Game theoretic analysis of pricing and vertical cooperative advertising of a retailer-duopoly with a common manufacturer

Abstract

This paper considers competition of duopolistic retailers, who sell substitutable products supplied by a single manufacturer offering a vertical cooperative advertising program. The price-dependent component of the demand function is derived from the consumers’ utility function in order to avoid logically inconsistent results. Additionally, each supply chain member can increase the costumers’ demand by advertising. By means of game theory, we get the following results: (a) Retailer competition harms all players, but is beneficial to the consumers. (b) Stronger competition is followed by less advertising. (c) Channel-leadership is not always advantageous to the manufacturer, and—likewise—retailers can also be better off when accepting followership. However, as our analysis shows, the increased complexity of the model under consideration reaches the limits of an analytical solution. Therefore, we give a brief outlook on non-nalytical solution methods for Nash and Stackelberg games, that could be used in future research, in the end of our paper.

Appendix

Proof

(Proposition 1) We start with the manufacturer’s decision problem stated in Eq. (11) and set the first order partial derivatives $\partial \pi _{\mathrm {m}}/ \partial w$ and $\partial \pi _{\mathrm {m}}/ \partial A$ to zero:

$$\begin{aligned} \frac{\partial \pi _{\mathrm {m}}}{\partial w}&= \left( k_{\mathrm {r}}\sum _{j=1}^2 \sqrt{a_i} + k_{\mathrm {m}}\sqrt{A}\right) \left[ \sum _{j=1}^2 \alpha _j - \beta (2w+m_j)+\epsilon (2w+m_{3-j})\right] =0\nonumber \\\end{aligned}$$

(17)

$$\begin{aligned} \frac{\partial \pi _{\mathrm {m}}}{\partial A}&= \frac{k_{\mathrm {m}}w}{2\sqrt{A}}\left[ \sum _{j=1}^2 \alpha _j - \beta (w+m_j)+\epsilon (w+m_{3-j})\right] -1=0. \end{aligned}$$

(18)

Please note that participation rate $t$ equals zero due to its solely negative influence on the manufacturer’s profit function. From Eqs. (17) and (18), we derive:

$$\begin{aligned} w&= \frac{\alpha _1+\alpha _2-(\beta -\epsilon )(m_1+m_2)}{4(\beta -\epsilon )}\end{aligned}$$

(19)

$$\begin{aligned} A&= \frac{1}{4}k_{\mathrm {m}}^2w^2\left[ \alpha _1+\alpha _2-(\beta -\epsilon )(m_1+m_2)-2w(\beta -\epsilon )\right] . \end{aligned}$$

(20)

Likewise, we calculate the first order partial derivatives of the retailers’ decision problems (see Eq. (12)), $\partial \pi _{\mathrm {r}i}/\partial m_i$ and $\partial \pi _{\mathrm {r}i}/ \partial a_i$, and set them to zero:

$$\begin{aligned} \frac{\partial \pi _{\mathrm {r}i}}{\partial m_i}&= \left( k_{\mathrm {r}}\sum _{j=1}^2 \sqrt{a_i} + k_{\mathrm {m}}\sqrt{A}\right) \left[ \alpha _i-\beta (w+m_i)+\epsilon (w+m_{3-i})-\beta m_i\right] =0\nonumber \\ \end{aligned}$$

(21)

$$\begin{aligned} \frac{\partial \pi _{\mathrm {r}i}}{\partial a_i}&= \frac{k_{\mathrm {r}}m_i}{2\sqrt{a_i}} \left[ \alpha _i - \beta (w+m_i)+\epsilon (w+m_{3-i})\right] -(1-t)=0. \end{aligned}$$

(22)

From these equations, we derive:

$$\begin{aligned} m_i&= \frac{\alpha _i-(\beta -\epsilon )w+\epsilon m_i}{2\beta }\end{aligned}$$

(23)

$$\begin{aligned} a_i&= \frac{k_{\mathrm {r}}^2m_i^2\left[ \alpha _i - (\beta -\epsilon )w-\beta m_i + \epsilon m_{3-i}\right] }{4(1-t)^2}. \end{aligned}$$

(24)

With $t=0$, we can now solve the system of equations described by Eqs. (17)–(20), which leads us to the expressions stated in Proposition 1. This completes proof of Proposition 1.

Proof

(Proposition 2) The retailers’ decision problems in a Manufacturer Stackelberg—Horizontal Nash game are identical to Eq. (12) and have the solutions stated in Eqs. (23) and (24). These expressions can be rearranged to

$$\begin{aligned} m_i&= \frac{2\alpha _i\beta +\alpha _{3-i}\epsilon +(-2\beta ^2+\beta \epsilon +\epsilon ^2)w}{(2\beta -\epsilon )(2\beta +\epsilon )}\end{aligned}$$

(25)

$$\begin{aligned} a_i&= \frac{\beta ^2k_{\mathrm {r}}^2\left[ 2\alpha _i\beta +\alpha _{3-i}\epsilon +(-2\beta ^2+\beta \epsilon +\epsilon ^2)w\right] ^4}{4(2\beta -\epsilon )^4(2\beta +\epsilon )^4(1-t)^2}. \end{aligned}$$

(26)

Constituting the constraints of the manufacturer’s decision problem [see Eq. (15)], these response functions have to be inserted into the manufacturer’s profit function. In order to reduce the complexness of this problem, we set $\varLambda _1 = \varLambda _2 = \varLambda$, which leads to $\alpha _1 = \alpha _2 = \alpha$. Hence, we can rewrite Eqs. (25) and (26) as follows:

$$\begin{aligned} m_1&= m_2 = \frac{\alpha -(\beta -\epsilon )w}{2\beta -\epsilon }\end{aligned}$$

(27)

$$\begin{aligned} a_1&= a_2 = \frac{\beta ^2k_{\mathrm {r}}^2\left[ \alpha -(\beta -\epsilon )w\right] ^4}{4(2\beta -\epsilon )^4(1-t)^2}. \end{aligned}$$

(28)

Inserting these equations into the profit function stated in Eq. (15), we get:

$$\begin{aligned} \pi _{\mathrm {m}}&= \frac{2\beta w\left[ \alpha -(\beta -\epsilon )w\right] }{2\beta -\epsilon }\left\{ \frac{\beta k_{\mathrm {r}}^2\left[ \alpha -(\beta -\epsilon )w\right] ^2}{(2\beta -\epsilon )^2(1-t)}+k_{\mathrm {m}}\sqrt{A}\right\} \nonumber \\&-\frac{\beta ^2k_{\mathrm {r}}^2\left[ \alpha -(\beta -\epsilon )w\right] ^4t}{2(2\beta -\epsilon )^4(1-t)^2}-A-C_{\mathrm {m}}. \end{aligned}$$

(29)

By setting the first order partial derivative $\partial \pi _{\mathrm {m}}/ \partial A$ to zero,

$$\begin{aligned} \frac{\partial \pi _{\mathrm {m}}}{\partial A} = \frac{\beta k_{\mathrm {m}}w \left[ \alpha -(\beta -\epsilon )w\right] }{(2\beta -\epsilon )\sqrt{A}}-1 =0, \end{aligned}$$

(30)

we can determine the optimal global advertising expenditures as a function of $w$:

$$\begin{aligned} A = \frac{\beta ^2k_{\mathrm {m}}^2w^2\left[ \alpha -(\beta -\epsilon )w\right] ^2}{(2\beta -\epsilon )^2}. \end{aligned}$$

(31)

Setting the first order partial derivative $\partial \pi _{\mathrm {m}}/ \partial t$ to zero,

$$\begin{aligned} \frac{\partial \pi _{\mathrm {m}}}{\partial t} = \frac{2\beta ^2 k_{\mathrm {r}}^2 w \left[ \alpha -(\beta -\epsilon )w\right] ^3}{(2\beta -\epsilon )^3(1-t)^2} - \frac{\beta ^2 k_{\mathrm {r}}^2 \left[ \alpha -(\beta -\epsilon )w\right] ^4(1+t)}{2(2\beta -\epsilon )^4(1-t)^3}=0 \end{aligned}$$

(32)

leads us to

$$\begin{aligned} t = \frac{9\beta w - 5 \epsilon w - \alpha }{7\beta w - 3 \epsilon w + \alpha }. \end{aligned}$$

(33)

As described in Sect. 2, the participation rate is only defined within $0\le t <1$. However, Eq. (33) can take negative values for $w<\alpha / (9\beta -5\epsilon )$. In this case, we have to set $t=0$ to avoid mathematical inconsistencies.

The first order partial derivative $\partial \pi _{\mathrm {m}}/ \partial w$ is

$$\begin{aligned} \frac{\partial \pi _{\mathrm {m}}}{\partial w}&= \frac{2\alpha \beta -4\beta (\beta -\epsilon )w}{2\beta -\epsilon }\left\{ \frac{\beta k_{\mathrm {r}}^2\left[ \alpha -(\beta -\epsilon )w\right] ^2}{(2\beta -\epsilon )^2(1-t)}+k_{\mathrm {m}}\sqrt{A}\right\} \nonumber \\&-\frac{4\beta ^2k_{\mathrm {r}}^2(\beta -\epsilon )w\left[ \alpha -(\beta -\epsilon )w\right] ^2}{(2\beta -\epsilon )^3(1-t)}+\frac{2\beta ^2k_{\mathrm {r}}^2(\beta -\epsilon )\left[ \alpha -(\beta -\epsilon )w\right] ^3t}{(2\beta -\epsilon )^4(1-t)^2}\nonumber \\ \end{aligned}$$

(34)

and is also set to zero. This equation can be simplified by inserting Eq. (31):

$$\begin{aligned}&k_{\mathrm {r}}^2(2\beta -\epsilon )\left[ \alpha -(\beta -\epsilon )w\right] \left[ \alpha -2(\beta -\epsilon )w\right] (1-t)\nonumber \\&\quad -2k_{\mathrm {r}}^2(\beta -\epsilon )(2\beta -\epsilon )w\left[ \alpha -(\beta -\epsilon )w\right] (1-t)\nonumber \\&\quad +k_{\mathrm {r}}^2(\beta -\epsilon )\left[ \alpha -(\beta -\epsilon )w\right] ^2t\nonumber \\&\quad -k_{\mathrm {m}}^2(2\beta -\epsilon )^2w\left[ \alpha -2(\beta -\epsilon )w\right] (1-t)^2=0. \end{aligned}$$

(35)

Due to the non-negativity restriction of $t$, we now have to conduct a case-by-case analysis. For $w\ge \alpha / (9\beta -5\epsilon )$, we insert Eq. (33) into Eq. (35):

$$\begin{aligned}&w^2\left[ k_{\mathrm {r}}^2\left( -49\beta ^3+91\beta ^2\epsilon -51\beta \epsilon ^2+9\epsilon ^3\right) +8k_{\mathrm {m}}^2\left( -4\beta ^3+8\beta ^2\epsilon -5\beta \epsilon ^2+\epsilon ^3\right) \right] \nonumber \\&\quad +w\left[ 2\alpha \beta k_{\mathrm {r}}^2(7\beta -3\epsilon )+4\alpha k_{\mathrm {m}}^2(2\beta -\epsilon )^2\right] + \alpha ^2k_{\mathrm {r}}^2(3\beta -\epsilon ) =0 \end{aligned}$$

(36)

The solution of this expression are given as $\tilde{w}_1$ and $\tilde{w}_2$ in Step 1 of the solution procedure stated in Proposition 2. For $w<\alpha / (9\beta -5\epsilon )$, we insert $t=0$ into Eq. (35):

$$\begin{aligned}&w^2\left[ 4k_{\mathrm {r}}^2(\beta -\epsilon )^2+2k_{\mathrm {m}}^2(-2\beta +\epsilon )(\beta -\epsilon ) \right] \nonumber \\&\quad +w\left[ 5\alpha k_{\mathrm {r}}^2(-\beta +\epsilon )+\alpha k_{\mathrm {m}}^2(2\beta -\epsilon ) \right] + \alpha ^2k_{\mathrm {r}}^2 =0. \end{aligned}$$

(37)

The solution of this expression are given as $\tilde{w}_3$ and $\tilde{w}_4$ in Step 1 of the solution procedure stated in Proposition 2. This completes proof of Proposition 2.

Proof

(Proposition 3) As defined in Sect. 2, $\alpha _i$ is a function of $\varLambda _i$ with

$$\begin{aligned} \frac{\partial \alpha _i}{\partial \varLambda _i} = \frac{1}{B+\varTheta }>0 \quad \text {and} \quad \frac{\partial \alpha _i}{\partial \varLambda _{3-i}} = 0. \end{aligned}$$

(38)

Due to the positive first order derivative $\partial \alpha _i/\partial \varLambda _i$ and the chain rule $\mathrm {d} f(h(x)) / \mathrm {d} x = \mathrm {d} f(h(x)) / \mathrm {d} h(x) \cdot \mathrm {d} h(x) / \mathrm {d} x$, the first order derivative with respect to $\alpha _i$ has the same prefix as the first order derivation with respect to $\varLambda _i$. Hence, one can easily make the conclusions given in Part (i) and Part (ii) of Proposition 3 with the following first order derivatives:

$$\begin{aligned} \frac{\partial m_i}{\partial \alpha _i}&= \frac{5\beta }{2(3\beta -\epsilon )(2\beta +\epsilon )}>0\end{aligned}$$

(39)

$$\begin{aligned} \frac{\partial m_i}{\partial \alpha _{3-i}}&= \frac{-\beta +2\epsilon }{2(3\beta -\epsilon )(2\beta +\epsilon )}<0\quad \text {for}\quad \beta >2\epsilon \Leftrightarrow B>2\varTheta \end{aligned}$$

(40)

$$\begin{aligned} \frac{\partial a_i}{\partial \alpha _i}&= \frac{5\beta ^3k_{\mathrm {r}}^2(5\alpha _i\beta +\alpha _{3-i}\beta +2\alpha _{3-i}\epsilon )^3}{16(3\beta -\epsilon )^4(2\beta +\epsilon )^4}>0\end{aligned}$$

(41)

$$\begin{aligned} \frac{\partial a_i}{\partial \alpha _{3-i}}&= \frac{\beta ^2k_{\mathrm {r}}^2(5\alpha _i\beta +\alpha _{3-i}\beta +2\alpha _{3-i}\epsilon )^3(\beta +2\epsilon )}{16(3\beta -\epsilon )^4(2\beta +\epsilon )^4}>0. \end{aligned}$$

(42)

Due to the complexity of the resulting first order derivatives $\partial \pi _{\mathrm {r}i}/ \partial \alpha _i$ and $\partial \pi _{\mathrm {r}i}/ \partial \alpha _{3-i}$, we are not able to proof Part (iii) of Proposition 3 analytically. Instead of that, we computed a numerical study with 3,000,000 randomly generated sets of parameters within the range $10\le \alpha _i\le 30$ and $0.1\le \beta ,\epsilon ,k_{\mathrm {m}},k_{\mathrm {r}}\le 10$ and could thereby show numerically that $\partial \pi _{\mathrm {r}i}/ \partial \alpha _i >0$ and $\partial \pi _{\mathrm {r}i}/ \partial \alpha _{3-i} >0$ holds for each considered combination of parameters—except 18 cases with $\beta \approx 0.1$ and $\epsilon >5$, which violate the condition $\beta >\epsilon$ resulting from $B>\varTheta$ given in Sect. 2, though.

Hence, we are confident that the given inequalities hold for feasible parameter combinations. This completes proof of Proposition 3.