Information leakage in a cross-border logistics supply chain considering demand uncertainty and signal inference

Abstract

In cross-border logistics, mainline carriers (MCs) confronting demand uncertainty have to share their information with the overseas regional carrier (RC) via information system or electronic data interchange. However, does the RC or the incumbent MC have incentives to disclose the demand information to a new MC? The new MC competes with the incumbent MC but has no demand information, while the incumbent MC has accumulated big data and known demand uncertainty well. In this paper, we study the supply chain parties’ preferences over information leakage in a system consisting of an incumbent MC, a new MC and an overseas RC. Interestingly, we show that, when there is no information leakage, the new MC’s signal inference dramatically restricts the RC’s pricing flexibility. This induces the RC to disclose the incumbent MC’s information. We also show that, information leakage alters the MCs’ competition and their profits by balancing the RC’s service price and the MCs’ order sizes.

Appendix

Proof of Table 2.

In Scenario N, the incumbent MC and RC have the accurate demand signal, but the new MC does not have this information. We use backward induction to solve this problem. Note that ${\mathrm{E}}\left[ {\epsilon_{\mathrm{o}} | {{\Gamma }}} \right] = \frac{{{{\upsigma }}_{\mathrm{o}}^{2} {{\Gamma }}_{1} }}{{{{\upsigma }}_{\mathrm{o}}^{2} + {{\upsigma }}_{1}^{2} }} = \frac{{{{\rm{s}\upsigma }}_{\mathrm{o}}^{2} }}{{1 + {\mathrm{s}}}}$ and ${\mathrm{s}} = \frac{{{{\upsigma }}_{\mathrm{o}}^{2} }}{{{{\upsigma }}_{1}^{2} }}$. We have the supply chain parties’ profit functions as follows, respectively.

$$ {\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{R}} \left| {{\Gamma }} \right.} \right] = {\mathrm{E}}\left[ {{\mathrm{w}}\left( {{\mathrm{q}}_{\mathrm{E}} + {\mathrm{q}}_{\mathrm{I}} } \right)\left| {{\Gamma }} \right.} \right]. $$

$$ {\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{E}} \left| {\mathrm{w}} \right.} \right] = {\mathrm{E}}\left[ {\left( {{\mathrm{p}}_{\mathrm{E}} - {\mathrm{w}}} \right){\mathrm{q}}_{\mathrm{E}} \left| {\mathrm{w}} \right.} \right]. $$

$$ {\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{I}} \left| {{\Gamma }} \right.} \right] = {\mathrm{E}}\left[ {\left( {{\mathrm{p}}_{\mathrm{I}} - {\mathrm{w}}} \right){\mathrm{q}}_{\mathrm{I}} \left| {{\Gamma }} \right.} \right]. $$

The new MC and the incumbent MC decide their order sizes simultaneously after observing the RC’s service price.

Remember that the new MC can infer the updated demand signal based on the service price decided by RC. The new MC conjectures the updated demand signal as a linear function of the service price: ${\mathrm{E}}\left[ {\epsilon_{\mathrm{o}} | {{\Gamma }}} \right] = {{\upalpha }}_{0} + {{\upalpha }}_{1} {\mathrm{w}}$. Let ${\mathrm{E}}\left[ {\epsilon_{\mathrm{o}} | {{\Gamma }}} \right] = {\mathrm{E}}\left[ {\epsilon_{\mathrm{o}} | {\mathrm{w}}} \right]$. Before we substitute this equation to the new MC’s profit function, we explain why the updated demand signal can be restricted to a linear function of ${\mathrm{E}}\left[ {\epsilon_{\mathrm{o}} | {{\Gamma }}} \right]$, which has been widely assumed in previous literature such as Gal-Or et al. (2008), Li and Zhang (2008) and Niu et al. (2019).

Note that ${\mathrm{E}}\left[ {\epsilon_{\mathrm{o}} | {{\Gamma }}} \right] = {{\upalpha }}_{0} + {{\upalpha }}_{1} {\mathrm{w}}$, where ${{\upalpha }}_{1} > 0$. Given the RC’s service price, both MCs’ optimal order sizes are:

$$ {\mathrm{q}}_{\mathrm{E}} \left( {\mathrm{w}} \right) = {\mathrm{A}}_{2}^{\mathrm{E}} {\mathrm{E}}\left[ {\epsilon_{\mathrm{o}} | {{\Gamma }}} \right] + {\mathrm{A}}_{1}^{\mathrm{E}} {\mathrm{w}} + {\mathrm{A}}_{0}^{\mathrm{E}} ,\;{\mathrm{q}}_{\mathrm{I}} \left( {\mathrm{w}} \right) = {\mathrm{A}}_{2}^{\mathrm{I}} {\mathrm{E}}\left[ {\epsilon_{\mathrm{o}} | {{\Gamma }}} \right] + {\mathrm{A}}_{1}^{\mathrm{I}} {\mathrm{w}} + {\mathrm{A}}_{0}^{\mathrm{I}} , $$

where ${\mathrm{A}}_{\mathrm{i}}^{\mathrm{k}}$ is associated with ${{\upmu }}_{\mathrm{o}} \;{\mathrm{and}}\;{{\uplambda }}$, ${\mathrm{k}} \in \left\{ {{\mathrm{E}},{\mathrm{I}}} \right\},\;{\mathrm{i}} \in \left\{ {0,1,2} \right\}$. Substituting ${\mathrm{q}}_{\mathrm{E}} \left( {\mathrm{w}} \right)\;{\mathrm{and}}\;{\mathrm{q}}_{\mathrm{I}} \left( {\mathrm{w}} \right)$ into the RC’s profit function ${\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{R}} | {{\Gamma }}} \right] = {\mathrm{E}}[{\mathrm{w}}\left( {{\mathrm{q}}_{\mathrm{E}} + {\mathrm{q}}_{\mathrm{I}} } \right)|{{\Gamma }}]$, we have

$$ {\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{R}} | {{\Gamma }}} \right]\left( {\mathrm{w}} \right) = \left[ {\left( {{\mathrm{A}}_{2}^{\mathrm{E}} + {\mathrm{A}}_{2}^{\mathrm{I}} } \right){\mathrm{E}}\left[ {\epsilon_{\mathrm{o}} | {{\Gamma }}} \right] + \left( {{\mathrm{A}}_{1}^{\mathrm{E}} + {\mathrm{A}}_{1}^{\mathrm{I}} } \right){\mathrm{w}} + {\mathrm{A}}_{0}^{\mathrm{E}} + {\mathrm{A}}_{0}^{\mathrm{I}} } \right]{\mathrm{w}} . $$

Therefore, the manufacturer cannot use nonlinear functions to decide the optimal service price, and the results will be the same as those without demand signal inference. However, the new MC can infer the demand signal as a linear function of the service price: ${\mathrm{E}}\left[ {\epsilon_{\mathrm{o}} | {{\Gamma }}} \right] = {{\upalpha }}_{0} + {{\upalpha }}_{1} {\mathrm{w}}$. Such a function accounts for the fact that the service price signals the RC’s demand forecast: The RC tends to decide a high (low) service price to signal to the new MC that the demand is high (low).

We substitute this into the new MC’s profit function, and have the optimal order sizes as follows, given the RC’s service price ${\mathrm{w}}$.

$$ {\mathrm{q}}_{\mathrm{E}} \left( {\mathrm{w}} \right) = \frac{1}{3}\left( { - {\mathrm{w}} + \left( {{{\upalpha }}_{0} + {\mathrm{w\upalpha }}_{1} } \right)\left( { - 1 + 2{{\uplambda }}} \right) + {{\upmu }}_{\mathrm{o}} } \right). $$

$$ {\mathrm{q}}_{\mathrm{I}} \left( {\mathrm{w}} \right) = \frac{1}{6}\left( {{{\upalpha }}_{0} + 3{\mathrm{E}}\left[ {\epsilon_{\mathrm{o}} | {{\Gamma }}} \right] - 2{{\upalpha }}_{0} {{\uplambda }} + {\mathrm{w}}\left( { - 2 + {{\upalpha }}_{1} - 2{{\upalpha }}_{1} {{\uplambda }}} \right) + 2{{\upmu }}_{\mathrm{o}} } \right). $$

Substituting ${\mathrm{q}}_{\mathrm{E}} \;{\mathrm{and}}\;{\mathrm{q}}_{\mathrm{I}}$ into the RC’s profit function, we have

$$ {\mathrm{w}} = \frac{{{{\upalpha }}_{0} - 2{{\upalpha }}_{0} {{\uplambda }} - 4{{\upmu }}_{\mathrm{o}} }}{{ - 8 + {{\upalpha }}_{1} \left( { - 2 + 4{{\uplambda }}} \right)}} - \frac{{3{\mathrm{E}}\left[ {\epsilon_{\mathrm{o}} | {{\Gamma }}} \right]}}{{ - 8 + {{\upalpha }}_{1} \left( { - 2 + 4{{\uplambda }}} \right)}}. $$

The service price ${\mathrm{w}} = \frac{{{{\upalpha }}_{0} - 2{{\upalpha }}_{0} {{\uplambda }} - 4{{\upmu }}_{\mathrm{o}} }}{{ - 8 + {{\upalpha }}_{1} \left( { - 2 + 4{{\uplambda }}} \right)}} - \frac{{3{\mathrm{E}}\left[ {\epsilon_{\mathrm{o}} | {{\Gamma }}} \right]}}{{ - 8 + {{\upalpha }}_{1} \left( { - 2 + 4{{\uplambda }}} \right)}}$ is an equilibrium if and only if the new MC’s conjecture is fulfilled. That is

$$ \frac{{{{\upalpha }}_{0} - 2{{\upalpha }}_{0} {{\uplambda }} - 4{{\upmu }}_{\mathrm{o}} }}{{ - 8 + {{\upalpha }}_{1} \left( { - 2 + 4{{\uplambda }}} \right)}} = - \frac{{{{\upalpha }}_{0} }}{{{{\upalpha }}_{1} }}.\; - \frac{{3{\mathrm{E}}\left[ {\epsilon_{\mathrm{o}} | {{\Gamma }}} \right]}}{{ - 8 + {{\upalpha }}_{1} \left( { - 2 + 4{{\uplambda }}} \right)}} = \frac{{{\mathrm{E}}\left[ {\epsilon_{\mathrm{o}} | {{\Gamma }}} \right]}}{{{{\upalpha }}_{1} }}. $$

The service price ${\mathrm{w}} = - \frac{{{{\upalpha }}_{0}^{ *} }}{{{{\upalpha }}_{1}^{ *} }} + \frac{{{\mathrm{E}}\left[ {\epsilon_{\mathrm{o}} | {{\Gamma }}} \right]}}{{{{\upalpha }}_{1}^{ *} }}$ is an equilibrium in the following sense. As long as the new MC believes the updated demand signal will be the form ${\mathrm{E}}\left[ {\epsilon_{\mathrm{o}} | {{\Gamma }}} \right] = {{\upalpha }}_{0} + {{\upalpha }}_{1} {\mathrm{w}}$, the new MC uses ${{\upalpha }}_{0} + {{\upalpha }}_{1} {\mathrm{w}}$ to substitute ${\mathrm{E}}\left[ {\epsilon_{\mathrm{o}} | {{\Gamma }}} \right]$ for profit maximization. It is optimal for the RC to set the service price to ${\mathrm{w}} = - \frac{{{{\upalpha }}_{0}^{ *} }}{{{{\upalpha }}_{1}^{ *} }} + \frac{{{\mathrm{E}}\left[ {\epsilon_{\mathrm{o}} | {{\Gamma }}} \right]}}{{{{\upalpha }}_{1}^{ *} }}$. Then we have

$$ {{\upalpha }}_{0} = - \frac{{2{{\upmu }}_{\mathrm{o}} }}{{1 + {{\uplambda }}}}.\;{{\upalpha }}_{1} = \frac{8}{{4{{\uplambda }} + 1}}. $$

$$ {\mathrm{w}}^{\mathrm{N}} = \frac{{\left( {1 + 4{{\uplambda }}} \right){{\upmu }}_{\mathrm{O}} }}{{4\left( {1 + {{\uplambda }}} \right)}} + \frac{{\left( {1 + 4{{\uplambda }}} \right){\mathrm{s\Gamma }}}}{{8\left( {1 + {\mathrm{s}}} \right)}}. $$

$$ {\mathrm{q}}_{\mathrm{E}}^{\mathrm{N}} = \frac{{{{\upmu }}_{\mathrm{O}} }}{{4\left( {1 + {{\uplambda }}} \right)}} + \frac{{\left( {4{{\uplambda }} - 3} \right){\mathrm{s\Gamma }}}}{{8\left( {1 + {\mathrm{s}}} \right)}}. $$

$$ {\mathrm{q}}_{\mathrm{I}}^{\mathrm{N}} = \frac{{{{\upmu }}_{\mathrm{O}} }}{{4\left( {1 + {{\uplambda }}} \right)}} + \frac{{\left( {5 - 4{{\uplambda }}} \right){\mathrm{s\Gamma }}}}{{8\left( {1 + {\mathrm{s}}} \right)}}. $$

Then we derive the Bayesian equilibrium profits: ${\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{E}}^{\mathrm{N}} | {{\Gamma }}} \right],\;{\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{R}}^{\mathrm{N}} | {{\Gamma }}} \right]$ and ${\mathrm{E}}[{{\uppi }}_{\mathrm{I}}^{\mathrm{N}} |{{\Gamma }}]$.

Because $\epsilon_{\mathrm{o}}$ and $\epsilon_{1}$ are mutually independent of each other, we have ${\mathrm{g}}\left( {\epsilon_{\mathrm{o}} ,\epsilon_{1} } \right) = {\mathrm{g}}\left( {\epsilon_{\mathrm{o}} } \right){\mathrm{g}}\left( {\epsilon_{1} } \right)$, where ${\mathrm{g}}\left( {\epsilon_{\mathrm{o}} } \right)$ is the normal probability density function of $\epsilon_{\mathrm{o}}$. ${\mathrm{g}}\left( {\epsilon_{1} } \right)$ is the normal probability density function of $\epsilon_{1}$, and ${\mathrm{g}}\left( {\epsilon_{\mathrm{o}} ,\epsilon_{1} } \right)$ is the bi-variate normal probability density function of $\epsilon_{\mathrm{o}}$ and $\epsilon_{1}$.

Recall ${\mathrm{E}}\left[ {\epsilon_{\mathrm{o}} | {{\Gamma }}} \right] = \frac{{{{\upsigma }}_{\mathrm{o}}^{2} {{\Gamma }}_{1} }}{{{{\upsigma }}_{\mathrm{o}}^{2} + {{\upsigma }}_{1}^{2} }}$. We have

$$ \mathop \smallint \limits_{ - \infty }^{\infty } \mathop \smallint \limits_{ - \infty }^{\infty } \left( {{\mathrm{E}}\left[ {\epsilon_{\mathrm{o}} | {{\Gamma }}} \right]} \right)^{2} {\mathrm{g}}\left( {\epsilon_{\mathrm{o}} ,\epsilon_{1} } \right){\mathrm{d}}\epsilon_{\mathrm{o}} {\mathrm{d}}\epsilon_{1} = \frac{{{{\upsigma }}_{\mathrm{o}}^{4} }}{{{{\upsigma }}_{\mathrm{o}}^{2} + {{\upsigma }}_{1}^{2} }} . $$

$$ \mathop \smallint \limits_{ - \infty }^{\infty } \mathop \smallint \limits_{ - \infty }^{\infty } {\mathrm{E}}\left[ {\epsilon_{\mathrm{o}} | {{\Gamma }}} \right]{\mathrm{g}}\left( {\epsilon_{\mathrm{o}} ,\epsilon_{1} } \right){\mathrm{d}}\epsilon_{\mathrm{o}} {\mathrm{d}}\epsilon_{1} = 0 . $$

Therefore, the supply chain parties’ expected profits are specified as

$$ {\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{E}}^{\mathrm{N}} } \right] = \frac{{{{\upmu }}_{\mathrm{O}}^{2} }}{{16\left( {1 + {{\uplambda }}} \right)^{2} }} + \frac{{\left( {3 - 4{{\uplambda }}} \right)^{2} {{\rm{s}\upsigma }}_{\mathrm{O}}^{2} }}{{64\left( {1 + {\mathrm{s}}} \right)}}. $$

$$ {\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{I}}^{\mathrm{N}} } \right] = \frac{{{{\upmu }}_{\mathrm{O}}^{2} }}{{16\left( {1 + {{\uplambda }}} \right)^{2} }} + \frac{{\left( {5 - 4{{\uplambda }}} \right)^{2} {{{\rm{s}}\upsigma }}_{\mathrm{O}}^{2} }}{{64\left( {1 + {\mathrm{s}}} \right)}}. $$

$$ {\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{R}}^{\mathrm{N}} } \right] = \frac{{\left( {1 + 4{{\uplambda }}} \right){{\upmu }}_{\mathrm{O}}^{2} }}{{8\left( {1 + {{\uplambda }}} \right)^{2} }} + \frac{{\left( {1 + 4{{\uplambda }}} \right){{{\rm{s}}\upsigma }}_{\mathrm{O}}^{2} }}{{32\left( {1 + {\mathrm{s}}} \right)}}. $$

Different from Scenario N, all the supply chain parties have the updated demand signal in Scenario L. Thus, their conditional expected profit functions are as follows, respectively.

$$ E\left[ {\pi_{R} \left| {{{\Gamma }}\left] { = E} \right[w\left( {q_{E} + q_{I} } \right)} \right|{{\Gamma }}} \right]. $$

$$ E\left[ {\pi_{E} \left| {w\left] { = E} \right[\left( {p_{E} - w} \right)q_{E} } \right|{{\Gamma }}} \right]. $$

$$ E\left[ {\pi_{I} \left| {{{\Gamma }}\left] { = E} \right[\left( {p_{I} - w} \right)q_{I} } \right|{{\Gamma }}} \right]. $$

Similarly, we first derive MCs’ optimal order sizes given the RC’s ${\mathrm{w}}$, as follows.

$$ {\mathrm{q}}_{\mathrm{E}} \left( w \right) = \frac{1}{3}\left( { - w - {\mathrm{E}}\left[ {\epsilon_{\mathrm{o}} | {{\Gamma }}} \right]\left( {1 - 2\lambda } \right) + \mu_{o} } \right). $$

$$ {\mathrm{q}}_{\mathrm{I}} \left( w \right) = \frac{1}{3}\left( { - w + \left( {2 - \lambda } \right){\mathrm{E}}\left[ {\epsilon_{\mathrm{o}} | {{\Gamma }}} \right] + \mu_{o} } \right) $$

Substituting $q_{E} \left( w \right)\;and\;{\mathrm{q}}_{\mathrm{I}} \left( w \right)$ into the RC’s expected profit $E[\pi_{R} |{{\Gamma }}]$, we have

$$ w^{L} = \frac{{\mu_{o} }}{2} + \frac{{\left( {1 + \lambda } \right)s{{\Gamma }}}}{{4\left( {1 + s} \right)}}. $$

$$ q_{E}^{L} = \frac{{\mu_{o} }}{6} + \frac{{\left( {7\lambda - 5} \right)s{{\Gamma }}}}{{12\left( {1 + s} \right)}}. $$

$$ q_{I}^{L} = \frac{{\mu_{o} }}{6} + \frac{{\left( {7 - 5\lambda } \right)s{{\Gamma }}}}{{12\left( {1 + s} \right)}}. $$

Similarly, the expected profits of all the supply chain parties are specified as

$$ E\left[ {\pi_{R}^{L} } \right] = \frac{{\mu_{o}^{2} }}{6} + \frac{{\left( {1 + \lambda } \right)^{2} s\sigma_{o}^{2} }}{{24\left( {1 + s} \right)}}. $$

$$ E\left[ {\pi_{E}^{L} } \right] = \frac{{\mu_{o}^{2} }}{36} + \frac{{\left( {5 - 7\lambda } \right)^{2} s\sigma_{o}^{2} }}{{144\left( {1 + s} \right)}}. $$

$$ E\left[ {\pi_{I}^{L} } \right] = \frac{{\mu_{o}^{2} }}{36} + \frac{{\left( {7 - 5\lambda } \right)^{2} s\sigma_{o}^{2} }}{{144\left( {1 + s} \right)}}. $$

Proof of Lemma 2

Comparing the RC’s service prices in two scenarios, we have ${\mathrm{E}}\left[ {{\mathrm{w}}^{\mathrm{N}} \left] { - {\mathrm{E}}} \right[{\mathrm{w}}^{\mathrm{L}} } \right] = \frac{{\left( {2{{\uplambda }} - 1} \right){{\upmu }}_{\mathrm{o}} }}{{4\left( {1 + {{\uplambda }}} \right)}}$. Subjecting to ${{\upmu }}_{\mathrm{o}} \in \left( {0, + \infty } \right)$ and ${{\uplambda }} \in \left( {0, + \infty } \right)$, we know the sign of ${\mathrm{E}}\left[ {{\mathrm{w}}^{\mathrm{N}} \left] -{{\mathrm{E}}} \right[{\mathrm{w}}^{\mathrm{L}} } \right]$ depends on $\left( {2{{\uplambda }} - 1} \right)$. Therefore, we have ${\mathrm{E}}\left[ {{\mathrm{w}}^{\mathrm{N}} } \right] > E\left[ {{\mathrm{w}}^{\mathrm{L}} } \right] {\quad\mathrm{if}}\quad{{ \lambda }} > {{\uplambda }}_{1}$, where ${{\uplambda }}_{1} = 1$. Otherwise, we have ${\mathrm{E}}\left[ {{\mathrm{w}}^{\mathrm{N}} } \right] \le {\mathrm{E}}\left[ {{\mathrm{w}}^{\mathrm{L}} } \right]$. By investigating the first derivative of ${\mathrm{w}}^{\mathrm{N}}$ and ${\mathrm{w}}^{\mathrm{L}}$ with respect to ${{\Gamma }}$, we obtain $\frac{{\partial {\mathrm{w}}^{\mathrm{N}} }}{{\partial {{\Gamma }}}} = \frac{{{\mathrm{s}}\left( {1 + 4{{\uplambda }}} \right)}}{{8\left( {1 + {\mathrm{s}}} \right)}} > 0,\;and\;\frac{{\partial {\mathrm{w}}^{\mathrm{L}} }}{{\partial {{\Gamma }}}} = \frac{{{\mathrm{s}}\left( {1 + {{\uplambda }}} \right)}}{{4\left( {1 + {\mathrm{s}}} \right)}} > 0$ for any feasible ${\mathrm{s}} > 0\;and\;\lambda > 0$. Therefore, Lemma 2 becomes immediate.

Proof of Lemma 3

Comparing MC’s order sizes in two scenarios, we have ${\mathrm{E}}\left[ {{\mathrm{q}}_{\mathrm{E}}^{\mathrm{N}} } \right] - {\mathrm{E}}\left[ {{\mathrm{q}}_{\mathrm{E}}^{\mathrm{L}} } \right] = \frac{{\left( {1 - 2{{\uplambda }}} \right){{\upmu }}_{\mathrm{o}} }}{{12\left( {1 + {{\uplambda }}} \right)}}$, and ${\mathrm{E}}\left[ {{\mathrm{q}}_{\mathrm{I}}^{\mathrm{N}} } \right] - {\mathrm{E}}\left[ {{\mathrm{q}}_{\mathrm{I}}^{\mathrm{L}} } \right] = \frac{{\left( {1 - 2{{\uplambda }}} \right){{\upmu }}_{\mathrm{o}} }}{{12\left( {1 + {{\uplambda }}} \right)}}$. Similar to the proof of Lemma 2, for any feasible ${{\upmu }}_{\mathrm{O}} > 0\;and\;\lambda > 0$, we obtain ${\mathrm{E}}\left[ {{\mathrm{q}}_{\mathrm{E}}^{\mathrm{N}} } \right] > E\left[ {{\mathrm{q}}_{\mathrm{E}}^{\mathrm{L}} } \right]\quad{\mathrm{if}}\quad{{\uplambda }} < {{\uplambda }}_{1}$. Otherwise, we have ${\mathrm{E}}\left[ {{\mathrm{q}}_{\mathrm{E}}^{\mathrm{N}} } \right] \le {\mathrm{E}}\left[ {{\mathrm{q}}_{\mathrm{E}}^{\mathrm{L}} } \right]$. Similarly, we can prove ${\mathrm{E}}\left[ {{\mathrm{q}}_{\mathrm{I}}^{\mathrm{N}} } \right] > E\left[ {{\mathrm{q}}_{\mathrm{I}}^{\mathrm{L}} } \right]\;{\mathrm{if}}\;{{\uplambda }} < {{\uplambda }}_{1}$. Otherwise, we have ${\mathrm{E}}\left[ {{\mathrm{q}}_{\mathrm{I}}^{\mathrm{N}} } \right] \le {\mathrm{E}}\left[ {{\mathrm{q}}_{\mathrm{I}}^{\mathrm{L}} } \right]$.

Proof of Proposition 1

• (a) Comparing the new MC’s deterministic values in two scenarios, we obtain ${\mathrm{DV}}_{\mathrm{E}}^{\mathrm{N}} - {\mathrm{DV}}_{\mathrm{E}}^{\mathrm{L}} = \frac{1}{144}\left( {\frac{9}{{\left( {1 + {{\uplambda }}} \right)^{2} }} - 4} \right){{\upmu }}_{\mathrm{o}}^{2}$. It is easy to show that, for any feasible ${{\upmu }}_{\mathrm{o}} > 0\;and\;\lambda > 0$, the sign of ${\mathrm{DV}}_{\mathrm{E}}^{\mathrm{N}} - {\mathrm{DV}}_{\mathrm{E}}^{\mathrm{L}}$ depends on the sign of $\frac{9}{{\left( {1 + {{\uplambda }}} \right)^{2} }} - 4$. Therefore, we have ${\mathrm{DV}}_{\mathrm{E}}^{\mathrm{N}} > D{\mathrm{V}}_{\mathrm{E}}^{\mathrm{L}}$ if ${{\uplambda }} < {{\uplambda }}_{1}$. Otherwise, we have ${\mathrm{DV}}_{\mathrm{E}}^{\mathrm{N}} \le {\mathrm{DV}}_{\mathrm{E}}^{\mathrm{L}}$.

• (b) Similarly, we show ${\mathrm{DV}}_{\mathrm{I}}^{\mathrm{N}} - {\mathrm{DV}}_{\mathrm{I}}^{\mathrm{L}} = \frac{1}{144}\left( {\frac{9}{{\left( {1 + {{\uplambda }}} \right)^{2} }} - 4} \right){{\upmu }}_{\mathrm{o}}^{2}$, and we have ${\mathrm{DV}}_{\mathrm{I}}^{\mathrm{N}} > D{\mathrm{V}}_{\mathrm{I}}^{\mathrm{L}} {\mathrm{if \lambda }} < {{\uplambda }}_{1}$. Otherwise, we have ${\mathrm{DV}}_{\mathrm{I}}^{\mathrm{N}} \le {\mathrm{DV}}_{\mathrm{I}}^{\mathrm{L}}$.

• (c) Comparing the RC’s deterministic values in two scenarios, we have ${\mathrm{DV}}_{\mathrm{R}}^{\mathrm{N}} - {\mathrm{DV}}_{\mathrm{R}}^{\mathrm{L}} = - \frac{{\left( {1 - 2{{\uplambda }}} \right)^{2} {{\upmu }}_{\mathrm{o}}^{2} }}{{24\left( {1 + {{\uplambda }}} \right)^{2} }} < 0$ for any feasible ${{\upmu }}_{\mathrm{o}} \;{\mathrm{and}}\;{{\uplambda }}$. Then Proposition 1 is proven.

Proofs of Proposition 2 and Proposition 3

Comparing the new MC’ information values in two scenarios, we have ${\mathrm{IV}}_{\mathrm{E}}^{\mathrm{N}} - {\mathrm{IV}}_{\mathrm{E}}^{\mathrm{L}} = - \frac{{{\mathrm{s}}\left( {19 - 64{{\uplambda }} + 52{{\uplambda }}^{2} } \right){{\upsigma }}_{\mathrm{o}}^{2} }}{{576\left( {1 + {\mathrm{s}}} \right)}}$. It can be shown that,${\mathrm{IV}}_{\mathrm{E}}^{\mathrm{N}} > {\mathrm{IV}}_{\mathrm{E}}^{\mathrm{L}}$ depends on a quadratic function of ${{\uplambda }}$. Solving the function with respect to ${{\uplambda }}$, we obtain ${\mathrm{IV}}_{\mathrm{E}}^{\mathrm{N}} > {\mathrm{IV}}_{\mathrm{E}}^{\mathrm{L}} \;{\mathrm{if}}\;{{\uplambda }} \in \left( {{{\uplambda }}_{1} ,{{\uplambda }}_{2} } \right))$. Otherwise, we have ${\mathrm{IV}}_{\mathrm{E}}^{\mathrm{N}} \le {\mathrm{IV}}_{\mathrm{E}}^{\mathrm{L}}$. Note that, ${{\uplambda }}_{2} = \frac{19}{26}$.

Similarly, the comparison result of the incumbent MC’s information values is ${\mathrm{IV}}_{\mathrm{I}}^{\mathrm{N}} - {\mathrm{IV}}_{\mathrm{I}}^{\mathrm{L}} = \frac{{{\mathrm{s}}\left( {29 - 80{{\uplambda }} + 44{{\uplambda }}^{2} } \right){{\upsigma }}_{\mathrm{o}}^{2} }}{{576\left( {1 + {\mathrm{s}}} \right)}}$. We have ${\mathrm{IV}}_{\mathrm{I}}^{\mathrm{N}} > I{\mathrm{V}}_{\mathrm{I}}^{\mathrm{L}}$ ${\mathrm{if}}\;{{\uplambda }} \in \left( {0,{{\uplambda }}_{1} } \right) \cup \left( {{{\uplambda }}_{3} , + \infty } \right)$. Otherwise, we have ${\mathrm{IV}}_{\mathrm{I}}^{\mathrm{N}} \le {\mathrm{IV}}_{\mathrm{I}}^{\mathrm{L}}$. Note that, ${{\uplambda }}_{3} = \frac{29}{22}$.

Therefore, the results in Proposition 2 and Proposition 3 are obtained.

Proof of Proposition 4

Comparing the RC’s information values in two scenarios, we have ${\mathrm{IV}}_{\mathrm{R}}^{\mathrm{N}} - {\mathrm{IV}}_{\mathrm{R}}^{\mathrm{L}} = - \frac{{{\mathrm{s}}\left( {1 - 2{{\uplambda }}} \right)^{2} {{\upsigma }}_{\mathrm{o}}^{2} }}{{96\left( {1 + {\mathrm{s}}} \right)}} < 0$ for any feasible ${\mathrm{s}},\;{{\uplambda }}\;{\mathrm{and}}\;{{\upsigma }}_{\mathrm{o}}^{2}$. Thus, the results in Proposition 4 are obtained.

Proof of Proposition 5

It is easy to verify that ${\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{E}}^{\mathrm{N}} } \right] - {\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{E}}^{\mathrm{L}} } \right] = \frac{{\left( {1 - 2{{\uplambda }}} \right)\left[ {4\left( {1 + {\mathrm{s}}} \right)\left( {5 + 2{{\uplambda }}} \right){{\upmu }}_{\mathrm{o}}^{2} + {\mathrm{s}}\left( {1 + {{\uplambda }}} \right)^{2} \left( {26{{\uplambda }} - 19} \right){{\upsigma }}_{\mathrm{o}}^{2} } \right]}}{{576\left( {1 + {\mathrm{s}}} \right)\left( {1 + {{\uplambda }}} \right)^{2} }}$. Clearly, we have $576\left( {1 + {\mathrm{s}}} \right)\left( {1 + {{\uplambda }}} \right)^{2} > 0$. Hence, the symbol of ${\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{E}}^{\mathrm{N}} } \right] - {\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{E}}^{\mathrm{L}} } \right]$ depends on $\left( {1 - 2{{\uplambda }}} \right)[4\left( {1 + {\mathrm{s}}} \right)\left( {5 + 2{{\uplambda }}} \right){{\upmu }}_{\mathrm{o}}^{2} + {\mathrm{s}}\left( {1 + {{\uplambda }}} \right)^{2} \left( {26{{\uplambda }} - 19} \right){{\upsigma }}_{\mathrm{o}}^{2}$. When $\left( {1 - 2{{\uplambda }}} \right) > 0$, i.e., ${{\uplambda }} < {{\uplambda }}_{1}$, we have $\left( {26{{\uplambda }} - 19} \right) < 0$. Therefore, given ${\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{E}}^{\mathrm{N}} } \right] > E\left[ {{{\uppi }}_{\mathrm{E}}^{\mathrm{L}} } \right],$ the inequation ${\mathrm{s}}\left( {1 + {{\uplambda }}} \right)^{2} \left( {19 - 26{{\uplambda }}} \right){{\upsigma }}_{\mathrm{o}}^{2} < 4\left( {1 + {\mathrm{s}}} \right)\left( {5 + 2{{\uplambda }}} \right){{\upmu }}_{\mathrm{o}}^{2}$ needs to be satisfied. Rewriting it as $\frac{{{{\upsigma }}_{\mathrm{o}}^{2} }}{{{{\upmu }}_{\mathrm{o}}^{2} }} < \frac{{4\left( {1 + {\mathrm{s}}} \right)\left( {5 + 2{{\uplambda }}} \right)}}{{{\mathrm{s}}\left( {1 + {{\uplambda }}} \right)^{2} \left( {19 - 26{{\uplambda }}} \right)}},$ where $\frac{{4\left( {1 + {\mathrm{s}}} \right)\left( {5 + 2{{\uplambda }}} \right)}}{{{\mathrm{s}}\left( {1 + {{\uplambda }}} \right)^{2} \left( {19 - 26{{\uplambda }}} \right)}} > 0$. Hence, we have $\frac{{{{\upsigma }}_{\mathrm{o}} }}{{{{\upmu }}_{\mathrm{o}} }} < {{\uptheta }}_{1}$. Note that ${{\uptheta }}_{1} = 2\sqrt {\frac{{\left( {1 + {\mathrm{s}}} \right)\left( {5 + 2{{\uplambda }}} \right)}}{{{\mathrm{s}}\left( {1 + {{\uplambda }}} \right)^{2} \left( {19 - 26{{\uplambda }}} \right)}}}$.

Similarly, when $\left( {1 - 2{{\uplambda }}} \right) < 0$, i.e., ${{\uplambda }} > {{\uplambda }}_{1}$, we show that $\left[ {{{\uppi }}_{\mathrm{E}}^{\mathrm{N}} } \right] > E\left[ {{{\uppi }}_{\mathrm{E}}^{\mathrm{L}} } \right]$ requires ${\mathrm{s}}\left( {1 + {{\uplambda }}} \right)^{2} \left( {19 - 26{{\uplambda }}} \right){{\upsigma }}_{\mathrm{o}}^{2} > 4\left( {1 + {\mathrm{s}}} \right)\left( {5 + 2{{\uplambda }}} \right){{\upmu }}_{\mathrm{o}}^{2}$. Further, when $\left( {19 - 26{{\uplambda }}} \right) > 0$, i.e., ${{\uplambda }} < {{\uplambda }}_{2} , {{\uplambda }}_{2} = \frac{19}{26}$ we obtain $\frac{{{{\upsigma }}_{\mathrm{o}} }}{{{{\upmu }}_{\mathrm{o}} }} > {{\uptheta }}_{1}$; While $\left( {19 - 26{{\uplambda }}} \right) < 0$, we obtain $\left( {1 - 2{{\uplambda }}} \right)[4\left( {1 + {\mathrm{s}}} \right)\left( {5 + 2{{\uplambda }}} \right){{\upmu }}_{\mathrm{o}}^{2} + {\mathrm{s}}\left( {1 + {{\uplambda }}} \right)^{2} \left( {26{{\uplambda }} - 19} \right){{\upsigma }}_{\mathrm{o}}^{2} < 0$, i.e., ${\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{E}}^{\mathrm{N}} } \right] < E\left[ {{{\uppi }}_{\mathrm{E}}^{\mathrm{L}} } \right]$.

Therefore, we have ${\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{E}}^{\mathrm{N}} } \right] > E\left[ {{{\uppi }}_{\mathrm{E}}^{\mathrm{L}} } \right]$ if one of the following conditions occurs,

• (a) ${{\uplambda }} < {{\uplambda }}_{1}$ and $\frac{{{{\upsigma }}_{\mathrm{o}} }}{{{{\upmu }}_{\mathrm{o}} }} < {{\uptheta }}_{1}$;

• (b) ${{\uplambda }}_{1} < \lambda < {{\uplambda }}_{2}$ and $\frac{{{{\upsigma }}_{\mathrm{o}} }}{{{{\upmu }}_{\mathrm{o}} }} > {{\uptheta }}_{1}$. Note that, ${{\uptheta }}_{1} = 2\sqrt {\frac{{\left( {1 + {\mathrm{s}}} \right)\left( {5 + 2{{\uplambda }}} \right)}}{{{\mathrm{s}}\left( {1 + {{\uplambda }}} \right)^{2} \left( {19 - 26{{\uplambda }}} \right)}}}$.

Otherwise, we have ${\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{E}}^{\mathrm{N}} } \right] \le {\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{E}}^{\mathrm{L}} } \right]$. Then, the results in Proposition 5 are obtained.

Proof of Proposition 6

Similarly, when comparing the incumbent MC’s profits, we have ${\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{I}}^{\mathrm{L}} } \right] - {\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{I}}^{\mathrm{N}} } \right] = \frac{{\left( {1 - 2{{\uplambda }}} \right)\left[ { - 4\left( {1 + {\mathrm{s}}} \right)\left( {5 + 2{{\uplambda }}} \right){{\upmu }}_{\mathrm{o}}^{2} + {\mathrm{s}}\left( {1 + {{\uplambda }}} \right)^{2} \left( {22{{\uplambda }} - 29} \right){{\upsigma }}_{0}^{2} } \right]}}{{576\left( {1 + {\mathrm{s}}} \right)\left( {1 + {{\uplambda }}} \right)^{2} }}$. Whether ${\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{I}}^{\mathrm{L}} } \right] > E\left[ {{{\uppi }}_{\mathrm{I}}^{\mathrm{N}} } \right]$ depends on the symbol of $\left( {1 - 2{{\uplambda }}} \right)\left[ { - 4\left( {1 + {\mathrm{s}}} \right)\left( {5 + 2{{\uplambda }}} \right){{\upmu }}_{\mathrm{o}}^{2} + {\mathrm{s}}\left( {1 + {{\uplambda }}} \right)^{2} \left( {22{{\uplambda }} - 29} \right){{\upsigma }}_{\mathrm{o}}^{2} } \right]$. We discuss it as follows: when $\left( {1 - 2{{\uplambda }}} \right) > 0$, i.e., ${{\uplambda }} < {{\uplambda }}_{1}$, we obtain $\left( {1 - 2{{\uplambda }}} \right)\left[ { - 4\left( {1 + {\mathrm{s}}} \right)\left( {5 + 2{{\uplambda }}} \right){{\upmu }}_{\mathrm{o}}^{2} + {\mathrm{s}}\left( {1 + {{\uplambda }}} \right)^{2} \left( {22{{\uplambda }} - 29} \right){{\upsigma }}_{\mathrm{o}}^{2} } \right] < 0$, that is ${\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{I}}^{\mathrm{L}} } \right] < E\left[ {{{\uppi }}_{\mathrm{I}}^{\mathrm{N}} } \right]$. When $\left( {1 - 2{{\uplambda }}} \right) < 0$, i.e., ${{\uplambda }} > {{\uplambda }}_{1}$, we know ${\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{I}}^{\mathrm{L}} } \right] > E\left[ {{{\uppi }}_{\mathrm{I}}^{\mathrm{N}} } \right]$ requires $\left[ { - 4\left( {1 + {\mathrm{s}}} \right)\left( {5 + 2{{\uplambda }}} \right){{\upmu }}_{\mathrm{o}}^{2} + {\mathrm{s}}\left( {1 + {{\uplambda }}} \right)^{2} \left( {22{{\uplambda }} - 29} \right){{\upsigma }}_{\mathrm{o}}^{2} } \right] < 0$. Clearly, when $\left( {22{{\uplambda }} - 29} \right) < 0$, i.e., ${{\uplambda }} < {{\uplambda }}_{3}$, ${{\uplambda }}_{3} = \frac{29}{22}$, we have $\left[ { - 4\left( {1 + {\mathrm{s}}} \right)\left( {5 + 2{{\uplambda }}} \right){{\upmu }}_{\mathrm{o}}^{2} + {\mathrm{s}}\left( {1 + {{\uplambda }}} \right)^{2} \left( {22{{\uplambda }} - 29} \right){{\upsigma }}_{\mathrm{o}}^{2} } \right] < 0$ and ${\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{I}}^{\mathrm{L}} } \right] > E\left[ {{{\uppi }}_{\mathrm{I}}^{\mathrm{N}} } \right]$. When $\left( {22{{\uplambda }} - 29} \right) > 0$, i.e., ${{\uplambda }} > {{\uplambda }}_{3}$, we know ${\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{I}}^{\mathrm{L}} } \right] > E\left[ {{{\uppi }}_{\mathrm{I}}^{\mathrm{N}} } \right]$ requires ${\mathrm{s}}\left( {1 + {{\uplambda }}} \right)^{2} \left( {22{{\uplambda }} - 29} \right){{\upsigma }}_{\mathrm{o}}^{2} < 4\left( {1 + {\mathrm{s}}} \right)\left( {5 + 2{{\uplambda }}} \right){{\upmu }}_{\mathrm{o}}^{2}$, and we obtain $\frac{{{{\upsigma }}_{\mathrm{o}}^{2} }}{{{{\upmu }}_{\mathrm{o}}^{2} }} < \frac{{4\left( {1 + {\mathrm{s}}} \right)\left( {5 + 2{{\uplambda }}} \right)}}{{{\mathrm{s}}\left( {1 + {{\uplambda }}} \right)^{2} \left( {22{{\uplambda }} - 29} \right)}}$, where $\frac{{4\left( {1 + {\mathrm{s}}} \right)\left( {5 + 2{{\uplambda }}} \right)}}{{{\mathrm{s}}\left( {1 + {{\uplambda }}} \right)^{2} \left( {22{{\uplambda }} - 29} \right)}} > 0.$ Hence, we have ${\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{I}}^{\mathrm{L}} } \right] > E\left[ {{{\uppi }}_{\mathrm{I}}^{\mathrm{N}} } \right]$ if ${{\uplambda }} > {{\uplambda }}_{3}$ and $\frac{{{{\upsigma }}_{\mathrm{o}} }}{{{{\upmu }}_{\mathrm{o}} }} < {{\uptheta }}_{2}$. Note that, ${{\uptheta }}_{2} = 2\sqrt {\frac{{\left( {1 + {\mathrm{s}}} \right)\left( {5 + 2{{\uplambda }}} \right)}}{{{\mathrm{s}}\left( {1 + {{\uplambda }}} \right)^{2} \left( {22{{\uplambda }} - 29} \right)}}}$. The results in Proposition 6 are obtained.

Proof of Proposition 7

Comparing the RC’s expected profits in two scenarios, we have ${\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{R}}^{\mathrm{N}} } \right] - {\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{R}}^{\mathrm{L}} } \right] = - \frac{{\left( {1 - 2{{\uplambda }}} \right)^{2} \left[ {4\left( {1 + {\mathrm{s}}} \right){{\upmu }}_{\mathrm{o}}^{2} + {\mathrm{s}}\left( {1 + {{\uplambda }}} \right)^{2} {{\upsigma }}_{\mathrm{o}}^{2} } \right]}}{{96\left( {1 + {\mathrm{s}}} \right)\left( {1 + {{\uplambda }}} \right)^{2} }} < 0$ for any feasible ${\mathrm{s}},{{\uplambda }},{{\upsigma }}_{\mathrm{o}} \;{\mathrm{and}}\;{{\upmu }}_{\mathrm{o}}$.

Therefore, Proposition 7 is proven.

Proof of Extensions

We present the equilibrium outcomes in each extension as follows.

1.1 MCs have their exclusive RC

We obtain the equilibrium outcomes summarized in Table 3.

Table 3 The outcomes and equilibrium profits in each scenario

Proof of Proposition 8

Comparing the new MC and its exclusive profits in two scenarios, we have ${\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{E}}^{{{\mathrm{L}}1}} } \right] - {\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{E}}^{{{\mathrm{N}}1}} } \right] = \frac{{4\left( {2 - 7{{\uplambda }}} \right)^{2} {{{\rm{s}}\upsigma }}_{\mathrm{o}}^{2} }}{{2025\left( {1 + {\mathrm{s}}} \right)}} > 0$, and ${\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{RE}}^{{{\mathrm{L}}1}} } \right] - {\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{RE}}^{{{\mathrm{N}}1}} } \right] = \frac{{2\left( {2 - 7{{\uplambda }}} \right)^{2} {{{\rm{s}}\upsigma }}_{\mathrm{o}}^{2} }}{{675\left( {1 + {\mathrm{s}}} \right)}} > 0$ for any ${\mathrm{s}},{{\uplambda }},{{\upsigma }}_{\mathrm{o}} \;{\mathrm{and}}\;{{\upmu }}_{\mathrm{o}}$.

Proof of Proposition 9

Comparing the incumbent MC’ profits in two scenarios, we have ${\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{I}}^{{{\mathrm{L}}1}} } \right] - {\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{I}}^{{{\mathrm{N}}1}} } \right] = \frac{{\left( {197 - 3584{{\uplambda }} + 512{{\uplambda }}^{2} } \right){{{\rm{s}}\upsigma }}_{\mathrm{o}}^{2} }}{{64800\left( {1 + {\mathrm{s}}} \right)}}$. Clearly, whether ${\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{I}}^{{{\mathrm{L}}1}} } \right] > E\left[ {{{\uppi }}_{\mathrm{I}}^{{{\mathrm{N}}1}} } \right]$ depends on the quadratic function $197 - 3584{{\uplambda }} + 512{{\uplambda }}^{2}$ of ${{\uplambda }}$. Solving the inequality $197 - 3584{{\uplambda }} + 512{{\uplambda }}^{2} > 0$, we obtain $0 < \lambda \left<{\frac{1}{32}\left( {112 - 45\sqrt 6 } \right) {\mathrm{and \lambda }}} \right> \frac{1}{32}(112 + 45\sqrt 6$). Similarly, comparing the its exclusive RC’s profits, we have ${\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{RI}}^{{{\mathrm{L}}1}} } \right] - {\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{RI}}^{{{\mathrm{N}}1}} } \right] = \frac{{\left( {1111 - 1792{{\uplambda }} + 256{{\uplambda }}^{2} } \right){{{\rm{s}}\upsigma }}_{\mathrm{o}}^{2} }}{{21600\left( {1 + {\mathrm{s}}} \right)}}$. Solving the inequality $1111 - 1792{{\uplambda }} + 256{{\uplambda }}^{2} > 0$, we have $0 < \lambda < \frac{11}{16}\;{\mathrm{or}}\;\lambda > \frac{101}{16}$. Therefore, Proposition 9 is proven.

1.2 Impact of logistics disruption

We obtain the equilibrium outcomes in Table 4.

Table 4 The outcomes and equilibrium profits in each scenario

Proof of Proposition 10

Comparing the new MC’s profits in two scenarios, we obtain ${\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{E}}^{{{\mathrm{N}}2}} } \right] - {\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{E}}^{{{\mathrm{L}}2}} } \right] = \frac{{\left( {1 - {{\uppsi }}} \right)^{2} \left( {1 - 2{{\uplambda }}} \right)\left[ {4{{\upmu }}_{\mathrm{o}}^{2} \left( {5 - 2{{\uplambda }}\left( { - 1 + {{\uppsi }}} \right) + 3{{\uppsi }}} \right) + \frac{{{{{\rm{s}}\upsigma }}_{\mathrm{o}}^{2} }}{{\left( {1 + {\mathrm{s}}} \right)}}\left( {1 + {{\uplambda }} + {{\uppsi }} - {{\uplambda \uppsi }}} \right)^{2} \left( { - 19 - 5{{\uppsi }} + {{\uplambda }}\left( {26 + 6{{\uppsi }}} \right)} \right)} \right]}}{{64\left( {3 + {{\uppsi }}} \right)^{2} \left( { - 1 - {{\uplambda }} - {{\uppsi }} + {{\uplambda \uppsi }}} \right)^{2} }}$. Clearly, the sign of ${\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{E}}^{{{\mathrm{N}}2}} } \right] - {\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{E}}^{{{\mathrm{L}}2}} } \right]$ depends on $\left( {1 - 2{{\uplambda }}} \right)\left[ {4{{\upmu }}_{\mathrm{o}}^{2} \left( {5 - 2{{\uplambda }}\left( { - 1 + {{\uppsi }}} \right) + 3{{\uppsi }}} \right) + \frac{{{{{\rm{s}}\upsigma }}_{\mathrm{o}}^{2} }}{{\left( {1 + {\mathrm{s}}} \right)}}\left( {1 + {{\uplambda }} + {{\uppsi }} - {{\uplambda \uppsi }}} \right)^{2} \left( { - 19 - 5{{\uppsi }} + {{\uplambda }}\left( {26 + 6{{\uppsi }}} \right)} \right)} \right]$ for any feasible ${\mathrm{s}},{{\uplambda }},{{\upmu }}_{\mathrm{o}} \;{\mathrm{and}}\;{{\upsigma }}_{\mathrm{o}}^{2} .$ Note that $\left( {5 - 2{{\uplambda }}\left( { - 1 + {{\uppsi }}} \right) + 3{{\uppsi }}} \right) > 0$ is always larger than zero for any ${{\uplambda }}$ and ${{\uppsi }}$. Similar to Proposition 5, when $\left( {1 - 2{{\uplambda }}} \right) > 0$, i.e., ${{\uplambda }} < {{\uplambda }}_{1}$, we obtain $- 19 - 5{{\uppsi }} + {{\uplambda }}\left( {26 + 6{{\uppsi }}} \right) < 0.$ Therefore, that ${\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{E}}^{{{\mathrm{N}}2}} } \right] > E\left[ {{{\uppi }}_{\mathrm{E}}^{{{\mathrm{L}}2}} } \right]$ requires $\frac{{{{\upsigma }}_{\mathrm{o}}^{2} }}{{{{\upmu }}_{\mathrm{o}}^{2} }} < \frac{{ - 4\left( {5 - 2{{\uplambda }}\left( { - 1 + {{\uppsi }}} \right) + 3{{\uppsi }}} \right)\left( {1 + {\mathrm{s}}} \right)}}{{{\mathrm{s}}\left( {1 + {{\uplambda }} + {{\uppsi }} - {{\uplambda \uppsi }}} \right)^{2} \left[ { - 19 - 5{{\uppsi }} + {{\uplambda }}\left( {26 + 6{{\uppsi }}} \right)} \right]}}$, where $\frac{{ - 4\left( {5 - 2{{\uplambda }}\left( { - 1 + {{\uppsi }}} \right) + 3{{\uppsi }}} \right)\left( {1 + {\mathrm{s}}} \right)}}{{{\mathrm{s}}\left( {1 + {{\uplambda }} + {{\uppsi }} - {{\uplambda \uppsi }}} \right)^{2} \left[ { - 19 - 5{{\uppsi }} + {{\uplambda }}\left( {26 + 6{{\uppsi }}} \right)} \right]}} > 0$. Hence, we have $\frac{{{{\upsigma }}_{\mathrm{o}} }}{{{{\upmu }}_{\mathrm{o}} }} < {{\uptheta }}_{3}$. Note that ${{\uptheta }}_{3} = \frac{2}{{\left( {1 + {{\uplambda }} + {{\uppsi }} - {{\uplambda \uppsi }}} \right)}}\sqrt {\frac{{\left[ { - 5 + 2{{\uplambda }}\left( { - 1 + {{\uppsi }}} \right) - 3{{\uppsi }}} \right]\left( {1 + {\mathrm{s}}} \right)}}{{{\mathrm{s}}\left[ { - 19 - 5{{\uppsi }} + {{\uplambda }}\left( {26 + 6{{\uppsi }}} \right)} \right]}}}$.

Similarly, when $\left( {1 - 2{{\uplambda }}} \right) < 0$, i.e., ${{\uplambda }} > {{\uplambda }}_{1}$, $\left[ {{{\uppi }}_{\mathrm{E}}^{{{\mathrm{N}}2}} } \right] > E\left[ {{{\uppi }}_{\mathrm{E}}^{{{\mathrm{L}}2}} } \right]$ requires $\frac{{{{{\rm{s}}\upsigma }}_{\mathrm{o}}^{2} }}{{\left( {1 + {\mathrm{s}}} \right)}}\left( {1 + {{\uplambda }} + {{\uppsi }} - {{\uplambda \uppsi }}} \right)^{2} \left( { - 19 - 5{{\uppsi }} + {{\uplambda }}\left( {26 + 6{{\uppsi }}} \right)} \right) < 4{{\upmu }}_{\mathrm{o}}^{2} \left( {5 - 2{{\uplambda }}\left( { - 1 + {{\uppsi }}} \right) + 3{{\uppsi }}} \right)$. Subjecting it into ${{\uplambda }} > {{\uplambda }}_{1} \;{\mathrm{and}}\;\left( { - 19 - 5{{\uppsi }} + {{\uplambda }}\left( {26 + 6{{\uppsi }}} \right)} \right) < 0$, we have ${{\uplambda }}_{1} < \lambda \left< {{{\uplambda }}_{4} ,\;{\mathrm{and}}\;{{\uplambda }}_{4} = \frac{{19 + {{\uppsi }}}}{{26 + {{\uppsi }}}}} \right> {{\uplambda }}_{2}$. Therefore, we obtain the condition of ${\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{E}}^{{{\mathrm{N}}2}} } \right] > E\left[ {{{\uppi }}_{\mathrm{E}}^{{{\mathrm{L}}2}} } \right]$: ${{\uplambda }}_{1} < \lambda \left< {{{\uplambda }}_{4} \;{\mathrm{and}}\;\frac{{{{\upsigma }}_{\mathrm{O}} }}{{{{\upmu }}_{\mathrm{O}} }}} \right> {{\uptheta }}_{3} .$ When ${{\uplambda }} > {{\uplambda }}_{1} \;{\mathrm{and}}\;\left( { - 19 - 5{{\uppsi }} + {{\uplambda }}\left( {26 + 6{{\uppsi }}} \right)} \right) > 0$, we obtain ${\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{E}}^{\mathrm{N}} } \right] < E\left[ {{{\uppi }}_{\mathrm{E}}^{\mathrm{L}} } \right]$. Therefore, it can be shown that ${\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{E}}^{{{\mathrm{N}}2}} } \right] > E\left[ {{{\uppi }}_{\mathrm{E}}^{{{\mathrm{L}}2}} } \right]$ if one of the following conditions occurs:

• (a) ${{\uplambda }} < {{\uplambda }}_{1}$ and $\frac{{{{\upsigma }}_{\mathrm{O}} }}{{{{\upmu }}_{\mathrm{O}} }} < {{\uptheta }}_{3}$;

• (b) ${{\uplambda }}_{1} < \lambda < {{\uplambda }}_{4}$ and $\frac{{{{\upsigma }}_{\mathrm{O}} }}{{{{\upmu }}_{\mathrm{O}} }} > {{\uptheta }}_{3}$.

Proof of Proposition 11

The difference between the incumbent MC’s profits in two scenarios is ${\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{I}}^{{{\mathrm{L}}2}} } \right] - {\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{I}}^{{{\mathrm{N}}2}} } \right] = \frac{{\left( {1 - 2{{\uplambda }}} \right)\left( {{{\uppsi }}^{2} - 1} \right)\left\{ {\frac{{{{{\rm{s}}\upsigma }}_{\mathrm{o}}^{2} }}{{\left( {1 + {\mathrm{s}}} \right)}}\left( {1 + {{\uplambda }} + {{\uppsi }} - {{\uplambda \uppsi }}} \right)^{2} \left[ {29 + 2{{\uplambda }}\left( { - 1 + {{\uppsi }}} \right)\left( {11 + 3{{\uppsi }}} \right) - {{\uppsi }}\left( {8 + 5{{\uppsi }}} \right)} \right] + 4{{\upmu }}_{\mathrm{o}}^{2} \left[ {\left( {1 + {{\uppsi }}} \right)\left( {5 + 3{{\uppsi }}} \right) + 2{{\uplambda }}\left( {1 - {{\uppsi }}^{2} } \right)} \right]} \right\}}}{{64\left( {3 + {{\uppsi }}} \right)^{2} \left( { - 1 - {{\uplambda }} - {{\uppsi }} + {{\uplambda \uppsi }}} \right)^{2} }}$. Obviously, we have $\left( {{{\uppsi }}^{2} - 1} \right) < 0$, $\left( {1 + {{\uplambda }} + {{\uppsi }} - {{\uplambda \uppsi }}} \right)^{2} > 0$, $\left( {1 + {{\uppsi }}} \right)\left( {5 + 3{{\uppsi }}} \right) + 2{{\uplambda }}\left( {1 - {{\uppsi }}^{2} } \right) > 0$ and $64\left( {3 + {{\uppsi }}} \right)^{2} \left( { - 1 - {{\uplambda }} - {{\uppsi }} + {{\uplambda \uppsi }}} \right)^{2} > 0$. The symbol of ${\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{I}}^{{{\mathrm{L}}2}} } \right] - {\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{I}}^{{{\mathrm{N}}2}} } \right]$ depends on $\left( {1 - 2{{\uplambda }}} \right)\Big\{ \frac{{{{{\rm{s}}\upsigma }}_{\mathrm{o}}^{2} }}{{\left( {1 + {\mathrm{s}}} \right)}}\left( {1 + {{\uplambda }} + {{\uppsi }} - {{\uplambda \uppsi }}} \right)^{2} \left[ {29 + 2{{\uplambda }}\left( { - 1 + {{\uppsi }}} \right)\left( {11 + 3{{\uppsi }}} \right) - {{\uppsi }}\left( {8 + 5{{\uppsi }}} \right)} \right]\break + 4{{\upmu }}_{\mathrm{o}}^{2} \left[ {\left( {1 + {{\uppsi }}} \right)\left( {5 + 3{{\uppsi }}} \right) + 2{{\uplambda }}\left( {1 - {{\uppsi }}^{2} } \right)} \right] \Big\}$. We discuss it as follows: when $\left( {1 - 2{{\uplambda }}} \right) < 0$ and $\left[ {29 + 2{{\uplambda }}\left( { - 1 + {{\uppsi }}} \right)\left( {11 + 3{{\uppsi }}} \right) - {{\uppsi }}\left( {8 + 5{{\uppsi }}} \right)} \right] > 0$, i.e., ${{\uplambda }}_{1} < \lambda < {{\uplambda }}_{5}$, where ${{\uplambda }}_{5} = \frac{{ - 29 + 8{{\uppsi }} + 5{{\uppsi }}^{2} }}{{ - 22 + 16{{\uppsi }} + 6{{\uppsi }}^{2} }} > {{\uplambda }}_{3}$, we have ${\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{I}}^{{{\mathrm{L}}2}} } \right] > E\left[ {{{\uppi }}_{\mathrm{I}}^{{{\mathrm{N}}2}} } \right]$. When $\left( {1 - 2{{\uplambda }}} \right) < 0$ and $\left[ {29 + 2{{\uplambda }}\left( { - 1 + {{\uppsi }}} \right)\left( {11 + 3{{\uppsi }}} \right) - {{\uppsi }}\left( {8 + 5{{\uppsi }}} \right)} \right] < 0$, i.e., ${{\uplambda }} > {{\uplambda }}_{5}$, we solve the inequality $\frac{{{{{\rm{s}}\upsigma }}_{\mathrm{o}}^{2} }}{{\left( {1 + {\mathrm{s}}} \right)}}\left( {1 + {{\uplambda }} + {{\uppsi }} - {{\uplambda \uppsi }}} \right)^{2} \left[ {29 + 2{{\uplambda }}\left( { - 1 + {{\uppsi }}} \right)\left( {11 + 3{{\uppsi }}} \right) - {{\uppsi }}\left( {8 + 5{{\uppsi }}} \right)} \right] + 4{{\upmu }}_{\mathrm{o}}^{2} \left[ {\left( {1 + {{\uppsi }}} \right)\left( {5 + 3{{\uppsi }}} \right) + 2{{\uplambda }}\left( {1 - {{\uppsi }}^{2} } \right)} \right] < 0$ to satisfy ${\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{I}}^{{{\mathrm{L}}2}} } \right] > E\left[ {{{\uppi }}_{\mathrm{I}}^{{{\mathrm{N}}2}} } \right]$, and find that $\frac{{{{\upsigma }}_{\mathrm{o}}^{2} }}{{{{\upmu }}_{\mathrm{o}}^{2} }} > \frac{{ - 4\left[ {\left( {1 + {{\uppsi }}} \right)\left( {5 + 3{{\uppsi }}} \right) + 2{{\uplambda }}\left( {1 - {{\uppsi }}^{2} } \right)} \right]\left( {1 + {\mathrm{s}}} \right)}}{{{\mathrm{s}}\left( {1 + {{\uplambda }} + {{\uppsi }} - {{\uplambda \uppsi }}} \right)^{2} \left[ {29 + 2{{\uplambda }}\left( { - 1 + {{\uppsi }}} \right)\left( {11 + 3{{\uppsi }}} \right) - {{\uppsi }}\left( {8 + 5{{\uppsi }}} \right)} \right]}}$, where ${{\uptheta }}_{4} = \frac{{ - 4\left[ {\left( {1 + {{\uppsi }}} \right)\left( {5 + 3{{\uppsi }}} \right) + 2{{\uplambda }}\left( {1 - {{\uppsi }}^{2} } \right)} \right]\left( {1 + {\mathrm{s}}} \right)}}{{{\mathrm{s}}\left( {1 + {{\uplambda }} + {{\uppsi }} - {{\uplambda \uppsi }}} \right)^{2} \left[ {29 + 2{{\uplambda }}\left( { - 1 + {{\uppsi }}} \right)\left( {11 + 3{{\uppsi }}} \right) - {{\uppsi }}\left( {8 + 5{{\uppsi }}} \right)} \right]}} > 0$. Therefore, Proposition 11 is proven.

Proof of Proposition 12

By comparing the RC’s Profits in two scenarios, we show that $E\left[ {\pi_{R}^{L2} } \right] > E\left[ {\pi_{R}^{N2} } \right]$. The following proof is similar to that of the Proposition 7. We omit the details here.

1.3 Impact of the RC’s differentiated service prices

We obtain the equilibrium outcomes in Table 5.

Table 5 The outcomes and equilibrium profits in each scenario

Proof of Proposition 13

Comparing the incumbent MC’s profits in two scenarios, we have ${\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{I}}^{{{\mathrm{L}}3}} } \right] - {\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{I}}^{{{\mathrm{N}}3}} } \right] = \frac{{\left[ {7 + 4\left( { - 4 + {{\uplambda }}} \right){{\uplambda }}} \right]{{{\rm{s}}\upsigma }}_{\mathrm{o}}^{2} }}{{144\left( {1 + {\mathrm{s}}} \right)}}$. Clearly, the sign of ${\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{I}}^{{{\mathrm{L}}3}} } \right] - {\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{I}}^{{{\mathrm{N}}3}} } \right]$ depends on $\left[ {7 + 4\left( { - 4 + {{\uplambda }}} \right){{\uplambda }}} \right]$, which is a quadratic function of ${{\uplambda }}$. Solving the inequality $\left[ {7 + 4\left( { - 4 + {{\uplambda }}} \right){{\uplambda }}} \right] > 0$, we obtain $0 < \lambda < \frac{1}{2}$ or ${{\uplambda }} > \frac{7}{2}$. Therefore, Proposition 13 is proven.

Proof of Proposition 14

By comparing the profits of the new MC and RC in two scenarios, we have ${\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{E}}^{{{\mathrm{L}}3}} } \right] - {\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{E}}^{{{\mathrm{N}}3}} } \right] = \frac{{\left( {1 - 2{{\uplambda }}} \right)^{2} {{{\rm{s}}\upsigma }}_{\mathrm{o}}^{2} }}{{36\left( {1 + {\mathrm{s}}} \right)}} > 0$ and ${\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{R}}^{{{\mathrm{L}}3}} } \right] - {\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{R}}^{{{\mathrm{N}}3}} } \right] = \frac{{\left( {1 - 2{{\uplambda }}} \right)^{2} {{{\rm{s}}\upsigma }}_{\mathrm{o}}^{2} }}{{24\left( {1 + {\mathrm{s}}} \right)}} > 0$ for any ${{\uplambda }},{\mathrm{s}}\;{\mathrm{and}}\;{{\upsigma }}_{\mathrm{o}}^{2}$.

1.4 Alternative model of the new MC’s brand image

We obtain the equilibrium outcomes in Table 6.

Table 6 The outcomes and equilibrium profits in each scenario

Proof of Proposition 15

To investigate the new MC’s preference over information leakage, we compare its expected profits in two scenarios, and find that $E\left[ {\pi_{E}^{N4} } \right] - E\left[ {\pi_{E}^{L4} } \right] = \frac{{\left( { - 21\lambda + 8\upmu_{o} } \right)s\sigma_{o}^{2} }}{{2304\left( {1 + s} \right)}}$. It is easy to verify $E\left[ {\pi_{E}^{N4} } \right] > E\left[ {\pi_{E}^{L4} } \right]$ if $\lambda < \frac{{8\upmu_{o} }}{21}$, so Proposition 15 is proven.

Proof of Proposition 16

Comparing the incumbent MC’s profits in two scenarios, we have ${\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{I}}^{{{\mathrm{L}}4}} } \right] - {\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{I}}^{{{\mathrm{N}}4}} } \right] = \frac{{41{{\uplambda }}\left( {{{\uplambda }} + 2{{\upmu }}_{\mathrm{o}} } \right) - \frac{{28{{{\rm{s}}\upsigma }}_{\mathrm{o}}^{2} }}{{1 + {\mathrm{s}}}}}}{2304}$. Clearly, the symbol of ${\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{I}}^{{{\mathrm{L}}4}} } \right] - {\mathrm{E}}\left[ {{{\uppi }}_{\mathrm{I}}^{{{\mathrm{N}}4}} } \right]$ depends on $41{{\uplambda }}\left( {{{\uplambda }} + 2{{\upmu }}_{\mathrm{o}} } \right) - \frac{{28{{{\rm{s}}\upsigma }}_{\mathrm{o}}^{2} }}{{1 + {\mathrm{s}}}}$. We discuss it as follows: when ${{\uplambda }} > 0$, solving the inequality $41{{\uplambda }}\left( {{{\uplambda }} + 2{{\upmu }}_{\mathrm{o}} } \right) - \frac{{28{{{\rm{s}}\upsigma }}_{\mathrm{o}}^{2} }}{{1 + {\mathrm{s}}}} > 0$, we have ${{\upsigma }}_{\mathrm{o}}^{2} < \frac{{41{{\uplambda }}\left( {{{\uplambda }} + 2{{\upmu }}_{\mathrm{o}} } \right)\left( {1 + {\mathrm{s}}} \right)}}{{28{\mathrm{s}}}}$. When ${{\uplambda }} < 0$ and $\left( {{{\uplambda }} + 2{{\upmu }}_{\mathrm{o}} } \right) < 0$, similarly, we have ${{\upsigma }}_{\mathrm{o}}^{2} < \frac{{41{{\uplambda }}\left( {{{\uplambda }} + 2{{\upmu }}_{\mathrm{o}} } \right)\left( {1 + {\mathrm{s}}} \right)}}{{28{\mathrm{s}}}}$. Thus, Proposition 16 is proven.

Proof of Proposition 17

The difference of $E\left[ {\pi_{R}^{L4} } \right] - E\left[ {\pi_{R}^{N4} } \right]$ is $\frac{{3\lambda^{2} + 48\lambda \upmu_{o} + \frac{{12s\sigma_{o}^{2} }}{1 + s}}}{1152}$. It is easy to find that $E\left[ {\pi_{R}^{L4} } \right] > E\left[ {\pi_{R}^{N4} } \right]$ for any feasible $\lambda , \upmu_{o} , s$ and $\sigma_{o}^{2}$, so Proposition 17 is proven.