Contracting and competing on a food delivery platform

Abstract

Food delivery platforms increase their revenue by charging the restaurants a commission fee for using their service. This study aims to understand the restaurant’s behaviour when they face competition from other restaurants while in a contract with the food delivery platform. We compare the behaviour of the restaurant when they face no competition to their behaviour when there is competition. Specifically, we analyse the impact of competition on the commission rate and the marketing level. Both the online and offline demand for the restaurant is considered to be stochastic. We find that the platform benefits when the restaurants compete. Our study suggests the platform to decide the required marketing index for the restaurant based on their unserved online demand and commission fee, and the restaurants to fix the commission rate based on their unserved offline demand and the marketing index required to meet their capacity.

Appendix

1.1 Symmetric case

Proof of proposition 1:

$$\begin{aligned} & \Pi_{{p_{i} }} = b_{i} p_{i} q_{ni} {-}\left( {\eta m_{i}^{2} } \right)/2 \\ & {\text{s}}{\text{.t}}{.},\,\, m_{i} \le 1 \\ \end{aligned}$$

(8)

Initially, we solved the platform utility function without considering the constraint, and the result of the optimum marketing index is within our limit (${0<m}_{i}\le 1)$. By taking the second order derivative of (8) with respect to ${m}_{i}$, we have $\frac{{\partial }^{2}{\Pi }_{{p}_{i}}}{{{\partial m}_{i}}^{2}}= -\frac{{b}_{i}{p}_{i} {\left({C}_{i}-{\mu }_{i}\right)}^{2} }{{B}_{n}{\left({m}_{i}-\lambda {r}_{i}\right)}^{3}}- \eta$ < 0, which means the marketing index provided by the platform has a unique optimal solution. The optimum value of the marketing index (${m}_{i}^{*}$) is obtained by equating the first order derivative of (8) with respect to ${m}_{i}$ equal to 0.

By the first order condition,$\frac{{\partial \Pi _{{p_{i} }} }}{{\partial m_{i} }} = \frac{{b_{i} p_{i} \left( {C_{i} - \mu _{i} } \right)^{2} }}{{2B_{n} (m_{i} - \lambda r_{i} )^{2} }} - \eta m_{i} = 0$, we get

$${m}_{i}^{*}= \lambda {r}_{i}+ \sqrt{\frac{{b}_{i}{p}_{i} }{2{B}_{n}\eta} {\left({C}_{i}-{\mu }_{i}\right)}^{2}}$$

Proof of corollary 1:

$$\frac{\partial {m}_{i}^{*}}{\partial {b}_{i}}>0 ,\boldsymbol{ }\frac{\partial {m}_{i}^{*}}{\partial {p}_{i} }>0, \frac{\partial {m}_{i}^{*}}{\partial \left({C}_{i}-{\mu }_{i}\right) }>0$$

$$\frac{\partial {m}_{i}^{*}}{\partial {b}_{i}}= \frac{{p}_{i}{({c}_{i}-{\mu }_{i})}^{2}}{2\sqrt{2}\eta {B}_{n}\sqrt{\frac{{b}_{i}{p}_{i}{({c}_{i}-{\mu }_{i})}^{2}}{\eta {B}_{n}}}} >0$$

$$\frac{\partial {m}_{i}^{*}}{\partial {p}_{i} }= \frac{{b}_{i}{({c}_{i}-{\mu }_{i})}^{2}}{2\sqrt{2}\eta {B}_{n}\sqrt{\frac{{b}_{i}{p}_{i}{({c}_{i}-{\mu }_{i})}^{2}}{\eta {B}_{n}}}} >0$$

$$\frac{\partial {m}_{i}^{*}}{\partial \left({C}_{i}-{\mu }_{i}\right) }= \frac{{b}_{i}{p}_{i}\left({C}_{i}-{\mu }_{i}\right)}{ \eta {B}_{n}\sqrt{\frac{2{b}_{i}{p}_{i} }{{B}_{n}\eta} {\left({C}_{i}-{\mu }_{i}\right)}^{2}}}>0$$

1.2 Proof for concavity of restaurant utility function

$$\begin{aligned} \Pi_{{r_{i} }} & = p_{i} q_{fi} + p_{i} q_{ni} - b_{i} p_{i} q_{ni} \\ & {\text{s}}{\text{.t}}{.}, 0 < b_{i} < 1 \\ \end{aligned}$$

(9)

As the second order derivative of (9) with respect to ${b}_{i }$ is negative (i.e., Less than zero), the unique optimal solution for the commission rate exists. By using the first-order necessary condition, $\frac{\partial {\Pi }_{{r}_{i}}}{\partial {b}_{i}}=0$, we can find the optimum commission rate provided by the restaurant.

$$\frac{\partial {\Pi }_{{r}_{i}}}{\partial {b}_{i}}= {p}_{i} \left({C}_{i}-{\mu }_{i}\right) \left[ \frac{{\mu }_{i}}{4 {B}_{n}{ B}_{f} {{b}_{i}}^{3/2}\sqrt{\frac{p}{2{B}_{n}\eta}}} + \frac{1}{4 {B}_{n}\sqrt{\frac{bp}{2{B}_{n}\eta}}} - 1+ \frac{\left({C}_{i}-{\mu }_{i}\right)}{4 {B}_{n}{ B}_{f} {{b}_{i}}^{2}\sqrt{\frac{p}{\eta}}} \right]=0$$

$$\frac{{\partial }^{2}{\Pi }_{{r}_{i}}}{\partial {{b}_{i}}^{2}}= {- p}_{i} \left({C}_{i}-{\mu }_{i}\right) \left[\frac{3{ \mu }_{i}}{8 {B}_{n}{ B}_{f} {{b}_{i}}^{5/2}\sqrt{\frac{p}{2{B}_{n}\eta}}} + \frac{1}{8 {B}_{n}{{b}_{i}}^{3/2}\sqrt{\frac{p}{2{B}_{n}\eta}}} + \frac{ \left({C}_{i}-{\mu }_{i}\right)}{2 {B}_{n}{ B}_{f} {{b}_{i}}^{3}\sqrt{\frac{p}{\eta}}} \right] <0$$

Since obtaining the closed form solution for the optimum commission rate is difficult, we obtained the numerical value of the optimum commission rate by performing the numerical analysis.

1.3 Asymmetric case

Proof of proposition 2:

$$\begin{aligned} \Pi_{{p_{i} }} & = b_{i} p_{i} q_{ni} {-}\left( {\eta \left( {m_{i} {-}\beta \left( {1 - \frac{{b_{i} p_{i} }}{{\Sigma_{i = 1}^{2} b_{i} p_{i} }}} \right)} \right)^{2} } \right)/2 \\ & {\text{ s}}{\text{.t}}{.},{\Sigma }_{i = 1}^{2} m_{i} \le 1 \\ \end{aligned}$$

(10)

Equation (10) can be written as shown below by considering the Lagrangian multiplier $\theta ,$

$$\Pi_{{p_{i} }} = b_{i} p_{i} q_{ni} {-}\left( {\eta \left( {m_{i} {-} \beta \left( {1 - \frac{{b_{i} p_{i} }}{{\Sigma_{i = 1}^{2} b_{i} p_{i} }}} \right)} \right)^{2} } \right)/2 - \theta (\mathop \sum \limits_{i = 1}^{2} m_{i} - 1)$$

(10.1)

The second derivative of Eq. (10.1) with respect to ${m}_{i}$ is negative. Therefore function is concave in nature and has a unique optimum solution. By using the first-order necessary condition, $\frac{\partial {\Pi }_{p}}{\partial {m}_{i}}= 0,$ we can obtain the optimum marketing index under the asymmetric information case.

$$\frac{{\partial \Pi _{p} }}{{\partial m_{i} }} = \frac{{b_{i} p_{i} \left( {C_{i} - \mu _{i} } \right)^{2} }}{{2B_{n} (m_{i} - \lambda r_{i} )^{2} }} - \eta \left( {m_{i} \eta \beta \left( {\Sigma _{{i = 1}}^{2} 1 - \frac{{b_{i} p_{i} }}{{\Sigma _{{i = 1}}^{2} b_{i} p_{i} }}} \right)} \right) - \theta$$

$$\frac{{\partial }^{2}{\Pi }_{{p}_{i}}}{{{\partial m}_{i}}^{2}}= -\frac{{b}_{i}{p}_{i} {\left({C}_{i}-{\mu }_{i}\right)}^{2} }{{B}_{n}{({m}_{i}-\lambda {r}_{i})}^{3}}- \eta$$

$$\frac{{\partial }^{2}{\Pi }_{{p}_{i}}}{{{\partial m}_{i}}^{2}}<0$$

By considering,$\frac{\partial {\Pi }_{p}}{\partial {m}_{i}}= 0$, we get

$${m}_{{i}_{a}}^{*}=\left(\frac{{b}_{i}{p}_{i} }{2{B}_{n}}{\left({C}_{i}-{\mu }_{i}\right)}^{2}-\theta \right)\left(\frac{1}{ \eta}\right)+ \beta \left(1-\frac{{b}_{i}{p}_{i}}{{\Sigma }_{i=1}^{2}{b}_{i}{p}_{i}}\right)$$

Proof of corollary 2:

$\frac{\partial {m}_{{i}_{a}}^{*}}{\partial {b}_{i}}>0 ,\boldsymbol{ }\frac{\partial {m}_{{i}_{a}}^{*}}{\partial {p}_{i} }>0, \frac{\partial {m}_{{i}_{a}}^{*}}{\partial \left({C}_{i}-{\mu }_{i}\right) }>0,$ Only when $\eta< \frac{{\left({C}_{1}-{\mu }_{1}\right)}^{2}{({b}_{1}{p}_{1}+{b}_{2}{p}_{2})}^{2}}{2{B}_{n}\beta {b}_{2}{p}_{2}}$

As we have considered only two restaurants in the asymmetric case, we explain corollary 2 with respect to restaurant 1

$$\frac{\partial {m}_{{1}_{a}}^{*}}{\partial {b}_{1}}= \frac{-\beta {b}_{2}{p}_{2}{p}_{1}}{{({b}_{1}{p}_{1}+{b}_{2}{p}_{2})}^{2}}+ \frac{{p}_{1}{\left({C}_{1}-{\mu }_{1}\right)}^{2}}{2{B}_{n}\eta} >0$$

$$\frac{\partial {m}_{{1}_{a}}^{*}}{\partial {p}_{1}}= \frac{-\beta {b}_{2}{p}_{2}{b}_{1}}{{({b}_{1}{p}_{1}+{b}_{2}{p}_{2})}^{2}}+ \frac{{b}_{1}{\left({C}_{1}-{\mu }_{1}\right)}^{2}}{2{B}_{n}\eta} >0$$

$$\frac{\partial {m}_{{1}_{a}}^{*}}{\partial \left({C}_{1}-{\mu }_{1}\right)}= \frac{{p}_{1}{b}_{1}\left({C}_{1}-{\mu }_{1}\right)}{{B}_{n}\eta} >0$$

From the optimum marketing index, we can find the maximum number of restaurants that can contract with the platform, until which the utility function of the platform increases under the assumption of equal commission fees; and is given as

$$N-N\left(\frac{{m}_{{i}_{a}}^{*}-\frac{- \frac{{b}_{i}{p}_{i} }{2{B}_{n}}{\left({C}_{i}-{\mu }_{i}\right)}^{2}-\theta }{\eta}}{\beta }\right)-1=0$$

The minimum value of $N$ obtained from the above equation corresponds to the maximum number of restaurants that can contract with the platform.

1.4 Proof for concavity of restaurant utility function

Because of asymmetric information, the optimum marketing index in terms of the restaurant is given as,

$$m_{{i_{a} }}^{*} = \beta \left( {1 - \frac{{2b_{i} p_{i} }}{{\left( {h_{i} + l_{i} } \right)n_{a} p_{a} }}} \right) + \frac{{ - \theta + \frac{{b_{i} p_{i} \left( {c_{i} - \mu_{i} } \right)^{2} }}{{2B_{n} }}}}{\eta }$$

$$\begin{aligned} \Pi_{{r_{i} }} & = p_{i} q_{fi} + p_{i} q_{ni} - b_{i} p_{i} q_{ni} \\ & s.t., 0 < b_{i} < 1 \\ \end{aligned}$$

(11)

As the second order derivative of Eq. (11) with respect to ${b}_{i }$ is negative (i.e., Less than zero), the unique optimal solution for the commission rate exists. By using the first-order necessary condition, $\frac{\partial {\Pi }_{{r}_{i}}}{\partial {b}_{i}}=0$, we can find the optimum commission rate provided by the restaurant.

$${m}_{{i}_{a}}=\beta \left(1-\frac{2{b}_{i}{p}_{i}}{({h}_{i}+{l}_{i}){n}_{a}{p}_{a}}\right)+\frac{-\theta +\frac{{b}_{i}{p}_{i}{({c}_{i}-{\mu }_{i})}^{2}}{2{B}_{n}}}{\eta }$$

$${q}_{ni}=(({c}_{i}-{\mu }_{i})(1-(({c}_{i}-{\mu }_{i})/(2{B}_{n}*({{m}_{i}}_{a}-(\lambda *{r}_{i}))))$$

$$\frac{\partial {m}_{i}}{\partial {b}_{i}}=-\frac{2\beta {p}_{i}}{({h}_{i}+{l}_{i}){n}_{a}{p}_{a}}+\frac{{p}_{i}{({c}_{i}-{\mu }_{i})}^{2}}{2\eta {B}_{n}}$$

$$\frac{\partial {q}_{ni}}{\partial {b}_{i}}= \frac{\frac{\partial {m}_{i}}{\partial {b}_{i}}{({c}_{i}-{\mu }_{i})}^{2}}{2{B}_{n}(-\lambda {r}_{i}+{m}_{i})}$$

$$\frac{\partial {\pi }_{ri}}{\partial {b}_{i}}={p}_{i}\left(-{q}_{ni}\left(1+\frac{\frac{\partial {q}_{ni}}{\partial {b}_{i}}}{{ 2B}_{f}}\right)+\frac{\partial {q}_{ni}}{\partial {b}_{i}}\left(-{b}_{i}+\frac{-{q}_{ni}+{c}_{i}}{{ 2B}_{f}}\right)+\frac{\frac{\partial {q}_{ni}}{\partial {b}_{i}}*{c}_{i}}{{ 2B}_{f}}\right)$$

$$\frac{{\partial }^{2}{m}_{{i}_{a}}}{\partial {{b}_{i}}^{2}}=0$$

$$\frac{{\partial }^{2}{q}_{ni}}{\partial {{b}_{i}}^{2}}=\frac{{({c}_{i}-{\mu }_{i})}^{2}\left(-\frac{2\frac{\partial {m}_{i}}{\partial {b}_{i}}}{{(-\lambda {r}_{i}+{m}_{i})}^{3}}+\frac{\frac{{\partial }^{2}{m}_{i}}{\partial {{b}_{i}}^{2}}}{{(-\lambda {r}_{i}+{m}_{i})}^{2}}\right)}{2{B}_{n}}$$

$$\frac{{\partial }^{2}{\pi }_{ri}}{\partial {{b}_{i}}^{2}}= {p}_{i}\left(-\frac{{\partial }^{2}{q}_{ni}}{\partial {{b}_{i}}^{2}}*{b}_{i}-2\frac{\partial {q}_{ni}}{\partial {b}_{i}}\left(1+\frac{\frac{\partial {q}_{ni}}{\partial {b}_{i}}}{{ 2B}_{f}}\right)+\frac{2*\frac{{\partial }^{2}{q}_{ni}}{\partial {{b}_{i}}^{2}}*{c}_{i}}{{ 2B}_{f}}\right) <0$$

Since obtaining the closed form solution for the optimum commission rate is difficult, we obtained the numerical value of the optimum commission rate by performing the numerical analysis.