first threshold defines the incentive of firm $i$ to deviate from $K$ to $NK$ when its sign is negative (and vice versa when its sign is positive) when the rival, firm $j$, is playing $NK$. The second threshold defines the incentive of firm $i$ to deviate from $NK$ to $K$ when its sign is negative (and vice versa when its sign is positive) when the rival, firm $j$, is playing $K$. The third threshold determines the Pareto efficiency/inefficiency of a symmetric SPNE.

From (31), the sign of $\Delta \Pi_{A} \left( n \right)$ is positive (resp. negative) if $Z < Z_{A} \left( n \right)$ (resp. $Z > Z_{A} \left( n \right)$), where

$$Z_{A} \left( n \right): = \frac{{n\left( {2 - n} \right)\left( {n^{2} - 4n + 6} \right)}}{{\left[ {3\left( {1 - n} \right) + n^{2} } \right]^{2} \left( {3 - n} \right)^{2} }},$$

(34)

is the threshold value of the quasi-fixed cost of compatibility (as a function of the extent of the network externality) such that $\Delta \Pi_{A} \left( n \right) = 0$.

From (32), the sign of $\Delta \Pi_{B} \left( n \right)$ is positive (resp. negative) if $Z < Z_{B} \left( n \right)$ (resp. $Z > Z_{B} \left( n \right)$), where

$$Z_{B} \left( n \right): = \frac{{n\left( {2 - n} \right)\left( {3n^{2} - 8n + 6} \right)}}{{\left[ {3\left( {1 - n} \right) + n^{2} } \right]^{2} \left( {3 - 2n} \right)^{2} }},$$

(35)

is the threshold value of the quasi-fixed cost of compatibility (as a function of the extent of the network externality) such that $\Delta \Pi_{B} \left( n \right) = 0$.

From (33), the sign of $\Delta \Pi_{C} \left( n \right)$ is positive (resp. negative) if $Z < Z_{C} \left( n \right)$ (resp. $Z > Z_{C} \left( n \right)$), where

$$Z_{C} \left( n \right): = \frac{{3n\left( {2 - n} \right)}}{{\left( {3 - n} \right)^{2} \left( {3 - 2n} \right)^{2} }},$$

(36)

is the threshold value of the quasi-fixed cost of compatibility (as a function of the extent of the network externality), such that $\Delta \Pi_{C} \left( n \right) = 0$. In addition, we note that $Z_{TH}^{K/K} \left( n \right) > Z_{B} \left( n \right) > Z_{A} \left( n \right) > Z_{C} \left( n \right)$ for any $n > 0$, and $Z_{B} \left( n \right) = Z_{A} \left( n \right) = Z_{C} \left( n \right) = 0$ and $Z_{TH}^{K/K} \left( n \right) = 1/9$ if $n = 0$.

Then, the following proposition holds:

Proposition 1

If $n > 0$, the endogenous market structure of the CDG when $k$ is endogenous is as follows.

• If $0 \le Z < Z_{C} \left( n \right)$ then $\left( {K,K} \right)$ is the unique Pareto-efficient SPNE, and the CDG is an anti-prisoner’s dilemma in which self-interest and the mutual benefit of product compatibility do not conflict.

• If $Z_{C} \left( n \right) < Z < Z_{A} \left( n \right)$ then $\left( {K,K} \right)$ is the unique Pareto-inefficient SPNE, and the CDG is a prisoner’s dilemma in which self-interest and the mutual benefit of product compatibility conflict.

• If $Z_{A} \left( n \right) < Z < Z_{B} \left( n \right),$ then $\left( {K,K} \right)$ and $\left( {NK,NK} \right)$ are two symmetric SPNE, and the CDG is a coordination game, but $NK$ payoff dominates $K$.

• If $Z_{B} \left( n \right) < Z < Z_{TH}^{K/K} \left( n \right),$ then $\left( {NK,NK} \right)$ is the unique Pareto-efficient SPNE, and the CDG is an anti-prisoner’s dilemma in which self-interest and mutual benefit of product incompatibility do not conflict.

Proof

See the online appendix.

The results summarised in Proposition 1 are driven exclusively by the relative size of the quasi-fixed cost of product compatibility and depend on how this cost affects the profits of the $K$-firms. These results go in an unexpected direction than those already pinpointed by the existing literature on the optimal product compatibility (Economides 1989; Kim and Choi 2015; Toshimitsu 2018; Stadler et al. 2022). There exists an optimal value of the extent of product compatibility, but we can discover the existence of different paradigms emerging in the CDG by studying the incentives of profit-maximising firms choosing non-cooperatively whether to produce compatible goods (incurring a quasi-fixed cost) or non-compatible goods (not incurring a quasi-fixed cost). This holds regardless of whether $K$-firms have the incentive to produce fully compatible goods in the sub-games $K/K$ and $K/NK$ (see Fig. 1 for a geometrical representation of Proposition 1).

Fig. 1

The CDG when $k$ is endogenous: SPNE in the space $\left( {n,Z} \right)$. The sand-coloured region represents unfeasibility. Area $A$: $\left( {K,K} \right)$ is the unique Pareto-efficient SPNE. Area $B$: $\left( {K,K} \right)$ is the unique Pareto-inefficient SPNE. Area $C$: $\left( {K,K} \right)$ and $\left( {NK,NK} \right)$ are two symmetric SPNE ($NK$ payoff dominates $K$). Area $D$: $\left( {NK,NK} \right)$ is the unique Pareto-efficient SPNE. The area of Pareto efficiency of $\left( {K,K} \right)$ (resp. $\left( {NK,NK} \right)$) increases (resp. reduces) when the strength of the network effect increases

If the quasi-fixed cost is low or zero, each firm has a dominant strategy represented by $K$ (fully compatible goods), and the $K$-firm produces more than the $NK$-firm, which leads to a Pareto-efficient outcome (each firm has a unilateral incentive to play $K$ and there is no conflict between self-interest and the mutual benefit of product compatibility). This is because the price that consumers are willing to pay for compatible goods is higher than the price that consumers are willing to pay for incompatible goods due to the positive network (bandwagon) effect. The set of signs of the profit differentials is $\Delta \Pi_{A} \left( n \right) > 0$, $\Delta \Pi_{B} \left( n \right) < 0$ and $\Delta \Pi_{C} \left( n \right) < 0$. This case is represented in Area $A$ of Fig. 1. The sign of the profit differentials reveals that firm $i$ has an incentive to play (1) $K$ when the rival (firm $j$) plays $NK$, and (2) $K$ when the rival (firm $j$) plays $K$. This outcome is Pareto efficient (the sign of $\Delta \Pi_{C} \left( n \right)$ is negative) so that the unilateral interest (to produce compatible goods) coincides with the social interest.

If the quasi-fixed cost increases, each firm continues to have $K$ as a dominant strategy, but profits of the $K$-firms reduce due to the direct negative effect exerted by the fixed cost (that does not change output and price), causing a Pareto-inefficient outcome (there is a conflict between self-interest and the mutual benefit of product compatibility). The set of signs of the profit differentials becomes $\Delta \Pi_{A} \left( n \right) > 0$, $\Delta \Pi_{B} \left( n \right) < 0$ and $\Delta \Pi_{C} \left( n \right) > 0$. This case is represented in Area $B$ of Fig. 1. The sign of the profit differentials reveals that firm $i$ has an incentive to play (1) $K$ when the rival (firm $j$) plays $NK$, and (2) $K$ when the rival (firm $j$) plays $K$. There is no strategic incentive to deviate from $K$ but unlike the previous scenario the SPNE is Pareto inefficient (the sign of ${{\Delta \Pi }}_{C} \left( n \right)$ becomes positive) so that the unilateral interest (to produce compatible goods) contrasts the social interest, and each firm remains entrapped in a dilemma.

Further increases in the quasi-fixed cost of compatibility reduce the profits of the $K$-firms further avoiding the existence of a dominant strategy, thereby incentivizing each firm to play the same strategy as the rival. This is because the increased quasi-fixed cost makes profits of the $K$-firm when the rival plays $NK$ lower than those that can be obtained by playing $NK$ without affecting output and price. These circumstances eventually lead to a coordination game in which $NK$ payoff dominates $K$. The set of signs of the profit differentials is ${{\Delta \Pi }}_{A} \left( n \right) < 0$, ${{\Delta \Pi }}_{B} \left( n \right) < 0$ and ${{\Delta \Pi }}_{C} \left( n \right) > 0$. This case is represented in Area $C$ of Fig. 1. The main difference from the previous two cases is that there is now no dominant strategy, and the sign of ${{\Delta \Pi }}_{A} \left( n \right)$ changes and becomes negative. This threshold tunes the incentive of firm $i$ to deviate from $K$ to $NK$ when its sign is negative when the rival, firm $j$, is playing $NK$. Combining a negative sign of ${{\Delta \Pi }}_{A} \left( n \right)$ with a negative sign of ${{\Delta \Pi }}_{B} \left( n \right)$, which tunes the incentive of firm $i$ to deviate from $NK$ to $K$ when its sign is negative when the rival, firm $j$, is playing $K$ generates no unambiguous incentives for each firm to play $K$ or $NK$. Therefore, in this scenario there is no clear deviation incentive for each player, there are multiple SPNE in pure strategies and there is a priori indeterminacy. The CDG becomes a coordination game. By evaluating the sign of ${{\Delta \Pi }}_{C} \left( n \right)$, however, every firm is better off by playing $NK$. Therefore, depending on the relative degree of the individual risk aversion a firm can be incentivised to coordinate (non-cooperatively) to play $NK$. Indeed, there exists a simple criterion to select equilibria by introducing mixed strategies showing that a (unique) mixed-strategy Nash equilibrium does exist. We do not proceed further into the mathematical details but simply note that a Nash equilibrium in mixed strategies can be obtained by defining probabilities $x_{i}$ and $1 - x_{i}$ (resp. $x_{j}$ and $1 - x_{j}$) that firm $i$ (resp. firm $j$) plays either $K$ or $NK$ so that the mixed-strategy Nash equilibrium is given by $x_{i} = x_{*}^{K/NK} = \left[ {{\Pi }_{i}^{*K/NK} \left( n \right) - {\Pi }_{i}^{*NK/NK} \left( n \right)\left] / \right[{\Pi }_{i}^{*K/NK} \left( n \right) - {\Pi }_{i}^{*NK/NK} \left( n \right) + {\Pi }_{i}^{*NK/K} \left( n \right) - {\Pi }_{i}^{*K/K} \left( n \right)} \right]$, where $i,j = \left\{ {1,2} \right\}$, $i \ne j$. Then, each firm will choose to play $K$ (resp. $NK$) as a pure strategy if the rival plays $K$ (resp. $NK$) with a probability $x > x_{*}^{K/NK}$ (resp. $x < x_{*}^{K/NK}$). More narratively, in this scenario, there is no incentive to deviate from the rival’s choice. As the fixed cost only relates to the introduction of compatibility, experiencing a sufficiently high increase in it implies, on the one hand, that it is no longer convenient to introduce compatibility unilaterally because each firm does not want to be the only one to bear the fixed cost of compatibility and obtain the worst possible outcome [see Eqs. (18) and (19)]. Thus, the possibility of multiple equilibria arises. On the other hand, choosing non-compatibility is, in turn, unprofitable as it causes a reduction in demand due to the reduced network effect on its product: the reduction in profits associated with the decrease in the network effect caused by the choice of non-compatibility is, at that level of fixed cost, greater than the increase in the product caused by giving up compatibility which means not bearing the fixed cost. Further increases in the fixed cost can imply the positive effect on profits due to the lack of fixed costs when a firm chooses the incompatibility strategy more than counterbalances the reduction of profits due to the reduced network effect. In other words, the growth in fixed costs of compatibility is the crucial factor that makes it no longer worthwhile choosing the compatibility strategy, and this eventually leads to non-compatibility being the dominant strategy, as also described below. Compatibility increases revenues (via the own-product compatibility positive network effect) and increases costs (via the compatibility cost effect). The conflict or balance between these two forces is responsible for the richness of the SPNE and for the switching amongst all Nash equilibria of the CDG.

Therefore, other increases in the quasi-fixed cost modify the incentive of each firm to have $NK$ as a dominant strategy by reducing profits of the $K$-firms, bringing them well below those of the $NK$-firms. This is because the compatibility cost effect is relatively high and the CDG becomes an anti-prisoner’s dilemma, in which there is no conflict between self-interest and mutual benefit of product incompatibility so that $\left( {NK,NK} \right)$ is Pareto efficient. The set of signs of the profit differentials is ${{\Delta \Pi }}_{A} \left( n \right) < 0$, ${{\Delta \Pi }}_{B} \left( n \right) > 0$ and ${{\Delta \Pi }}_{C} \left( n \right) > 0$. This case is represented in Area $D$ of Fig. 1. In this last scenario, the sign of the profit differentials reveals that firm $i$ has an incentive to play (1) $NK$ when the rival (firm $j$) plays $NK$, and (2) $NK$ when the rival (firm $j$) plays $K$. This outcome is Pareto efficient (the sign of ${{\Delta \Pi }}_{C} \left( n \right)$ is positive) so that the unilateral interest (to produce incompatible goods) coincides with the social interest.

These results hold under positive network externality (bandwagon effect). Unlike this case, under negative network externality (snob effect), the emerging SPNE of the CDG is trivial; no firm has an incentive to let its products become compatible, and any deviations from $NK$ leads each of them to be worse off. Therefore, the unique Pareto-efficient SPNE emerging for any $n < 0$ is $\left( {NK,NK} \right),$ and the CDG is a deadlock in which there is no conflict between self-interest and the mutual benefits of producing incompatible goods.

Interestingly, Proposition 1 (and Fig. 1) predicts that, for any value of the quasi-fixed cost of compatibility, the CDG changes the paradigm for increasing values of the extent of the (positive) consumption externality: (1) when the externality is low, self-interest and mutual benefit of product incompatibility do not conflict, and the game becomes an anti-prisoner’s dilemma; (2) if the extent of the network externality becomes larger, there is indeterminacy (multiple Nash equilibria), but there is an incentive for firms to play $NK$; (3) when the externality increases further, self-interest and mutual benefit of product compatibility conflict and the game is a prisoner’s dilemma; and (4) when the externality is high, self-interest and the mutual benefit of product compatibility do not conflict, and the game is an anti-prisoner’s dilemma. Therefore, the network externality favours product compatibility by increasing firms’ profits through the increase in output and price that the bandwagon effect generates.

As the only feasibility constraint is $Z < Z_{TH}^{K/K} \left( n \right)$, there is a one-to-one correspondence between $Z$ and $n$, such that, for each value of $Z$ (resp. $n$), there exists a corresponding value of $n$ (resp. $Z$) above (resp. below), upon which the CDG is meaningful, and the higher the quasi-fixed cost of compatibility, the higher the strength of the network effect required for feasibility (i.e., the network externality should be high enough to adequately sustain the market demand and total revenues to avoid negative profits). This knowledge is useful to tackle the issue of the social welfare outcomes corresponding to the SPNE and the possible policy implications. In this regard, Figs. 2 and 3 allow us to examine the welfare outcomes by overlapping the SPNE detailed in Fig. 1, with the consumer surplus and social welfare differentials that can be obtained from the equilibrium analysis discussed so far. These differentials are defined as follows:

$${\Delta }CS\left( n \right): = CS^{*K/K} \left( n \right) - CS^{*K/NK} \left( n \right): = \frac{{n\left( {3 - 6n + 2n^{2} } \right)}}{{(3 - n)^{2} (3 - 2n)^{2} }},$$

(37)

and

$${\Delta }W\left( n \right): = W^{*K/K} \left( n \right) - W^{*NK/NK} \left( n \right): = \frac{{n\left( {15 - 12n + 2n^{2} } \right)}}{{(3 - n)^{2} (3 - 2n)^{2} }} - Z.$$

(38)

Fig. 2

The CDG when $k$ is endogenous: consumer surplus and social welfare in the space $\left( {n,Z} \right)$. The blue (resp. red) line is the loci of points in the space $\left( {n,Z} \right),$ such that ${\Delta }CS\left( n \right) = 0$ (resp. ${\Delta }W\left( n \right) = 0$)

Fig. 3

The CDG when $k$ is endogenous: SPNE, consumer surplus, and social welfare in the space $\left( {n,Z} \right)$. Win–win results: area $X$ and area $Y$

From (37), ${\Delta }CS\left( n \right) > 0$ if $n < n^{CS}$ and ${\Delta }CS\left( n \right) < 0$ if $n > n^{CS}$, where $n^{CS} : = \frac{3 - \sqrt 3 }{2} \cong 0.64$ is the threshold value of $n$ such that ${\Delta }CS\left( n \right) = 0$. Under $K/K$, there is an inverted U-shaped behaviour of the consumer when $n$ increases. This is because a larger value of the (positive) network effect shifts outward the market demand, thus increasing the quantity and price that consumers are willing to pay. On the one hand, the increase in the quantity makes consumers better off. On the other hand, the increase in the market price makes consumers worse off. When $n$ is high enough, the latter effect dominates the former, and the consumer surplus reduces with $n$ under $K/K$. In contrast, under $NK/NK$, the consumer surplus monotonically increases with $n$, as the former effect always dominates the latter. This eventually implies that, for any $n < n^{CS}$, consumers are better under full compatibility, and for any $n > n^{CS}$, consumers are better off under no compatibility.

From (38), ${\Delta }W\left( n \right) > 0$ if $Z < Z^{W} \left( n \right)$ and ${\Delta }W\left( n \right) < 0$ if $Z > Z^{W} \left( n \right)$, where $Z^{W} : = \frac{{n\left( {15 - 12n + 2n^{2} } \right)}}{{2(3 - n)^{2} (3 - 2n)^{2} }}$ is the threshold value of $Z$, such that ${\Delta }W\left( n \right) = 0$, in which the numerator is positive for any $n$. This allows us to pinpoint the existence of a one-to-one relationship between $n$ and $Z$, such that social welfare under $K/K$ is larger or smaller than the social welfare under $NK/NK$. Figure 2 clarifies this point. Figure 3 combines SPNE and social welfare outcome by pinpointing the areas for which there are win–win results of the CDG from a societal perspective. This depends on the mutual relationship between the strength of the network effect and the fixed cost of compatibility.

• Area $X$: $\left( {NK,NK} \right)$ is the unique Pareto-efficient SPNE (there is no conflict between self-interest and mutual benefit of incompatibility); ${\Delta }CS\left( n \right) < 0$ and ${\Delta }W\left( n \right) < 0$. The strength of the network effect and the fixed cost of compatibility are significant. Consumers and firms are better off: win–win result under incompatibility.

• Area $Y$: $\left( {K,K} \right)$ is the unique Pareto-efficient SPNE (there is no conflict between self-interest and the mutual benefit of full compatibility); ${\Delta }CS\left( n \right) > 0$ and ${\Delta }W\left( n \right) > 0$. The strength of the network effect and the fixed cost of compatibility are small. Consumers and firms are better off: win–win result under full compatibility.

The conflict between consumers’ and producers’ goals disappears when $n$ and $Z$ are either small or large. In the first case, compatibility is cheap on the firm side, and this situation favours the production of fully compatible goods. As a result of this case, consumers are also better off, as the positive effect of the increase in consumption is not overweighted by the increase in the price. In the second case, compatibility is expensive on the firm side, and this favours the production of incompatible goods from which also consumers are better off, as the positive effect of the increase in consumption of compatible goods would have been more than offset by a sharp increase in the price.

The policy implications emerging from these scenarios are far-reaching. First, there are two public authorities involved in designing specific policies to protect the interests of economic agents, including the anti-trust authority, which takes account of consumer interests, and the government, which takes account of society interests. The objectives of these two authorities may diverge or converge. In the absence of interventions, the outcome for society strictly depends on the SPNE emerging endogenously in the market and therefore on the production of compatible or incompatible goods.

When the strength of the positive network consumption externality is sufficiently low (resp. high), i.e., $n < n^{CS}$ (resp. $n > n^{CS}$), consumers are better off if firms produce compatible (resp. incompatible) products. This is because the beneficial effect on consumer surplus due to the increase in quantity, caused by the joint effects of network externality and compatibility, more than compensates for (is more than compensated by) the negative effect on consumer surplus due to the price increase.

If $n$ is relatively low (resp. high) and the fixed costs of producing compatible goods are too high (resp. low), the government’s goal of maximising social welfare and the anti-trust authority’s goal of maximising consumer surplus can converge through the introduction of a lump-sum subsidy (resp. tax) to firms with the aim of incentivising the production of compatible (resp. incompatible) goods. This subsidy (resp. tax), would reduce (resp. increase) the fixed costs, thereby favouring the emergence of the Pareto-efficient SPNE $\left( {K,K} \right)$ (resp. $\left( {NK,NK} \right)$), and could be financed with (could finance) indirect taxes (resp. subsidies) on the consumption of $q_{i}$ and $q_{j}$ levied (resp. given) at a homogeneous rate to avoid a distorting effect on the consumer side.

The government’s goal could be to incentivise the production of incompatible (resp. compatible) goods, even when the positive network externalities are low (resp. high). In fact, in this case, social welfare would be the highest, even if consumers would be worse off, as they would be better off if firms produced compatible (resp. incompatible) goods. The anti-trust authority, therefore, could ask the government for an additional compensatory intervention to improve the consumer welfare, which however may in principle generates a deadweight loss in other markets.

4 Exogenous product compatibility

This section considers the alternative case of exogenous product compatibility so that firms cannot choose $k$ to maximise profits because of technological constraints. The non-cooperative CDG, therefore, develops like two (instead of) three stages, as was already mentioned in Sect. 2. The Nash equilibrium values ($*$) of quantity and profits in each sub-game are respectively given by Eqs. (5) and (6) under $K/K$; (14) and (15) under $NK/NK$; and (20), (21), (22), and (23) under $K/NK$. For clarity, we report the equations below by stressing their dependency on $n$ and $k$:

$$q_{i}^{*K/K} \left( {n,k} \right) = \frac{1}{{3 - n\left( {1 + k} \right)}},$$

(39)

$${\Pi }_{i}^{*K/K} \left( {n,k} \right) = \frac{1}{{\left[ {3 - n\left( {1 + k} \right)} \right]^{2} }} - Z,$$

(40)

and

$$q_{i}^{*NK/NK} \left( {n,0} \right) = \frac{1}{3 - n},$$

(41)

$${\Pi }_{i}^{*NK/NK} \left( {n,0} \right) = \frac{1}{{\left( {3 - n} \right)^{2} }},$$

(42)

$$q_{i}^{*K/NK} \left( {n,k} \right) = \frac{{1 - n\left( {1 - k} \right)}}{{\left( {3 - n} \right)\left( {1 - n} \right) + nk}},$$

(43)

$$q_{j}^{*K/NK} \left( {n,k} \right) = \frac{1 - n}{{\left( {3 - n} \right)\left( {1 - n} \right) + nk}},$$

(44)

and

$${\Pi }_{i}^{*K/NK} \left( {n,k} \right) = \frac{{\left[ {1 - n\left( {1 - k} \right)} \right]^{2} }}{{\left[ {\left( {3 - n} \right)\left( {1 - n} \right) + nk} \right]^{2} }} - Z,$$

(45)

$${\Pi }_{j}^{*K/NK} \left( {n,k} \right) = \frac{{\left( {1 - n} \right)^{2} }}{{\left[ {\left( {3 - n} \right)\left( {1 - n} \right) + nk} \right]^{2} }}.$$

(46)

Before proceeding with the analysis of the SPNE, we investigate how the degree of compatibility and the intensity of the network externality affect the output and the corresponding values of the profits, knowing that the cost of compatibility is fixed and the competition in each sub-game occurs in strategic substitutes. This is done by considering the case of unilateral deviation from the situations of universal compatibility and incompatibility. The results are summarised in the following lemmas.

Lemma 1

If $n > 0$, the output reduces (resp. increases) with $k$ in the case of unilateral deviation from $K$ to $NK$ (resp. from $NK$ to $K$).

Proof

See the online appendix.

Lemma 1 is intuitive, as switching towards $NK$ reduces the demand via a reduced size of the network of the firm upon considering the case of positive externality. However, the network externality may reduce or increase the effects of the increasing degree of compatibility, depending on whether the firm is cheating from the equilibrium with or without compatibility.

Lemma 2

If $n > 0$, the output-reducing (resp. increasing) effect of the increasing degree of compatibility in the case of unilateral deviation from $K$ to $NK$ [resp. $NK$ to $K$] is mitigated (for sufficiently low values of $k$) [resp. for sufficiently high values of $k$] or magnified (for sufficiently high values of $k$) [resp. for sufficiently low values of $k$] by the intensity of the network effect.

Proof

See the online appendix.

We now investigate how the intensity of the network externality affects output and profits in both cases of unilateral deviation.

Lemma 3

The positive network effect monotonically increases (resp. can increase or reduce depending on whether $k$ and $n$ are sufficiently low or high) output and profits in the case of unilateral deviation from $NK$ to $K$ (resp. from $K$ to $NK$).

Proof

See the online appendix.

Lemma 3 implies that, when $k$ and $n$ are sufficiently high, an increase in the network effect always reduces the convenience for one firm to choose to play $NK$. This is counterintuitive, as an existing important level of both compatibility and network externality should cause a detrimental deviation from $K$.

Lemma 4

In both cases of deviation, the output-reducing (resp. increasing) effect of an increasing degree of the network effect in the case of unilateral deviation from $K$ to $NK$ [resp. $NK$ to $K$] is mitigated (for sufficiently low values of $k$) [resp. for sufficiently high values of $k$] or magnified (for sufficiently high values of $k$) [resp. for sufficiently low values of $k$] by an increasing degree of compatibility.

Proof

See the online appendix.

The complicated interactions of the effects of the levels of the compatibility and the network externality on the incentive to cheat from a symmetric Nash equilibrium will drive, together with the levels of the quasi-fixed cost, the occurrence of a rich set of possible SPNE.

Therefore, at the first stage of the game, each firm chooses the convenience to produce compatible (at an exogenous degree) or incompatible products. The main variables of the problem are summarised in Table 3 (payoff matrix), which summarises the equilibrium profit functions given by the expressions (40), (42), (45), and (46). These functions now depend on the extent of the network externality ($n$) and the degree of product compatibility ($k$).

Table 3 The compatibility decision game (payoff matrix) when $k$ is exogenous

The technical restrictions that must be satisfied to have well-defined equilibria in pure strategies for every strategic profile (one for each player) are the same as those detailed so far in Sect. 3. However, for reasons of tractability, we re-write these conditions as a function of $n$ and $Z$ as follows: $k > k_{TH}^{K/K} \left( {n,Z} \right)$ and $k > k_{TH}^{K/NK} \left( {n,Z} \right)$, where $k_{TH}^{K/K} \left( {n,Z} \right) > k_{TH}^{K/NK} \left( {n,Z} \right)$. Therefore, the unique feasibility condition that must hold to guarantee meaningful Nash equilibria is $k > k_{TH}^{K/K} \left( {n,Z} \right)$. Then, to derive all of the possible SPNE of this non-cooperative game, one must study the sign of the profit differentials for $i = \left\{ {1,2} \right\}, i \ne j$, that is⁸:

$${{\Delta \Pi }}_{A} \left( {n,k} \right): = {\Pi }_{i}^{*K/NK} \left( {n,k} \right) - {\Pi }_{i}^{*NK/NK} \left( {n,k} \right),$$

(47)

$${{\Delta \Pi }}_{B} \left( {n,k} \right): = {\Pi }_{i}^{*NK/K} \left( {n,k} \right) - {\Pi }_{i}^{*K/K} \left( {n,k} \right),$$

(48)

and

$${{\Delta \Pi }}_{C} \left( {n,k} \right): = {\Pi }_{i}^{*NK/NK} \left( {n,k} \right) - {\Pi }_{i}^{*K/K} \left( {n,k} \right).$$

(49)

From (47), the sign of ${{\Delta \Pi }}_{A} \left( {n,k} \right)$ is positive (resp. negative) if $k > k_{A} \left( {n,Z} \right)$ (resp. $k < k_{A} \left( {n,Z} \right)$, where $k_{A} \left( {n,Z} \right)$ is the threshold value of the degree of product compatibility (as a function of the extent of the network externality and the fixed cost of compatibility), such that ${{\Delta \Pi }}_{A} \left( {n,k} \right) = 0$.

From (48), the sign of ${{\Delta \Pi }}_{B} \left( {n,k} \right)$ is positive (resp. negative) if $k < k_{B} \left( {n,Z} \right)$ (resp. $k > k_{B} \left( {n,Z} \right)$, where $k_{B} \left( {n,Z} \right)$ is the threshold value of the degree of product compatibility (as a function of the extent of the network externality and the fixed cost of compatibility), such that ${{\Delta \Pi }}_{B} \left( {n,k} \right) = 0$.

From (49), the sign of ${{\Delta \Pi }}_{C} \left( {n,k} \right)$ is positive (resp. negative) if $k < k_{C} \left( {n,Z} \right)$ (resp. $k > k_{C} \left( {n,Z} \right)$, where $k_{C} \left( {n,Z} \right)$ is the threshold value of the degree of product compatibility (as a function of the extent of the network externality and the fixed cost of compatibility), such that ${{\Delta \Pi }}_{C} \left( {n,k} \right) = 0$.

The SPNE of the CDG when $k$ is exogenous are illustrated in Figs. 4, 5 and 6 that report the Nash equilibrium values emerging in the space $\left( {k,n} \right)$ for $Z = 0.1$, $Z = 0.25$ and $Z = 0.5$, respectively. These figures fully replace the related propositions that we do not present here as the outcomes emerging when $k$ is exogenous are the same as those pinpointed in Sect. 3 when $k$ is endogenous (areas $A$, $B$, $C$, and $D$ in Figs. 4, 5, 6) with one relevant exception, which is reported in Result 1. There are also several analytical complications that make it difficult to characterise and disentangle the different scenarios emerging in the game, as the SPNE outcomes now depend on the interactions amongst three parameters $n$, $k$ and $Z$.

Fig. 4

The CDG when $k$ is exogenous: SPNE in the space $\left( {k,n} \right)$ for $Z = 0.1$. The sand-coloured region represents unfeasibility. Area $A$: $\left( {K,K} \right)$ is the unique Pareto-efficient SPNE. Area $B$: $\left( {K,K} \right)$ is the unique Pareto-inefficient SPNE. Area $C$: $\left( {K,K} \right)$ and $\left( {NK,NK} \right)$ are two symmetric SPNE ($NK$ payoff dominates $K$). Area $D$: $\left( {NK,NK} \right)$ is the unique Pareto-efficient SPNE. Area $E$: $\left( {NK,K} \right)$ and $\left( {K,NK} \right)$ are two asymmetric Pareto-efficient SPNE

Fig. 5

The CDG when $k$ is exogenous: SPNE in the space $\left( {k,n} \right)$ for $Z = 0.25$. The sand-coloured region represents unfeasibility. Area $A$: $\left( {K,K} \right)$ is the unique Pareto-efficient SPNE. Area $B$: $\left( {K,K} \right)$ is the unique Pareto-inefficient SPNE. Area $C$: $\left( {K,K} \right)$ and $\left( {NK,NK} \right)$ are two symmetric SPNE ($NK$ payoff dominates $K$). Area $D$: $\left( {NK,NK} \right)$ is the unique Pareto-efficient SPNE. Area $E$: $\left( {NK,K} \right)$ and $\left( {K,NK} \right)$ are two asymmetric Pareto-efficient SPNE

Fig. 6

The CDG when $k$ is exogenous: SPNE in the space $\left( {k,n} \right)$ for $Z = 0.5$. The sand-coloured region represents unfeasibility. Area $A$: $\left( {K,K} \right)$ is the unique Pareto-efficient SPNE. Area $B$: $\left( {K,K} \right)$ is the unique Pareto-inefficient SPNE. Area $C$: $\left( {K,K} \right)$ and $\left( {NK,NK} \right)$ are two symmetric SPNE ($NK$ payoff dominates $K$). Area $D$: $\left( {NK,NK} \right)$ is the unique Pareto-efficient SPNE. Area $E$: $\left( {NK,K} \right)$ and $\left( {K,NK} \right)$ are two asymmetric Pareto-efficient SPNE

Result 1

When $k$ is exogenous, the CDG can be an anti-coordination game in which $\left( {NK,K} \right)$ and $\left( {K,NK} \right)$ are two asymmetric Pareto efficient SPNE.

When product are partially compatible it is possible also to observe one-way compatibility (area $E$ in Figs. 4, 5, 6). This result holds irrespective of the fixed cost of compatibility and is favoured by low (but not too much low) values of $Z$ (see area $E$ of Fig. 5 in comparison with area $E$ of Figs. 4, 6). This outcome can explain several cases of compatibility in actual markets, the most popular and remarkable of which is Apple versus Microsoft in the computer market: Apple products are based on macOS but can also be used with Windows OS; Microsoft products are based on Windows OS and cannot be used with macOS.

The complete set of SPNE of the CDG when $k$ is exogenous is the following.

• Area $A$: $\left( {K,K} \right)$ is the unique Pareto efficient SPNE (${{\Delta \Pi }}_{A} \left( {n,k} \right) > 0$, ${{\Delta \Pi }}_{B} \left( {n,k} \right) < 0$ and ${{\Delta \Pi }}_{C} \left( {n,k} \right) < 0$).

• Area $B$: $\left( {K,K} \right)$ is the unique Pareto inefficient SPNE (${{\Delta \Pi }}_{A} \left( {n,k} \right) > 0$, ${{\Delta \Pi }}_{B} \left( {n,k} \right) < 0$ and ${{\Delta \Pi }}_{C} \left( {n,k} \right) > 0$).

• Area $C$: $\left( {K,K} \right)$ and $\left( {NK,NK} \right)$ are two symmetric SPNE but $NK$ payoff dominates $K$ (${{\Delta \Pi }}_{A} \left( {n,k} \right) < 0$, ${{\Delta \Pi }}_{B} \left( {n,k} \right) < 0$ and ${{\Delta \Pi }}_{C} \left( {n,k} \right) > 0$).

• Area $D$: $\left( {NK,NK} \right)$ is the unique Pareto efficient SPNE (${{\Delta \Pi }}_{A} \left( {n,k} \right) < 0$, ${{\Delta \Pi }}_{B} \left( {n,k} \right) > 0$ and ${{\Delta \Pi }}_{C} \left( {n,k} \right) > 0$).

• Area $E$: $\left( {NK,K} \right)$ and $\left( {K,NK} \right)$ are two asymmetric Pareto efficient SPNE (${{\Delta \Pi }}_{A} \left( {n,k} \right) > 0$, ${{\Delta \Pi }}_{B} \left( {n,k} \right) > 0$ and ${{\Delta \Pi }}_{C} \left( {n,k} \right) > 0$).

The analytical difficulties encountered in characterizing the equilibrium outcomes can however be overcome by a discussion of the economic reasons why the anti-coordination scenario emerges when $k$ is exogenous but it cannot be observed when $k$ is endogenous. In the former case, strategic interactions between firms can lead to the production of partially compatible goods (the degree of compatibility depends on the prevailing technological characteristics). In the latter case, however, the strategic interactions between firms lead to the production of perfectly compatible or incompatible goods (the degree of compatibility is chosen by the firms as a profit maximizing variable in each sub-game).

Once the size of the fixed cost of compatibility ($Z$) is given, the production of partially compatible goods by a single firm as an SPNE depends on the extent of (1) the bandwagon effect generated by the network ($n$) and (2) the degree of compatibility ($k$). Given the size of $n$, the prevailing paradigm of the game depends on $k$.

• If $n$ is high enough, low values of $k$ make the bandwagon effect weak and the prevailing SPNE is $\left( {NK,NK} \right)$ as a Pareto efficient result. An increase in $k$ strengthens the bandwagon effect and then increases the output of the $K$-firms in the symmetric and asymmetric sub-games allowing also for an increase in the marginal willingness to pay of consumers for the products of the $K$-firms. However, the $NK$-firms in the asymmetric sub-games reduce the output due to the stronger bandwagon effect, which subsequently causes a reduction in the marginal willingness to pay of consumers for their products. These combined effects imply an increase in profits of the $K$-firms in all the sub-games and a reduction in profits of the $NK$-firms in the asymmetric sub-games. If the increase in $k$ is high enough, the percentage increase in the output of the $K$-firm when the rival plays $NK$ is sufficiently high to increase the profits of the $K$-firm beyond what it would get by playing $NK$ when the rival plays $NK$. This contributes to changing the paradigm of the game, which becomes an anti-coordination scenario according to which each firm has the incentive to play the opposite strategy to that played by the rival.

• If $n$ is low enough, low values of $k$ make the bandwagon effect weak and the prevailing SPNE is $\left( {NK,NK} \right)$ as a Pareto efficient result. An increase in $k$ strengthens the bandwagon effect and then increases the output of the $K$-firms in the symmetric and asymmetric sub-games allowing also for an increase in the marginal willingness to pay of consumers for the products of the $K$-firms. The $NK$-firms in the asymmetric sub-games reduce the output due to the stronger bandwagon effect, which subsequently causes a reduction in the marginal willingness to pay of consumers for their products. These effects resemble those discussed above. However, unlike the previous case, the percentage increase in the output of the $K$-firm when the rival plays $NK$ is lower than when the network effect was stronger. This causes the $K$-firm’s profits to increase beyond what it would get if it played $NK$ when the rival is playing $K$ and, given the initial situation in which $\left( {NK,NK} \right)$ was the Pareto efficient SPNE of the game, this induces each firm to play the same strategy as the rival to avoid being worse off allowing for the emergence of a coordination scenario. High values of $k$ favours this outcome. This also explains why an anti-coordination scenario cannot emerge when $k$ is endogenous.

The welfare outcomes related to the SPNE when $k$ is exogenous resemble those found when $k$ is endogenous. Therefore, we do not report a detailed analysis of the shape of the consumer surplus and the social welfare. We simply report that the social welfare values related to the asymmetric SPNE are intermediate compared to those related to $\left( {K,K} \right)$ and $\left( {NK,NK} \right)$ so that there are not relevant differences about the win–win results found in the previous section.

5 Conclusions

This research has studied the non-cooperative multi-stage compatibility decision game (CDG), based on Cournot rivalry, with network (consumption) externalities and fixed costs of compatibility, like Katz and Shapiro (1985). The main innovation is to consider the degree of product compatibility as a strategic variable. Although some existing contributions already studied the problem of choosing the degree of compatibility to maximise profits (e.g., Economides 1989; Kim and Choi 2015; Toshimitsu 2018; Stadler et al. 2022), to the best of our knowledge, no one has addressed the endogenous emergence of the strategic incentives to let products become (in)compatible in a non-cooperative game. The article has pinpointed the emergence of (1) the benefits and costs of compatibility determining a wide spectrum of SPNE, ranging from a situation in which self-interest and mutual benefit of producing compatible goods conflict to situations in which self-interest and mutual benefit of producing (in)compatible goods do not conflict, (2) the social welfare outcomes corresponding to the SPNE, (3) the win–win results emerging under full compatibility and incompatibility and the subsequent policy implications, and (4) a strategic explanation for the existence of one-way compatibility.

To the best of our knowledge, this article represents the first attempt to model a non-cooperative game in which product compatibility is a strategic device in a quantity-setting duopoly and aims to spark a debate on this issue. Potential future research can focus on differences in product quality, managerial delegation, and corporate social responsibility. Another line of development is to follow Suleymanova and Wey (2012) in considering the different timing at which consumers make decisions relative to firms’ decisions and studying the CDG also when firms commit themselves to an announced output level.