Advertising bidding involving consumer information sharing

Abstract

As of recently, consumers can actively manage their level of individual-level information sharing, making the advertising bidding decision more complicated. In this paper, we study the advertising bidding decisions under the assumption that consumers can manage the level of their information sharing. Using a Salop city model to capture consumers’ prior product preferences, we firstly find that whether a targeted advertisement or a mass advertisement is the optimal advertising type in a duopoly hinges on the neighbor seller’s price and the level of consumer information sharing. Secondly, when we consider several sellers in an oligopoly, we find that the level of consumer information sharing affects the advertising bidding among different sellers. Thirdly, we compare the performance of a mass advertisement versus a targeted advertisement by numerical analysis in an oligopoly. We find that the targeted advertisements always perform better than mass advertisements in achieving optimal profit for both dominant sellers and non-dominant sellers in an oligopoly. Finally, we also discuss the regulatory implications for firms using consumer information as well as the managerial implications.

Appendices

Appendix A

Proof of Lemma 1

We consider the situation where the seller places mass advertisements. Given that $n$ sellers will join the auction, each seller knows its own bidding $b_{\left( . \right)}^{M}$ but does not know their opponent sellers’ biddings. However, they can have a belief, and therefore an estimation about these biddings. To make our discussion simple, the opponents’ biddings are assumed to be drawn independently from the cumulative distribution function $F\left( x \right)$ with density $f\left( x \right) \in \left[ {0,\infty } \right)$.$F_{T} \left( t \right) = P\left( {T \le t} \right)$ represents the probability that the random variable $T$ is less than or equal to a certain bidding $t$. Without loss of generality, we look at seller 1, who sells the product with price $p_{1}^{M}$ and chooses a bid $b_{1}^{M}$ to maximize its expected profits. Assume that seller $i$ places the highest bid among seller 2, 3, …, n. With the rule of the second-price auction, seller 1’s expected profit can be determined by:

$$\pi _{1}^{M} \left( {p_{1}^{M} ,b_{i}^{M} ,b_{{\left( . \right)}}^{M} } \right) = \left\{ {\begin{array}{*{20}l} {p_{1}^{M} - b_{i}^{M} \;\;,\quad if\;b_{1}^{M} > b_{i}^{M} > \max \left\{ {b_{2}^{M} ,b_{3}^{M} ,...,b_{n}^{M} } \right\}} \hfill \\ {0\quad ,\quad if\;b_{1}^{M} < \max \left\{ {b_{2}^{M} ,b_{3}^{M} ,...,b_{n}^{M} } \right\}} \hfill \\ \end{array} } \right.$$

(A.1)

where the probability of $b_{1}^{M} > b_{i}^{M} > \max \left\{ {b_{2}^{M} ,b_{3}^{M} ,...,b_{n}^{M} } \right\}$ is $\int_{0}^{{b_{1}^{M} }} d F\left( x \right)^{n - 1} = \int_{0}^{{b_{1}^{M} }} {\left( {n - 1} \right)f\left( x \right)} F\left( x \right)^{n - 2} dx$. Thus, the expected profit can be defined by:

$$\pi_{1}^{M} \left( {p_{1}^{M} ,b_{i}^{M} ,b_{\left( . \right)}^{M} } \right) = \int_{0}^{{b_{1}^{M} }} {\left( {p_{1} - x} \right)} \left( {n - 1} \right)f\left( x \right)F\left( x \right)^{n - 2} dx$$

(A.2)

On the basis of the profit function, if $b_{1}^{M} > p_{1}^{M}$ then we have:

$$\pi_{1}^{M} \left( {p_{1}^{M} ,b_{i}^{M} ,b_{\left( . \right)}^{M} } \right) = \int_{0}^{{p_{1}^{M} }} {\left( {p_{1} - x} \right)} \left( {n - 1} \right)f\left( x \right)F\left( x \right)^{n - 2} dx + \int_{{p_{1}^{M} }}^{{b_{1}^{M} }} {\left( {p_{1} - x} \right)} \left( {n - 1} \right)f\left( x \right)F\left( x \right)^{n - 2} dx$$

(A.3)

where the second part of the profit function may be negative. Thus, the expected profit will increase when $b_{1}^{M}$ decreases to $p_{1}^{M}$.

Conversely, if $b_{1}^{M} < p_{1}^{M}$ then the second part of the profit function may be positive while it does not reach the maximum. When $b_{1}^{M}$ increases to $p_{1}^{M}$, the expected profit will increase by $\int_{{b_{1}^{M} }}^{{p_{1}^{M} }} {\left( {p_{1} - x} \right)} \left( {n - 1} \right)f\left( x \right)F\left( x \right)^{n - 2} dx$. Therefore, $b_{1}^{M} = p_{1}^{M}$ is the dominant optimal strategy.

Consequently, $b_{1}^{M} = p_{1}^{M}$ is the dominant optimal strategy for any seller $i$ when the seller places a mass advertisement. When the seller places a targeted advertisement, we use superscript ‘T’ instead of ‘M,’ and the proof method is the same.

Proof of Proposition 1

Based on Eqs. (7), we can easily find that $\partial b_{i}^{{M^{*} }} /\partial v = 1/2 > 0$; based on Eqs. (10), we can find for any $0 \le E\left( Y \right) \le E\left( Y \right)^{*}$, we have $\partial p_{i}^{{M^{*} }} /\partial E\left( Y \right) \ge 0$; for any $E\left( Y \right)^{*} < E\left( Y \right) \le 1$, we have $\partial p_{i}^{{M^{*} }} /\partial E\left( Y \right) \le 0$. Specifically, $E\left( Y \right)^{*} = \alpha_{i} /2c_{i}$.

Proof of Proposition 2

4.1 (1) Demand comparison

When a seller uses consumer information, the targeted advertisement yields the optimal demand $D_{i}^{{T^{*} }}$; If not, the seller places a mass advertisement and yields the optimal demand $D_{i}^{{M^{*} }}$. Then, we compare the optimal demand: $D_{i}^{{M^{*} }} - D_{i}^{{T^{*} }} { = }\frac{v}{t} - \frac{{E\left( Y \right)\left[ {\alpha_{i} - c_{i} E\left( Y \right)} \right] + v - p_{i + 1}^{T} }}{t}$, we can find:

(a) when $p_{i + 1} \le \alpha_{i}^{2} /4c_{i}$, for any $E\left( Y \right) \in \left( {\frac{{a - \sqrt {a^{2} - 4cp_{i + 1} } }}{{2c_{i} }},\frac{{a + \sqrt {a^{2} - 4cp_{i + 1} } }}{{2c_{i} }}} \right)$, $D_{i}^{{M^{*} }} \le D_{i}^{{T^{*} }}$; for any $E\left( Y \right) \in \left[ {0,\frac{{a - \sqrt {a^{2} - 4cp_{i + 1} } }}{{2c_{i} }}} \right]$ and $E\left( Y \right) \in \left[ {\frac{{a + \sqrt {a^{2} - 4cp_{i + 1} } }}{{2c_{i} }},1} \right]$, $D_{i}^{{M^{*} }} \ge D_{i}^{{T^{*} }}$;

(b) when $p_{i + 1} > \alpha_{i}^{2} /4c_{i}$, for any $E\left( Y \right)$, we have $D_{i}^{{M^{*} }} > D_{i}^{{T^{*} }}$.

(2) Profit comparison

Next, comparing the optimal profit under different advertisements we have $\pi_{i}^{{M^{*} }} - \pi_{i}^{{T^{*} }} = \frac{{v^{2} - 2tp_{i + 1}^{M} - \left\{ {v - p_{i + 1}^{T} + E\left( Y \right)\left[ {\alpha_{i} - E\left( Y \right)c_{i} } \right]} \right\}^{2} }}{2t}$. When $\pi_{i}^{{M^{*} }} - \pi_{i}^{{T^{*} }} = 0$, we can obtain $E\left( Y \right) = \frac{{\alpha_{i} \pm \sqrt { - 4c_{i} \left( { - v + p_{i + 1}^{T} \pm \sqrt {v^{2} - 2tp_{i + 1}^{M} } } \right) + \alpha_{i}^{2} } }}{{2c_{i} }}$. Since $v^{2} - 2tp_{i + 1}^{M} > 0$, $v > p_{i + 1}^{T}$, $0 \le E\left( Y \right) \le 1$, we can find:

• (a) when $p_{i + 1} \le \alpha_{i}^{2} /4c_{i}$, for any $E\left( Y \right)$ we have $\pi_{i}^{{M^{*} }} \ge \pi_{i}^{{T^{*} }}$;

• (b) when $p_{i + 1} > \alpha_{i}^{2} /4c_{i}$, for any $E\left( Y \right) \in \left[ {0,\frac{{\alpha_{i} + \sqrt { - 4c_{i} \left( { - v + p_{i + 1}^{T} - \sqrt {v^{2} - 2tp_{i + 1}^{M} } } \right) + \alpha_{i}^{2} } }}{{2c_{i} }}} \right]$, we have $\pi_{i}^{{M^{*} }} \le \pi_{i}^{{T^{*} }}$; for any $E\left( Y \right) \in \left( {\frac{{\alpha_{i} + \sqrt { - 4c_{i} \left( { - v + p_{i + 1}^{T} - \sqrt {v^{2} - 2tp_{i + 1}^{M} } } \right) + \alpha_{i}^{2} } }}{{2c_{i} }},1} \right]$, $\pi_{i}^{{M^{*} }} \ge \pi_{i}^{{T^{*} }}$.

Proof of Proposition 4

According to the equilibrium result in Proposition 4 and the constraint condition $0 \le \alpha_{i} - \alpha_{{i{ + }1}} \le 2\left( {c_{i} - c_{i + 1} } \right)$, we can easily find that for any $0 \le E\left( Y \right) \le \left( {\alpha_{i} - \alpha_{i + 1} } \right)/2\left( {c_{i} - c_{i + 1} } \right)$, we have $\partial b_{i}^{{T^{*} }} /\partial E\left( Y \right) \ge 0$; for any $\left( {\alpha_{i} - \alpha_{i + 1} } \right)/2\left( {c_{i} - c_{i + 1} } \right) < E\left( Y \right) \le 1$, we have $\partial b_{i}^{{T^{*} }} /\partial E\left( Y \right) \le 0$.

Proof of Proposition 5

Based on the equilibrium result in Proposition 4, we can easily find that: (a)$\partial b_{i}^{{T^{*} }} /\partial E\left( Y \right)\partial c_{i} = - E\left( Y \right) < 0$, $\partial b_{i}^{{T^{*} }} /\partial E\left( Y \right)\partial c_{i + 1} = E\left( Y \right) > 0$; (b) $\partial b_{i}^{{T^{*} }} /\partial E\left( Y \right)\partial \alpha_{i} { = }1/2 > 0$; $\partial b_{i}^{{T^{*} }} /\partial E\left( Y \right)\partial \alpha_{i + 1} = - 1/2 < 0$.

Proof of Proposition 6

As $\partial D_{i}^{{T^{*} }} /\partial E\left( y \right) = \frac{{2E\left( Y \right)\left( {c_{i + 1} - c_{i} } \right) + a_{i} - a_{i + 1} }}{2nt}$, it’s easy to see if $0 \le E\left( Y \right) \le \frac{{\alpha_{i} - \alpha_{i + 1} }}{{2\left( {c_{i} - c_{i + 1} } \right)}}$, we have $\partial D_{i}^{{T^{*} }} /\partial E\left( y \right) \ge 0$; if $\frac{{\alpha_{i} - \alpha_{i + 1} }}{{2\left( {c_{i} - c_{i + 1} } \right)}} < E\left( Y \right) \le 1$, we have $\partial D_{i}^{{T^{*} }} /\partial E\left( Y \right) \le 0$. Similarly, when $0 \le E\left( Y \right) \le \frac{{\alpha_{i} - \alpha_{i + 1} }}{{2\left( {c_{i} - c_{i + 1} } \right)}}$, $\partial \pi_{i}^{{T^{*} }} /\partial E\left( y \right) \ge 0$; when $\frac{{\alpha_{i} - \alpha_{i + 1} }}{{2\left( {c_{i} - c_{i + 1} } \right)}} < E\left( Y \right) \le 1$, $\partial \pi_{i}^{{T^{*} }} /\partial E\left( y \right) \le 0$.

Proof of Proposition 7

First, we analyze the effect of the level of consumer information sharing on the advertising bidding. Based on the equilibrium result in Proposition 4, it is easy to get $\partial b_{i + 1}^{{T^{*} }} /\partial E\left( y \right) = \frac{{2E\left( y \right)\left( {c_{i} - 2c_{i + 1} + c_{i + 2} } \right) - a_{i} + 2a_{i + 1} - a_{i + 2} }}{4}$. Therefore, when $0 \le E\left( Y \right) \le \frac{{\alpha_{i} - 2\alpha_{i + 1} + \alpha_{i + 2} }}{{2\left( {c_{i} - 2c_{i + 1} + c_{i + 2} } \right)}}$, we have $\partial b_{i + 1}^{{T^{*} }} /\partial E\left( Y \right) \le 0$; when $\frac{{\alpha_{i} - 2\alpha_{i + 1} + \alpha_{i + 2} }}{{2\left( {c_{i} - 2c_{i + 1} + c_{i + 2} } \right)}} < E\left( Y \right) \le 1$, we have $\partial b_{i + 1}^{{T^{*} }} /\partial E\left( Y \right) \ge 0$. We then analyzed the effect of the level of consumer information sharing on the optimal demand and optimal profit. The demand function $D_{i + 1}^{{T^{*} }}$ yields the optimal level of consumer information sharing $E\left( Y \right)^{*} = \frac{{\alpha_{i} - 2\alpha_{i + 1} + \alpha_{i + 2} }}{{2\left( {c_{i} - 2c_{i + 1} + c_{i + 2} } \right)}}$. To support $E\left( Y \right)^{*} \in [0,1]$, we assert that $0 \le \alpha_{i} - 2\alpha_{i + 1} + \alpha_{i + 2} \le 2\left( {c_{i} - 2c_{i + 1} + c_{i + 2} } \right)$. As $\frac{{\partial D_{i + 1}^{{T^{*} }} }}{\partial E\left( Y \right)} = \frac{{2E\left( Y \right)\left( {c_{i} - 2c_{i + 1} + c_{i + 2} } \right) - a_{i} + 2a_{i + 1} - a_{i + 2} }}{4t}$, we can find when $0 \le E\left( Y \right) \le \frac{{\alpha_{i} - 2\alpha_{i + 1} + \alpha_{i + 2} }}{{2\left( {c_{i} - 2c_{i + 1} + c_{i + 2} } \right)}}$, $\partial D_{i + 1}^{{T^{*} }} /\partial E\left( y \right) \le 0$; when $\frac{{\alpha_{i} - 2\alpha_{i + 1} + \alpha_{i + 2} }}{{2\left( {c_{i} - 2c_{i + 1} + c_{i + 2} } \right)}} < E\left( Y \right) \le 1$,$\partial D_{i + 1}^{{T^{*} }} /\partial E\left( Y \right) \ge 0$. Similarly, when $0 \le E\left( Y \right) \le \frac{{\alpha_{i} - 2\alpha_{i + 1} + \alpha_{i + 2} }}{{2\left( {c_{i} - 2c_{i + 1} + c_{i + 2} } \right)}}$, $\partial \pi_{i + 1}^{{T^{*} }} /\partial E\left( Y \right) \le 0$; when $\frac{{\alpha_{i} - 2\alpha_{i + 1} + \alpha_{i + 2} }}{{2\left( {c_{i} - 2c_{i + 1} + c_{i + 2} } \right)}} < E\left( y \right) < 1$,$\partial \pi_{i + 1}^{{T^{*} }} /\partial E\left( Y \right) \ge 0$.