Reactive or proactive? An online retailer’s omnichannel strategy for managing consumer returns

Abstract

This study investigates two popular omnichannel strategies for managing consumer returns. The reactive strategy is online–offline return partnership, which offers eco-friendly and cost-effective reverse logistics. The proactive strategy involves conveying fit information through showrooms, in order to reduce returns. We apply a game-theoretic model to explore online retailers’ optimal choice among four strategies, namely, the benchmark strategy of pure online channel, the reactive strategy of return partnership, the proactive strategy of fit information, and the hybrid strategy of joint implementation. Our main findings are as follows. First, online retailers should not implement any omnichannel strategy on extremely low-end products. Second, offering fit information is essential for online retailers who sell sufficiently high-end products. Third, the single reactive strategy is optimal in terms of standardized products with moderate valuation. Finally, implementing both omnichannel strategies simultaneously may hurt online retailers, especially those owning an efficient logistics system.

Appendices

Appendix: Proofs for main results

1.1 Proof of Proposition 1

With Eq. (1), ${U}_{BORO}\ge 0\iff {h}_{o}\le \frac{\theta (v-p)}{2-\theta }\iff d=\frac{\theta (v-p)}{2-\theta }$. Substitute $d$ into the profit function in Eq. (4). Thus, we have $\pi (p)$ being concave in $p$, since $\frac{{\partial }^{2}\pi }{\partial {p}^{2}}=\frac{2{\theta }^{2}}{-2+\theta }\le 0$. Using the first order condition (FOC), we can obtain ${p}^{B*}=\frac{c+v}{2}+\frac{(1-\theta ){{\text{c}}}_{o}}{2\theta }$. Then ${d}^{B*}={d}_{BORO}^{B*}=\frac{\theta (v-c)-(1-\theta ){c}_{o}}{2(2-\theta )}$, and ${\pi }^{B*}=\frac{(\theta (v-c)-(1-\theta ){c}_{o}{)}^{2}}{4(2-\theta )}$. The condition $0\le {d}^{B*}\le 1$ yields the feasible area ${\underline{v}}^{B}\le v\le {\overline{v} }^{B},$ where ${\underline{v}}^{B}=c+\frac{(1-\theta ){{\text{c}}}_{o}}{\theta }$ and ${\overline{v} }^{B}=c+\frac{4+\left(1-\theta \right){{\text{c}}}_{o}}{\theta }-2$.

1.2 Proof of Lemma 1

Clearly, we have $\frac{\partial {p}^{B*}}{\partial v}=\frac{1}{2}$, $\frac{\partial {\pi }^{B*}}{\partial v}=\frac{\theta (\theta (v-c)-(1-\theta ){c}_{o})}{2(2-\theta )}\ge 0$, $\frac{\partial {p}^{B*}}{\partial \theta }=-\frac{{{\text{c}}}_{o}}{2{\theta }^{2}}<0$, $\frac{\partial {\pi }^{B*}}{\partial \theta }=\frac{(\theta (v-c)-(1-\theta ){c}_{o})(\left(4-\theta \right)\left(v-c\right)+(3-\theta ){c}_{o})}{4{(2-\theta )}^{2}}\ge 0$, $\frac{\partial {p}^{B*}}{\partial {c}_{o}}=\frac{1-\theta }{2\theta }\ge 0$, and $\frac{\partial {\pi }^{B*}}{\partial {c}_{o}}=\frac{-(1-\theta )(\theta (v-c)-(1-\theta ){c}_{o})}{2(2-\theta )}\le 0$. Thus, Lemma 1 is true.

1.3 Proofs of Propositions 2–4

The derivations of equilibrium outcomes for other strategies follow in a similar way. For brevity, we omit the similar analysis. In the reactive strategy, we have ${d}_{BORO}^{R*}={h}_{s}$, ${d}_{BORS}^{R*}=\frac{\theta (v-c)-(3-\theta ){h}_{s}-(1-\theta ){c}_{s}}{2}$ and ${d}^{R*}=\frac{\theta \left(v-c\right)-(1-\theta )({c}_{s}+{h}_{s})}{2}$. The conditions ${d}_{BORS}^{R*}\ge 0$ and ${d}^{R*}\le 1$ yield the feasible area ${\underline{v}}^{R}\le v\le {\overline{v} }^{R},$ where ${\underline{v}}^{R}=c+\frac{\left(1-\theta \right){{\text{c}}}_{s}+(3-\theta ){h}_{s}}{\theta }$ and ${\overline{v} }^{R}=c+\frac{2+\left(1-\theta \right){({\text{c}}}_{s}+{h}_{s})}{\theta }$. Furthermore, ${\underline{v}}^{R}\le {\overline{v} }^{R}$ requires ${h}_{s}\le 1$.

In the proactive strategy, we have ${d}_{BORO}^{P*}=\frac{{h}_{s}}{2(1-\theta )}$, ${d}_{ESBO}^{P*}=\frac{v-c}{2}-\frac{{h}_{s}}{2\theta (1-\theta )}$ and ${d}^{P*}=\frac{v-c}{2}-\frac{{h}_{s}}{2\theta }$. The feasible area for the proactive strategy is ${\underline{v}}^{P}\le v\le {\overline{v} }^{P},$ where ${\underline{v}}^{P}=c+\frac{{h}_{s}}{(1-\theta )\theta }$ and ${\overline{v} }^{P}=c+\frac{{h}_{s}}{\theta }+2$. Moreover, ${\underline{v}}^{P}\le {\overline{v} }^{P}$ requires ${h}_{s}\le 2\left(1-\theta \right)$.

In the hybrid strategy, we have ${d}_{BORO}^{H*}={h}_{s}$, ${d}_{BORS}^{H*}=\frac{(2\theta -1){h}_{s}}{1-\theta }$, ${d}_{ESBO}^{H*}=\frac{v-c}{2}-\frac{(1-\theta +2{\theta }^{2}){h}_{s}}{2\theta (1-\theta )}$ and ${d}^{H*}=\frac{v-c}{2}-\frac{{h}_{s}}{2\theta }$. Moreover, ${d}_{BORS}^{H*}=\frac{(2\theta -1){h}_{s}}{1-\theta }>0$ requires $\theta >0.5$, and $0\le {d}^{H*}\le 1$ yields ${\underline{v}}^{H}\le v\le {\overline{v} }^{H},$ where ${\underline{v}}^{H}=c+\frac{\left(1-\theta +2{\theta }^{2}\right){h}_{s}}{(1-\theta )\theta }$ and ${\overline{v} }^{H}=c+\frac{{h}_{s}}{\theta }+2$. Moreover, ${\underline{v}}^{H}\le {\overline{v} }^{H}$ requires ${h}_{s}\le \frac{1-\theta }{\theta }$.

1.4 Proofs of Lemmas 2–4

The proofs are similar to that of Lemma 1. For brevity, we omit the similar analyses.

1.5 Proof of Lemma 5

First, we have ${p}^{B*}-{p}^{R*}=\frac{\left(1-\theta \right)\left({{\text{c}}}_{o}-{{\text{c}}}_{s}+{h}_{s}\right)}{2\theta }>0$, ${p}^{B*}-{p}^{P*}=\frac{\left(1-\theta \right){{\text{c}}}_{o}+{h}_{s}}{2\theta }>0$, and ${p}^{R*}-{p}^{P*}=\frac{\left(1-\theta \right){{\text{c}}}_{s}+\theta {h}_{s}}{2\theta }>0$. Obviously, ${{p}^{H*}=p}^{P*}<{p}^{R*}<{p}^{B*}$.

Second, we have $\Delta {d}^{RB}={d}^{R*}-{d}^{B*}={d}_{BORO}^{R*}+{d}_{BORS}^{R*}-{d}^{B*}=\frac{\left(1-\theta \right)\left(\theta \left(v-c\right)-\left(2-\theta \right)\left({{\text{c}}}_{s}+{h}_{s}\right)+{{\text{c}}}_{o}\right)}{2\left(2-\theta \right)}$, $\Delta {d}^{PB}={d}^{P*}-{d}^{B*}={d}_{BORO}^{P*}+{d}_{ESBO}^{P*}-{d}^{B*}=\frac{2\theta \left(1-\theta \right)\left(v-c\right)+\theta \left(1-\theta \right){{\text{c}}}_{o}-\left(2-\theta \right){h}_{s}}{2\theta \left(2-\theta \right)}$, and $\Delta {d}^{PR}={d}^{P*}-{d}^{R*}={d}_{BORO}^{P*}+{d}_{ESBO}^{P*}-{d}_{BORO}^{R*}-{d}_{BORS}^{R*}=\frac{\theta \left(1-\theta \right)\left(v-c+{{\text{c}}}_{s}\right)-(1-\theta +{\theta }^{2}){h}_{s}}{2\theta }$.

Obviously, all of the demand differences above increase with $v$. Since $\Delta {d}^{RB}\left({\underline{v}}^{R}\right)=\frac{\left(1-\theta \right)\left({{\text{c}}}_{o}-{{\text{c}}}_{s}+{h}_{s}\right)}{2\left(2-\theta \right)}>0$, $\Delta {d}^{PB}\left({\underline{v}}^{P}\right)=\frac{\left(1-\theta \right){{\text{c}}}_{o}+{h}_{s}}{2\left(2-\theta \right)}>0$, and $\Delta {d}^{PR}\left({\underline{v}}^{P}\right)=\frac{\left(1-\theta \right)\left({{\text{c}}}_{s}+{h}_{s}\right)}{2}>0$, it follows that $\Delta {d}^{RB}$, $\Delta {d}^{PB}$, and $\Delta {d}^{PR}$ are always positive. Finally, we have ${{d}^{H*}=d}^{P*}>{d}^{R*}>{d}^{B*}$.

1.6 Proof of Proposition 5

The common feasible domain for the two strategies is $\theta >0.5$, ${h}_{s}\le {\text{min}}(2\left(1-\theta \right),\frac{1-\theta }{\theta })$ and ${\text{max}}({\underline{v}}^{P},{\underline{v}}^{H})\le v\le {\text{min}}({\overline{v} }^{P},{\overline{v} }^{H})$. Given that ${\pi }^{H*}-{\pi }^{P*}=\frac{(2\theta -1){h}_{s}{(c}_{o}-{2c}_{s})}{2}$, we have ${\pi }^{H*}\le {\pi }^{P*}$ if ${c}_{o}\le 2{c}_{s}$ and ${\pi }^{H*}>{\pi }^{P*}$ if ${c}_{o}>2{c}_{s}$ in the common feasible domain.

1.7 Proof of Proposition 6

The common feasible area of the three strategies is ${h}_{s}\le {\text{min}}(2\left(1-\theta \right),1)$ and ${\text{max}}\left({\underline{v}}^{B},{\underline{v}}^{R},{\underline{v}}^{P}\right)\le v\le {{\text{min}}({\overline{v} }^{B},\overline{v} }^{R},{\overline{v} }^{P})$.

The profit differences are ${\pi }^{R*}-{\pi }^{B*}=\frac{(1-\theta )}{4(2-\theta )}{A}_{1}$, ${\pi }^{P*}-{\pi }^{B*}=\frac{{A}_{2}}{4\theta \left(2-\theta \right)}$, and ${\pi }^{P*}-{\pi }^{R*}=\frac{{A}_{3}}{4\theta }$, where ${A}_{1}={\theta }^{2}{(v-c)}^{2}+2\theta \left({c}_{o}-\left(2-\theta \right)\left({c}_{s}+{h}_{s}\right)\right)\left(v-c\right)-\left(2-\theta \right)\left(4{c}_{o}{h}_{s}-2{c}_{s}{h}_{s}\left(3-\theta \right)-\left({{c}_{s}}^{2}+{{h}_{s}}^{2}\right)\left(1-\theta \right)\right)-\left(1-\theta \right){{c}_{o}}^{2}$, ${A}_{2}=2\left(1-\theta \right){\theta }^{2}{\left(v-c\right)}^{2}+2\theta \left(\left(1-\theta \right)\theta {c}_{o}-{h}_{s}\left(2-\theta \right)\right)\left(v-c\right)-{{c}_{o}}^{2}\theta +(2-\theta ){({h}_{s}-{c}_{o}\theta )}^{2}$, and ${A}_{3}=\left(1-\theta \right){\theta }^{2}{\left(v-c\right)}^{2}+2\theta \left(\left(1-\theta \right)\theta {c}_{s}-{h}_{s}(1-\theta +{\theta }^{2})\right)\left(v-c\right)-{{c}_{s}}^{2}{(1-\theta )}^{2}\theta +2{h}_{s}\theta ({c}_{o}(1-2\theta )-{c}_{s}(3-\theta )(1-\theta ))-{{h}_{s}}^{2}({(1-\theta )}^{2}\theta -1)$.

Obviously, ${A}_{1}$, ${A}_{2}$, and ${A}_{3}$ are convex in $v$. Letting ${A}_{1}=0$ yields ${v}_{RB1}=c+\frac{\left(2-\theta \right)\left({c}_{s}+{h}_{s}\right)-{c}_{o}-\sqrt{2-\theta }\left({c}_{o}-{c}_{s}+{h}_{s}\right)}{\theta }$ and ${v}_{RB2}=c+\frac{\left(2-\theta \right)\left({c}_{s}+{h}_{s}\right)-{c}_{o}+\sqrt{2-\theta }\left({c}_{o}-{c}_{s}+{h}_{s}\right)}{\theta }$. Since ${v}_{RB1}-{\underline{v}}^{R}=-\frac{\left({c}_{o}-{c}_{s}+{h}_{s}\right)\left(1+\sqrt{2-\theta }\right)}{\theta }<0$, we have ${A}_{1}\le 0$ if $v\le {v}_{RB2}$, and ${A}_{1}>0$ if $v>{v}_{RB2}$. Denote ${v}_{RB2}$ with ${\widehat{v}}_{RB}$. We then have ${\pi }^{R*}\le {\pi }^{B*}$ if $v\le {\widehat{v}}_{RB}$, and ${\pi }^{R*}>{\pi }^{B*}$ if $v>{\widehat{v}}_{RB}$ in the common feasible domain.

In a similar way, we obtain thresholds ${\widehat{v}}_{PB}=c+\frac{\left(2-\theta \right){h}_{s}-\theta \left(1-\theta \right){c}_{o}+\sqrt{\theta (2-\theta )}\left(\left(1-\theta \right){c}_{o}+{h}_{s}\right)}{2(1-\theta )\theta }$ and ${\widehat{v}}_{PR}=c+\frac{(1-\theta +{\theta }^{2}){h}_{s}-\theta (1-\theta ){c}_{s}+\sqrt{\theta ({{c}_{s}}^{2}{(1-\theta )}^{2}+2{c}_{s}{h}_{s}(2-5\theta +3{\theta }^{2})+{h}_{s}({\theta }^{2}{h}_{s}+2(3\theta -2{\theta }^{2}-1){c}_{o}))}}{(1-\theta )\theta }$, such that ${\pi }^{P*}\le {\pi }^{B*}$ if $v\le {\widehat{v}}_{PB}$ and ${\pi }^{P*}\le {\pi }^{R*}$ if $v\le {\widehat{v}}_{PR}$ in the common feasible domain.

Given that $\frac{\partial {\widehat{v}}_{RB}}{\partial \theta }=\frac{(-4+2\sqrt{2-\theta }+\theta )({c}_{o}-{c}_{s}+{h}_{s})-2\sqrt{2-\theta }{c}_{s}-6\sqrt{2-\theta }{h}_{s}}{2\sqrt{2-\theta }{\theta }^{2}}<0$, ${\widehat{v}}_{RB}$ is decreasing in $\theta$. Meanwhile, $\frac{{\partial }^{2}{\widehat{v}}_{PB}}{\partial {\theta }^{2}}=\frac{{\left(1-\theta \right)}^{3}\theta \left(3-2\theta \right){c}_{o}+(3\theta -12{\theta }^{2}+21{\theta }^{3}-12{\theta }^{4}+2{\theta }^{5}+(8-28\theta +36{\theta }^{2}-16{\theta }^{3}+2{\theta }^{4})\sqrt{(2-\theta )\theta }){h}_{s}}{2{(1-\theta )}^{3}{\theta }^{2}{((2-\theta )\theta )}^{3/2})}\break >0$, which indicates that ${\widehat{v}}_{PB}$ is convex in $\theta$. Moreover, ${\widehat{v}}_{RB}\left(\theta =0\right)=+\infty ={\widehat{v}}_{PB}\left(\theta =0\right)$, ${\widehat{v}}_{RB}\left(\theta =0.1\right)=3.78{c}_{o}+5.21{c}_{s}+32.78{h}_{s}>{\widehat{v}}_{PB}\left(\theta =0.1\right)=1.68{c}_{o}+12.98{h}_{s}$, and ${\widehat{v}}_{RB}\left(\theta =1\right)=c+2{c}_{s}<{\widehat{v}}_{PB}\left(\theta =1\right)=+\infty$. Thus, there exists a unique $\widehat{\theta }$ for ${\widehat{v}}_{RB}={\widehat{v}}_{PB}$, which is the root of $2{c}_{s}\left(1-\theta \right)\left(\sqrt{2-\theta }+\theta -2\right)-2{h}_{s}+{h}_{s}\left(\left(\sqrt{\theta }-2\right)\sqrt{2-\theta }+\left(5+2\sqrt{2-\theta }-2\theta \right)\theta \right)-(1-\theta )(2\sqrt{2-\theta }-2+\theta -\sqrt{(2-\theta )\theta }{)c}_{o}=0$.

When $\theta <\widehat{\theta }$, we have ${\widehat{v}}_{RB}>{\widehat{v}}_{PB}$; otherwise, we have ${\widehat{v}}_{RB}\le {\widehat{v}}_{PB}$. If ${\widehat{v}}_{RB}>{\widehat{v}}_{PB}$, we have ${\pi }^{P*}>{\pi }^{B*}>{\pi }^{R*}$ for ${\widehat{v}}_{PB}<v<{\widehat{v}}_{RB}$, which indicates that ${\widehat{v}}_{PB}>{\widehat{v}}_{PR}$. Therefore, ${\widehat{v}}_{RB}>{\widehat{v}}_{PB}>{\widehat{v}}_{PR}$ always holds when $\theta <\widehat{\theta }$. Adversely, when ${\widehat{v}}_{RB}\le {\widehat{v}}_{PB}$, we have ${\pi }^{R*}\ge {\pi }^{B*}\ge {\pi }^{P*}$ for ${\widehat{v}}_{RB}\le v\le {\widehat{v}}_{PB}$, which indicates that ${\widehat{v}}_{PB}\le {\widehat{v}}_{PR}$. Thus, ${\widehat{v}}_{PR}\ge {\widehat{v}}_{PB}\ge {\widehat{v}}_{RB}$ always holds when $\theta \ge \widehat{\theta }$. In summary, ${\widehat{v}}_{PR}<{\widehat{v}}_{PB}<{\widehat{v}}_{RB}$ if $\theta <\widehat{\theta }$, and ${\widehat{v}}_{PR}\ge {\widehat{v}}_{PB}\ge {\widehat{v}}_{RB}$ if $\theta \ge \widehat{\theta }$.

Therefore, for $\theta \le 0.5$ or $\theta >0.5$ and ${c}_{o}\le 2{c}_{s}$ in the common feasible domain, when $v\ge {\text{max}}({\widehat{v}}_{PB},{\widehat{v}}_{PR})$, the proactive strategy is optimal for the retailer; when $\theta >\widehat{\theta }$ and ${\widehat{v}}_{RB}<v<{\widehat{v}}_{PR}$, the reactive strategy is optimal; when $v\le {\text{min}}({\widehat{v}}_{RB},{\widehat{v}}_{PB})$, the benchmark strategy is optimal.

1.8 Proof of Proposition 7

The common feasible area of the three strategies is $\theta >0.5$, ${h}_{s}\le {\text{min}}(\frac{1-\theta }{\theta },1)$, and ${\text{max}}\left({\underline{v}}^{B},{\underline{v}}^{R},{\underline{v}}^{H}\right)\le v\le {{\text{min}}({\overline{v} }^{B},\overline{v} }^{R},{\overline{v} }^{H})$.

The profit differences are ${\pi }^{H*}-{\pi }^{B*}=\frac{{B}_{1}}{4(2-\theta )\theta }$, ${\pi }^{H*}-{\pi }^{R*}=\frac{{B}_{2}}{4\theta }$, where ${B}_{1}=2\left(1-\theta \right){\theta }^{2}{\left(v-c\right)}^{2}-2\theta (\left(2-\theta \right){h}_{s}-\left(1-\theta \right)\theta {c}_{o})\left(v-c\right)-\theta {{c}_{o}}^{2}+(2-\theta )({{h}_{s}}^{2}+{\theta }^{2}{{c}_{o}}^{2}-4{\theta h}_{s}(\left(2\theta -1\right){c}_{s}+\left(1-\theta \right){c}_{o}))$ and ${B}_{2}=\left(1-\theta \right){\theta }^{2}{\left(v-c\right)}^{2}-2\theta ({h}_{s}-\left(1-\theta \right)\theta ({h}_{s}+{c}_{s}))\left(v-c\right)+{{h}_{s}}^{2}-{\left({h}_{s}+{c}_{s}\right)}^{2}\theta {(1+\theta }^{2})+2({{c}_{s}}^{2}+{{h}_{s}}^{2}){\theta }^{2}$.

Obviously, ${B}_{1}$ and ${B}_{2}$ are convex in $v$. Letting ${B}_{1}=0$ yields ${v}_{HB1}=c+\frac{\left(2-\theta \right){h}_{s}-\theta \left(1-\theta \right){{\text{c}}}_{o}-\sqrt{\theta (2-\theta )({{c}_{o}}^{2}{\left(1-\theta \right)}^{2}+2{c}_{o}{h}_{s}(3-7\theta +4{\theta }^{2})+{h}_{s}({h}_{s}+8(3\theta -2{\theta }^{2}-1){c}_{s}))}}{2(1-\theta )\theta }$ and ${v}_{HB2}=c+\frac{\left(2-\theta \right){h}_{s}-\theta \left(1-\theta \right){{\text{c}}}_{o}+\sqrt{\theta (2-\theta )({{c}_{o}}^{2}{\left(1-\theta \right)}^{2}+2{c}_{o}{h}_{s}(3-7\theta +4{\theta }^{2})+{h}_{s}({h}_{s}+8(3\theta -2{\theta }^{2}-1){c}_{s}))}}{2(1-\theta )\theta }$. Since ${v}_{HB1}-{\underline{v}}^{H}=-\frac{\theta \left(1-\theta \right){{\text{c}}}_{o}+\theta \left(4\theta -1\right){h}_{s}+\sqrt{\theta (2-\theta )({{c}_{o}}^{2}{\left(1-\theta \right)}^{2}+2{c}_{o}{h}_{s}(3-7\theta +4{\theta }^{2})+{h}_{s}({h}_{s}+8(3\theta -2{\theta }^{2}-1){c}_{s}))}}{2(1-\theta )\theta }<0$ for $\theta >0.5$, we have ${B}_{1}\le 0$ if $v\le {v}_{HB2}$ and ${B}_{1}>0$ if ${v>v}_{HB2}$. Denote ${v}_{HB2}$ with ${\widehat{v}}_{HB}$. We then have ${\pi }^{H*}\le {\pi }^{B*}$ if $v\le {\widehat{v}}_{HB}$, and ${\pi }^{H*}>{\pi }^{B*}$ if $v>{\widehat{v}}_{HB}$ in the common feasible domain.

Following a similar line of reasoning, we can obtain ${\pi }^{H*}\le {\pi }^{R*}$ if $v\le {\widehat{v}}_{HR}$, and ${\pi }^{H*}>{\pi }^{R*}$ if $v>{\widehat{v}}_{HR}$ in the common feasible domain, where ${\widehat{v}}_{HR}=c-{c}_{s}+\frac{\left(1-\theta +{\theta }^{2}\right){h}_{s}+({c}_{s}-{c}_{s}\theta +{h}_{s}\theta )\sqrt{\theta }}{(1-\theta )\theta }$.

With a proof similar to that of Proposition 6, we can see that ${\widehat{v}}_{RB}$ is decreasing in $\theta$ and ${\widehat{v}}_{HR}$ is convex in $\theta$, since $\frac{{\partial }^{2}{\widehat{v}}_{HR}}{\partial {\theta }^{2}}=\frac{3\sqrt{\theta }{(1-\sqrt{\theta })}^{3}{c}_{s}+(8-24\sqrt{\theta }+24\theta -9\theta \sqrt{\theta }-3{\theta }^{2}){h}_{s}}{4{\theta }^{3}{(1-\sqrt{\theta })}^{3}}>0$. Moreover, we have ${\widehat{v}}_{RB}\left(\theta =0\right)=+\infty ={\widehat{v}}_{HR}\left(\theta =0\right)$, ${\widehat{v}}_{RB}\left(\theta =0.5\right)=0.44{c}_{o}+0.55{c}_{s}+5.44{h}_{s}>{\widehat{v}}_{HR}\left(\theta =0.5\right)=0.42{c}_{s}+4.42{h}_{s}$, and ${\widehat{v}}_{RB}\left(\theta =1\right)=c+2{c}_{s}<{\widehat{v}}_{HR}\left(\theta =1\right)=+\infty$. Thus, there exists a unique ${\widehat{\theta }}_{1}$ for ${\widehat{v}}_{RB}={\widehat{v}}_{HR}$, which is the root of $(1-\sqrt{2-\theta }-\sqrt{\theta }+\sqrt{\left(2-\theta \right)\theta }){c}_{o}-(2-\sqrt{2-\theta }-3\sqrt{\theta }+\theta +\sqrt{\left(2-\theta \right)\theta }){c}_{s}-(1+\sqrt{2-\theta }-\sqrt{\theta }-\theta -\sqrt{\left(2-\theta \right)\theta }){h}_{s}=0$. Consequently, for $0.5<\theta <1$, we have ${\widehat{v}}_{HR}<{\widehat{v}}_{HB}<{\widehat{v}}_{RB}$ if $\theta <{\widehat{\theta }}_{1}$, and ${\widehat{v}}_{HR}\ge {\widehat{v}}_{HB}\ge {\widehat{v}}_{RB}$ if $\theta \ge {\widehat{\theta }}_{1}$.

In summary, for $\theta >0.5$ and ${c}_{o}>2{c}_{s}$ in the common feasible domain, when $v\ge {\text{max}}({\widehat{v}}_{HB},{\widehat{v}}_{HR})$, the hybrid strategy is optimal; when $\theta >{\widehat{\theta }}_{1}$ and ${\widehat{v}}_{RB}<v<{\widehat{v}}_{HR}$, the reactive strategy is optimal; when $v\le {\text{min}}({\widehat{v}}_{RB},{\widehat{v}}_{HB})$, the benchmark strategy is optimal.

1.9 Proof of Proposition 8

Let us define $\Delta {\pi }^{HPr*}={\pi }^{Hr*}-{\pi }^{Pr*}=\frac{1}{8}(\left(1-\theta \right){{c}_{o}}^{2}+2{c}_{o}\left(2\left({c}_{s}+{h}_{s}\right)\theta -\left(2{c}_{s}+{h}_{s}\right)\right)+2{\left({c}_{s}+{h}_{s}\right)}^{2}-\frac{{{h}_{s}}^{2}}{1-\theta }-2({{c}_{s}}^{2}+6{c}_{s}{h}_{s}+{{h}_{s}}^{2})\theta )$, which is convex in ${c}_{o}$. Letting $\Delta {\pi }^{HPr*}=0$ yields ${c}_{o1}=2{c}_{s}+\frac{\left(1-2\theta \right){h}_{s}-\sqrt{2}({c}_{s}-\theta {c}_{s}+\theta {h}_{s})}{1-\theta }$ and ${c}_{o2}=2{c}_{s}+\frac{\left(1-2\theta \right){h}_{s}+\sqrt{2}({c}_{s}-\theta {c}_{s}+\theta {h}_{s})}{1-\theta }$. Table 2 shows that the feasible domain for the hybrid strategy is ${c}_{o}{<h}_{s}+{c}_{s}$ and ${c}_{o}>{c}_{s}-\frac{(3\theta -1){h}_{s}}{1-\theta }$. Since ${c}_{o1}-{c}_{s}+\frac{\left(3\theta -1\right){h}_{s}}{1-\theta }=-\frac{\left(\sqrt{2}-1\right)\left(\left(1-\theta \right){c}_{s}+\theta {h}_{s}\right)}{1-\theta }<0$, and ${c}_{o2}-\left({h}_{s}+{c}_{s}\right)=\frac{{h}_{s}+\frac{\left(3+2\sqrt{2}\right)\left(1-\theta \right){c}_{s}}{\theta }}{1-\theta }>0$, both roots are outside of the feasible domain. Therefore, we have ${\pi }^{Hr*}<{\pi }^{Pr*}$ in the common feasible domain.

Table 2 Equilibrium outcomes in the case of partial refunds

The profit differences among other strategies are:

${\pi }^{Rr*}-{\pi }^{Br*}=\frac{\left(1-\theta \right){\left({c}_{o}-2\left({c}_{s}+{h}_{s}\right)+\left({c}_{s}+{h}_{s}+v-c\right)\theta \right)}^{2}}{4\left(2-\theta \right)}>0$,

${\pi }^{Pr*}-{\pi }^{Br*}=\frac{{(\left({c}_{o}+2(v-c)\right)\left(1-\theta \right)\theta -{h}_{s}(2-\theta ))}^{2}}{8(2-\theta )(1-\theta )\theta }>0$, and

${\pi }^{Pr*}-{\pi }^{Rr*}=\frac{1}{8\left(1-\theta \right)\theta }(2{\theta }^{2}{\left(1-\theta \right)}^{2}(v-c{)}^{2}-4\left(1-\theta \right)\theta \left({h}_{s}-\left({c}_{s}+{h}_{s}\right)\left(1-\theta \right)\theta \right)\left(v-c\right)+{\left(1-\theta \right)}^{2}\theta \left(2{{c}_{s}}^{2}\theta -{\left({c}_{o}-2{c}_{s}\right)}^{2}\right)+{{h}_{s}}^{2}\left(2-\theta \right)\left(1-2{\left(1-\theta \right)}^{2}\theta \right)-2{h}_{s}(1-\theta )\theta (2{c}_{s}(2-\theta )(1-\theta )+{c}_{o}(2\theta -1)))$.

Letting ${\pi }^{Pr*}={\pi }^{Rr*}$ yields $v=c+\frac{1}{2\left(1-\theta \right)\theta }(2({h}_{s}-({c}_{s}+{h}_{s})\theta +({c}_{s}+{h}_{s}){\theta }^{2})\pm \sqrt{2\theta {({h}_{s}\left(1-2\theta \right)-\left(1-\theta \right)\left({c}_{o}-2{c}_{s}\right))}^{2}})$. Since the smaller root minus ${\underline{v}}^{Pr}=-\frac{\theta \left({h}_{s}\left(1-2\theta \right)-\left(1-\theta \right)\left({c}_{o}-2{c}_{s}\right)\right)\sqrt{2\theta {\left({h}_{s}\left(1-2\theta \right)-\left(1-\theta \right)\left({c}_{o}-2{c}_{s}\right)\right)}^{2}}}{2\left(1-\theta \right)\theta }<0$, we have ${\pi }^{Pr*}\le {\pi }^{Rr*}$ if $v<{\widehat{v}}_{PRr}$, and ${\pi }^{Pr*}>{\pi }^{Rr*}$ otherwise, where ${\widehat{v}}_{PRr}=c+\frac{1}{2\left(1-\theta \right)\theta }(2\left({h}_{s}-\left({c}_{s}+{h}_{s}\right)\theta +\left({c}_{s}+{h}_{s}\right){\theta }^{2}\right)+\sqrt{2\theta {({h}_{s}\left(1-2\theta \right)-\left(1-\theta \right)\left({c}_{o}-2{c}_{s}\right))}^{2}})$.

In summary, when $v\ge {\widehat{v}}_{PRr}$, the proactive strategy is optimal; otherwise, the reactive strategy is optimal.

Equilibrium outcomes and comparisons for extensions

See Tables 2, 3, and

Table 3 Equilibrium outcomes when consumers absorb ship** fees

Table 4 Equilibrium outcomes in cases of hassle cost differences

4.

The procedure of profit comparisons in extensions is similar to those of the main model. For brevity, we omit the detailed analyses. The final results are as follows.

2.1 Consumers absorb ship** fees of online returns

For ${c}_{o}\le 2{c}_{s}+f$, in the common feasible domain, when $v\ge {\text{max}}({\widehat{v}}_{PBf},{\widehat{v}}_{PRf})$, the proactive strategy is optimal; when $\theta >{\widehat{\theta }}_{f1}$ and ${\widehat{v}}_{RBf}<v<{\widehat{v}}_{PRf}$, the reactive strategy is optimal; when $v\le {\text{min}}({\widehat{v}}_{RBf},{\widehat{v}}_{PBf})$, the benchmark strategy is optimal. For ${c}_{o}>2{c}_{s}+f$, in the common feasible domain, when $v\ge {\text{max}}({\widehat{v}}_{HBf},{\widehat{v}}_{HRf})$, the hybrid strategy is optimal; when $\theta >{\widehat{\theta }}_{f2}$ and ${\widehat{v}}_{RBf}<v<{\widehat{v}}_{HRf}$, the reactive strategy is optimal; when $v\le {\text{min}}({\widehat{v}}_{RBf},{\widehat{v}}_{HBf})$, the benchmark strategy is optimal. The thresholds are:${\widehat{v}}_{RBf}=c+\frac{({c}_{s}+{h}_{s})(2-\theta )-{c}_{o}+\sqrt{{({c}_{o}-{c}_{s}+{h}_{s}-2f)}^{2}(2-\theta )}}{\theta }$, ${\widehat{v}}_{PBf}=c+\frac{{c}_{o}\theta (1-\theta )-{h}_{s}(2-\theta )+(({c}_{o}-2f)(1-\theta )+{h}_{s})\sqrt{(2-\theta )\theta }}{2(1-\theta )\theta }$, ${\widehat{v}}_{PRf}=c+\frac{\left(\begin{array}{c}{h}_{s}\left(1-\theta +{\theta }^{2}\right)-{c}_{s}\theta \left(1-\theta \right)\\ +\sqrt{(2\left({c}_{o}-f\right)\left(f-{h}_{s}\right)+{{c}_{s}}^{2}{\left(1-\theta \right)}^{2}+2\left(3{h}_{s}-2f\right)\left({c}_{o}-f\right)\theta +\left({{h}_{s}}^{2}+4{h}_{s}f-2{f}^{2}+2{c}_{o}\left(f-2{h}_{s}\right)\right){\theta }^{2}-2{c}_{s}(1-\theta )(2f(1-\theta )+{h}_{s}(3\theta -2)))\theta }\end{array}\right)}{(1-\theta )\theta }$, ${\widehat{v}}_{HRf}=c+\frac{{h}_{s}\left(1-\theta +{\theta }^{2}\right)-{c}_{s}\theta \left(1-\theta \right)+({c}_{s}-{c}_{s}\theta +{h}_{s}\theta )\sqrt{\theta }}{(1-\theta )\theta }$, and

$${\widehat{v}}_{HBf}=c+\frac{\left(\begin{array}{c}{c}_{o}\theta \left(1-\theta \right)-{h}_{s}\left(2-\theta \right)\\ +\sqrt{(2-\theta )\theta ({{h}_{s}}^{2}-8{h}_{s}f+8{f}^{2}+{{c}_{o}}^{2}{\left(1-\theta \right)}^{2}+16\left({h}_{s}-f\right)f\theta +8f\left(f-{h}_{s}\right){\theta }^{2}-8{c}_{s}\left(1-\theta \right)\left({h}_{s}-\left(1-\theta \right)f-2{h}_{s}\theta \right)-2{c}_{o}(1-\theta )(4(1-\theta )f+{h}_{s}(4\theta -3)))}\end{array}\right)}{2(1-\theta )\theta }$$

In addition, ${\widehat{\theta }}_{f1}$ and ${\widehat{\theta }}_{f2}$ are the roots for ${\widehat{v}}_{RBf}={\widehat{v}}_{PBf}$ and ${\widehat{v}}_{RBf}={\widehat{v}}_{HBf}$, respectively.

2.2 Cost differences

For ${c}_{o}\le 2{c}_{s}$, in the common feasible domain, when $v\ge {\text{max}}({\widehat{v}}_{PBd},{\widehat{v}}_{PRd})$, the proactive strategy is optimal for the retailer; when $\theta >{\widehat{\theta }}_{d1}$ and ${\widehat{v}}_{RBd}<v<{\widehat{v}}_{PRd}$, the reactive strategy is optimal; when $v\le {\text{min}}({\widehat{v}}_{RBd},{\widehat{v}}_{PBd})$, the benchmark strategy is optimal. For ${c}_{o}>2{c}_{s}$, in the common feasible domain, when $v\ge {\text{max}}({\widehat{v}}_{HBd},{\widehat{v}}_{HRd})$, the hybrid strategy is optimal for the retailer; when $\theta >{\widehat{\theta }}_{d2}$ and ${\widehat{v}}_{RBd}<v<{\widehat{v}}_{HRd}$, the reactive strategy is optimal; when $v\le {\text{min}}({\widehat{v}}_{RBd},{\widehat{v}}_{HBd})$, the benchmark strategy is optimal. The thresholds are:${\widehat{v}}_{RBd}=c+\frac{\left(2-\theta \right)\left({c}_{s}+{h}_{s}+{\alpha }_{s}\right)-{c}_{o}-{\alpha }_{o}+\sqrt{2-\theta }\left({c}_{o}-{c}_{s}+{h}_{s}-{\alpha }_{o}+{\alpha }_{s}\right)}{\theta }$, ${\widehat{v}}_{PBd}=c+\frac{\left(2-\theta \right){h}_{s}-\theta \left(1-\theta \right)({c}_{o}+{\alpha }_{o})+\sqrt{\theta (2-\theta )}\left(\left(1-\theta \right)({c}_{o}-{\alpha }_{o})+{h}_{s}\right)}{2(1-\theta )\theta }$, ${\widehat{v}}_{PRd}=c+\frac{\left(\begin{array}{c}\left(1-\theta +{\theta }^{2}\right){h}_{s}-\theta \left(1-\theta \right)\left({c}_{s}+{\alpha }_{s}\right)\\ +\sqrt{\theta ({{c}_{s}}^{2}{(1-\theta )}^{2}+2{c}_{o}(1-\theta )(({\alpha }_{o}-2{\alpha }_{s})(1-\theta )+{h}_{s}(2\theta -1))+{({\alpha }_{o}(1-\theta )-{h}_{s}\theta )}^{2}+2{c}_{s}(1-\theta )((3{\alpha }_{s}-2{\alpha }_{o})(1-\theta )+{h}_{s}(2-3\theta ))}\end{array}\right)}{(1-\theta )\theta }$, ${\widehat{v}}_{HBd}=c+\frac{\left(\begin{array}{c}\left(2-\theta \right){h}_{s}-\theta \left(1-\theta \right)\left({c}_{o}+{\alpha }_{o}\right)\\ +\sqrt{\theta (2-\theta )({({h}_{s}+{\alpha }_{o}(1-\theta ))}^{2}+{{c}_{o}}^{2}{(1-\theta )}^{2}-8{c}_{s}(1-\theta )({h}_{s}-({\alpha }_{o}-2{\alpha }_{s})(1-\theta )-2{h}_{s}\theta )+2{c}_{o}(1-\theta )((4{\alpha }_{s}-3{\alpha }_{o})(1-\theta )+{h}_{s}(3-4\theta )))}\end{array}\right)}{2(1-\theta )\theta }$, and ${\widehat{v}}_{HRd}=c+\frac{(1-\theta +{\theta }^{2}){h}_{s}-\theta (1-\theta )({c}_{s}+{\alpha }_{s})+\sqrt{\theta {(({c}_{s}-{\alpha }_{s})(1-\theta )+{h}_{s}\theta )}^{2}}}{(1-\theta )\theta }$.

In addition, ${\widehat{\theta }}_{d1}$ and ${\widehat{\theta }}_{d2}$ are the roots for ${\widehat{v}}_{RBd}={\widehat{v}}_{PBd}$ and ${\widehat{v}}_{RBd}={\widehat{v}}_{HBd}$, respectively.