Distinct role of targeting precision of Internet-based targeted advertising in duopolistic e-business firms’ heterogeneous consumers market

Abstract

Advances in information technologies and e-commerce enable e-business firms to collect detailed consumers’ data and send Internet-based targeted advertising to their potential consumers accurately, thereby performing B2C sales. However, the perfect targeting for an e-business firm is always a challenge due to the asymmetric information between the firm and consumers. This essay starts by setting up a model with two horizontally differentiated firms competing in prices and targeted advertising at a variable targeting precision in an initially uninformed heterogeneous consumers market. The results indicate that both e-business firms should target only their advantage markets with optimal targeting precision. Otherwise, each e-business firm’s equilibrium profit might be lower or higher with Internet-based targeted advertising comparing to mass advertising, which depends on the effective choice of distinct targeting precision. Finally, the effective investment on the optimal targeting precision for e-business firms pursuing the maximum profit in the B2C sales model is generated.

Appendix

1.1 Proof of Lemma 1

Proof

With imperfect targeting precision $\alpha_{i}$, the e-business firm i’s profit is shown in below.

$$\pi_{i} = (p_{i} - c)\left[ {\int_{0}^{{\widehat{x}}} {\phi_{i} (x)\alpha_{i} dx + \int_{{\widehat{x}}}^{1} {\phi_{i} (x)\alpha_{i} (1 - \phi_{j} (x))dx} } } \right] - \frac{\lambda }{2}\int_{0}^{1} {(\phi_{i} (x))^{2} dx}$$

(18)

We define $\phi_{i} = \phi_{i} (x)$ and $\phi_{j} = \phi_{j} (x)$, $(i,j = 1,2)$. E-business firm $i$’s profit function in perceived advantage market is given by

$$\pi_{is} (\phi_{i} ) = (p_{i} - c)\widehat{x}\alpha_{i} \phi_{i} (x) - \frac{\lambda }{2}(\phi_{i} (x))^{2} \widehat{x}$$

(19)

Then, we choose $\phi_{i} (x)$ that maximizes firm $i$’s profit respectively, when $x_{i} \in [0,\widehat{x}]$, which implies that

$$\frac{{\partial \pi_{i} }}{{\partial \phi_{i} (x)}} = (p_{i} - c)\alpha_{i} - \lambda \phi_{i} (x) = 0$$

(20)

Likewise, $\phi_{i} (x)$ that maximizes firm $i$’s profit respectively, when $x_{i} \in [\widehat{x},1]$, which implies that

$$\frac{{\partial \pi_{i} }}{{\partial \phi_{i} (x)}} = (p_{i} - c)\alpha_{i} \left( {1 - \phi_{j} (x)} \right) - \lambda \phi_{i} (x) = 0$$

(21)

And symmetrically for Firm $j$, which implies

$$\left\{ {\begin{array}{*{20}{l}} {\frac{{\partial \pi_{j} }}{{\partial \phi_{j} (x)}} = (p_{j} - c)\alpha_{j} \left( {1 - \phi_{i} (x)} \right) - \lambda \phi_{j} (x) = 0\begin{array}{*{20}{l}} {} & {} \\ \end{array} } \\ {\frac{{\partial \pi_{j} }}{{\partial \phi_{j} (x)}} = (p_{j} - c)\alpha_{j} - \lambda \phi_{j} (x) = 0\begin{array}{*{20}{l}} {\begin{array}{*{20}c} {} & {} \\ \end{array} } & {\begin{array}{*{20}c} {} & {} \\ \end{array} } & {} \\ \end{array} } \\ \end{array} } \right.$$

(22)

From these conditions in Eq. (20) and (23), we deduce that $\phi_{i} (x)$ is a variable and equals to 1, or satisfied as $\phi_{i} = \hbox{min} \left\{ {\frac{{(p_{i} - c)\alpha_{i} }}{\lambda },1} \right\}$ on $[0,\widehat{x}]$.

Likewise, $\phi_{j} (x)$ is also variable and equals to $\phi_{j} = \hbox{min} \left\{ {\frac{{(p_{j} - c)\alpha_{j} }}{\lambda },1} \right\}$ on $[\widehat{x},1]$. Consequently, $\psi_{i} (x)$ is variable and equals to $\psi_{i} = \frac{{(p_{i} - c)\alpha_{i} (1 - \phi_{j} )}}{\lambda }$ on $[\widehat{x},1]$ and $\psi_{j} (x)$ is variable and equals to $\psi_{j} = \frac{{(p_{j} - c)\alpha_{j} (1 - \phi_{j} )}}{\lambda }$ on $[0,\widehat{x}]$. □

1.2 Proof of Proposition 1

Proof

Here, based on the model of both e-business firms competing in targeted advertising with different targeting precision, F.O.C. condition on advertising intensity implies that

$$\left\{ {\begin{array}{*{20}l} {\frac{{\partial \pi _{i} }}{{\partial \phi _{i} }} = m_{i} \hat{x}\alpha _{i} - \lambda \phi _{i} \hat{x} = 0} \hfill \\ {\frac{{\partial \pi _{j} }}{{\partial \psi _{i} }} = m_{i} (1 - \hat{x})\alpha _{i} (1 - \phi _{j} ) - \lambda \psi _{i} (1 - \hat{x}) = 0} \hfill \\ \end{array} } \right.$$

(23)

• When $m_{i} < m_{j} - t$, from expression (4), $\pi_{i1} = m_{i} \phi_{i} \alpha_{i} - \frac{\lambda }{2}\phi_{i}^{2}$. For a fixed $m_{i}$, F.O.C. condition on $\phi_{i}$ yields $\phi_{i}^{*} = \frac{{m_{i} \alpha_{i} }}{\lambda }$, thus the result.

• When $m_{j} - t < m_{i} < m_{j} + t$, the profit is given by Eq. (4) as a function of $(\phi_{i} ,\psi_{i} )$. The Hessian matrix relative to these two variables writes with the following expression.

  $$\left( {\begin{array}{*{20}c} {\frac{{\partial^{2} \pi_{i} }}{{\partial \phi_{i}^{2} }}} & {\frac{{\partial^{2} \pi_{i} }}{{\partial \phi_{i} \partial \psi_{i} }}} \\ {\frac{{\partial^{2} \pi_{i} }}{{\partial \psi_{i} \partial \phi_{i} }}} & {\frac{{\partial^{2} \pi_{i} }}{{\partial \psi_{i}^{2} }}} \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {\frac{{ - \lambda (t - m_{i} + m_{j} )}}{2t}} & 0 \\ 0 & {\frac{{ - \lambda (t - m_{i} + m_{j} )}}{2t}} \\ \end{array} } \right)$$

  (24)

Note that this Hessian matrix is a negative definite matrix. Thus, if $(\phi_{i}^{*} ,\psi_{i}^{*} ) \in [0,1] \times [0,1]$, the global maximum profit of the e-business firm might be realized. Otherwise, the global maximum corresponds to a corner solution.

When $\phi_{i}^{*} > 1$, we can get $\frac{{\partial \pi_{i} }}{{\partial \phi_{i} }} > 0$, for all $\phi_{i} \in [0,1]$. Therefore, the firm’s max profit $\pi_{i}$ must be realized at $\phi_{i} = 1$. Hence, $\phi_{i}^{*} = \hbox{min} \left\{ {\frac{{m_{i} \alpha_{i} }}{\lambda },1} \right\}$ realizes the max profit of $\pi_{i}$.

Finally, we consider the optimal value of $\phi_{j}^{*}$. For $\phi_{j}^{*} = 1$(for instance, $m_{j} [1 - \psi_{i} (1 - \alpha_{j} )] > \lambda$), $\frac{{\partial \pi_{i} }}{{\partial \psi_{i} }} < 0$ thus necessarily $\psi_{i} = 0$. For $\psi_{i} > \psi_{i}^{*}$, the maximum advertising intensity in the weak market is considered as $\psi_{i}^{*} = \frac{{m_{i} \alpha_{i} }}{\lambda }(1 - \frac{{m_{j} \alpha_{j} }}{\lambda })$.

• When $m_{i} > m_{j} + t$, $\pi_{i3} = m_{i} \alpha_{i} \psi_{i} (1 - \phi_{j} ) - \frac{\lambda }{2}\psi_{i}^{2}$. For a fixed $m_{i}$, F.O.C. condition on the advertising intensity in the firm’s weak market yields $\psi_{i}^{*} = \frac{{m_{i} \alpha_{i} (1 - \phi_{j} )}}{\lambda }$. As $\phi_{i}^{*} = \frac{{m_{i} \alpha_{i} }}{\lambda }$, that means $\phi_{j}^{*} = \frac{{m_{j} \alpha_{j} }}{\lambda }$. Therefore, $\psi_{i}^{*} = \frac{{m_{i} \alpha_{i} }}{\lambda }(1 - \frac{{m_{j} \alpha_{j} }}{\lambda })$. □

1.3 Prof of Proposition 2

Proof

According to Eq. (2) and (4), the e-business firm’s profit can be easily expressed as follows.

$$\pi_{i} = m_{i} \left[ {\frac{{t - m_{i} + m_{j} }}{2t}\phi_{i} \alpha_{i} + \psi_{i} \frac{{t + m_{i} - m_{j} }}{2t}\alpha_{i} (1 - \phi_{j} )} \right] - \frac{\lambda }{2}\phi_{i}^{2} \frac{{t - m_{i} + m_{j} }}{2t} - \frac{\lambda }{2}\psi_{i}^{2} \frac{{t + m_{i} - m_{j} }}{2t}$$

(25)

Then, we simplify this expression (25) and can easily acquire the following result.

$$\pi_{i} = \frac{{\alpha_{i}^{2} }}{4t\lambda }\left[ {(m_{i}^{2} t - m_{i}^{3} + m_{i}^{2} m_{j} ) + (1 - \frac{{m_{j} \alpha_{j} }}{\lambda })^{2} (m_{i}^{2} t + m_{i}^{3} - m_{i}^{2} m_{j} )} \right]$$

(26)

That means the expression (26) can be simplified in below.

$$\pi_{i}^{*} = \frac{{\alpha_{i}^{2} }}{4t\lambda }(2\lambda^{2} m_{i}^{2} t - 2\lambda m_{j} \alpha_{j} m_{i}^{2} t - 2\lambda m_{j} \alpha_{j} m_{i}^{3} + 2\lambda m_{j}^{2} \alpha_{j} m_{i}^{2} + m_{j}^{2} \alpha_{j}^{2} m_{i}^{2} t + m_{j}^{2} \alpha_{j}^{2} m_{i}^{3} - m_{j}^{3} \alpha_{j}^{2} m_{i}^{2} )$$

(27)

According to F.O.C. condition,

$$\frac{{\partial \pi_{i} }}{{\partial m_{i} }} = 4\lambda^{2} m_{i} t - 4\lambda m_{j} \alpha_{j} m_{i} t - 6\lambda m_{j} \alpha_{j} m_{i}^{2} + 4\lambda m_{j} \alpha_{j} m_{i} m_{j} + 2m_{j}^{2} \alpha_{j}^{2} m_{i} t + 3m_{j}^{2} \alpha_{j}^{2} m_{i}^{2} - 2m_{j}^{3} \alpha_{j}^{2} m_{i} = 0$$

(28)

In equilibrium, $m_{i} = m_{j} = m$, that means $P(m)$ can be expressed as Eq. (7).

$$P(m) = \alpha_{j}^{2} m^{3} - 2\alpha_{j} m^{2} (\lambda - \alpha_{j} t) - 4\alpha_{j} \lambda mt + 4\lambda^{2} t$$

Note that the F.O.C. condition on m is given in below.

$$\frac{\partial P(m)}{\partial m} = 3m^{2} \alpha_{j}^{2} - 4m\alpha_{j} (\lambda - \alpha_{j} t) - 4\alpha_{j} \lambda t$$

(29)

$$P(2t) = 4t(2\alpha_{j} t - \lambda )^{2} > 0$$

(30)

$$P\left( {t + \frac{\lambda }{2}} \right) = \alpha_{j}^{2} \left( {t + \frac{\lambda }{2}} \right)^{3} - 2\alpha_{j} \left( {t + \frac{\lambda }{2}} \right)^{2} (\lambda - \alpha_{j} t) - 4\alpha_{j} \lambda \left( {t + \frac{\lambda }{2}} \right)t + 4\lambda^{2} t < 0$$

(31)

Hence, $P(m)$ admits two positive roots. One of them is considered as $2t < m^{*} < t + \frac{\lambda }{2}$. On the other hand,

$$\begin{aligned} P'(\lambda ) = \alpha_{j}^{2} \lambda^{3} - 2\alpha_{j} \lambda^{2} (\lambda - \alpha_{j} t) - 4\alpha_{j} \lambda^{2} t + 4\lambda^{2} t \\ = \lambda^{2} \left[ {(\lambda + 2t)\left( {1 - \alpha_{j} } \right)^{2} - \lambda + 2t} \right] \\ \end{aligned}$$

(32)

As $3\lambda \alpha_{j} - 4\lambda < 0$ and $4\alpha_{j} t - 4t < 0$, that means $\frac{\partial P(\lambda )}{\partial \lambda } < 0$. The two positive roots of $P(m)$, they exist, are larger than the advertising cost parameter $\lambda$. As a result, $P(m) > 0$ for all $m \in [0,\lambda ]$ and shows no effect on the rival’s targeting precision. The Proposition 2 can be obtained after solving the game. □

1.4 Proof of Corollary 1 and Proposition 3

Proof

According to F.O.C. condition,

$$\frac{{\partial \pi_{i}^{*} }}{\partial m} = \frac{{m\alpha_{i}^{2} }}{{2\lambda^{3} }}[2(\lambda - m\alpha_{j} )^{2} + m\alpha_{j} \lambda ] > 0$$

(33)

Here, the implicit function theorem is used to seek the variation of the margin profit at equilibrium $m^{*}$ as well as the advertising cost parameter $\lambda$.

$$\frac{{dm^{*} }}{d\lambda } = - \frac{{P_{\lambda }^{'} }}{{P_{m}^{'} }}$$

(34)

$$\frac{\partial P(m)}{\partial \lambda } = 2(4\lambda t - 2\alpha_{j} mt - \alpha_{j} m^{2} )$$

(35)

Then, the following expression can be deduced.

$$P(\sqrt {2\lambda t/\alpha_{j} } ) = 2\lambda (2\lambda t - 2\alpha_{j} m^{2} ) < 0$$

(36)

Finally, replacing in this expression m by $m = \sqrt {2\lambda t/\alpha_{j} }$, and then $P(\sqrt {2\lambda t/\alpha_{j} } ) = - \frac{{m\alpha_{j} }}{2}\frac{{\partial P(\sqrt {2\lambda t/\alpha_{j} } )}}{\partial \lambda } < 0$.

Consequently,$\sqrt {2\lambda t/\alpha_{j} } > m^{*}$, which implies $2\lambda (2\lambda t - 2\alpha_{j} m^{2} ) > 0$. Hence, according to Eq. (36), for $m = m^{*}$,$\frac{{\partial P(m = m^{*} )}}{\partial \lambda } < 0$, thus $\frac{{dm^{*} }}{d\lambda } > 0$. □

1.5 Proof of Proposition 6 and Proposition 7

Proof

In equilibrium, $m_{i} = m_{j} = m$, and then it is easy to obtain both firms’ optimal profits in below.

$$\pi_{i}^{*} = \frac{{m^{2} \alpha_{i}^{2} }}{2\lambda }\left[ - \frac{{m^{3} \alpha_{i} }}{2\lambda } - \left(\frac{{\alpha_{i} }}{2\lambda } - 1 \right)m^{2} - m + \frac{9}{8}\right]$$

(37)

$$\pi_{j}^{*} = \frac{{m^{2} (2\lambda - m\alpha_{i} )^{2} }}{{8\lambda^{3} }}$$

(38)

Consequently, the related conclusion of Proposition 6 can be obtained. □

Proof

Then, we compare the firm $i$’s equilibrium targeting intensity in its advantage market and weak market as proof of the aforementioned Lemma 1.

$$\frac{{\varphi_{i}^{*} }}{{\psi_{i}^{*} }} = \frac{{\alpha_{is} \lambda }}{{\alpha_{iw} (\lambda - m_{j} \alpha_{js} )}}$$

(39)

Consequently, the related conclusion of Proposition 7 can be obtained. □