Analysis of market competition and information asymmetry on selling strategies

Abstract

In this paper, we consider the seller’s selling strategies in a supply chain consisting of one supplier and multiple retailers, who compete in the same consumer market. The production lead time is relatively long compared to the selling season. Therefore, the supplier can choose to sell the products either before production begins (i.e., advance sale) or after production finishes (i.e. regular sale). Different to existing literature, we analyze the selling strategies in a more realistic environment where each buyer has a private demand signal about the consumer market at the beginning of the selling season. Three different scenarios (i.e., the monopoly retailer, the competing retailers with information sharing, and the competing retailers without information sharing) are analyzed to study the effect of competition and information asymmetry on the supplier’s selling strategy. In each scenario, we establish the equilibrium result and investigate how the competition and information asymmetry affect the supply chain and its each party’s profits with the underlying reason explained. Our main findings include: (1) the supplier is optimal to adopt regular sale when the retailers form as a monopoly retailer, or when the retailers competing with each other with demand information sharing; (2) he prefers advance sale when facing competing retailers with private information; (3) competition and private information also influence the retailers’ and the supply chain’s preferences on the supplier’s sale strategies. Moreover, when the retailers have private information under competition, we find interestingly that (1) the supplier’s profit is increasing in the information precision under advance sale, while it is not affected by information precision under regular sale; and (2) the retailers’ profits may decrease in the information precision under regular sale.

Appendix

1.1 Proof to Lemma 1

Proof

Given the true state being $\theta _H$, the fraction of retailers who get the high signal is $l$, and $(1-l)$ fraction of retailers get the low signal. On the other hand, given the true state being $\theta _L$, only $1-l$ fraction of retailers getting the high signal. Thus, this produces the first part of the lemma.

Now, given the information set $\{s_i\}, i\in [0,1]$ and the fraction of retailers getting high demand signal, by the Bayesion updating formulation, we have

$$\begin{aligned} P(\theta _H|\{s_i\})=\frac{P(\{s_i\}|\theta _H)\frac{1}{2}}{P(\{s_i\}|\theta _H)\frac{1}{2}+P(\{s_i\}|\theta _L)\frac{1}{2}} \end{aligned}$$

in which $\{s_i\}$ have a fraction of $\gamma$ with high demand signal $s_H$. This produces the desired result. $\square$

1.2 Proof to Lemma 2

Proof

We first analyze the retailer’s optimal decisions on $K$ to maximize $\pi ^R(K)$. We can explicitly express $\pi ^R(K)$ as

$$\begin{aligned} \pi ^R(K)&= -\omega _AK+\frac{1}{2}\left( \left( \theta _L-\min \left\{ K,\frac{\theta _L}{2}\right\} \right) \min \left\{ K,\frac{\theta _L}{2}\right\} \right. \nonumber \\&\left. +\left( \theta _H-\min \left\{ K,\frac{\theta _H}{2}\right\} \right) \min \left\{ K,\frac{\theta _H}{2}\right\} \right) \end{aligned}$$

Depending on the value of $K$, we can further simplify $\pi ^R(K)$ as

$$\begin{aligned} \pi ^R(K)=-\omega _AK+\left\{ \begin{array}{l@{\quad }l} \frac{1}{2}((\theta _L-K)K+(\theta _H-K)K), &{} \hbox {if}\, K\le \frac{\theta _L}{2};\\ \frac{1}{2}\left( \frac{\theta _L^2}{4}+(\theta _H-K)K\right) , &{} \hbox {if}\, \frac{\theta _L}{2}<K\le \frac{\theta _H}{2};\\ \frac{1}{2}\left( \frac{\theta _L^2}{4}+\frac{\theta _H^2}{4}\right) , &{} \hbox {if}\, K>\frac{\theta _H}{2} \end{array} \right. \end{aligned}$$

Then, by optimizing the above formulation, we can get

$$\begin{aligned} K(\omega _A)=\left\{ \begin{array}{l@{\quad }l} \frac{\theta _H}{2}-\omega _A, &{} \hbox {if}\, \omega _A\le \frac{\theta _H-\theta _L}{2};\\ \frac{\theta _H+\theta _L}{4}-\frac{\omega _A}{2}, &{} \hbox {if}\, \omega _A>\frac{\theta _H-\theta _L}{2} \end{array} \right. \end{aligned}$$

Now we turn back to the decision of the supplier and can express $\pi ^S(\omega _A)$ as

$$\begin{aligned} \pi ^S(\omega _A)=\left\{ \begin{array}{l@{\quad }l} (\omega _A-c)\left( \frac{\theta _H}{2}-\omega _A\right) , &{} \hbox {if}\, \omega _A\le \frac{\theta _H-\theta _L}{2};\\ (\omega _A-c)\left( \frac{\theta _H+\theta _L}{4}-\frac{\omega _A}{2}\right) , &{} \hbox {if}\, \omega _A> \frac{\theta _H-\theta _L}{2} \end{array} \right. \end{aligned}$$

Optimizing the above formulation, we get

$$\begin{aligned} \omega ^{MA}=\left\{ \begin{array}{l@{\quad }l} \frac{1}{2}\left( \frac{\theta _H}{2}+c\right) , &{} \hbox {if}\, c\le \frac{\theta _H-(1+\sqrt{2})\theta _L}{2};\\ \frac{1}{2}\left( \frac{\theta _H+\theta _L}{2}+c\right) , &{} \hbox {if}\, c> \frac{\theta _H-(1+\sqrt{2})\theta _L}{2} \end{array} \right. \end{aligned}$$

This completes the proof. $\square$

1.3 Proof to Lemma 3

Proof

In the second stage, we can express the supplier’s profit function as

$$\begin{aligned} \tilde{\pi }^S(\omega _R)=\left\{ \begin{array}{l@{\quad }l} \min \left\{ \frac{\theta _L-\omega _R}{2},Q\right\} \omega _R, &{} \hbox {if}\, \gamma =1-l;\\ \min \left\{ \frac{\theta _H-\omega _R}{2},Q\right\} \omega _R, &{} \hbox {if}\, \gamma =l \end{array} \right. \end{aligned}$$

Thus, the optimal wholesale price, as a function of $\gamma$ and $Q$, is given by

$$\begin{aligned} \omega ^{MR}=\left\{ \begin{array}{l@{\quad }l} \frac{\theta _L}{2}, &{} \hbox {if}\, \gamma =1-l\, \hbox {and}\, Q\ge \frac{\theta _L}{4};\\ \theta _L-2Q, &{} \hbox {if}\, \gamma =1-l\, \hbox {and}\, Q<\frac{\theta _L}{4};\\ \frac{\theta _H}{2}, &{} \hbox {if}\, \gamma =l\, \hbox {and}\, Q\ge \frac{\theta _H}{4};\\ \theta _H-2Q, &{} \hbox {if}\, \gamma =l\, \hbox {and}\, Q<\frac{\theta _H}{4} \end{array} \right. \end{aligned}$$

Given this, the supplier’s profit for the whole planning horizon is given by

$$\begin{aligned} \pi ^S(Q)=-cQ+\left\{ \begin{array}{l@{\quad }l} \frac{1}{2}\left( \frac{\theta _L^2}{8}+\frac{\theta _H^2}{8}\right) , &{} \hbox {if}\, Q\ge \frac{\theta _H}{4};\\ \frac{1}{2}\left( \frac{\theta _L^2}{8}+(\theta _H-2Q)Q\right) , &{} \hbox {if}\, \frac{\theta _L}{4}\le Q<\frac{\theta _H}{4};\\ \frac{1}{2}(Q(\theta _L-2Q)+Q(\theta _H-2Q)), &{} \hbox {if}\, Q< \frac{\theta _L}{4} \end{array} \right. \end{aligned}$$

By optimizing the above objective function, we get the desired result. $\square$

1.4 Proof to Theorem 1

Proof

If $c\le \frac{\theta _H-(1+\sqrt{2})\theta _L}{2}$, then, we have $\pi _R^{MA}-\pi _R^{MR}=\frac{3}{32}\theta _L^2>0$, $\pi _S^{MA}-\pi _S^{MR}=-\frac{\theta _L^2}{16}<0$ and $\varPi ^{MA}-\varPi ^{MR}=\frac{\theta _L^2}{32}>0$, thus, we have $\pi _R^{MA}>\pi _R^{MR}$,$\pi _S^{MA}<\pi _S^{MR}$,$\varPi ^{MA}>\varPi ^{MR}$.

If $\frac{\theta _H-(1+\sqrt{2})\theta _L}{2}<c\le \frac{\theta _H-\theta _L}{2}$, then, we have

$$\begin{aligned} \pi _R^{MA}-\pi _R^{MR}&= \frac{1}{16}\left( \frac{\theta _H+\theta _L}{2}-c\right) ^2 -\frac{1}{8}\left( \frac{\theta _L^2}{4}+\left( \frac{\theta _H}{2}-c\right) \right) ^2\\&= \frac{1}{16}\left( -\frac{(\theta _H-\theta _L)^2}{4}+c(\theta _H-\theta _L)-c^2\right) = -\frac{1}{16}\left( \frac{\theta _H-\theta _L}{2}-c\right) ^2<0\\ \pi _S^{MA}-\pi _S^{MR}&= \frac{1}{8}\left( \frac{\theta _H+\theta _L}{2}-c\right) ^2 -\frac{1}{4}\left( \frac{\theta _L^2}{4}+\left( \frac{\theta _H}{2}-c\right) \right) ^2\\&= -\frac{1}{8}\left( \frac{\theta _H-\theta _L}{2}-c\right) ^2<0\\ \varPi ^{MA}-\varPi ^{MR}&= \frac{3}{16}\left( \frac{\theta _H+\theta _L}{2}-c\right) ^2 -\frac{3}{8}\left( \frac{\theta _L^2}{4}+\left( \frac{\theta _H}{2}-c\right) \right) ^2\\&= -\frac{3}{16}\left( \frac{\theta _H-\theta _L}{2}-c\right) ^2<0 \end{aligned}$$

Thus, we have $\pi _R^{MA}<\pi _R^{MR}$,$\pi _S^{MA}<\pi _S^{MR}$,$\varPi ^{MA}<\varPi ^{MR}$.

If $c>\frac{\theta _H-\theta _L}{2}$, we have $\pi _R^{MA}-\pi _R^{MR}=0$, $\pi _S^{MA}-\pi _S^{MR}=0$, $\varPi ^{MA}-\varPi ^{MR}=0$, which implies the results. $\square$

1.5 Proof to Lemma 4

Proof

We first analyze the retailer’s optimal decisions on $K$ to maximize $\pi _i^R(K)$. We can explicitly express $\pi _i^R(K)$ as

$$\begin{aligned} \pi ^R_i(K_i)=-\omega _AK_i+\frac{1}{2}((\theta _L-\min \{K_i^{-},\theta _L\})\min \{K_i,\theta _L\}+(\theta _H-\min \{K_i^{-},\theta _H\})\min \{K_i,\theta _H\}) \end{aligned}$$

in which $K_i^{-}=\int _0^1K_jdj=K_i$ in value as we only focus on $K_i=K_j$, for any $i\ne j$. Then, depending on the value of $K_i$, we can further simplify $\pi _i^R(K_i)$ as

$$\begin{aligned} \pi ^R_i(K_i)=-\omega _AK_i+\left\{ \begin{array}{l@{\quad }l} \frac{1}{2}((\theta _L-K_i^{-})K_i+(\theta _H-K_i^{-})K_i), &{} \hbox {if}\, K_i\le \theta _L;\\ \frac{1}{2}((\theta _H-K_i^{-})K_i), &{} \hbox {if}\, \theta _L<K_i\le \theta _H;\\ 0, &{} \hbox {if}\, K_i>\theta _H \end{array} \right. \end{aligned}$$

Then, by optimizing the above formulation, we can get that

$$\begin{aligned} \begin{array}{l@{\quad }l} -\omega _A+\frac{\theta _H+\theta _L}{2}=\int _0^1K_idi, &{} \hbox {if}\, K_i\le \theta _L;\\ -\omega _A+\frac{1}{2}(\theta _H-\int _0^1K_idi)=0, &{} \hbox {if}\, \theta _L<K_i\le \theta _H \end{array} \end{aligned}$$

which induces to

$$\begin{aligned} K^{CA}=\left\{ \begin{array}{l@{\quad }l} \theta _H-2\omega _A, &{} \hbox {if}\, \omega _A\le \frac{\theta _H-\theta _L}{2};\\ \frac{\theta _H+\theta _L}{2}-\omega _A, &{} \hbox {if}\, \omega _A>\frac{\theta _H-\theta _L}{2} \end{array} \right. \end{aligned}$$

Now we turn back to the decision of the supplier and can express $\pi ^S(\omega _A)$ as

$$\begin{aligned} \pi ^S(\omega _A)=\left\{ \begin{array}{l@{\quad }l} 2(\omega _A-c)\left( \frac{\theta _H}{2}-\omega _A\right) , &{} \hbox {if}\, \omega _A\le \frac{\theta _H-\theta _L}{2};\\ 2(\omega _A-c)\left( \frac{\theta _H+\theta _L}{4}-\frac{\omega _A}{2}\right) , &{} \hbox {if}\, \omega _A> \frac{\theta _H-\theta _L}{2} \end{array} \right. \end{aligned}$$

Optimizing the above formulation, we get

$$\begin{aligned} \omega ^{CA}=\left\{ \begin{array}{l@{\quad }l} \frac{1}{2}\left( \frac{\theta _H}{2}+c\right) , &{} \hbox {if}\, c\le \frac{\theta _H-(1+\sqrt{2})\theta _L}{2};\\ \frac{1}{2}\left( \frac{\theta _H+\theta _L}{2}+c\right) , &{} \hbox {if}\, c> \frac{\theta _H-(1+\sqrt{2})\theta _L}{2} \end{array} \right. \end{aligned}$$

This completes the proof. $\square$

1.6 Proof to Lemma 5

Proof

In the second stage, we can express the supplier’s value function as

$$\begin{aligned} \tilde{\pi }^S(\omega _R)=\left\{ \begin{array}{l@{\quad }l} \min \{\theta _L-\omega _R,Q\}\omega _R, &{} \hbox {if}\, \gamma =1-l;\\ \min \{\theta _H-\omega _R,Q\}\omega _R, &{} \hbox {if}\, \gamma =l \end{array} \right. \end{aligned}$$

Thus, the optimal wholesale price, as a function of $\gamma$ and $Q$, is given by

$$\begin{aligned} \omega ^{CR}=\left\{ \begin{array}{l@{\quad }l} \frac{\theta _L}{2}, &{} \hbox {if}\, \gamma =1-l\, \hbox {and}\, Q\ge \frac{\theta _L}{2};\\ \theta _L-Q, &{} \hbox {if}\, \gamma =1-l\, \hbox {and}\, Q<\frac{\theta _L}{2};\\ \frac{\theta _H}{2}, &{} \hbox {if}\, \gamma =l\, \hbox {and}\, Q\ge \frac{\theta _H}{2};\\ \theta _H-Q, &{} \hbox {if}\, \gamma =l\, \hbox {and}\, Q<\frac{\theta _H}{2} \end{array} \right. \end{aligned}$$

Given this, the supplier’s profit is given by

$$\begin{aligned} \pi ^S(Q)=-cQ+\left\{ \begin{array}{l@{\quad }l} \frac{1}{2}\left( \frac{\theta _L^2}{4}+\frac{\theta _H^2}{4}\right) , &{} \hbox {if}\, Q\ge \frac{\theta _H}{2};\\ \frac{1}{2}\left( \frac{\theta _L^2}{4}+(\theta _H-Q)Q\right) , &{} \hbox {if}\, \frac{\theta _L}{2}\le Q<\frac{\theta _H}{2};\\ \frac{1}{2}(Q(\theta _L-Q)+Q(\theta _H-Q)), &{} \hbox {if}\, Q< \frac{\theta _L}{2} \end{array} \right. \end{aligned}$$

By optimizing the above objective function, we get the desired result. $\square$

1.7 Proof to Theorem 2

Proof

The proof is similar to that in Theorem 1 and is omitted here. $\square$

1.8 Proof to Lemma 6

Proof

Given all firms have the same order quantity, we analyze retailer $i$ optimal decision on selling quantity $q$ by maximizing

$$\begin{aligned} \tilde{\pi }_i^R(q)=\hbox {E}\left[ \left( \theta -\int _0^1q_jdj\right) q|s_i\right] =q\hbox {E}[\theta |s_i]-\hbox {E}\left[ \int _0^1q_jdj|s_i\right] \end{aligned}$$

Depending on whether $s_i=s_H$ or $s_i=s_L$, we have

$$\begin{aligned} \tilde{\pi }_i^R(q)=\left\{ \begin{array}{l@{\quad }l} q_H(\hbox {E}[\theta |s_H]-\zeta q_H^{-}-(1-\zeta )q_L^{-}), &{} \hbox {if}\, s_i=s_H;\\ q_L(\hbox {E}[\theta |s_L]-(1-\zeta )q_H^{-}-\zeta q_L^{-}), &{} \hbox {if}\, s_i=s_L. \end{array} \right. \end{aligned}$$

in which $q_H^{-}, q_L^{-}$ denotes the other retailers’ total order quantity for both states with the same value as $q_H$ or $q_L$. Then, in equilibrium,

(Case i) if the unconstraint selling quantities for both states are within the on hand inventory, we have

$$\begin{aligned} \hbox {E}[\theta |s_H]-\zeta q_H-(1-\zeta )q_L&= 0\\ \hbox {E}[\theta |s_L]-(1-\zeta )q_H-\zeta q_L&= 0 \end{aligned}$$

That is, $q_H=\bar{y}$ and $q_L=\underline{y}$ if $K\ge \bar{y}$.

(Case ii) if only the unconstraint selling quantity for the low demand signal state is larger than $K$, we have

$$\begin{aligned} q_H&= K\\ \hbox {E}[\theta |s_L]-(1-\zeta )q_H-\zeta q_L&= 0 \end{aligned}$$

Note that the second equation holds for $q_L\le K$.

(Case iii) if the unconstraint selling quantities for both demand signal states are within $K$, we have $q_L=q_H=K$.

Combining the three cases, we get the result. $\square$

1.9 Proof to Lemma 7

Proof

The monotone properties of $\underline{y}$ and $\bar{y}$ are straightforward by taking the first order derivative with respect to $l$. $\square$

1.10 Proof to Lemma 8

Proof

The retailer $i$’s profit is

$$\begin{aligned} \pi _i^R(K_i)&= -\omega _AK_i\\&\quad +\left\{ \begin{array}{l@{\quad }l} 0, &{} \hbox {if}\, K_i\ge \bar{y};\\ \frac{1}{2}\left( \left( \hbox {E}[\theta |s_H]\!-\!\frac{1-\zeta }{\zeta } \hbox {E}[\theta |s_L]\!-\!\frac{2\zeta -1}{\zeta }\int _0^1K_jdj\right) K_i\right) , &{} \hbox {if}\, \bar{y}>K_i\ge \hbox {E}[\theta |s_L];\\ \frac{1}{2}\left( \left( \hbox {E}[\theta |s_H]+\hbox {E}[\theta |s_L]-2\int _0^1K_jdj\right) K_i\right) , &{} \hbox {if}\, K_i<\hbox {E}[\theta |s_L]. \end{array} \right. \end{aligned}$$

in which $\int _0^1K_jdj$ represents the other retailers’ total order quantity. Optimizing $\pi _i^R(K_i)$ with respect to $K_i$, we get

$$\begin{aligned} K(\omega _A)=\left\{ \begin{array}{l@{\quad }l} \bar{y}-\frac{2\zeta }{2\zeta -1}\omega _A, &{} \hbox {if}\, c\le \frac{2l-1}{2}(\theta _H-\theta _L);\\ \frac{\theta _H+\theta _L}{2}-\omega _A, &{} \hbox {if}\, c> \frac{2l-1}{2}(\theta _H-\theta _L) \end{array} \right. \end{aligned}$$

Then, the supplier’s objective function is given by

$$\begin{aligned} \pi ^S(\omega _A)=\left\{ \begin{array}{l@{\quad }l} \frac{2\zeta }{2\zeta -1}(\omega _A-c)\left( \frac{2\zeta -1}{2\zeta }\bar{y}-\omega _A\right) , &{} \hbox {if}\, \omega _A\le (2l-1)\frac{\theta _H-\theta _L}{2};\\ (\omega _A-c)\left( \frac{\theta _H+\theta _L}{2}-\omega _A\right) , &{} \hbox {if}\, \omega _A> (2l-1)\frac{\theta _H-\theta _L}{2} \end{array} \right. \end{aligned}$$

Optimizing the above formulation gives out the second part. $\square$

1.11 Proof to Lemma 9

Proof

We have

$$\begin{aligned} \pi ^S(Q(\omega _R), \omega _R)= \left\{ \begin{array}{l@{\quad }l} -c(\theta _H-\omega _R)+\omega _R\left( \frac{\theta _H+\theta _L}{2}-\omega _R\right) , &{} \hbox {if}\, \omega _R\ge 2c;\\ (\omega _R-c)(\theta _L-\omega _R), &{} \hbox {if}\, c\le \omega _R<2c \end{array} \right. \end{aligned}$$

Then, we have

$$\begin{aligned} \omega _R=\left\{ \begin{array}{l@{\quad }l} \frac{1}{2}\left( \frac{\theta _H+\theta _L}{2}+c\right) , &{} \hbox {if}\, c\le \frac{\theta _H+3\theta _L}{12};\\ \frac{1}{2}(\theta _L+c), &{} \hbox {if}\, c>\frac{\theta _H+3\theta _L}{12} \end{array} \right. \end{aligned}$$

This leads to the result. $\square$

1.12 Proof to Theorem 3

Proof

The proof of the first part is straightforward by noting that both $-c\theta _H+\frac{1}{4}(c+\frac{\theta _H+\theta _L}{2})^2$ and $\frac{1}{4}((\theta _L-c)^+)^2$ are less than $\frac{1}{4}(\frac{\theta _H+\theta _L}{2}-c)^2$. We omit the detail here. For the second part, when $l\in [1-\frac{1}{2}\frac{\theta _L-c}{\theta _H-c},1]$, we can verify that $\pi _R^{OR}$ is convex in $l$. Thus, we only need to compare the supply chain profits for the cases when $l=1$ and $l=1-\frac{1}{2}\frac{\theta _L-c}{\theta _H-c}$. A simple calculation reveals the desired results. $\square$