Modeling the relationship between fairness concern and customer loyalty in dual distribution channel

Abstract

With the rapid growth of the online market, the e-tailer platform has accumulated loyal consumers, which has caused e-tailers to be more concerned about fairness, especially by paying more attention to manufacturers’ profits earned from the e-commerce platform. We build a dual-channel model and consider whether an e-tailer and manufacturer should be fair minded. We find that the platform will care about fairness only when the proportion of loyal customers is at a medium level. When the proportion of loyal customers is greater, whether e-tailers are fair minded will make no difference. Given that the e-tailer cares about fairness, the manufacturer will also be fair-minded when there is a moderate proportion of loyal customers. When the proportion of loyal customers is greater, the manufacturer will not be fair minded. When only the e-tailer is concerned about fairness, the loyal consumer will be worse off if there is a larger proportion of loyal consumers, and a case where both the e-tailer and manufacturer care about fairness shows the opposite trend. When only the e-tailer is concerned about fairness, the normal consumer will always be better off. When both supply chain members care about fairness, the normal consumer will be worse off if the proportion of loyal consumers is greater.

Appendix

Proof of Proposition 1

The manufacturer’s profit function can be written as: $\pi_{m} = p_{m} \left( {\alpha \left( {\frac{{p_{r} - p_{m} }}{1 - \rho } - \frac{{p_{m} }}{\rho }} \right) + 1 - \alpha } \right) + w\alpha \left( {1 - \frac{{p_{r} - p_{m} }}{1 - \rho }} \right)$, which is derived from two segments: the first is from the marketplace channel, in which she directly sells to consumers; and the second is from the reseller channel, in which she sells products at the wholesale price. Then the reseller’s profit function can be written as: $\pi_{r} = \left( {p_{r} - w} \right)\alpha \left( {1 - \frac{{p_{r} - p_{m} }}{1 - \rho }} \right);$ according to the game timing and solving the above optimization problem, we obtain the manufacturer’s optimal wholesale price and direct channel price, and the reseller’s optimal price.

This completes the proof of Proposition 1.

Proof of Proposition 2

As the reseller’s price decision can be written as $p_{r}^{C} = \left\{ {\begin{array}{*{20}l} {p_{r1}^{C*} } \hfill & {{\text{if}}\quad w^{C} > w_{1} } \hfill \\ {\left( {1 + \gamma } \right)w^{C} } \hfill & {{\text{if}}\quad w_{2} \le w^{C} \le w_{1} } \hfill \\ {p_{r2}^{C*} } \hfill & {{\text{if}}\quad w^{C} \le w_{2} } \hfill \\ \end{array} } \right.$ as in Eq. (12), If the manufacturer chooses a wholesale price from range $w^{C} > w_{1}$, then the manufacturer’s optimization problem is given by

$$ \mathop {{\text{max}}}\limits_{{w^{C} }} p_{m}^{C} \left( {\alpha \left( {\frac{{p_{r}^{C} - p_{m}^{C} }}{1 - \rho } - \frac{{p_{m}^{C} }}{\rho }} \right) + 1 - \alpha } \right) + w^{C} \alpha \left( {1 - \frac{{p_{r}^{C} - p_{m}^{C} }}{1 - \rho }} \right) $$

$$ s.t.\left\{ {\begin{array}{*{20}l} {p_{m}^{C} = \frac{{\rho \left( {1 - \rho + \alpha \left( { - 1 + p_{r}^{C} + w^{C} + \rho } \right)} \right)}}{2\alpha }} \hfill \\ {p_{r}^{C} = p_{r1}^{C*} } \hfill \\ {w^{C} > w_{1} } \hfill \\ \end{array} } \right. $$

By solving the first-order conditions, we can get

$${w}_{1}^{C*}=-\frac{(1+\beta )(-1+\rho )(\alpha (-2+\rho )(2(4-5\rho +{\rho }^{2})+\beta (8-2(5+\gamma )\rho +(2+\gamma ){\rho }^{2}))-\rho (2(8-5\rho +{\rho }^{2})+\beta (\gamma (8-6\rho +{\rho }^{2})+2(8-5\rho +{\rho }^{2}))))}{\alpha (4(-8+13\rho -6{\rho }^{2}+{\rho }^{3})+4\beta (-1+\rho )(\gamma (8-6\rho +{\rho }^{2})+2(8-5\rho +{\rho }^{2}))+{\beta }^{2}({\gamma }^{2}{(-2+\rho )}^{2}\rho +4\gamma (-8+14\rho -7{\rho }^{2}+{\rho }^{3})+4(-8+13\rho -6{\rho }^{2}+{\rho }^{3})))}$$

$${\alpha }_{1}=\frac{\gamma \rho ({\beta }^{2}(4-2\gamma (-2+\rho ))+2(8-5\rho +{\rho }^{2})+\beta (\gamma (8-6\rho +{\rho }^{2})+2(10-5\rho +{\rho }^{2})))}{4{\beta }^{2}(-2+\gamma (-2+\rho ))(-1+\rho )+\beta (-16(-1+\rho )+2\gamma {(-2+\rho )}^{2}(-1+\rho )+{\gamma }^{2}{(-2+\rho )}^{2}\rho )+2(-1+\rho )(-4+\gamma (8-6\rho +{\rho }^{2}))}$$

The optimal wholesale price is given as: $w^{C} = \left\{ {\begin{array}{*{20}c} {w_{1}^{C*} } & {{\upalpha } \le \alpha_{1} } \\ {w_{1} } & {otherwise} \\ \end{array} } \right.$.

If the manufacturer chooses a wholesale price from range $w_{2} \le w^{C} \le w_{1}$, then the manufacturer’s optimization problem is given by

$$ \mathop {\text{max }}\limits_{{w^{C} }} p_{m}^{C} \left( {\alpha \left( {\frac{{p_{r}^{C} - p_{m}^{C} }}{1 - \rho } - \frac{{p_{m}^{C} }}{\rho }} \right) + 1 - \alpha } \right) + w^{C} \alpha \left( {1 - \frac{{p_{r}^{C} - p_{m}^{C} }}{1 - \rho }} \right) $$

$$ s.t.\left\{ {\begin{array}{*{20}l} {p_{m}^{C} = \frac{{\rho \left( {1 - \rho + \alpha \left( { - 1 + p_{r}^{C} + w^{C} + \rho } \right)} \right)}}{2\alpha }} \hfill \\ {p_{r}^{C} = \left( {1 + \gamma } \right)w^{C} } \hfill \\ {w_{2} \le w^{C} \le w_{1} } \hfill \\ \end{array} } \right. $$

By solving the first-order conditions, we can get$w_{2}^{C*} = - \frac{{\left( { - 1 + \rho } \right)\left( { - \left( {2 + \gamma } \right)\rho + \alpha \left( { - 2 + \left( {2 + \gamma } \right)\rho } \right)} \right)}}{{\alpha \left( {4\left( { - 1 + \rho } \right) + 4\gamma \left( { - 1 + \rho } \right) + \gamma^{2} \rho } \right)}}$; $\alpha_{2} = \frac{{\gamma \left( { - 2\beta \left( {1 + \gamma } \right) + \gamma \left( { - 4 + \rho } \right) + 2\left( { - 3 + \rho } \right)} \right)\rho }}{{ - 4 + 4\beta \left( {1 + \gamma } \right)\left( { - 1 + \rho } \right) - 2\left( { - 2 + \gamma + \gamma^{2} } \right)\rho + \gamma \left( {2 + \gamma } \right)\rho^{2} }}$; $\alpha_{3} = \frac{{\gamma \left( {\gamma \left( { - 4 + \rho } \right) + 2\left( { - 3 + \rho } \right)} \right)\rho }}{{ - 4 - 2\left( { - 2 + \gamma + \gamma^{2} } \right)\rho + \gamma \left( {2 + \gamma } \right)\rho^{2} }}$.

The optimal wholesale price is given as: $w^{C} = \left\{ {\begin{array}{*{20}c} {w_{1} } & {{\upalpha } \le \alpha_{2} } \\ {w_{2}^{C*} } & {\alpha_{2} \le {\upalpha } \le \alpha_{3} } \\ {w_{2} } & {{\upalpha } > \alpha_{3} } \\ \end{array} } \right.$.

If the manufacturer chooses a wholesale price from range $w^{C} \le w_{2}$, then the manufacturer’s optimization problem is given by

$$ \mathop {\text{max }}\limits_{{w^{C} }} p_{m}^{C} \left( {\alpha \left( {\frac{{p_{r}^{C} - p_{m}^{C} }}{1 - \rho } - \frac{{p_{m}^{C} }}{\rho }} \right) + 1 - \alpha } \right) + w^{C} \alpha \left( {1 - \frac{{p_{r}^{C} - p_{m}^{C} }}{1 - \rho }} \right) $$

$$ s.t.\left\{ {\begin{array}{*{20}l} {p_{m}^{C} = \frac{{\rho \left( {1 - \rho + \alpha \left( { - 1 + p_{r} + w^{C} + \rho } \right)} \right)}}{2\alpha }} \hfill \\ {p_{r} = p_{r2}^{C*} } \hfill \\ {w^{C} \le w_{2} } \hfill \\ \end{array} } \right. $$

By solving the first-order conditions, we can get $w_{3}^{C*} = \frac{{\rho \left( {8 - 5\rho + \rho^{2} } \right) - \alpha \left( { - 8 + 14\rho - 7\rho^{2} + \rho^{3} } \right)}}{{2\alpha \left( {8 - 5\rho + \rho^{2} } \right)}}$; $\alpha_{4} = \frac{{\gamma \rho \left( {8 - 5\rho + \rho^{2} } \right)}}{{\left( { - 1 + \rho } \right)\left( { - 4 + \gamma \left( {8 - 6\rho + \rho^{2} } \right)} \right)}}$.

The optimal wholesale price is given as: $w^{C} = \left\{ {\begin{array}{*{20}l} {w_{2} } \hfill & {{\upalpha } \le \alpha_{4} } \hfill \\ {w_{3}^{C*} } \hfill & {{\text{otherwise}}} \hfill \\ \end{array} } \right.$.

Therefore, the manufacturer will compare the resulting payoffs to determine the globally optimal payoff. The globally optimal wholesale price and profits are given as Proposition2.

This completes the proof of Proposition 2.

Proof of Proposition 3

In this case, the manufacturer’s optimization problem is given by

$$ \begin{gathered} \mathop {{\text{max}}}\limits_{{p_{m}^{CC} }} p_{m}^{CC} \left( {\alpha \left( {\frac{{p_{r}^{CC} - p_{m}^{CC} }}{1 - \rho } - \frac{{p_{m}^{CC} }}{\rho }} \right) + 1 - \alpha } \right) + w\alpha \left( {1 - \frac{{p_{r}^{CC} - p_{m}^{CC} }}{1 - \rho }} \right) - \mu \hfill \\ {\text{max}}\left\{ {\varepsilon \left( {p_{r}^{CC} - w} \right)\alpha \left( {1 - \frac{{p_{r}^{CC} - p_{m}^{CC} }}{1 - \rho }} \right) - w\alpha \left( {1 - \frac{{p_{r}^{CC} - p_{m}^{CC} }}{1 - \rho }} \right),0} \right\} \hfill \\ \end{gathered} $$

And the reseller’s optimization problem is given by

$$ \begin{gathered} \mathop {\max }\limits_{{p_{r} }} \left( {p_{r}^{CC} - w} \right)\alpha \left( {1 - \frac{{p_{r}^{CC} - p_{m}^{CC} }}{1 - \rho }} \right) - \beta \hfill \\ \max \left\{ {\gamma w\alpha \left( {1 - \frac{{p_{r}^{CC} - p_{m}^{CC} }}{1 - \rho }} \right) - \left( {p_{r}^{CC} - w} \right)\alpha \left( {1 - \frac{{p_{r}^{CC} - p_{m}^{CC} }}{1 - \rho }} \right),0} \right\} \hfill \\ \end{gathered} $$

According to the game timing, the manufacturer should decide the direct channel price first:

• 1. When $p_{r}^{C} > \left( {1 + \frac{1}{\varepsilon }} \right)w$, manufacturer’s problem can be denoted as:

  $$ \begin{gathered} \mathop {{\text{max}}}\limits_{{p_{m}^{CC} }} p_{m}^{CC} \left( {\alpha \left( {\frac{{p_{r}^{CC} - p_{m}^{CC} }}{1 - \rho } - \frac{{p_{m}^{CC} }}{\rho }} \right) + 1 - \alpha } \right) + w\alpha \left( {1 - \frac{{p_{r}^{CC} - p_{m}^{CC} }}{1 - \rho }} \right) \hfill \\ - \mu \left[ {\varepsilon \left( {p_{r}^{CC} - w} \right)\alpha \left( {1 - \frac{{p_{r}^{CC} - p_{m}^{CC} }}{1 - \rho }} \right) - w\alpha \left( {1 - \frac{{p_{r}^{CC} - p_{m}^{CC} }}{1 - \rho }} \right)} \right], \hfill \\ \end{gathered} $$

  By solving the first-order conditions, we can get $p_{m1}^{CC} = \frac{{\rho \left( {1 - \rho + \alpha \left( { - 1 + p_{r}^{CC} + w + w\mu - p_{r}^{CC} \varepsilon \mu + w\varepsilon \mu + \rho } \right)} \right)}}{2\alpha }$.

• 2. When $p_{r}^{CC} \le \left( {1 + \frac{1}{\varepsilon }} \right)w$, manufacturer’s problem can be denoted as:

  $$ \mathop {{\text{max}}}\limits_{{p_{m}^{CC} }} p_{m}^{CC} \left( {\alpha \left( {\frac{{p_{r}^{CC} - p_{m}^{CC} }}{1 - \rho } - \frac{{p_{m}^{CC} }}{\rho }} \right) + 1 - \alpha } \right) + w\alpha \left( {1 - \frac{{p_{r}^{CC} - p_{m}^{CC} }}{1 - \rho }} \right), $$

By solving the first-order conditions, we can get $p_{m2}^{CC} = \frac{{\rho \left( {1 - \rho + \alpha \left( { - 1 + p_{r}^{C} + w + \rho } \right)} \right)}}{2\alpha }$.

Take $p_{m1}^{CC}$ and $p_{m2}^{CC}$ into the reseller’s profit function respectively:

When $p_{r}^{CC} \le \left( {1 + \gamma } \right)w$, the reseller’s profit function can be written as:

$$ \pi_{r1}^{CC} = \left( {p_{r}^{CC} - w} \right)\alpha \left( {1 - \frac{{p_{r}^{CC} - p_{m}^{CC} }}{1 - \rho }} \right) - \beta \left[ {\gamma w\alpha \left( {1 - \frac{{p_{r}^{CC} - p_{m}^{CC} }}{1 - \rho }} \right) - \left( {p_{r}^{CC} - w} \right)\alpha \left( {1 - \frac{{p_{r}^{CC} - p_{m}^{CC} }}{1 - \rho }} \right)} \right]; $$

When $p_{r}^{CC} \ge \left( {1 + \gamma } \right)w$, the reseller’s profit function can be written as:$\pi_{r2}^{CC} = \left( {p_{r}^{CC} - w} \right)\alpha \left( {1 - \frac{{p_{r}^{CC} - p_{m}^{CC} }}{1 - \rho }} \right)$

Combine the two cases, we can get:

(1) when $\gamma \varepsilon > 1\left( {1 + \frac{1}{\varepsilon }} \right)w \le \left( {1 + \gamma } \right)w$ (acrimonious channel),

(1.1) when $p_{r}^{CC} > \left( {1 + \gamma } \right)w$, take $p_{m1}^{CC}$ into $\pi_{r2}^{CC}$, we can get:

$$ \pi_{r2}^{CC} = \mathop {\max }\limits_{{p_{r} }} \frac{{\left( {p_{r}^{CC} - w} \right)\left( {\left( { - 1 + \rho } \right)\rho - \alpha \left( {2 + \left( { - 3 + w + w\mu + w\varepsilon \mu } \right)\rho + \rho^{2} + p_{r}^{CC} \left( { - 2 + \rho - \varepsilon \mu \rho } \right)} \right)} \right)}}{{2\left( { - 1 + \rho } \right)}} $$

$$ s.t.\quad p_{r}^{CC} > \left( {1 + \gamma } \right)w $$

By solving the first-order conditions, we can get $p_{r11}^{CC*} = \frac{{ - \left( { - 1 + \rho } \right)\rho + \alpha \left( {2 - 3\rho + \rho^{2} + w\left( {2 + \mu \rho + 2\varepsilon \mu \rho } \right)} \right)}}{{2\alpha \left( {2 + \left( { - 1 + \varepsilon \mu } \right)\rho } \right)}}$, then the price in reseller channel can be denoted as: $p_{r}^{CC} = \left\{ {\begin{array}{*{20}c} {\left( {1 + \gamma } \right)w} & {w > w_{A} } \\ {p_{r11}^{CC*} } & {w \le w_{A} } \\ \end{array} } \right.,$ where $w_{A} = \frac{{\left( {\alpha \left( { - 2 + \rho } \right) - \rho } \right)\left( { - 1 + \rho } \right)}}{{\alpha \left( {2 - \left( {2 + \mu } \right)\rho + 2\gamma \left( {2 + \left( { - 1 + \varepsilon \mu } \right)\rho } \right)} \right)}}.$

(1.2) when $\left( {1 + \frac{1}{\varepsilon }} \right)w < p_{r}^{CC} \le \left( {1 + \gamma } \right)w,$ take $p_{m1}^{CC}$ into $\pi_{r1}^{CC}$, we can get:

$$ \pi_{r1}^{CC} = \mathop {\max }\limits_{{p_{r} }} \frac{{\left( {p_{r}^{CC} \left( {1 + \beta } \right) - w\left( {1 + \beta + \beta \gamma } \right)} \right)\left( {\left( { - 1 + \rho } \right)\rho - \alpha \left( {2 + \left( { - 3 + w + w\mu + w\varepsilon \mu } \right)\rho + \rho^{2} + p_{r}^{CC} \left( { - 2 + \rho - \varepsilon \mu \rho } \right)} \right)} \right)}}{{2\left( { - 1 + \rho } \right)}} $$

$$ s.t.\left( {1 + \frac{1}{\varepsilon }} \right)w < p_{r}^{CC} \le \left( {1 + \gamma } \right)w $$

By solving the first-order conditions, we can get

$$ p_{r12}^{CC*} = \frac{{ - \left( {1 + \beta } \right)\left( { - 1 + \rho } \right)\rho + \alpha \left( {\left( {1 + \beta } \right)\left( {2 - 3\rho + \rho^{2} } \right) + w\left( {2 + \mu \rho + 2\varepsilon \mu \rho + \beta \left( {2 + \mu \rho + 2\varepsilon \mu \rho + \gamma \left( {2 - \rho + \varepsilon \mu \rho } \right)} \right)} \right)} \right)}}{{2\alpha \left( {1 + \beta } \right)\left( {2 + \left( { - 1 + \varepsilon \mu } \right)\rho } \right)}}, $$

then the price in reseller channel can be denoted as: $p_{r}^{CC} = \left\{ {\begin{array}{*{20}l} {\left( {1 + \gamma } \right)w} \hfill & {w \le w_{B1} } \hfill \\ {p_{r12}^{CC*} } \hfill & {w_{B1} < w \le w_{B2} } \hfill \\ {\left( {1 + \frac{1}{\varepsilon }} \right)w} \hfill & {w > w_{B2} } \hfill \\ \end{array} } \right.,$ where $w_{B1} = \frac{{\left( {1 + \beta } \right)\left( {\alpha \left( { - 2 + \rho } \right) - \rho } \right)\left( { - 1 + \rho } \right)}}{{\alpha \left( {2 - 2\rho - \mu \rho + 2\gamma \left( {2 + \left( { - 1 + \varepsilon \mu } \right)\rho } \right) + \beta \left( {2 - \left( {2 + \mu } \right)\rho + \gamma \left( {2 + \left( { - 1 + \varepsilon \mu } \right)\rho } \right)} \right)} \right)}};$ $w_{B2} = - \frac{{\left( {1 + \beta } \right)\varepsilon \left( {\alpha \left( { - 2 + \rho } \right) - \rho } \right)\left( { - 1 + \rho } \right)}}{{\alpha \left( {2\left( { - 2 + \rho } \right) - \varepsilon \left( {2 + \left( { - 2 + \mu } \right)\rho } \right) + \beta \left( {2\left( { - 2 + \rho } \right) + \gamma \varepsilon^{2} \mu \rho - \varepsilon \left( {2 + \gamma \left( { - 2 + \rho } \right) + \left( { - 2 + \mu } \right)\rho } \right)} \right)} \right)}},$ what’s more ${w}_{B1}\le {w}_{B2}$ always holds.

(1.3) when $p_{r}^{CC} \le \left( {1 + \frac{1}{\varepsilon }} \right)w$, take $p_{m2}^{CC}$ into $\pi_{r1}^{CC}$, we can get:

$$ \pi_{r1}^{CC} = - \frac{{\left( {p_{r}^{CC} \left( {1 + \beta } \right) - w\left( {1 + \beta + \beta \gamma } \right)} \right)\left( { - \left( { - 1 + \rho } \right)\rho + \alpha \left( {2 + p_{r}^{CC} \left( { - 2 + \rho } \right) + \left( { - 3 + w} \right)\rho + \rho^{2} } \right)} \right)}}{{2\left( { - 1 + \rho } \right)}} $$

$$ s.t.\quad p_{r}^{CC} \le \left( {1 + \frac{1}{\varepsilon }} \right)w $$

By solving the first-order conditions, we can get $p_{r13}^{CC*} = \frac{{\left( {1 + \beta } \right)\left( { - 1 + \rho } \right)\rho + \alpha \left( {w\left( { - 2 + \beta \left( { - 2 + \gamma \left( { - 2 + \rho } \right)} \right)} \right) - \left( {1 + \beta } \right)\left( {2 - 3\rho + \rho^{2} } \right)} \right)}}{{2\alpha \left( {1 + \beta } \right)\left( { - 2 + \rho } \right)}}$, then the price in reseller channel can be denoted as: $p_{r}^{CC} = \left\{ {\begin{array}{*{20}c} {p_{r13}^{CC*} } & {w > w_{C} } \\ {\left( {1 + \frac{1}{\varepsilon }} \right)w} & {w \le w_{C} } \\ \end{array} } \right.$, where $w_{C} = \frac{{\left( {1 + \beta } \right)\varepsilon \left( {\alpha \left( { - 2 + \rho } \right) - \rho } \right)\left( { - 1 + \rho } \right)}}{{\alpha \left( {\beta \left( {4 + \varepsilon \left( {2 + \gamma \left( { - 2 + \rho } \right) - 2\rho } \right) - 2\rho } \right) - 2\left( { - 2 + \varepsilon \left( { - 1 + \rho } \right) + \rho } \right)} \right)}}.$

$w_{A} \le w_{B1} \le w_{B2} \le w_{C}$ always holds, then we can combine (1.1) to (1.3) as:

When $\gamma \varepsilon > 1\left( {1 + \frac{1}{\varepsilon }} \right)w \le \left( {1 + \gamma } \right)w,$ $p_{r}^{CC} = \left\{ {\begin{array}{*{20}c} {p_{r11}^{CC*} } & {w \le w_{A} } \\ {\left( {1 + \gamma } \right)w} & {w_{A} < w \le w_{B1} } \\ {\begin{array}{*{20}c} {p_{r12}^{CC*} } \\ {\left( {1 + \frac{1}{\varepsilon }} \right)w} \\ {p_{r13}^{CC*} } \\ \end{array} } & {\begin{array}{*{20}c} {w_{B1} < w \le w_{B2} } \\ {w_{B2} < w \le w_{C} } \\ {otherwise} \\ \end{array} } \\ \end{array} } \right.$

Then as the proof of proposition 2, we can get $w^{*} = \left\{ {\begin{array}{*{20}c} {w_{5}^{*} } & {\alpha \le \alpha_{7} } \\ {w_{B2} } & {\alpha_{7} \le \alpha \le \alpha_{4} } \\ {\begin{array}{*{20}c} {w_{3}^{*} } \\ {w_{A} } \\ \end{array} } & {\begin{array}{*{20}c} {\alpha_{4} \le \alpha \le \alpha_{3} } \\ {otherwise} \\ \end{array} } \\ \end{array} } \right.$, where $w_{3}^{*} = \frac{{\rho \left( {8 - 5\rho + \rho^{2} } \right) - \alpha \left( { - 8 + 14\rho - 7\rho^{2} + \rho^{3} } \right)}}{{2\alpha \left( {8 - 5\rho + \rho^{2} } \right)}}$;

$w_{5}^{*} \!=\! - \frac{{\left( {1 + \beta } \right)\left( { - 1 + \rho } \right)\left( {\alpha \left( { - 2 + \rho } \right)\left( {2\left( {4 - 5\rho + \rho^{2} } \right) + \beta \left( {8 - 2\left( {5 + \gamma } \right)\rho + \left( {2 + \gamma } \right)\rho^{2} } \right)} \right) - \rho \left( {2\left( {8 - 5\rho + \rho^{2} } \right) + \beta \left( {\gamma \left( {8 - 6\rho + \rho^{2} } \right) + 2\left( {8 - 5\rho + \rho^{2} } \right)} \right)} \right)} \right)}}{{\alpha \left( {4\left( { - 8 + 13\rho - 6\rho^{2} + \rho^{3} } \right) + 4\beta \left( { - 1 + \rho } \right)\left( {\gamma \left( {8 - 6\rho + \rho^{2} } \right) + 2\left( {8 - 5\rho + \rho^{2} } \right)} \right) + \beta^{2} \left( {\gamma^{2} \left( { - 2 + \rho } \right)^{2} \rho + 4\gamma \left( { - 8 + 14\rho - 7\rho^{2} + \rho^{3} } \right) + 4\left( { - 8 + 13\rho - 6\rho^{2} + \rho^{3} } \right)} \right)} \right)}}$; $\alpha_{3} = \frac{{\gamma \rho \left( {8 - 5\rho + \rho^{2} } \right)}}{{\left( { - 1 + \rho } \right)\left( { - 4 + \gamma \left( {8 - 6\rho + \rho^{2} } \right)} \right)}}$; $\alpha_{4} = \frac{{\rho \left( {8 - 5\rho + \rho^{2} } \right)}}{{\left( { - 1 + \rho } \right)\left( {8 - 4\varepsilon - 6\rho + \rho^{2} } \right)}}$;$\alpha_{7} = \frac{{\gamma \rho \left( {\beta^{2} \left( {4 - 2\gamma \left( { - 2 + \rho } \right)} \right) + 2\left( {8 - 5\rho + \rho^{2} } \right) + \beta \left( {\gamma \left( {8 - 6\rho + \rho^{2} } \right) + 2\left( {10 - 5\rho + \rho^{2} } \right)} \right)} \right)}}{{4\beta^{2} \left( { - 2 + \gamma \left( { - 2 + \rho } \right)} \right)\left( { - 1 + \rho } \right) + \beta \left( { - 16\left( { - 1 + \rho } \right) + 2\gamma \left( { - 2 + \rho } \right)^{2} \left( { - 1 + \rho } \right) + \gamma^{2} \left( { - 2 + \rho } \right)^{2} \rho } \right) + 2\left( { - 1 + \rho } \right)\left( { - 4 + \gamma \left( {8 - 6\rho + \rho^{2} } \right)} \right)}}$.

(2) when $\gamma \varepsilon \le 1\left( {1 + \frac{1}{\varepsilon }} \right)w > \left( {1 + \gamma } \right)w$ (acrimonious channel), the proof is as the situation where $\gamma \varepsilon > 1$.

This completes the proof of Proposition 3.