Merchant selection and pricing strategy for a platform firm in the online group buying market

Abstract

The online group-buying market is characterized by intense competition between brokers, called platform firms, which function as intermediaries between merchants and consumers. In an environment of intense competition, merchant selection and pricing strategies are critical for platform firms. This paper employs business analytics to support strategy formulation for these firms by forecasting market demand and analyzing competitive environments. We apply the proposed decision framework, which relies on business analytics, to a study of the online group-buying market in Japan.

Appendix: proof of Theorem 1

First, we point out that our proof relies on the strategy in Amemiya (1985, pp. 366–372). For simplicity of notation, we denote $f_d (d_{kt}|{\varvec{z}}_{kt},{\varvec{\gamma }},\sigma _v^2)$ and $g(Q_{kt}|{\varvec{x}}_{kt},{\varvec{z}}_{kt},{\varvec{\beta }},\sigma _\varepsilon ^2,\sigma _v^2,\rho )$ as $f_d (d_{kt},{\varvec{\theta }}_1)$ and $g(Q_{kt}|{\varvec{\theta }}_1,{\varvec{\theta }}_2)$, respectively. Here ${\varvec{\theta }}_1=({\varvec{\gamma }}',\sigma _v^2)'$, and ${\varvec{\theta }}_2=({\varvec{\beta }}',\sigma _\varepsilon ^2,\rho )'$. Noting that $\partial \ell (\hat{{\varvec{\theta }}}_1,\hat{{\varvec{\theta }}}_2)/\partial {\varvec{\theta }}_2={\varvec{0}}$, we have

$$\begin{aligned} {\varvec{0}}= & {} \frac{1}{\sqrt{n}} \sum _{t=1}^T \sum _{k=1}^{N_t} \frac{\partial g(Q_{kt}|\hat{{\varvec{\theta }}}_1,\hat{{\varvec{\theta }}}_2)}{\partial {\varvec{\theta }}_2}\nonumber \\= & {} \frac{1}{\sqrt{n}} \sum _{t=1}^T \sum _{k=1}^{N_t} \frac{\partial g(Q_{kt}|{\varvec{\theta }}_1^0,{\varvec{\theta }}_2^0)}{\partial {\varvec{\theta }}_2} + \frac{1}{n} \sum _{t=1}^T \sum _{k=1}^{N_t} \frac{\partial ^2 g(Q_{kt}|{\varvec{\theta }}_1^0,{\varvec{\theta }}_2^0)}{\partial {\varvec{\theta }}_2\partial {\varvec{\theta }}_2'} \sqrt{n}(\hat{{\varvec{\theta }}}_2-{\varvec{\theta }}_2^0)\nonumber \\&+ \frac{1}{n} \sum _{t=1}^T \sum _{k=1}^{N_t} \frac{\partial ^2 g(Q_{kt}|{\varvec{\theta }}_1^0,{\varvec{\theta }}_2^0)}{\partial {\varvec{\theta }}_2\partial {\varvec{\theta }}_1'} \sqrt{n}(\hat{{\varvec{\theta }}}_1-{\varvec{\theta }}_1^0) +o_p(1)\nonumber \\= & {} \frac{1}{\sqrt{n}} \sum _{t=1}^T \sum _{k=1}^{N_t} \frac{\partial g(Q_{kt}|{\varvec{\theta }}_1^0,{\varvec{\theta }}_2^0)}{\partial {\varvec{\theta }}_2} -\hat{R}_{\theta _2\theta _2}\sqrt{n}(\hat{{\varvec{\theta }}}_2-{\varvec{\theta }}_2^0) -\hat{R}_{\theta _2\theta _1}\sqrt{n}(\hat{{\varvec{\theta }}}_1-{\varvec{\theta }}_1^0) +o_p(1), \nonumber \\ \end{aligned}$$

(7)

where $\hat{R}_{\theta _2\theta _2}$ and $\hat{R}_{\theta _2\theta _1}$ are the empirical versions of $R_{\theta _2\theta _2}$ and $R_{\theta _2\theta _1}$, respectively. In a similar manner, we expand the first-order conditions for $\hat{{\varvec{\theta }}}_1$ as

$$\begin{aligned} {\varvec{0}}= & {} \frac{1}{\sqrt{n}} \sum _{t=1}^T \sum _{k=1}^{N_t} \frac{\partial \log f_d (d_{kt},\hat{{\varvec{\theta }}}_1)}{\partial {\varvec{\theta }}_1} = \frac{1}{\sqrt{n}} \sum _{t=1}^T \sum _{k=1}^{N_t} \frac{\partial \log f_d (d_{kt},{\varvec{\theta }}_1^0)}{\partial {\varvec{\theta }}_1}\\&+ \frac{1}{n} \sum _{t=1}^T \sum _{k=1}^{N_t} \frac{\partial ^2 \log f_d (d_{kt},{\varvec{\theta }}_1^0)}{\partial {\varvec{\theta }}_2\partial {\varvec{\theta }}_1'} \sqrt{n}(\hat{{\varvec{\theta }}}_1-{\varvec{\theta }}_1^0) +o_p(1), \end{aligned}$$

which implies

$$\begin{aligned} \sqrt{n}(\hat{{\varvec{\theta }}}_1-{\varvec{\theta }}_1^0)= & {} \left[ - \frac{1}{n} \sum _{t=1}^T \sum _{k=1}^{N_t} \frac{\partial ^2 \log f_d (d_{kt},{\varvec{\theta }}_1^0)}{\partial {\varvec{\theta }}_2\partial {\varvec{\theta }}_1'} \right] ^{-1} \frac{1}{\sqrt{n}} \sum _{t=1}^T \sum _{k=1}^{N_t} \frac{\partial \log f_d (d_{kt},{\varvec{\theta }}_1^0)}{\partial {\varvec{\theta }}_1} +o_p(1)\nonumber \\= & {} \hat{R}_{\theta _1\theta _1}^{-1} \frac{1}{\sqrt{n}} \sum _{t=1}^T \sum _{k=1}^{N_t} \frac{\partial \log f_d (d_{kt},{\varvec{\theta }}_1^0)}{\partial {\varvec{\theta }}_1} +o_p(1), \end{aligned}$$

(8)

where $\hat{R}_{\theta _1\theta _1}$ are the empirical versions of $R_{\theta _1\theta _1}$.

Substituting (8) into (7), we have

$$\begin{aligned} \hat{R}_{\theta _2\theta _2} \sqrt{n}(\hat{{\varvec{\theta }}}_2-{\varvec{\theta }}_2^0)= & {} \frac{1}{\sqrt{n}} \sum _{t=1}^T \sum _{k=1}^{N_t} \frac{\partial g(Q_{kt}|{\varvec{\theta }}_1^0,{\varvec{\theta }}_2^0)}{\partial {\varvec{\theta }}_2}\\&- \hat{R}_{\theta _2\theta _1} \hat{R}_{\theta _1\theta _1}^{-1} \frac{1}{\sqrt{n}} \sum _{t=1}^T \sum _{k=1}^{N_t} \frac{\partial \log f_d (d_{kt},{\varvec{\theta }}_1^0)}{\partial {\varvec{\theta }}_1} +o_p(1), \end{aligned}$$

or equivalently

$$\begin{aligned} \sqrt{n}(\hat{{\varvec{\theta }}}_2-{\varvec{\theta }}_2^0) = \hat{R}_{\theta _2\theta _2}^{-1} \left( I,-\hat{R}_{\theta _2\theta _1} \hat{R}_{\theta _1\theta _1}^{-1} \right) \frac{1}{\sqrt{n}} \sum _{t=1}^T \sum _{k=1}^{N_t} \left( \begin{array}{cc} \displaystyle \frac{\partial g(Q_{kt}|{\varvec{\theta }}_1^0,{\varvec{\theta }}_2^0)}{\partial {\varvec{\theta }}_2}\\ \displaystyle \frac{\partial \log f_d (d_{kt}|{\varvec{\theta }}_1^0)}{\partial {\varvec{\theta }}_1} \end{array} \right) +o_p(1). \end{aligned}$$

Together with (5), $\sqrt{n}(\hat{{\varvec{\theta }}}_2-{\varvec{\theta }}_2^0)$ is asymptotically normal with mean ${\varvec{0}}$ and variance matrix V. This completes the proof.