Conspicuous consumption is often characterized by the number of goods sold increasing with the price, which is known as the Veblen effect. Veblen (1899) proposes that it is a consumption activity in which people show their wealth to the public, seek a certain identity, or highlight social status through ultra-practical or wasteful consumption. Since then, an increasing body of literature on conspicuous consumption.

Many scholars focus on the factors that influence conspicuous consumption, such as subjective socioeconomic status (Wang et al., 2022b), non-material social comparison (Zheng et al., 2018), and narcissism (Neave et al., 2020; Sedikides and Hart, 2022). Wang and Tian (2022) revealed the deeper connection between face culture (the composite and unified expression of honor, dignity, self-worth, and prestige given by others and felt by the self in social situations) and conspicuous consumption. Wang et al. (2022a) showed that moral, cognitive, and interpersonal self-uncertainty contribute to a stronger tendency to engage in conspicuous consumption. People are more likely to shift intention when they perceive social pressure from their surroundings (Liu et al., 2023); therefore, the external environment also greatly affects people's tendency toward conspicuous consumption.

Existing studies concentrate on investigating how the firm's pricing decisions are influenced by conspicuous consumption. For instance, Tereyağoğlu and Veeraraghavan (2012) used a rational expectations framework to study the pricing and production decisions of firms facing highly uncertain market demand, strategic consumer behavior, and conspicuous consumption behavior. Zhou et al. (2018) evaluated the firm's pricing and production decisions on conspicuous consumers in the presence of discount sensitivity behavior. Zhu et al. (2022) assumed that conspicuous consumers are less sensitive to price than non-conspicuous consumers, and they developed a game model involving a monopoly firm selling conspicuous products and these two types of consumers and studied how conspicuous consumption affects firm pricing, quality decisions, and revenue. Rao and Schaefer (2013) considered the consumer's pursuit of the intrinsic quality and status effect of durable goods and studied the pricing and product management decisions in the market for conspicuous durable goods. Li (2019) examined how firms selling status goods make vertical line extension decisions when they take consumers' status preferences into account. Wei and Li (2020) investigated how conspicuous behavior and concerns of stock availability influence a luxury firm's operational decisions.

The existing research and models on physical goods cannot effectively capture the features of virtual goods. Some research explores the pricing decisions of virtual goods. Liu et al. (2011) studied the optimal pricing strategy of spreadsheet software products under the influence of piracy and word-of-mouth. Hao and Fan (2014) developed a game theory model to study the pricing of e-books and e-readers under both wholesale and agency pricing models. Chen et al. (2020) priced and designed loot boxes to maximize revenue for video game companies. However, these study does not take into account the psychology of conspicuous consumers. Therefore, we develope this analytical model, and our findings and implications contribute to a better understanding of the conspicuous consumption of virtual goods.

Many researchers explored the pricing strategy of game companies, for instance, Lee et al. (2006) proposed a basic strategy that tries to earn revenue by developing premium goods or services to market created by their free goods or services and then they analyzed pricing and service quality strategies for e-business companies providing information services to customers. Civelek et al. (2018) established a Stackelberg game model to discuss the game provider's determination of the optimal pricing of virtual goods and designing game challenge level for free-to-play mobile games in the presence of heterogeneous players and copycat competitors. Chen et al. (2021) considered the snobbery of players and analyzed a mixed revenue model to investigate the effect of introducing advertising incentives to the premium subscription.

Some research focuses on the freemium model. Freemium is a combination of the words “free” and “premium.” It is a business model that offers services or products for free to attract users and then charges for premium features or products (Liu et al., 2014; Hsiao and Chen, 2016). Offering a free trial version and a paid version of the software, as well as setting up in-game paid items in F2P games, are both kinds of the freemium model. Some scholars have studied the former. For example, Haruvy and Prasad (2001) modeled the effect of two decision variables price and freeware quality on the adoption of software using static and evolutionary game theory. Cheng and Tang (2010) examined the trade-off between network effects and the cannibalization effect and aimed to disclose the circumstances under which companies should introduce the free trial product. Geng and Chen (2019) established a model to facilitate this trade-off between increasing the total user volume and maintaining scarcity by considering a reference-based utility shift associated with conspicuous consumption, offering the optimal pricing strategy for a monopoly firm, given different levels of snobbery. Feng et al. (2022) discussed the influence of the positive network externality and the negative security externality on software vendors' optimal freemium versioning strategy and a subsequent security-patching strategy.

Our study concentrates on the pricing of premium items within F2P games. It analyzes the impact of in-game mechanics on game companies, such as the time needed to acquire the grinding version item and the quality differences among various versions. In addition, this study aims to provide insights into the version strategy and pricing decisions of the game company facing consumers with conspicuous consumption psychology.

The monopoly game company offers a paid version of a specific item (such as a character's skin) with price p∈P=[0,+∞) and quality S∈S=(0,+∞), where the quality of the item refers to its gorgeous degree. This study considers the decorative item and not the functional item; that is, whatever S is, it does not affect fairness. Noting that the decorative item that is bought in the game are only valuable in the game itself. We assume that the marginal cost of the firm is zero and the reason is once virtual goods are developed, they are available in unlimited quantities, and the cost of reproduction of digital content is low or close to zero (Shapiro et al., 1999; Na et al., 2017). We assume that the perceived value of the item is heterogeneous among players. θ denotes the preference for quality of the in-game items by an individual player, where θ is uniformly distributed over [0, 1] (Feng et al., 2022). Without the loss of generality, the market is normalized to 1. Conspicuous consumers aspire to be distinct from all other consumers (Arifoğlu et al., 2020; Zhu et al., 2022), so the utility is that a consumer with conspicuous psychology not only relies on the basic valuation of the item but also on the user base. The social value decreases linearly with the quantity of consumers using the product. Similar to Geng and Chen (2019), Wei and Li (2020), and Zhu et al. (2022), we denote α > 0 as the conspicuous intensity, which is homogeneous among consumers. It is crucial for game companies as it helps them determine version strategy and adjust optimal prices to maximize their revenue. All notations are summarized in Table 1.

First, we considered that the game company does not introduce the grinding version that only offers the paid version item to consumers. In this case, the consumer's valuation for the quality of the paid version item is represented by θS, and we used −αQ_H to represent the conspicuous utility, where Q_H is the quantity of customers who purchase the item. It implies that the more buyers there are, the less show-off the customer feels. Thus, the utility of the consumer who buys the item with the preference θ can be expressed as

Customers buy the item if and only if Eq. 1 ≥ 0; that is, θ is larger than or equal to a threshold θ~, where θ~=αQH+pS. Since θ is uniformly distributed within the interval [0, 1], thus, from θ~ to 1 is the quantity of buyers, as shown in Figure 1. Therefore, the quantity of buyers is indicated by QH=1-θ~. Thus, we can get the demand of the item as

The firm seeks to maximize the revenue

Proposition 1. In the case of single-version strategy:

(1) The optimal price and the corresponding demand of the paid version item are p*=12S, QH *=S2(α+S). The maximum revenue of the firm is π*=S24(α+S).

(2) ∂p*∂S>0, ∂QH *∂S>0, ∂π*∂S>0, ∂p*∂α=0, ∂QH *∂α<0, and ∂π*∂α<0.

Proposition 1 shows the optimal price, and the corresponding demand and revenue, and they all increase with the quality of the version because the consumer's utility increases with the quality of the paid version of the item. However, the optimal price does not change with α, the demand and the revenue decrease with the conspicuous intensity. If players do not have conspicuous psychology, the utility is not affected by the behavior of other buyers. Because of the psychology of showing off, the higher the sales, the greater the negative effect on utility, thus consumers' willingness to pay decreases, and the higher the conspicuous intensity is, the greater the impact of sales on consumers' utility, leading to a decline in the quantity of buyers, which in turn is detrimental to the revenue.

In this subsection, we consider the scenario under which the game company not only provides the paid version item but also a relatively low-quality version, which can be obtained by grinding. We assume that the time needed to get the grinding version is t, and the consumers' time cost coefficient is η. The quality of the grinding version item is S − Δ, where 0 < Δ < S. Note that the quality gap Δ between two versions of the in-game item also represents the difference in the gorgeous degree. Then, the consumer valuation for the quality of the paid version and the grinding version item can be expressed by θS and θ(S − Δ), respectively.

How players choose is based on the utility they can obtain from the two versions. Due to the conspicuousness, we can see that the utility of buyers, who purchase the paid version item, is not only related to the quantity of buyers Q_H but also the quantity of grinding version users Q_L. The conspicuous term of utility can be indicated by −α(Q_H − λQ_L), where the cross-quantity sensitivity coefficient λ ≥ 0 means the extent to which the quantity of grinding version users affects conspicuous buyers, α is the intensity of conspicuousness. That means the utility from the show-off psychology decreases as the quantity of buyers increases but increases as the quantity of grinding version users increases. Therefore, the utility of a player, if he chooses the paid version item, is

and the utility of a player if he chooses the grinding version item is

Consumers with a preference for quality θ will choose the grinding version item if their utility is larger than or equal to zero and larger than buying the item. Two marginal values of preference of quality are denoted by θ_L and θ_H. Consumers with preference θ_L are indifferent between the grinding version item and not obtaining the item, that is, Eq. 5 = 0, while consumers with preference θ_H are indifferent between buying the item and obtaining the grinding version item, that is, Eq. 4 = 0. Figure 2 shows the demands of two versions. Accordingly, the sizes of the user base of the grinding version and the paid version are

We need to ensure that there are players who are willing to spend time to obtain the item in the market, that is, Q_L ≥ 0, hence, according to Eq. 6, we can conclude that

In other words, if the price of the paid version item is extremely low, there are only players who buy the paid version item and those who do not. That is, potential consumers with net utility greater than 0 will choose to buy the paid version item, which makes the game company offer the grinding version of the strategic product ineffective.

Similarly, to avoid triviality and ensure some players are willing to buy the item in the market, we assume that Q_H ≥ 0. According to Eq. 7, we can conclude that

Noting that we only consider the situation that P₂ > P₁, which is equivalent to 0≤t<S-Δη. Equations 8, 9 also imply that the cutoff price of the paid version item increases with the time needed to obtain the grinding version item.

We know that the purchasing behavior of paying players in the game can bring benefits to the company. However, non-paying players can also indirectly bring benefits to the company. The externality is defined as the indirect income or costs generated for other economic participants by the production and consumption of goods (Raz and Ovchinnikov, 2015; Chen et al., 2019; Xie et al., 2021). By considering positive externality, we depict the externality revenue generated by grinding version users here with μtQ_L, where η<μ<SS-Δη means the coefficient of externality revenue is greater than the time cost coefficient but falls below a specific threshold. Therefore, the revenue of the firm can be expressed as follows:

Solving the above optimization problem, we can obtain the optimal price of the paid version item when the company provides two versions in Proposition 2.

Proposition 2. The optimal price of the paid version item under the dual-version strategy is

where X = (αλ+Δ)(Δ+ηt), Y = Δ+ηt, Z = αλ+μt, L_t = X(S-Δ)αη(λ+2)+Δ(η+μ)+S(η-μ).

Corollary 1. If the company adopts the optimal price, the quantity of grinding version users is

where M = 2(S − Δ)(αλ + α + Δ), N = α(λ + 2) + Δ.

The quantity of buyers is

The maximum revenue under the dual-version strategy of the company is

Proposition 2 shows under what circumstances the company induces customers to choose the grinding version. When the time needed to acquire the grinding version item t is shorter than the threshold L_t, the company has the incentive to allow both versions of items to coexist in the market. That means the introduction of the grinding version decreases the quantity of buyers but increases the quantity of grinding version users, which leads to a higher utility for buyers due to the psychology of conspicuousness. However, when t is larger than or equal to L_t, the company sets a price to induce all players to either choose the paid version items or not use items at all.

In the following, we analyze how the optimal price and the corresponding demands (QL * and QH *) change with parameters t and α.

Proposition 3. In the case of dual-version strategy, p^*, QL *, and QH * have the following properties with t:

(1) If t ∈ [0, L_t), ∂QL *∂t<0, ∂QH *∂t<0. Furthermore, ∂p*∂t>0 when 0≤λ<(η+μ)(S-Δ)ηα, ∂p*∂t<0 when λ≥(η+μ)(S-Δ)ηα.

(2) If t∈[Lt,S-Δη), QL *=0, ∂QH *∂t<0, ∂p*∂t>0.

Proposition 3(1) depicts that when t is lower than the threshold L_t, as the longer time needed to obtain the grinding version item, the quantity of such non-paying players QL * decreases and then weakens the showoff of buyers which lead to the reduction of the quantity of such buyers QH *.

When λ is lower than a threshold (η+μ)(S-Δ)ηα, the reduction of QH * causes the utility of buyers increases, and the optimal price of the paid version item increases with the time needed to obtain the grinding version item. However, when λ is greater than or equal to this threshold, the gradual reduction of QL * results in the utility of buyers decreasing, thus dragging down the price of the paid version item.

Proposition 3(2) indicates that when t is larger or equal to L_t, the demand of the grinding version is 0, which leads to a negative effect on the conspicuousness of buyers; thus, the quantity of buyer reduces with t, and the utility of buyers increases. So, the firm has the motivation to raise the optimal price constantly.

Proposition 4. In the case of dual-version strategy, p^*, QL *, and QH * have the following properties with α when Δ(S-Δ)Δ(η+μ)+S(η-μ)≤t<S-Δη:

(1) If α ∈ [0, L_α), where Lα=Δ(μt+Y)-S(Δ+t(μ-η))(λ+2)(S-Y), QL *=0 and ∂QH *∂α=0, ∂p*∂α<0.

(2) If α ∈ [L_α, +∞), ∂QL *∂α>0 and ∂QH *∂α<0, ∂p*∂α>0.

Proposition 4 states that the optimal price of the paid version item p^* decreases first and then increases with the conspicuous intensity α, and this conclusion is consistent with the finding of Geng and Chen (2019). The demand of the grinding version QL * first remains unchanged and then increases, and the demand of the paid version QH * first remains unchanged and then decreases.

When α is less than the threshold L_α, QL * is equal to zero, buyers' utility decreases with α, which leads to the reduction of the optimal price. While when α is larger than or equal to the threshold L_α, the quantity of the grinding version users increases with α, but the quantity of buyers decreases with α, which leads to the larger utility of buyers, and then the company adjusts the optimal price upward. This reflects a common phenomenon in real life that excessive conspicuous intensity will lead to a continuous price increase.

According to Proposition 3, in the case of dual-version strategy, we can see that when λ is lower than a threshold, the optimal price of the paid version increases with t. Otherwise, the optimal price decreases first and then increases with t, as shown in Figure 3A. The optimal price of the single version is not affected by t. It can be observed from Figure 3B that the demand of the grinding version decreases as the time needed to obtain the grinding version item increases, reaching zero when t ≥ L_t. Simultaneously, the utility of buyers decreases as the quantity of grinding version users declines, resulting in a decrease in the demand of the paid version as well.

Figure 3C illustrates the maximum revenue of the firm under the single-version (the blue curve) and the dual-version (the orange curve) strategy respectively. It is easy to see that the revenue of providing a single version does not vary with the time needed to obtain the grinding version item; when t₁ < t < t₂ (t₁, t₂ are two thresholds), it is more profitable to provide dual-version items; otherwise, the single-version strategy is a better choice for the firm. In other words, the company should adjust the time needed to acquire the grinding version item and try to offer dual-version items to increase revenue. Furthermore, we can conclude that the introduction of the grinding version decreases the quantity of buyers but increases the utility of buyers which can improve the optimal price of the paid version item.

Figure 4A shows that the optimal price of the paid version item under the dual-version strategy first decreases and then increases with conspicuous intensity α. It is always higher than the price of the single-version. On the one hand, when the conspicuous intensity is weak, conspicuous psychology will lead to the reduction of the optimal price of the paid version items because the utility of players who choose the paid version item is not only larger or equal to zero but also larger or equal to the utility of players who choose the grinding version. To improve buyers' utility, the company has to lower the price. On the other hand, when the intensity of conspicuous is higher than or equal to a threshold L_α, the optimal price increases monotonously. That is, the behavior of conspicuousness raises unit prices. At this point, people are more likely to choose the grinding version items, and as a result, many players stop buying items, which can be seen in Figure 4B.

Figure 4C illustrates the maximum revenue of both single-version and dual-version. As the conspicuous intensity increases, the revenue of the dual-version strategy first declines and then shows an increasing trend. We can see that the price of the item under the single-version strategy remains constant, and the corresponding demand declines, so the firm's maximum revenue from offering a single version falls. The maximum revenue of the single-version strategy is first higher than that obtained by offering dual versions, and then it is exceeded by the dual-version strategy. Under this set of parameters, we suggest that when the conspicuous intensity is not that strong, it is more profitable for the company to adopt the single-version strategy; otherwise, the company should implement the dual-version strategy to increase its revenue.

The quality difference distinguishes the paid version from the grinding version. In the following, to expand the range of value differences for analysis, we increase the quality S of the paid version item to 10. The results can give suggestions on how to control the quality difference.

Similarly, we can also obtain the effect of the quality difference between the two versions on optimal strategy and pricing. The critical value L_Δ and the maximum revenue expression are shown in the Appendix. The comparison of the two strategies under the influence of quality difference is shown in Figure 5. Obviously, the single version item has no quality difference, so its equilibrium solutions do not change with Δ. Figure 5A shows that the optimal price of dual-version increases with the quality difference. As shown in Figure 5B, when the quality difference is lower than the threshold L_Δ, the demand of the paid version decreases, but the demand of the grinding version increases; while when Δ is larger than or equal to L_Δ, the quantity of grinding version users is 0 and buyers drop to 0, which lead to the company's revenue plummet. It can be seen from Figure 5C that the maximum revenue of providing dual-version items is lower than that of providing the single version item at first and then exceeds the single version, reaching the highest point and then decreasing. When the quality difference is within a certain range (Δ₁ < Δ < Δ₂), it is more profitable to provide dual-version items. Therefore, to maximize the revenue, the game company can try to adopt a dual-version strategy and choose a moderate level of difference.

The optimal strategy for the items' version provided by the game company is shown in Figure 6. Figure 6A shows that when conspicuous intensity and the time needed to obtain the grinding version are both at low level, or when the time needed to obtain the grinding version is extended, it is more profitable for the company to offer single-version. Otherwise, it is more profitable to offer dual-version.

Figure 6B shows the impact of the time needed to obtain the grinding version item and the quality difference on the firm's revenue under two different strategies. We can get the following findings. First, if there is a small difference between the two versions, and the time needed to obtain the grinding version is too long or too short, the single-version strategy is more profitable than the dual-version strategy. Second, when the time needed to obtain the grinding version item is short, and the version difference is too large or does not exceed a certain threshold, the single-version strategy is a better option for the game company. Finally, when the time required to obtain the grinding version item exceeds a certain threshold, no matter what the version difference is, the single-version strategy is always better than the dual-version strategy. If and only if the time needed to obtain the grinding version item and the difference between the two versions are at moderate level, the company will be able to make more profitable by adopting the dual-version strategy. It is worth noting that we compare the impact of these two parameters on the revenue satisfying the previously mentioned constraint S-ηSμ<Δ<S-ηt.