Time-locked free trial strategy in duopoly markets with switching costs

Abstract

Offering time-locked free trials is a common strategy in the software industry. Pioneering studies have mainly focused on time-locked free trial strategies in monopoly markets, and acknowledged that an overall positive effect on consumers’ WTP is a prerequisite for offering free trials. However, software markets are often competitive, and consumers typically follow a Bayesian learning principle. We develop a game-theoretic model to examine time-locked free trial strategies in duopoly markets with switching costs. The results show that, in the markets with small switching costs, firms pursue opposite free trial strategies when the horizontal difference between products is small, and neither offers otherwise. If the switching costs are larger than a threshold, then both firms offer free trials when the horizontal difference between products is moderate. We further demonstrate that in the presence of functionality heterogeneity between products, firms are unable to reach an equilibrium in some specific cases, and more likely to take opposite free trial strategies in equilibrium. Moreover, if the firms do choose opposite strategies, interestingly, the firm with superior product does not offer free trials when switching costs are low, and is more willing to offer as switching costs increase. Our results are robust to the inclusion of other factors in the model, such as the possibility of consumers trying only one product, correlated consumer learning across different products, consumer heterogeneity in usage time, and network effects.

Appendices

Appendix A

A1. Proof of Lemma 1

The profits of firms A and B are ${\pi }_{A}={p}_{A}\cdot {x}_{0}$ and ${\pi }_{B}={p}_{B}\cdot \left(1-{x}_{0}\right)$, respectively, where ${x}_{0}=\frac{\delta +{p}_{B}-{p}_{A}}{2\delta }$. Solving equations $\frac{\partial {\pi }_{A}}{\partial {p}_{A}}=0$ and $\frac{\partial {\pi }_{B}}{\partial {p}_{B}}=0$ simultaneously, we can derive the equilibrium solution as shown in Lemma 1. To ensure the market is fully covered by two competing firms, the utility of the marginal consumer at point ${x}_{0}$ should be no less than 0. That is, ${\left.{U}_{A}\right|}_{x={x}_{0}}={\left.{U}_{B}\right|}_{x={x}_{0}}=v-\frac{3\delta }{2}\ge 0$. It follows that $v\ge \frac{3\delta }{2}$.

A2. Proof of Lemma 2

From Eqs. (5) and (6), we derive the profit functions of the two firms ${\pi }_{i}={p}_{i}\cdot {D}_{i}\left({p}_{i},{p}_{-i}\right)$ and ${\pi }_{-i}={p}_{-i}\cdot {D}_{-i}\left({p}_{i},{p}_{-i}\right)$. If $\delta >1/2$, Case 1 cannot occur; otherwise, Case 3 cannot occur. It can be shown that either firm’s profit is concave in its price given the other parameters. Hence, from the first-order conditions, we get the equilibrium prices of the two firms in each case as shown in Lemma 2. Then, the corresponding equilibrium sales and profits for a given $\delta$ and $c$ are derived.

To ensure the market is fully covered by two competing firms, the utility of the marginal consumer at point ${x}_{0}$ should be no less than 0. That is, if $\delta \le \frac{1.5-{c}^{^{\prime}}}{3}$, ${\left.{U}_{A}\right|}_{x={x}_{0}}={\left.{U}_{B}\right|}_{x={x}_{0}}=\left(1-\tau \right)\left(v+\frac{{h}_{i}-1-\delta -{c}^{^{\prime}}}{2}\right)\ge 0$; if $\frac{1.5-{c}^{^{\prime}}}{3}<\delta \le \frac{1.5+{c}^{^{\prime}}}{3}$, ${\left.{U}_{A}\right|}_{x={x}_{0}}={\left.{U}_{B}\right|}_{x={x}_{0}}=\left(1-\tau \right)\left[v+\frac{{h}_{i}}{2}+\frac{1}{8}-\frac{\sqrt{{\left(\delta +0.5-{c}^{^{\prime}}\right)}^{2}+16\delta }}{4}-\frac{\delta +3{c}^{^{\prime}}}{4}\right]\ge 0$; and if $\delta >\frac{1.5+{c}^{^{\prime}}}{3}$, ${\left.{U}_{A}\right|}_{x={x}_{0}}={\left.{U}_{B}\right|}_{x={x}_{0}}=\left(1-\tau \right)\left(v+\frac{{h}_{i}-3\delta -{c}^{^{\prime}}}{2}\right)\ge 0$. Because $-0.5\le {h}_{i}\le 0.5$, it follows that $v\ge \frac{\delta +1.5+{c}^{^{\prime}}}{2}$ if $\delta \le \frac{1.5-{c}^{^{\prime}}}{3}$, $v\ge \frac{\sqrt{{\left(\delta +0.5-{c}^{^{\prime}}\right)}^{2}+16\delta }+\delta +0.5+3{c}^{^{\prime}}}{4}$ if $\frac{1.5-{c}^{^{\prime}}}{3}<\delta \le \frac{1.5+{c}^{^{\prime}}}{3}$, and $v\ge \frac{3\delta +0.5+{c}^{^{\prime}}}{2}$ if $\delta >\frac{1.5+{c}^{^{\prime}}}{3}$.

A3. Proof of Lemma 3

The profits of firms A and B are ${p}_{A}\cdot {x}_{0}$ and ${p}_{B}\cdot \left(1-{x}_{0}\right)$, respectively, where ${x}_{0}=\frac{1}{2}+\frac{{h}_{A}-{h}_{B}}{2\delta }+\frac{{p}_{B}-{p}_{A}}{2\delta \left(1-\tau \right)}$. Solving equations $\frac{\partial {\pi }_{A}}{\partial {p}_{A}}=0$ and $\frac{\partial {\pi }_{B}}{\partial {p}_{B}}=0$ simultaneously, we can derive the equilibrium solution as shown in Lemma 3. To ensure the market is fully covered by two competing firms, the utility of the marginal consumer at point ${x}_{0}$ should be no less than 0. That is, ${\left.{U}_{A}\right|}_{x={x}_{0}}={\left.{U}_{B}\right|}_{x={x}_{0}}=\left(1-\tau \right)\left(v+{h}_{i}-\frac{3\delta }{2}\right)\ge 0$. Because $-0.5\le {h}_{i}\le 0.5$, it follows that $v\ge \frac{3\delta +1}{2}$.

A4. Proof of Proposition 1

Let $c=0$. By Lemmas 1, 2, and 3, we know that ${\pi }_{i}^{NN}=\frac{\delta }{2}$, ${\pi }_{i}^{YY}=\frac{\left(1-\tau \right)\delta }{2}$, while ${\pi }_{i}^{YN/NY}=\frac{1-\tau }{4}$ if $\delta \le \frac{1}{2}$ and ${\pi }_{i}^{YN/NY}=\frac{\left(1-\tau \right)\delta }{2}$ otherwise.

First, we analyze the optimal response of firm A given the strategy of firm B. Suppose firm B does not offer free trials, we see that ${\pi }_{A}^{YN}\ge {\pi }_{A}^{NN}$ when $\delta \le \frac{1-\tau }{2}$ and ${\pi }_{A}^{YN}<{\pi }_{A}^{NN}$ otherwise. Suppose firm B offers free trials, we always have ${\pi }_{A}^{YY}\le {\pi }_{A}^{NY}$. Therefore, we derive that firm A adopt opposite strategy of firm B when $\delta \le \frac{1-\tau }{2}$, and must not offer free trials otherwise.

According to symmetry, we know that firm B adopt opposite strategy of firm A when $\delta \le \frac{1-\tau }{2}$, and must not offer free trials otherwise.

From the above results, we can get the equilibrium of the free trial game as shown in Proposition 1.

A5. Proof of Proposition 2

Following the same approach as in Proposition 1, we first analyze the optimal response of firm A given the strategy of firm B.

Suppose firm B does not offer time-locked free trials. By Lemma 1, we have ${\pi }_{A}^{NN}=\delta /2$, while by Lemma 2, we know that ${\pi }_{A}^{YN}$ increases with $c$. Therefore, by setting ${\pi }_{A}^{YN}={\pi }_{A}^{NN}$, we derive a threshold $\widehat{c}$, satisfying that ${\pi }_{A}^{YN}\ge {\pi }_{A}^{NN}$ when $c\ge \widehat{c}$ and ${\pi }_{A}^{YN}<{\pi }_{A}^{NN}$ otherwise.

$$\widehat{c}=\left\{\begin{array}{ll}\frac{3}{2}\left[\sqrt{2(1-\tau )\delta }-(1-\tau )\right] &\quad if\ \delta \le {\displaystyle \mathcalligra{g}}_{1}\left(\tau \right),\\ \left\{\left.c\right|{\pi }_{A}^{YN\left(b\right)}=\frac{\delta }{2}\right\} &\quad if\ {\mathcalligra{g}}_{1}\left(\tau \right)<\delta \le {\mathcalligra{g}}_{2}\left(\tau \right),\\ 3\delta \left[\sqrt{1-\tau }-(1-\tau )\right] &\quad if \ \delta >{\mathcalligra{g}}_{2}\left(\tau \right),\end{array}\right.$$

(A.1)

where ${\mathcalligra{g}}_{1}\left(\tau \right)=\frac{5-4\tau -\sqrt{9-8\tau }}{4\left(1-\tau \right)}$ and ${\mathcalligra{g}}_{2}\left(\tau \right)=\frac{\sqrt{1-\tau }}{4\sqrt{1-\tau }-2}$.

It can be seen that $\widehat{c}$ increases with $\delta$. Therefore, let $\widehat{\delta }={\widehat{c}}^{-1}$, it follows that ${\pi }_{A}^{YN}\ge {\pi }_{A}^{NN}$ when $\delta \le \widehat{\delta }$ and ${\pi }_{A}^{YN}<{\pi }_{A}^{NN}$ otherwise.

$$\widehat{\delta }=\left\{\begin{array}{c}\frac{{\left[3\left(1-\tau \right)+2c\right]}^{2}}{18\left(1-\tau \right)} if c\le {c}_{1}\left(\tau \right), \\ \left\{\left.\delta \right|{\pi }_{A}^{YN\left(b\right)}=\frac{\delta }{2}\right\} if {c}_{1}\left(\tau \right)<c\le {c}_{2}\left(\tau \right),\\ \frac{c}{3\left[\sqrt{1-\tau }-\left(1-\tau \right)\right]} if c>{c}_{2}\left(\tau \right),\end{array}\right.$$

(A.2)

where ${c}_{1}\left(\tau \right)=\frac{12\tau -15+\sqrt{225-216\tau }}{8}$ and ${c}_{2}\left(\tau \right)=\frac{3(1-\tau )\left[\sqrt{1-\tau }-\left(1-\tau \right)\right]}{4\left(1-\tau \right)-2\sqrt{1-\tau }}$.

Suppose firm B offers free trials. By Lemma 3, we have ${\pi }_{A}^{YY}=\frac{\left(1-\tau \right)\delta }{2}$, while by Lemma 2, we know that ${\pi }_{A}^{NY}$ decreases with $c$. Therefore, by setting ${\pi }_{A}^{YY}={\pi }_{A}^{NY}$, we derive a threshold $\widetilde{c}$, satisfying that ${\pi }_{A}^{YY}\ge {\pi }_{A}^{NY}$ if $c\ge \widetilde{c}$ and ${\pi }_{A}^{YY}<{\pi }_{A}^{NY}$ otherwise, where $\widetilde{c}=\frac{3\left(1-\tau \right)\left(1-\sqrt{2\delta }\right)}{2}<\frac{3\left(1-\tau \right)}{2}$.

It can be seen that $\widetilde{c}$ decreases with $\delta$. Therefore, let $\widetilde{\delta }={\widetilde{c}}^{-1}=\frac{{\left[3\left(1-\tau \right)-2c\right]}^{2}}{18{\left(1-\tau \right)}^{2}}$. It follows that ${\pi }_{A}^{YY}\ge {\pi }_{A}^{NY}$ if $\delta \ge \widetilde{\delta }$ and ${\pi }_{A}^{YY}<{\pi }_{A}^{NY}$ otherwise.

According to symmetry, we can get the optimal response of firm B given the strategy of firm A: suppose firm A does not offer free trials, ${\pi }_{B}^{NY}\ge {\pi }_{B}^{NN}$ when $\delta \le \widehat{\delta }$ and ${\pi }_{B}^{NY}<{\pi }_{B}^{NN}$ otherwise; suppose firm A offers free trials, ${\pi }_{B}^{YY}\ge {\pi }_{B}^{YN}$ if $\delta \ge \widetilde{\delta }$ and ${\pi }_{B}^{YY}<{\pi }_{B}^{YN}$ otherwise.

From the above results, we derive the following equilibria of the free trial game.

• (a) When $\delta \le \mathrm{min}\left\{\widehat{\delta }, \widetilde{\delta }\right\}$, the firms take opposite free trial strategies. That is, one firm offers free trials while the other firm does not.

• (b) When $\widetilde{\delta }<\delta \le \widehat{\delta }$, i.e., $c>\mathrm{max}\left\{\widehat{c},\widetilde{c}\right\}$, both firms offer free trials.

• (c) When $\delta >\widehat{\delta }$, neither firm offers free trials. Specifically, (i) if $\widehat{\delta }<\delta \le \widetilde{\delta }$, i.e., $c\le \mathrm{min}\left\{\widehat{c},\widetilde{c}\right\}$, either firm must not offer free trials; (ii) if $\delta >\mathrm{max}\left\{\widehat{\delta },\widetilde{\delta }\right\}$, i.e., $\widetilde{c}<c\le \widehat{c}$, there exist two equilibria, both-offering and neither-offering, but the neither-offering strategy always dominates.

We see that $\widehat{\delta }$ increases with $c$, while $\widetilde{\delta }$ decreases with $c$. Therefore, as switching costs become larger, the region where firms take the both-offering strategy expands, while the region where firms take the neither-offering strategy shrinks.

When $c$ is sufficiently large, $\widehat{\delta }$ will be so large that $\delta <\widehat{\delta }$ definitely hold as long as the two firms are competitive in nature. Then, the firms adopt opposite free trial strategies when $\delta \le \widetilde{\delta }$, and both offer free trials when $\delta >\widetilde{\delta }$.

A6. Proof of Proposition 3

Without loss of generality, we assume that $\Delta v={v}_{B}-{v}_{A}>0$.

First, we analyze the equilibrium pricing strategies under each subgame following the same approach as in Lemmas 1, 2, and 3. Then, we derive the equilibrium free trial strategies of firms following the same approach as in Propositions 1 and 2. For simplicity, we still use ${c}^{^{\prime}}$ to represent $\frac{c}{1-\tau }$.

1.1 I. Equilibrium pricing strategies under each subgame

1.1.1 I-1. Neither firm offers free trials

According to the consumer utility functions shown with formulas (1) and (2), we can derive the location of the marginal consumer who is indifferent between purchasing from either firm, ${x}_{0}=\frac{\delta +{p}_{B}-{p}_{A}-\Delta v}{2\delta }$. The profits of firms A and B are ${\pi }_{A}={p}_{A}\cdot {x}_{0}$ and ${\pi }_{B}={p}_{B}\cdot \left(1-{x}_{0}\right)$, respectively. Solving equations $\frac{\partial {\pi }_{A}}{\partial {p}_{A}}=0$ and $\frac{\partial {\pi }_{B}}{\partial {p}_{B}}=0$, we derive the equilibrium result as follows.

Lemma 1′

If neither firm offers free trials, then the equilibrium prices and profits are ${p}_{A}^{NN}=\frac{3\delta -\Delta v}{3}$, ${p}_{B}^{NN}=\frac{3\delta +\Delta v}{3}$, ${\pi }_{A}^{NN}=\frac{{\left(3\delta -\Delta v\right)}^{2}}{18\delta }$, and ${\pi }_{B}^{NN}=\frac{{\left(3\delta +\Delta v\right)}^{2}}{18\delta }$.

To ensure the market is fully covered by two competing firms, there must be ${\left.{U}_{A}\right|}_{x={x}_{0}}={\left.{U}_{B}\right|}_{x={x}_{0}}\ge 0$ and $0<{x}_{0}<1$. It follows that ${v}_{A}+{v}_{B}\ge 3\delta$ and $\Delta v<3\delta$.

1.1.2 I-2. Only firm $A$ with smaller product functionality offers free trials

According to the consumer utility functions shown with formulas (3) and (4), we can derive the location of the marginal consumer who is indifferent between purchasing from either firm, ${x}_{0}=\frac{\left(1-\tau \right)\left(\delta +{h}_{A}-\Delta v\right)+{p}_{B}-{p}_{A}+c}{2\left(1-\tau \right)\delta }$. As shown in Fig. 4, there may exist four cases of market segmentations.

Fig. 4

Market segmentations when only firm A offers free trials ($\Delta v={v}_{B}-{v}_{A}>0$)

Consumers below and above the indifference line ${x}_{0}$ purchase from firms A and B, respectively. Let ${c}^{^{\prime}}=\frac{c}{1-\tau }$, the demands of the two firms can then be expressed as follows:

$${D}_{A}\left({p}_{A},{p}_{B}\right)=\left\{\begin{array}{c}\frac{\left(1-\tau \right)\left(0.5+{c}^{^{\prime}}-\Delta v\right)+{p}_{B}-{p}_{A}}{1-\tau } in Case 1: \delta \le min\left\{\frac{1}{2}-\frac{{p}_{B}-{p}_{A}}{1-\tau }-{c}^{^{\prime}}+\Delta v, \frac{1}{2}+\frac{{p}_{B}-{p}_{A}}{1-\tau }+{c}^{^{\prime}}-\Delta v\right\},\\ \frac{{\left[\left(1-\tau \right)\left(\delta +0.5+{c}^{^{\prime}}-\Delta v\right)+{p}_{B}-{p}_{A}\right]}^{2}}{4{\left(1-\tau \right)}^{2}\delta } in Case 2: \frac{1}{2}+\frac{{p}_{B}-{p}_{A}}{1-\tau }+{c}^{^{\prime}}-\Delta v<\delta \le \frac{1}{2}-\frac{{p}_{B}-{p}_{A}}{1-\tau }-{c}^{^{\prime}}+\Delta v,\\ 1-\frac{{\left[\left(1-\tau \right)\left(\delta +0.5-{c}^{^{\prime}}+\Delta v\right)+{p}_{A}-{p}_{B}\right]}^{2}}{4{\left(1-\tau \right)}^{2}\delta } in Case 3: \frac{1}{2}-\frac{{p}_{B}-{p}_{A}}{1-\tau }-{c}^{^{\prime}}+\Delta v<\delta \le \frac{1}{2}+\frac{{p}_{B}-{p}_{A}}{1-\tau }+{c}^{^{\prime}}-\Delta v,\\ \frac{\left(1-\tau \right)\left(\delta +{c}^{^{\prime}}-\Delta v\right)+{p}_{B}-{p}_{A}}{2\left(1-\tau \right)\delta } in Case 4: \delta >max\left\{\frac{1}{2}-\frac{{p}_{B}-{p}_{A}}{1-\tau }-{c}^{^{\prime}}+\Delta v, \frac{1}{2}+\frac{{p}_{B}-{p}_{A}}{1-\tau }+{c}^{^{\prime}}-\Delta v\right\}.\end{array}\right.$$

(A.3)

$${D}_{B}\left({p}_{A},{p}_{B}\right)=1-{D}_{A}\left({p}_{A},{p}_{B}\right).$$

(A.4)

The two firms simultaneously set the prices of their commercial products to maximize their profits, ${\pi }_{A}={p}_{A}\cdot {D}_{A}\left({p}_{A},{p}_{B}\right)$ and ${\pi }_{B}={p}_{B}\cdot {D}_{B}\left({p}_{A},{p}_{B}\right)$. Solving equations $\frac{\partial {\pi }_{A}}{\partial {p}_{A}}=0$ and $\frac{\partial {\pi }_{B}}{\partial {p}_{B}}=0$, we derive the equilibrium result as follows.

Lemma 2′

If only firm A offers free trials, then the equilibrium prices and profits are:

• (a) If $\delta \le \mathrm{min}\left\{\frac{1.5-{c}^{^{\prime}}+\Delta v}{3},\frac{1.5+{c}^{^{\prime}}-\Delta v}{3}\right\}$, then ${p}_{A}^{YN}=\frac{\left(1-\tau \right)(1.5+{c}^{^{\prime}}-\Delta v)}{3}$, ${p}_{B}^{YN}=\frac{\left(1-\tau \right)(1.5-{c}^{^{\prime}}+\Delta v)}{3}$, ${\pi }_{A}^{YN}=\frac{\left(1-\tau \right){(1.5+{c}^{^{\prime}}-\Delta v)}^{2}}{9}$ and ${\pi }_{B}^{YN}=\frac{\left(1-\tau \right){(1.5-{c}^{^{\prime}}+\Delta v)}^{2}}{9}$;

• (b) If $\frac{1.5+{c}^{^{\prime}}-\Delta v}{3}<\delta \le \frac{1.5-{c}^{^{\prime}}+\Delta v}{3}$, then ${p}_{A}^{YN}=\frac{\left(1-\tau \right)\left(\sqrt{{{\eta }_{1}}^{2}+16\delta }+{\eta }_{1}\right)}{8}$, ${p}_{B}^{YN}=\frac{\left(1-\tau \right)\left(3\sqrt{{{\eta }_{1}}^{2}+16\delta }-5{\eta }_{1}\right)}{8}$, ${\pi }_{A}^{YN}=\frac{\left(1-\tau \right){\left(\sqrt{{{\eta }_{1}}^{2}+16\delta }+{\eta }_{1}\right)}^{3}}{512\delta }$, and ${\pi }_{B}^{YN}=\frac{\left(1-\tau \right)\left(3\sqrt{{{\eta }_{1}}^{2}+16\delta }-5{\eta }_{1}\right)\left[64\delta -{\left(\sqrt{{{\eta }_{1}}^{2}+16\delta }+{\eta }_{1}\right)}^{2}\right]}{512\delta }$, where ${\eta }_{1}=\delta +0.5+{c}^{^{\prime}}-\Delta v$;

• (c) If $\frac{1.5-{c}^{^{\prime}}+\Delta v}{3}<\delta \le \frac{1.5+{c}^{^{\prime}}-\Delta v}{3}$, then ${p}_{A}^{YN}=\frac{\left(1-\tau \right)\left(3\sqrt{{{\eta }_{2}}^{2}+16\delta }-5{\eta }_{2}\right)}{8}$, ${p}_{B}^{YN}=\frac{\left(1-\tau \right)\left(\sqrt{{{\eta }_{2}}^{2}+16\delta }+{\eta }_{2}\right)}{8}$, ${\pi }_{A}^{YN}=\frac{\left(1-\tau \right)\left(3\sqrt{{{\eta }_{2}}^{2}+16\delta }-5{\eta }_{2}\right)\left[64\delta -{\left(\sqrt{{{\eta }_{2}}^{2}+16\delta }+{\eta }_{2}\right)}^{2}\right]}{512\delta }$, and ${\pi }_{B}^{YN}=\frac{\left(1-\tau \right){\left(\sqrt{{{\eta }_{2}}^{2}+16\delta }+{\eta }_{2}\right)}^{3}}{512\delta }$, where ${\eta }_{2}=\delta +0.5-{c}^{^{\prime}}+\Delta v$;

• (d) If $\delta >\mathrm{max}\left\{\frac{1.5-{c}^{^{\prime}}+\Delta v}{3},\frac{1.5+{c}^{^{\prime}}-\Delta v}{3}\right\}$, then ${p}_{A}^{YN}=\frac{\left(1-\tau \right)(3\delta +{c}^{^{\prime}}-\Delta v)}{3}$, ${p}_{B}^{YN}=\frac{\left(1-\tau \right)(3\delta -{c}^{^{\prime}}+\Delta v)}{3}$, ${\pi }_{A}^{YN}=\frac{{\left(1-\tau \right)(3\delta +{c}^{^{\prime}}-\Delta v)}^{2}}{18\delta }$ and ${\pi }_{B}^{YN}=\frac{{\left(1-\tau \right)(3\delta -{c}^{^{\prime}}+\Delta v)}^{2}}{18\delta }$.

To ensure the market is fully covered by two competing firms, there must be ${\left.{U}_{A}\right|}_{x={x}_{0}}={\left.{U}_{B}\right|}_{x={x}_{0}}\ge 0$ and $0<{x}_{0}<1$. It follows that ${v}_{A}+{v}_{B}\ge \delta +1.5+{c}^{^{\prime}}$ if $\delta \le \mathrm{min}\left\{\frac{1.5-{c}^{^{\prime}}+\Delta v}{3},\frac{1.5+{c}^{^{\prime}}-\Delta v}{3}\right\}$, $3{v}_{A}+{v}_{B}\ge \sqrt{{{\eta }_{1}}^{2}+16\delta }+\delta +0.5+{c}^{^{\prime}}$ if $\frac{1.5+{c}^{^{\prime}}-\Delta v}{3}<\delta \le \frac{1.5-{c}^{^{\prime}}+\Delta v}{3}$, ${v}_{A}+3{v}_{B}\ge \sqrt{{{\eta }_{2}}^{2}+16\delta }+\delta +0.5+3{c}^{^{\prime}}$ if $\frac{1.5-{c}^{^{\prime}}+\Delta v}{3}<\delta \le \frac{1.5+{c}^{^{\prime}}-\Delta v}{3}$, and ${v}_{A}+{v}_{B}\ge 3\delta +0.5+{c}^{^{\prime}}$ if $\delta >\mathrm{max}\left\{\frac{1.5-{c}^{^{\prime}}+\Delta v}{3},\frac{1.5+{c}^{^{\prime}}-\Delta v}{3}\right\}$.

1.1.3 I-3. Only firm $B$ with larger product functionality offers free trials

Following similar approach to the case where only firm A offers free trials, we can derive the equilibrium result when only firm B offers free trials, shown as follows.

Lemma 2′′

If only firm B offers free trials, then the equilibrium prices and profits are:

• (a) If $\delta \le \frac{1.5-{c}^{^{\prime}}-\Delta v}{3}$, then ${p}_{A}^{NY}=\frac{\left(1-\tau \right)(1.5-{c}^{^{\prime}}-\Delta v)}{3}$, ${p}_{B}^{NY}=\frac{\left(1-\tau \right)(1.5+{c}^{^{\prime}}+\Delta v)}{3}$, ${\pi }_{A}^{NY}=\frac{\left(1-\tau \right){(1.5-{c}^{^{\prime}}-\Delta v)}^{2}}{9}$ and ${\pi }_{B}^{NY}=\frac{\left(1-\tau \right){(1.5+{c}^{^{\prime}}+\Delta v)}^{2}}{9}$;

• (b) If $\frac{1.5-{c}^{^{\prime}}-\Delta v}{3}<\delta \le \frac{1.5+{c}^{^{\prime}}+\Delta v}{3}$, then ${p}_{A}^{NY}=\frac{\left(1-\tau \right)\left(\sqrt{{{\eta }_{3}}^{2}+16\delta }+{\eta }_{3}\right)}{8}$, ${p}_{B}^{NY}=\frac{\left(1-\tau \right)\left(3\sqrt{{{\eta }_{3}}^{2}+16\delta }-5{\eta }_{3}\right)}{8}$, ${\pi }_{A}^{NY}=\frac{\left(1-\tau \right){\left(\sqrt{{{\eta }_{3}}^{2}+16\delta }+{\eta }_{3}\right)}^{3}}{512\delta }$, and ${\pi }_{B}^{NY}=\frac{\left(1-\tau \right)\left(3\sqrt{{{\eta }_{3}}^{2}+16\delta }-5{\eta }_{3}\right)\left[64\delta -{\left(\sqrt{{{\eta }_{3}}^{2}+16\delta }+{\eta }_{3}\right)}^{2}\right]}{512\delta }$, where ${\eta }_{3}=\delta +0.5-{c}^{^{\prime}}-\Delta v$;

• (c) If $\delta >\frac{1.5+{c}^{^{\prime}}+\Delta v}{3}$, then ${p}_{A}^{NY}=\frac{\left(1-\tau \right)(3\delta -{c}^{^{\prime}}-\Delta v)}{3}$, ${p}_{B}^{NY}=\frac{\left(1-\tau \right)(3\delta +{c}^{^{\prime}}+\Delta v)}{3}$, ${\pi }_{A}^{NY}=\frac{{\left(1-\tau \right)(3\delta -{c}^{^{\prime}}-\Delta v)}^{2}}{18\delta }$ and ${\pi }_{B}^{NY}=\frac{{\left(1-\tau \right)(3\delta +{c}^{^{\prime}}+\Delta v)}^{2}}{18\delta }$.

To ensure the market is fully covered by two competing firms, there must be ${\left.{U}_{A}\right|}_{x={x}_{0}}={\left.{U}_{B}\right|}_{x={x}_{0}}\ge 0$ and $0<{x}_{0}<1$. It follows that ${v}_{A}+{v}_{B}\ge \delta +1.5+{c}^{^{\prime}}$ if $\delta \le \frac{1.5-{c}^{^{\prime}}-\Delta v}{3}$, $3{v}_{A}+{v}_{B}\ge \sqrt{{{\eta }_{3}}^{2}+16\delta }+\delta +0.5+3{c}^{^{\prime}}$ if $\frac{1.5-{c}^{^{\prime}}-\Delta v}{3}<\delta \le \frac{1.5+{c}^{^{\prime}}+\Delta v}{3}$, and ${v}_{A}+{v}_{B}\ge 3\delta +0.5+{c}^{^{\prime}}$ if $\delta >\frac{1.5+{c}^{^{\prime}}+\Delta v}{3}$.

1.1.4 I-4. Both firms offer free trials

According to the consumer utility functions shown with formulas (7) and (8), we can derive the location of the marginal consumer who is indifferent between purchasing from either firm, ${x}_{0}=\frac{\left(1-\tau \right)\left(\delta -\Delta v\right)+{p}_{B}-{p}_{A}}{2\left(1-\tau \right)\delta }$. The profits of firms A and B are ${\pi }_{A}={p}_{A}\bullet {x}_{0}$ and ${\pi }_{B}={p}_{B}\bullet \left(1-{x}_{0}\right)$, respectively. Solving equations $\frac{\partial {\pi }_{A}}{\partial {p}_{A}}=0$ and $\frac{\partial {\pi }_{B}}{\partial {p}_{B}}=0$, we derive the equilibrium result as follows.

Lemma 3′

If both firms offer free trials, then the equilibrium prices and profits are ${p}_{A}^{YY}=\frac{\left(1-\tau \right)\left(3\delta -\Delta v\right)}{3}$, ${p}_{B}^{YY}=\frac{\left(1-\tau \right)\left(3\delta +\Delta v\right)}{3}$, ${\pi }_{A}^{YY}=\frac{{\left(1-\tau \right)\left(3\delta -\Delta v\right)}^{2}}{18\delta }$, and ${\pi }_{B}^{YY}=\frac{\left(1-\tau \right){\left(3\delta +\Delta v\right)}^{2}}{18\delta }$.

To ensure the market is fully covered by two competing firms, there must be ${\left.{U}_{A}\right|}_{x={x}_{0}}={\left.{U}_{B}\right|}_{x={x}_{0}}\ge 0$ and $0<{x}_{0}<1$. It follows that ${v}_{A}+{v}_{B}\ge 3\delta +1$ and $\Delta v<3\delta$.

From the above results, we can see that, to ensure the profits of firms A and B are positive irrespective of their free trial strategies, $\Delta v$ should be smaller than $3\delta$.

1.2 II. Equilibrium analysis of the free trial game

1.2.1 II-1. Firm A’s optimal response given the strategy of firm B

Suppose firm B does not offer free trials. From Lemmas 1′ and 2′, we know that ${\pi }_{A}^{NN}$ is independent of $c$, while ${\pi }_{A}^{YN}$ increases with $c$. Therefore, by setting ${\pi }_{A}^{YN}={\pi }_{A}^{NN}$, we derive a threshold ${\widehat{c}}^{a}$, satisfying that ${\pi }_{A}^{YN}\ge {\pi }_{A}^{NN}$ when $c\ge {\widehat{c}}^{a}$ and ${\pi }_{A}^{YN}<{\pi }_{A}^{NN}$ otherwise.

$$ \hat{c}^{a} = \left\{ {\begin{array}{*{20}l} {\left( {1 - \tau } \right)\left[ {\frac{3\delta }{{\sqrt {2\left( {1 - \tau } \right)\delta } }} - \frac{3}{2} - \left( {\frac{1}{{\sqrt {2\left( {1 - \tau } \right)\delta } }} - 1} \right)\Delta v} \right]} \hfill & {if} \hfill & {\delta \le {\text{min}}\left\{ {\frac{{{{1.5 - \hat{c}^{a} } \mathord{\left/ {\vphantom {{1.5 - \hat{c}^{a} } {\left( {1 - \tau } \right) + \Delta v}}} \right. \kern-0pt} {\left( {1 - \tau } \right) + \Delta v}}}}{3},\frac{{{{1.5 + \hat{c}^{a} } \mathord{\left/ {\vphantom {{1.5 + \hat{c}^{a} } {\left( {1 - \tau } \right) - \Delta v}}} \right. \kern-0pt} {\left( {1 - \tau } \right) - \Delta v}}}}{3}{ }} \right\},} \hfill \\ {\left\{ {\left. \delta \right|\pi_{A}^{YN\left( b \right)} = \pi_{A}^{NN} } \right\}} \hfill & {if} \hfill & {\frac{{{{1.5 + \hat{c}^{a} } \mathord{\left/ {\vphantom {{1.5 + \hat{c}^{a} } {\left( {1 - \tau } \right) - \Delta v}}} \right. \kern-0pt} {\left( {1 - \tau } \right) - \Delta v}}}}{3} < \delta \le \frac{{{{1.5 - \hat{c}^{a} } \mathord{\left/ {\vphantom {{1.5 - \hat{c}^{a} } {\left( {1 - \tau } \right)}}} \right. \kern-0pt} {\left( {1 - \tau } \right)}} + \Delta v}}{3},} \hfill \\ {\left\{ {\left. \delta \right|\pi_{A}^{YN\left( c \right)} = \pi_{A}^{NN} } \right\}} \hfill & {if} \hfill & {\frac{{{{1.5 - \hat{c}^{a} } \mathord{\left/ {\vphantom {{1.5 - \hat{c}^{a} } {\left( {1 - \tau } \right) + \Delta v}}} \right. \kern-0pt} {\left( {1 - \tau } \right) + \Delta v}}}}{3} < \delta \le \frac{{{{1.5 + \hat{c}^{a} } \mathord{\left/ {\vphantom {{1.5 + \hat{c}^{a} } {\left( {1 - \tau } \right) - \Delta v}}} \right. \kern-0pt} {\left( {1 - \tau } \right) - \Delta v}}}}{3},} \hfill \\ {\left[ {\sqrt {1 - \tau } - \left( {1 - \tau } \right)} \right]\left( {3\delta - \Delta v} \right)} \hfill & {if} \hfill & {\delta > {\text{max}}\left\{ {\frac{{{{1.5 - \hat{c}^{a} } \mathord{\left/ {\vphantom {{1.5 - \hat{c}^{a} } {\left( {1 - \tau } \right) + \Delta v}}} \right. \kern-0pt} {\left( {1 - \tau } \right) + \Delta v}}}}{3},\frac{{{{1.5 + \hat{c}^{a} } \mathord{\left/ {\vphantom {{1.5 + \hat{c}^{a} } {\left( {1 - \tau } \right) - \Delta v}}} \right. \kern-0pt} {\left( {1 - \tau } \right) - \Delta v}}}}{3}{ }} \right\}.} \hfill \\ \end{array} } \right. $$

(A.5)

Suppose firm B offers free trials. From Lemmas 2′′ and 3′, we know that ${\pi }_{A}^{YY}$ is independent of $c$, while ${\pi }_{A}^{NY}$ decreases with $c$. Therefore, by setting ${\pi }_{A}^{YY}={\pi }_{A}^{NY}$, we derive a threshold ${\widetilde{c}}^{a}$, satisfying that ${\pi }_{A}^{YY}\ge {\pi }_{A}^{NY}$ when $c\ge {\widetilde{c}}^{a}$ and ${\pi }_{A}^{YY}<{\pi }_{A}^{NY}$ otherwise.

$${\widetilde{c}}^{a}=\left\{\begin{array}{c}\left(1-\tau \right)\left[\frac{3\left(1-\sqrt{2\delta }\right)}{2}+\left(\frac{1}{\sqrt{2\delta }}-1\right)\Delta v\right] if \delta \le \frac{1.5-{\widetilde{c}}^{a}/\left(1-\tau \right)-\Delta v}{3}, \\ \left\{\left.\delta \right|{\pi }_{A}^{NY\left(b\right)}={\pi }_{A}^{YY}\right\} if if \delta >\frac{1.5-{\widetilde{c}}^{a}/\left(1-\tau \right)-\Delta v}{3}.\end{array}\right.$$

(A.6)

In summary, the optimal response of firm A is: (i) if $c\le \mathrm{min}\left\{{\widehat{c}}^{a},{\widetilde{c}}^{a}\right\}$, it must not offer free trials; (ii) if ${\widetilde{c}}^{a}<c\le {\widehat{c}}^{a}$, it adopts the same strategy with firm B; (iii) if ${\widehat{c}}^{a}<c\le {\widetilde{c}}^{a}$, it adopts opposite strategy of firm B; (iv) if $c>\mathrm{max}\left\{{\widehat{c}}^{a},{\widetilde{c}}^{a}\right\}$, it definitely offers free trials.

1.3 II-2. Firm B’s optimal response given the strategy of firm A

Suppose firm A does not offer free trials, as above, we derive a threshold ${\widehat{c}}^{b}$, satisfying that ${\pi }_{B}^{NY}\ge {\pi }_{B}^{NN}$ when $c\ge {\widehat{c}}^{b}$ and ${\pi }_{B}^{NY}<{\pi }_{B}^{NN}$ otherwise.

$${\widehat{c}}^{b}=\left\{\begin{array}{c}\left(1-\tau \right)\left[\frac{3\delta }{\sqrt{2\left(1-\tau \right)\delta }}-\frac{3}{2}+\left(\frac{1}{\sqrt{2\left(1-\tau \right)\delta }}-1\right)\Delta v\right] if \delta \le \frac{1.5-{\widehat{c}}^{b}/\left(1-\tau \right)-\Delta v}{3}, \\ \left\{\left.\delta \right|{\pi }_{B}^{NY\left(b\right)}={\pi }_{B}^{NN}\right\} if \frac{1.5-{\widehat{c}}^{b}/\left(1-\tau \right)-\Delta v}{3}<\delta \le \frac{1.5+{\widehat{c}}^{b}/\left(1-\tau \right)+\Delta v}{3},\\ \left[\sqrt{1-\tau }-\left(1-\tau \right)\right]\left(3\delta +\Delta v\right) if \delta >\frac{1.5+{\widehat{c}}^{b}/\left(1-\tau \right)+\Delta v}{3} .\end{array}\right.$$

(A.7)

Suppose firm A offers free trials, as above, we derive a threshold ${\widetilde{c}}^{b}$, satisfying that ${\pi }_{B}^{YY}\ge {\pi }_{B}^{YN}$ when $c\ge {\widetilde{c}}^{b}$ and ${\pi }_{B}^{YY}<{\pi }_{B}^{YN}$ otherwise.

$${\widetilde{c}}^{b}=\left\{\begin{array}{c}\left(1-\tau \right)\left[\frac{3\left(1-\sqrt{2\delta }\right)}{2}-\left(\frac{1}{\sqrt{2\delta }}-1\right)\Delta v\right] if \delta \le min\left\{\frac{1.5-{\widetilde{c}}^{b}/\left(1-\tau \right)+\Delta v}{3},\frac{1.5+{\widetilde{c}}^{b}/\left(1-\tau \right)-\Delta v}{3} \right\}, \\ \left\{\left.\delta \right|{\pi }_{B}^{YN\left(b\right)}={\pi }_{B}^{YY}\right\} if \frac{1.5+{\widetilde{c}}^{b}/\left(1-\tau \right)-\Delta v}{3}<\delta \le \frac{1.5-{\widetilde{c}}^{b}/\left(1-\tau \right)+\Delta v}{3},\\ \left\{\left.\delta \right|{\pi }_{B}^{YN\left(c\right)}={\pi }_{B}^{YY}\right\} if \frac{1.5-{\widetilde{c}}^{b}/\left(1-\tau \right)+\Delta v}{3}<\delta \le \frac{1.5+{\widetilde{c}}^{b}/\left(1-\tau \right)-\Delta v}{3} .\end{array}\right.$$

(A.8)

In summary, the optimal response of firm B is: (i) if $c\le \mathrm{min}\left\{{\widehat{c}}^{b},{\widetilde{c}}^{b}\right\}$, it must not offer free trials; (ii) if ${\widetilde{c}}^{b}<c\le {\widehat{c}}^{b}$, it adopts the same strategy with firm A; (iii) if ${\widehat{c}}^{b}<c\le {\widetilde{c}}^{b}$, it adopts opposite strategy of firm A; (iv) if $c>\mathrm{max}\left\{{\widehat{c}}^{b},{\widetilde{c}}^{b}\right\}$, it definitely offers free trials.

1.4 II-3. Equilibrium of the free trial game

It can be shown that ${\widehat{c}}^{b}\ge {\widehat{c}}^{a}$ and ${\widetilde{c}}^{b}\le {\widetilde{c}}^{a}$. Let $\hat{\delta }^{a} = \hat{c}^{a - 1}$, $\hat{\delta }^{b} = \hat{c}^{b - 1}$, $\tilde{\delta }^{a} = \tilde{c}^{a - 1}$ and $\tilde{\delta }^{b} = \tilde{c}^{b - 1}$. Then, from I-1 and I-2, we can derive the equilibrium of the free trial game as follows.

• (a) When ${\widehat{c}}^{a}<c\le {\widetilde{c}}^{b}$ or ${\widehat{c}}^{b}<c\le {\widetilde{c}}^{a}$, i.e., $\delta \le \mathrm{min}\left\{{\widehat{\delta }}^{a},{\widetilde{\delta }}^{a},\mathrm{max}\left\{{\widehat{\delta }}^{b},{\widetilde{\delta }}^{b}\right\}\right\}$, firms adopt opposite free trial strategies. Specifically, (i) if $c\le \mathrm{min}\left\{{\widehat{c}}^{b},{\widetilde{c}}^{b}\right\}$, firm A with smaller product functionality offers; (ii) if ${\widehat{c}}^{b}<c\le {\widetilde{c}}^{b}$, either firm can be the one offering free trials; (iii) if $c>\mathrm{max}\left\{{\widehat{c}}^{b},{\widetilde{c}}^{b}\right\}$, firm B with larger product functionality offers.

• (b) When $\mathrm{max}\left\{{\widehat{c}}^{a},{\widetilde{c}}^{b}\right\}<c\le \mathrm{min}\left\{{\widehat{c}}^{b},{\widetilde{c}}^{a}\right\}$, i.e., $\mathrm{max}\left\{{\widehat{\delta }}^{b},{\widetilde{\delta }}^{b}\right\}<\delta \le \mathrm{min}\left\{{\widehat{\delta }}^{a},{\widetilde{\delta }}^{a}\right\}$, there exists no equilibrium.

• (c) When $c>\mathrm{max}\left\{{\widehat{c}}^{a},{\widetilde{c}}^{a}\right\}$, i.e., ${\widetilde{\delta }}^{a}<\delta \le {\widehat{\delta }}^{a}$, both firms offer free trials.

• (d) When $c\le {\widehat{c}}^{a}$, i.e., $\delta >{\widehat{\delta }}^{a}$, neither firm offers free trials. More specifically, (i) if $c\le \mathrm{min}\left\{{\widehat{c}}^{a},{\widetilde{c}}^{a}\right\}$, either firm must not offer free trials; (ii) if ${\widetilde{c}}^{a}<c\le {\widehat{c}}^{a}$, there exist two equilibria, both-offering and neither-offering, but the neither-offering strategy always dominates.

It can be shown that ${\widehat{c}}^{a}$ decreases with $\Delta v$, while ${\widetilde{c}}^{a}$ increases with $\Delta v$. Therefore, as the two firms’ difference in product functionality becomes larger, the regions where firms take opposite free trial strategies or cannot reach an equilibrium expand, while the regions where firms take the both-offering strategy or the neither-offering strategy shrink.

A7. Proof of Proposition 4

By Proposition 3, we see that the thresholds ${\widehat{c}}^{a}$ and ${\widehat{c}}^{b}$ increase with $\tau$, while ${\widetilde{c}}^{a}$ and ${\widetilde{c}}^{b}$ decrease with $\tau$. Therefore, as $\tau$ decreases, the regions where firms take opposite free trial strategies expand, while the regions where firms take the neither-offering strategy shrink.

As $\tau$ approaches 0, ${\widehat{c}}^{a}$ and ${\widehat{c}}^{b}$ approach zero. Therefore, the equilibrium of the free trial game is reduced to as follows.

• (a) When $\delta \le {\widetilde{\delta }}^{a}$, firms adopt opposite free trial strategies. More specifically, either firm can be the one offering free trials if $\delta \le {\widetilde{\delta }}^{b}$, and firm B with larger product functionality unilaterally offers if ${\widetilde{\delta }}^{b}<\delta \le {\widetilde{\delta }}^{a}$.

• (b) When $\delta \le {\widetilde{\delta }}^{a}$, both firms offer free trials.

A8. Proof of Proposition 5

According to Lemmas 2, and 3, the demands for the two firms are as follows:

$${D}_{A}\left({p}_{A},{p}_{B}\right)=\left\{\begin{array}{c}\frac{\left(1-\tau \right)\delta +\left[\left(1-\lambda \right)+2\lambda \delta \right]\left({p}_{B}-{p}_{A}\right)}{2\left(1-\tau \right)\delta } if \delta \in Case 1,\\ \frac{\left(1-\tau \right)\delta +\left[\left(1-\lambda \right)+2\lambda \left(0.5-{c}^{^{\prime}}\right)\right]\left({p}_{B}-{p}_{A}\right)}{2\left(1-\tau \right)\delta } if \delta \in Cases 2 or 3.\end{array}\right.$$

(A.9)

$${D}_{B}\left({p}_{A},{p}_{B}\right)=1-{D}_{A}\left({p}_{A},{p}_{B}\right).$$

(A.10)

The profit functions of the two firms are ${\pi }_{i}={p}_{i}\bullet {D}_{i}\left({p}_{i},{p}_{-i}\right)$, $i=A,B$. Let the first derivatives of the profit functions with respect to each firm’s price equal to 0. Solving these two equations simultaneously yields equilibrium prices ${p}_{i}^{YY}=\frac{\left(1-\tau \right)\delta }{\left(1-\lambda \right)+2\lambda \delta }$ if $\delta \le 0.5-{c}^{^{\prime}}$ and ${p}_{i}^{YY}=\frac{\left(1-\tau \right)\delta }{\left(1-\lambda \right)+2\lambda \left(0.5-{c}^{^{\prime}}\right)}$ if $\delta >0.5-{c}^{^{\prime}}$. It follows that, the equilibrium demands are ${D}_{i}^{YY}=1/2$, and the equilibrium profits are ${\pi }_{i}^{YY}=\frac{\left(1-\tau \right)\delta }{2\left[\left(1-\lambda \right)+2\lambda \delta \right]}$ if $\delta \le 0.5-{c}^{^{\prime}}$ and ${\pi }_{i}^{YY}=\frac{\left(1-\tau \right)\delta }{2\left[\left(1-\lambda \right)+2\lambda \left(0.5-{c}^{^{\prime}}\right)\right]}$ otherwise.

Taking the first derivative of ${\pi }_{i}^{YY}$ with respect to $\lambda$, we have that ${\pi }_{i}^{YY}$ increases with $\lambda$. It follows that, ${\pi }_{A}^{YY}-{\pi }_{A}^{NY}$ (or ${\pi }_{B}^{YY}-{\pi }_{B}^{YN}$) increases with $\lambda$. As in the baseline model, by setting ${\pi }_{A}^{YY}={\pi }_{A}^{NY}$ (or ${\pi }_{B}^{YY}={\pi }_{B}^{YN}$), we derive the threshold $\widetilde{c}$ (i.e., $\widetilde{\delta }$) and it decreases with $\lambda$. Since $\lambda$ has no effect on the other equilibrium profits except ${\pi }_{i}^{YY}$, the other thresholds remain the same as in the baseline model. Therefore, the two firms are more willing to take the both-offering strategy as $\lambda$ increases.

Let $\lambda =1$. We can show that ${\pi }_{A}^{YY}-{\pi }_{A}^{NY}\ge 0$, and ${\pi }_{B}^{YY}-{\pi }_{B}^{YN}\ge 0$. Therefore, in this case, the opposite free trial strategies are never taken and both firms offer free trials when $\delta \le \widehat{\delta }$.

A9. Proof of Proposition 6

As stated in the baseline model, if a consumer consistently evaluates different firms’ products, ${h}_{A}-{h}_{B}={\mu }_{B}-{\mu }_{A}=0$. If consumer evaluations of different products are completely independent, define ${Z}_{0}={h}_{A}-{h}_{B}$, then it is easy to show that ${Z}_{0}$ follows a triangular distribution ${f}_{{Z}_{0}}\left({z}_{0}\right)$ with mean $0$:

$${f}_{{Z}_{0}}\left({z}_{0}\right)=\left\{\begin{array}{c}1+{z}_{0} if -1\le {z}_{0}\le 0,\\ 1-{z}_{0} if 0<{z}_{0}\le 1.\end{array}\right.$$

(A.11)

In the general case, when we denote the correlation of consumer learning across different products by $\alpha$, that means a consumer consistently evaluates different firms’ products with a probability of $\alpha$, and the probability that her evaluations are completely independent is $1-\alpha$. Define $Z={h}_{A}-{h}_{B}$ in the general case. Then, $Z=\alpha \bullet 0+\left(1-\alpha \right)\bullet {Z}_{0}$ and

$${f}_{Z}\left(z\right)=\left\{\begin{array}{c}\left(1-\alpha +z\right)/{\left(1-\alpha \right)}^{2} if -\left(1-\alpha \right)\le z\le 0,\\ \left(1-\alpha -z\right)/{\left(1-\alpha \right)}^{2} if 0<z\le \left(1-\alpha \right).\end{array}\right.$$

(A.12)

Substituting $Z$ into the indifference line ${x}_{0}$, then the demands for the two firms are as follows:

$${D}_{A}\left({p}_{A},{p}_{B}\right)=\left\{\begin{array}{c}\frac{1}{2}+\frac{{\left({p}_{A}-{p}_{B}\right)}^{3}}{6{\left(1-\tau \right)}^{3}{\left(1-\alpha \right)}^{2}\delta }+\frac{\left[\delta -2\left(1-\alpha \right)\right]\left({p}_{A}-{p}_{B}\right)}{2\left(1-\tau \right){\left(1-\alpha \right)}^{2}} if \delta \le 1-\alpha ,\\ \frac{\left(1-\tau \right)\delta +{p}_{B}-{p}_{A}}{2\left(1-\tau \right)\delta } if \delta >1-\alpha .\end{array}\right.$$

(A.13)

$${D}_{B}\left({p}_{A},{p}_{B}\right)=1-{D}_{A}\left({p}_{A},{p}_{B}\right).$$

(A.14)

The profit functions of the two firms are ${\pi }_{i}={p}_{i}\bullet {D}_{i}\left({p}_{i},{p}_{-i}\right)$, $i=A,B$. Let the first derivatives of the profit functions with respect to each firm’s price equal to 0. Solving these two equations simultaneously yields the equilibrium prices ${p}_{i}^{YY}=\frac{\left(1-\tau \right){\left(1-\alpha \right)}^{2}}{2\left(1-\alpha \right)-\delta }$ if $\delta \le 1-\alpha$ and ${p}_{i}^{YY}=\left(1-\tau \right)\delta$ if $\delta >1-\alpha$. It follows that, the equilibrium demands are ${D}_{i}^{YY}=1/2$, and the equilibrium profits are ${\pi }_{i}^{YY}=\frac{\left(1-\tau \right){\left(1-\alpha \right)}^{2}}{4\left(1-\alpha \right)-2\delta }$ if $\delta \le 1-\alpha$ and ${\pi }_{i}^{YY}=\frac{\left(1-\tau \right)\delta }{2}$ otherwise.

Taking the first derivative of ${\pi }_{i}^{YY}$ with respect to $\alpha$, we have that ${\pi }_{i}^{YY}$ weakly decreases with $\alpha$. It follows that, ${\pi }_{A}^{YY}-{\pi }_{A}^{NY}$ (or ${\pi }_{B}^{YY}-{\pi }_{B}^{YN}$) decreases with $\alpha$. As in the baseline model, by setting ${\pi }_{A}^{YY}={\pi }_{A}^{NY}$ (or ${\pi }_{B}^{YY}={\pi }_{B}^{YN}$), we derive the threshold $\widetilde{c}$ (i.e., $\widetilde{\delta }$) and it increases with $\alpha$. Since $\alpha$ has no effect on the other equilibrium profits except ${\pi }_{i}^{YY}$, it is clear that Propositions 1 and 2 and 4 still qualitatively hold.

When $\alpha$ equals 0, we have ${\pi }_{A}^{YY}-{\pi }_{A}^{NY}\ge 0$ and ${\pi }_{B}^{YY}-{\pi }_{B}^{YN}\ge 0$. Therefore, in this case, the opposite free trial strategies are never taken and both firms offer free trials when $\delta \le \widehat{\delta }$.

1.5 A10. Analysis of Extension 5.3

Assume that consumers’ usage time of a software product is uniformly distributed, and let ${p}_{i}^{^{\prime}}$, ${D}_{i}^{^{\prime}}$, ${U}_{i}^{^{\prime}}$, and ${\pi }_{i}^{^{\prime}}$ denote the corresponding price, demand, utility, and profit, respectively. Then ${\pi }_{i}^{^{\prime}}={\pi }_{i}/2$ when neither firm offers free trials.

If at least one firm offers free trials, then, for any consumer with usage time longer than $\tau$, the utility the consumer obtains when buying from firms A and B becomes ${U}_{A}^{^{\prime}}=\left(t-\tau \right)\left(v+{\varphi }_{A}{h}_{A}-\delta x\right)-\left(t-\tau \right){p}_{A}^{^{\prime}}-{\varphi }_{B}c$ and ${U}_{B}^{^{\prime}}=\left(t-\tau \right)\left[v+{\varphi }_{B}{h}_{B}-\delta \left(1-x\right)\right]-\left(t-\tau \right){p}_{B}^{^{\prime}}-{\varphi }_{A}c$, respectively.

When $c=0$, it is clear that the utility functions can be reformulated as ${U}_{A}^{^{\prime}}=\frac{t-\tau }{1-\tau }\bullet \left[\left(1-\tau \right)\left(v+{\varphi }_{A}{h}_{A}-\delta x\right)-\left(1-\tau \right){p}_{A}^{^{\prime}}\right]$ and ${U}_{B}^{^{\prime}}=\frac{t-\tau }{1-\tau }\bullet \left\{\left(1-\tau \right)\left[v+{\varphi }_{B}{h}_{B}-\delta \left(1-x\right)\right]-\left(1-\tau \right){p}_{B}^{^{\prime}}\right\}$. Let ${p}_{i}=\left(1-\tau \right){p}_{i}^{^{\prime}}$. The equilibrium price and market share of each firm in Lammas 2 and 3 can be directly applied. Since the potential market size is $\left(1-\tau \right)$ of that in the baseline model, we have ${D}_{i}^{^{\prime}}=\left(1-\tau \right){D}_{i}$. The average price for the product of firm $i$ is $\overline{{p }_{i}^{^{\prime}}}={\int }_{\tau }^{1}\left(t-\tau \right){p}_{i}^{^{\prime}}dt/\left(1-\tau \right)={p}_{i}/2$. Therefore, we have ${\pi }_{i}^{^{\prime}}=\left(1-\tau \right){\pi }_{i}/2$. Based on the above results, it is easy to show that similar to Proposition 1, firms take opposite free trial strategies when $\updelta \le \frac{{\left(1-\tau \right)}^{2}}{2}$, and neither offers otherwise. Moreover, Propositions 2 and 4 hold except that the thresholds $\widehat{c}$ and $\widehat{\delta }$ are respectively larger and smaller than that obtained in the baseline model.

1.6 A11. Analysis of Extension 5.5

Under each possible configuration of the two firms’ free trial strategies, if the market is not fully covered, there exist the following local monopoly cases.

When neither firm offers free trials, as shown in Fig. 5, the marginal consumer who is indifferent between purchasing from firm A or not is located at ${x}_{1}=\frac{v-{p}_{A}}{\delta }$, and the marginal consumer who is indifferent between purchasing from firm B or not is located at ${x}_{2}=\frac{\delta +{p}_{B}-v}{\delta }$. Consumers with $x\le {x}_{1}$ buy firm A’s product, whereas consumers with $x>{x}_{2}$ buy firm B’s product.

Fig. 5

Market segmentation when neither firm offers free trials

When only one firm (take firm A for example) offers free trials, as shown in Fig. 6, there may exist six cases of market segmentations. The marginal consumer who is indifferent between purchasing from firm A or not is located at ${x}_{1}=\frac{\left(1-\tau \right)\left(v+{h}_{A}\right)-{p}_{A}}{\left(1-\tau \right)\delta }$, and the marginal consumer who is indifferent between purchasing from firm B or not is located at ${x}_{2}=\frac{\left(1-\tau \right)\left(\delta -v\right)+{p}_{B}+c}{\left(1-\tau \right)\delta }$. The marginal consumer who is indifferent between purchasing from either firm is located at ${x}_{0}=\frac{\left(1-\tau \right)\left(\delta +{h}_{A}\right)+{p}_{B}-{p}_{A}+c}{2\left(1-\tau \right)\delta }$. Consumers with $x\le \mathrm{min}\left\{{x}_{1},{x}_{0}\right\}$ buy firm A’s product, whereas consumers with $x>\mathrm{max}\left\{{x}_{2},{x}_{0}\right\}$ buy firm B’s product.

Fig. 6

Market segmentation when only firm A offers free trials

When both firms offer free trials, as shown in Fig. 7, there may exist four cases of market segmentations. The marginal consumer who is indifferent between purchasing from firm A or not is located at ${x}_{1}=\frac{\left(1-\tau \right)\left(v+{h}_{A}\right)-{p}_{A}}{\left(1-\tau \right)\delta }$, and the marginal consumer who is indifferent between purchasing from firm B or not is located at ${x}_{2}=\frac{\left(1-\tau \right)\left(\delta -v-{h}_{B}\right)+{p}_{B}}{\left(1-\tau \right)\delta }$. The marginal consumer who is indifferent between purchasing from either firm is located at ${x}_{0}=\frac{\left(1-\tau \right)\delta +{p}_{B}-{p}_{A}}{2\left(1-\tau \right)\delta }$. Consumers with $x\le \mathrm{min}\left\{{x}_{1},{x}_{0}\right\}$ buy firm A’s product, whereas consumers with $x>\mathrm{max}\left\{{x}_{2},{x}_{0}\right\}$ buy firm B’s product.

Fig. 7

Market segmentation when both firms offer free trials

1.7 A12. Data

Table 2 records the data we collected for a sample of 77 Windows-based antivirus software products developed and sold by 29 companies. For each product, it includes the name of the product, the name of the product developer/seller, the trial length, the main markets, and the web site and date on which we access the data.

Table 2 Data for 77 Windows-based antivirus software products from 29 companies

Table 3 records the data we collected for a sample of 79 CRM software products developed and sold by 31 companies. For each product, it includes the name of the product, the name of the product developer/seller, the trial length, and the web site and date on which we access the data. Because some CRM software products of Salesforce, Oracle and SAP are sold in most countries, all the CRM markets are competitive. Therefore, the market environment of a CRM software product is not reported.

Table 3 Data for 79 CRM software products from 31 companies

Appendix B

As stated in Sect. 4.1.2, when only one firm offers free trials, consumers may face two options: try the firm’s product before deciding which firm to buy from (option 1), or directly buy from the competing firm which does not offer free trials (option 2). In the baseline model, we consider the case where all consumers prefer option 1. Now, we will analyze the complicated case where both options may exist, and compare the free trial strategies of firms in this case with the original results.

Because firms A and B are symmetric, we suppose only firm A offers free trials, and analyze consumers’ usage behaviors as follows.

For option 1, consumers expect to obtain utility $\tau (v-\delta x)$ for the duration of the trial. Upon trying, they will stick with A for relatively high values of ${h}_{A}$ and switch to B otherwise, obtaining utility $\left(1-\tau \right)(v+{h}_{A}-\delta x)-{p}_{A}$ and $\left(1-\tau \right)\left[v-\delta \left(1-x\right)\right]-{p}_{B}-c$, respectively. It follows that, the indifferent value of ${h}_{A}$ is, $\overline{{h }_{A}}=\frac{{p}_{A}-{p}_{B}-c-\left(1-\tau \right)\delta \left(1-2x\right)}{1-\tau }$.

As shown in Fig. 8, the indifference line $\overline{{h }_{A}}$ may cut across the vertical and horizontal axes in one of two ways.

Fig. 8

Location of the indifference line when only firm A offers free trials

Then, the expected utility for option 1 in each case is as follows.

• Case 1: ${p}_{B}-{p}_{A}+c+\left(1-\tau \right)\left(\delta -0.5\right)\le 0$

$$ \begin{aligned} U_{1}^{1} \left( x \right) & = \tau \left( {v - \delta x} \right) + \mathop \smallint \limits_{ - 0.5}^{{\overline{{h_{A} }} }} \left\{ {\left( {1 - \tau } \right)\left[ {v - \delta \left( {1 - x} \right)} \right] - p_{B} - c} \right\}dh_{A}\\&\quad + \mathop \smallint \limits_{{\overline{{h_{A} }} }}^{0.5} \left[ {\left( {1 - \tau } \right)\left( {v + h_{A} - \delta x} \right) - p_{A} } \right]dh_{A} \\ & = v - \delta x - p_{A} + \frac{{\left[ {p_{A} - p_{B} - c - \left( {1 - \tau } \right)\delta \left( {1 - 2x} \right)} \right]^{2} }}{{2\left( {1 - \tau } \right)}}\\&\quad + \frac{{p_{A} - p_{B} - c - \left( {1 - \tau } \right)\delta \left( {1 - 2x} \right)}}{2} + \frac{1 - \tau }{8} \\ & = v - \delta x - p_{A} + \frac{{\left\{ {2\left[ {p_{A} - p_{B} - c - \left( {1 - \tau } \right)\delta \left( {1 - 2x} \right)} \right] + \left( {1 - \tau } \right)} \right\}^{2} }}{{8\left( {1 - \tau } \right)}} \\ \end{aligned} $$

(B.1)

• Case 2: $p_{B} - p_{A} + c + \left( {1 - \tau } \right)\left( {\delta - 0.5} \right) > 0$

  $$ \begin{aligned} U_{1}^{2} \left( x \right) & = \tau \left( {v - \delta x} \right) + \frac{{\hat{x}}}{{\overline{x}}}\mathop \smallint \limits_{ - 0.5}^{0.5} \left[ {\left( {1 - \tau } \right)\left( {v + h_{A} - \delta x} \right) - p_{A} } \right]dh_{A} \\ & \quad + \frac{{\overline{x} - \hat{x}}}{{\overline{x}}}\left\{ {\mathop \smallint \limits_{ - 0.5}^{{\overline{{h_{A} }} }} \left\{ {\left( {1 - \tau } \right)\left[ {v - \delta \left( {1 - x} \right)} \right] - p_{B} - c} \right\}dh_{A}}\right.\\&\quad\left.{ + \mathop \smallint \limits_{{\overline{{h_{A} }} }}^{0.5} \left[ {\left( {1 - \tau } \right)\left( {v + h_{A} - \delta x} \right) - p_{A} } \right]dh_{A} } \right\} \\ & = v - \delta x - p_{A} + \frac{{\overline{x} - \hat{x}}}{{\overline{x}}}\left\{ {\frac{{\left[ {p_{A} - p_{B} - c - \left( {1 - \tau } \right)\delta \left( {1 - 2x} \right)} \right]^{2} }}{{2\left( {1 - \tau } \right)}}}\right.\\&\quad\left.{+ \frac{{p_{A} - p_{B} - c - \left( {1 - \tau } \right)\delta \left( {1 - 2x} \right)}}{2} + \frac{1 - \tau }{8}} \right\} \\ & = v - \delta x - p_{A} + \frac{{\overline{x} - \hat{x}}}{{\overline{x}}} \cdot \frac{{\left\{ {2\left[ {p_{A} - p_{B} - c - \left( {1 - \tau } \right)\delta \left( {1 - 2x} \right)} \right] + \left( {1 - \tau } \right)} \right\}^{2} }}{{8\left( {1 - \tau } \right)}} \\ \end{aligned} $$

  (B.2)

  where $\widehat{x}=\frac{{p}_{B}-{p}_{A}+c+\left(1-\tau \right)\left(\delta -0.5\right)}{2\left(1-\tau \right)\delta }$.

For option 2, the expected utility of directly buying firm B’s product is, ${U}_{2}\left(x\right)=v-\delta \left(1-x\right)-{p}_{B}$.

Consumers select the option that maximizes their expected utility. Setting ${U}_{1}^{k}\left(x\right)={U}_{2}\left(x\right)$ ($k=1, 2$), we can derive the location of the marginal consumer who is indifferent between option 1 and option 2, $\overline{x }$, as follows.

$$\overline{x }=\left\{\begin{array}{c}\left\{x\left|{p}_{B}-{p}_{A}-\delta \left(1-2x\right)+\frac{{\left\{2\left[{p}_{A}-{p}_{B}-c-\left(1-\tau \right)\delta \left(1-2x\right)\right]+\left(1-\tau \right)\right\}}^{2}}{8\left(1-\tau \right)}=0\right.\right\} in Case 1,\\ \left\{x\left|{p}_{B}-{p}_{A}-\delta \left(1-2x\right)+\frac{x-\widehat{x}}{x}\bullet \frac{{\left\{2\left[{p}_{A}-{p}_{B}-c-\left(1-\tau \right)\delta \left(1-2x\right)\right]+\left(1-\tau \right)\right\}}^{2}}{8\left(1-\tau \right)}\right.=0\right\} in Case 2.\end{array}\right.$$

(B.3)

Consumers with $x>\overline{x }$ will select option 2, i.e., directly buy firm B’s product without trying. Consumers with $x\le \overline{x }$ will select option 1, i.e., try firm A’s product before deciding to buy from which firm. Moreover, upon trying, those with ${h}_{A}\le \overline{{h }_{A}}$ will buy from firm B, whereas those with ${h}_{A}>\overline{{h }_{A}}$ will buy from firm A. Accordingly, we derive the demands for the two firms in each case as follows.

$${D}_{A}\left({p}_{A},{p}_{B}\right)=\left\{\begin{array}{c}\left[\frac{1}{2}-\frac{{p}_{A}-{p}_{B}-c-\left(1-\tau \right)\delta \left(1-\overline{x }\right)}{1-\tau }\right]\bullet \overline{x } in Case 1,\\ \overline{x }-\left[\frac{1}{4}+\frac{{p}_{A}-{p}_{B}-c-\left(1-\tau \right)\delta \left(1-2\overline{x }\right)}{2\left(1-\tau \right)}\right]\bullet \left(\overline{x }-\widehat{x}\right) in Case 2.\end{array}\right.$$

(B.4)

$${D}_{B}\left({p}_{A},{p}_{B}\right)=1-{D}_{A}\left({p}_{A},{p}_{B}\right).$$

(B.5)

The profit-maximization problem of firm $i$ ($i=A, B$) is ${\pi }_{i}={p}_{i}\bullet {D}_{i}\left({p}_{i},{p}_{-i}\right)$. It can be seen that the analytical solutions to the problem are exceedingly complicated to derive, let alone the further analysis of the free trial strategies of firms for this new model version. Therefore, we resort to numerical experiments. Parameters are set as follows: $\tau$ ranges from 0 to 0.25 with a step size of 0.05; $\delta$ ranges from 0 to 2 with a step size of 0.1; $c$ ranges from 0 to 1 with a step size of 0.05. Results show that the theoretical insights obtained in Propositions 1 and 2 still hold, except that the thresholds $\widehat{\delta }$ and $\widetilde{\delta }$ are smaller than those in the baseline model. It implies that, compared with the baseline model, firms are less likely to adopt opposite strategies, and more likely to adopt the both-offering or neither-offering strategy. The reason causing this difference lies in that, when we no longer assume that all consumers try before buying, the advantage of unilaterally offering free trials is diminished and the competition between firms is lessened.