Subdivided or aggregated online review systems: Which is better for online takeaway vendors?

Abstract

This paper examines the role of a subdivided or aggregated online review system to help online takeaway vendors select the most appropriate information strategy. First, we develop two models to depict the interaction between online vendors’ information strategies and consumers’ responses. Second, we take the multidimensional product attributes with their corresponding weights into consideration and illustrate that the sensitivity to product misfits, instead of the relative importance of product attributes, dominates profit maximization. Third, we make a comparison to find the most appropriate scenario to adopt a full or partial information strategy. When a large number of consumers satisfy the delivery time performance, an aggregated review system will be a better choice. Otherwise, vendors are advised to host a subdivided review system. Finally, we universally identify a variance boundary in the rating-star review system, which not only prevents consumers from expressing their real feelings but also makes observing consumer feedback and strategic adjustments inconvenient for online vendors.

Appendices

Appendix A

Summary of notations

Notation	Definition
$v$	Product valuation
$\hat{v}_{i}$	Customers’ expected product valuation in period i before consumption
αv	Product quality
(1 − α)v	Product delivery time
α	Relative importance of product attributes
τ	Customers’ unit misfit cost, where 0 ≤ τ ≤ v
$\hat{\tau }$	Customers’ expected unit misfit cost
β	Sensitivity to mismatched attributes
βτ	Quality misfit cost, where 0 ≤ βτ ≤ αv
(1 − β)τ	Time misfit cost, where 0 ≤ (1 − β)τ ≤ (1 − α)v
x t	Customers’ taste used to measure quality misfit, where 0 ≤ xt ≤ 1
x u	Customers’ time urgency used to measure time misfit, where 0 ≤ xu ≤ 1
f	The likelihood of customers dissatisfied with the delivery time
(1 − f)	The probability of customers satisfied with the delivery time
$u_{i}^{subdivided}$	Customers’ utility before consumption in period i for the subdivided review system
$u_{i}^{aggregated}$	Customers’ utility before consumption in period i for the aggregated review system
$D_{i}^{subdivided}$	Customers’ demand in period i for the subdivided review system
$D_{i}^{aggregated}$	Customers’ demand in period i for the aggregated review system
M q	Average rating of product quality in the subdivided review system
M t	Average rating of delivery time in the subdivided review system
M a	Average rating in the aggregated review system
V q	Rating variance in product quality in the subdivided review system
V t	Rating variance in delivery time in the subdivided review system
V a	Rating variance in the aggregated review system
V adt	Rating variance caused by accepted delivery time in the aggregated review system
$P_{i}^{subdivided}$	Product price in period i for the subdivided review system
$P_{i}^{aggregated}$	Product price in period i for the aggregated review system
$x_{t,i}^{u = 0}$	Customers’ taste with zero time misfit
$x_{u,i}^{t = 0}$	Customers’ time urgency with zero quality misfit
$\pi_{i}^{subdivided}$	Online vendors’ profit in period i for the subdivided review system
$\pi_{i}^{aggregated}$	Online vendors’ profit in period i for the aggregated review system

Appendix B

2.1 B.1 Proof of Proposition 1

To check the effects of the average rating and variance on the second-period equilibrium outcomes in the subdivided review system, we first calculate the first-order conditions of price. From Eq. (9), we have the following:

$$\frac{{\partial p_{2}^{subdivided*} }}{{\partial M_{q} }} = \frac{{\partial p_{2}^{subdivided*} }}{{\partial M_{t} }} = \frac{1}{3} > 0, \, \frac{{\partial p_{2}^{subdivided*} }}{{\partial V_{q} }} = \frac{1}{{2\sqrt {3V_{q} } }} > 0, \, \frac{{\partial p_{2}^{subdivided*} }}{{\partial V_{t} }} = \frac{1}{{2\sqrt {3V_{t} } }} > 0.$$

It shows that the best second-period price increases with the average ratings and variances of both product attributes.

Second, we solve the first-order conditions of demand. As shown below,

$$\begin{aligned} \frac{{\partial D_{2}^{subdivided*} }}{{\partial M_{q} }} & = \frac{{\partial D_{2}^{subdivided*} }}{{\partial M_{t} }} = \frac{{\left[ {\beta x_{q} + \left( {1 - \beta } \right)x_{t} } \right]^{2} \left( {M_{q} + M_{t} + \sqrt {3V_{q} } + \sqrt {3V_{t} } } \right)}}{{9\beta \left( {1 - \beta } \right)\left( {\sqrt {3V_{q} } + \sqrt {3V_{t} } } \right)^{2} }} > 0 \\ \frac{{\partial D_{2}^{subdivided*} }}{{\partial V_{q} }} & = \frac{{ - 3\left[ {\beta x_{q} + \left( {1 - \beta } \right)x_{t} } \right]^{2} \left( {M_{q} + M_{t} + \sqrt {3V_{q} } + \sqrt {3V_{t} } } \right)\left( {M_{q} + M_{t} } \right)}}{{18\beta \left( {1 - \beta } \right)\sqrt {3V_{q} } \left( {\sqrt {3V_{q} } + \sqrt {3V_{t} } } \right)^{3} }} < 0 \\ \frac{{\partial D_{2}^{subdivided*} }}{{\partial V_{t} }} & = \frac{{ - 3\left[ {\beta x_{q} + \left( {1 - \beta } \right)x_{t} } \right]^{2} \left( {M_{q} + M_{t} + \sqrt {3V_{q} } + \sqrt {3V_{t} } } \right)\left( {M_{q} + M_{t} } \right)}}{{18\beta \left( {1 - \beta } \right)\sqrt {3V_{t} } \left( {\sqrt {3V_{q} } + \sqrt {3V_{t} } } \right)^{3} }} < 0 \\ \end{aligned}$$

we know that the second-period demand increases with the average ratings caused by the quality and delivery time, but it decreases with the variances of both product attributes.

$$\begin{aligned} \frac{{\partial \pi_{2}^{subdivided*} }}{{\partial M_{q} }} & = \frac{{\partial \pi_{2}^{subdivided*} }}{{\partial M_{t} }} = \frac{{\left[ {\beta x_{q} + \left( {1 - \beta } \right)x_{t} } \right]^{2} \left( {M_{q} + M_{t} + \sqrt {3V_{q} } + \sqrt {3V_{t} } } \right)^{2} }}{{18\beta \left( {1 - \beta } \right)\left( {\sqrt {3V_{q} } + \sqrt {3V_{t} } } \right)^{2} }} > 0 \\ \frac{{\partial \pi_{2}^{subdivided*} }}{{\partial V_{q} }} & = \frac{{\left[ {\beta x_{q} + \left( {1 - \beta } \right)x_{t} } \right]^{2} \left( {M_{q} + M_{t} + \sqrt {3V_{q} } + \sqrt {3V_{t} } } \right)^{2} \left( {\sqrt {3V_{q} } + \sqrt {3V_{t} } - 2M_{q} - 2M_{t} } \right)}}{{36\sqrt {3V_{t} } \left( {\sqrt {3V_{q} } + \sqrt {3V_{t} } } \right)^{3} }} \\ \frac{{\partial \pi_{2}^{subdivided*} }}{{\partial V_{t} }} & = \frac{{\left[ {\beta x_{q} + \left( {1 - \beta } \right)x_{t} } \right]^{2} \left( {M_{q} + M_{t} + \sqrt {3V_{q} } + \sqrt {3V_{t} } } \right)^{2} \left( {\sqrt {3V_{q} } + \sqrt {3V_{t} } - 2M_{q} - 2M_{t} } \right)}}{{36\sqrt {3V_{t} } \left( {\sqrt {3V_{q} } + \sqrt {3V_{t} } } \right)^{3} }} \\ \end{aligned}$$

The first-order conditions of profit prove that the second-period profit increases with the average ratings of the quality and delivery time. Only when $\sqrt {3V_{q} } + \sqrt {3V_{t} }$ is larger than $2M_{q} + 2M_{t}$, the profit increases with the variances of both product attributes. However, $\sqrt {3V_{q} } + \sqrt {3V_{t} }$ cannot exceed $2M_{q} + 2M_{t}$. Assuming that a is the highest rating score and b is the lowest rating score in any rating scale, as we know, when customers publish extreme ratings, it results in the largest variance. Thus, we have $V_{q} = V_{t} = \left( {\frac{a - b}{2}} \right)^{2} = M_{q} = M_{t} \,$ and $3\sqrt {V_{q} } < M_{q} ,{ 3}\sqrt {V_{t} } < M_{t}$. This means that a variance boundary exists in current rating stars review systems.

2.2 B.2 Proof of Proposition 2

To check the impacts of the average ratings and variances on the second-period equilibrium outcomes in the aggregated review system, we first calculate the first-order conditions of price. From Eq. (13), we have the following:

$$\begin{aligned} \frac{{\partial p_{2}^{aggregated*} }}{{\partial M_{a} }} & = \frac{{\left( {M_{a}^{2} + V_{a} } \right)^{2} + V_{a} \sqrt {2V_{adt} \left( {M_{a}^{2} + V_{a} } \right)} }}{{3\left( {M_{a}^{2} + V_{a} } \right)^{2} }} > 0 \\ \frac{{\partial p_{2}^{aggregated*} }}{{\partial V_{a} }} & = - \frac{{M_{a} \sqrt {2V_{adt} \left( {M_{a}^{2} + V_{a} } \right)} }}{{6\left( {M_{a}^{2} + V_{a} } \right)^{2} }} < 0 \\ \frac{{\partial p_{2}^{aggregated*} }}{{\partial V_{adt} }} & = \frac{{M_{a} }}{{3\sqrt {2V_{adt} \left( {M_{a}^{2} + V_{a} } \right)} }} > 0. \\ \end{aligned}$$

It describes that the best second-period price increases with the aggregated average rating and the variance caused by accepted delivery time and decreases with the aggregated rating variance otherwise.

Second, we solve the first-order conditions of demand. As seen below,

$$\begin{aligned} \frac{{\partial D_{2}^{aggregated*} }}{{\partial M_{h} }} & = \frac{{8M_{a}^{3} \left[ {\beta x_{q} + \left( {1 - \beta } \right)x_{t} } \right]^{2} \left[ {V_{a} + M_{a}^{2} + \sqrt {2V_{adt} \left( {M_{a}^{2} + V_{a} } \right)} } \right]\left[ {V_{a} + M_{a}^{2} + \sqrt {2V_{adt} \left( {M_{a}^{2} + V_{a} } \right)} } \right]}}{{81\beta \left( {1 - \beta } \right)V_{a} V_{adt} \left( {M_{a}^{2} + V_{a} } \right)^{2} }} > 0 \\ \frac{{\partial D_{2}^{aggregated*} }}{{\partial V_{h} }} & = \frac{{ - M_{a}^{4} \left[ {\beta x_{q} + \left( {1 - \beta } \right)x_{t} } \right]^{2} \left[ {V_{a} + M_{a}^{2} + \sqrt {2V_{adt} \left( {M_{a}^{2} + V_{a} } \right)} } \right]}}{{81\beta \left( {1 - \beta } \right)}} \\ & \quad \cdot \frac{{\left[ {\left( {V_{a} + M_{a}^{2} } \right)\left( {M_{a}^{2} + 6V_{a} V_{adt} + 2V_{adt} M_{a}^{2} } \right) + V_{a} \left( {M_{a}^{2} - V_{a} } \right)} \right]}}{{V_{a} V_{adt}^{2} \left( {M_{a}^{2} + V_{a} } \right)}} < 0 \\ \frac{{\partial D_{2}^{aggregated*} }}{{\partial V_{adt} }} & = \frac{{ - M_{a}^{4} \left[ {\beta x_{q} + \left( {1 - \beta } \right)x_{t} } \right]^{2} \left[ {V_{a} + M_{a}^{2} + \sqrt {2V_{adt} \left( {M_{a}^{2} + V_{a} } \right)} } \right]}}{{81\beta \left( {1 - \beta } \right)V_{a} V_{adt}^{2} \left( {M_{a}^{2} + V_{a} } \right)}} < 0 \\ \end{aligned}$$

we know second-period demand increases with the aggregated average rating. However, it decreases with the aggregated rating variance and the variance caused by accepting delivery time.

$$\begin{aligned} \frac{{\partial \pi_{2}^{aggregated*} }}{{\partial M_{a} }} & = \frac{{M_{a}^{4} \left[ {\beta x_{q} + \left( {1 - \beta } \right)x_{t} } \right]^{2} \left[ {V_{a} + M_{a}^{2} + \sqrt {2V_{adt} \left( {M_{a}^{2} + V_{a} } \right)} } \right]^{2} }}{{243\beta \left( {1 - \beta } \right)}} \\ & \quad \cdot \frac{{\left[ {5\left( {M_{a}^{2} + V_{a} } \right)^{2} + 2M_{a}^{2} \sqrt {2V_{adt} \left( {M_{a}^{2} + V_{a} } \right)} + 5V_{a} \sqrt {2V_{adt} \left( {M_{a}^{2} + V_{a} } \right)} } \right]}}{{V_{a} V_{adt} \left( {M_{a}^{2} + V_{a} } \right)^{4} }} > 0 \\ \frac{{\partial \pi_{2}^{aggregated*} }}{{\partial V_{a} }} & = \frac{{ - M_{a}^{5} \left[ {\beta x_{q} + \left( {1 - \beta } \right)x_{t} } \right]^{2} \left[ {V_{a} + M_{a}^{2} + \sqrt {2V_{adt} \left( {M_{a}^{2} + V_{a} } \right)} } \right]^{2} }}{{243\beta \left( {1 - \beta } \right)}} \\ & \quad \cdot \frac{{\left\{ {\left. {\left( {M_{a} + V_{a} } \right)\left[ {V_{a} + M_{a}^{2} + \sqrt {2V_{adt} \left( {M_{a}^{2} + V_{a} } \right)} } \right] + \frac{3}{2}V_{a} \sqrt {2V_{adt} \left( {M_{a}^{2} + V_{a} } \right)} } \right\}} \right.}}{{V_{a}^{2} V_{adt} \left( {M_{a}^{2} + V_{a} } \right)^{4} }} < 0 \\ \frac{{\partial \pi_{2}^{aggregated*} }}{{\partial V_{adt} }} & = \frac{{ - M_{a}^{5} \left[ {\beta x_{q} + \left( {1 - \beta } \right)x_{t} } \right]^{2} \left[ {V_{a} + M_{a}^{2} + \sqrt {2V_{adt} \left( {M_{a}^{2} + V_{a} } \right)} } \right]^{2} \left[ {M_{a}^{2} + V_{a} - \frac{1}{2}\sqrt {2V_{adt} \left( {M_{a}^{2} + V_{a} } \right)} } \right]}}{{243\beta \left( {1 - \beta } \right)V_{a} V_{adt}^{2} \left( {M_{a}^{2} + V_{a} } \right)^{3} }} \\ \end{aligned}$$

The first-order conditions of profit illustrate that the second-period profit increases with the aggregated average rating and decreases with the aggregated rating variance. Only when the variance caused by the accepting delivery time is higher than a threshold, i.e., $V_{acc} > 2\left( {V_{a} + M_{a}^{2} } \right),$ the profit increase. However, $V_{acc}$ cannot exceed $2\left( {V_{a} + M_{a}^{2} } \right),$ which is restricted by the variance boundary.

2.3 B.3 Proof of Proposition 3

Regardless of which types of online review systems online vendors apply, the true value of the product is fixed. From Eqs. (6) and (12), we obtain the relationship between rating scores and variances as $\left( {M_{q} + M_{t} + \sqrt {3V_{q} } + \sqrt {3V_{t} } } \right)^{3} = \frac{{\left[ {V_{a} + M_{a}^{2} + \sqrt {2V_{adt} \left( {M_{a}^{2} + V_{a} } \right)} } \right]^{3} }}{{M_{a}^{3} }}.$ Given the rating distribution of the aggregated review system, the aggregated average rating M_q is $\left( {1 - f} \right)\left( {v - \frac{{\beta x_{t} \tau + \left( {1 - \beta } \right)x_{u} \tau }}{3}} \right)$, and the aggregated rating variance V_a is $\left( {1 - f} \right)f\left( {v - \frac{{\beta x_{t} \tau + \left( {1 - \beta } \right)x_{u} \tau }}{3}} \right)^{2} .$ Based on the results of M_q and V_a, $\frac{f}{1 - f}$ is equal to $\frac{{V_{a} }}{{V_{a} + M_{a}^{2} }}.$ Because $\beta x_{t} \tau + \left( {1 - \beta } \right)x_{u} \tau = 2\left( {\sqrt {3V_{q} } + \sqrt {3V_{t} } } \right),$ the variance caused by accepting the delivery time can be rewritten as $V_{acc} = \frac{{2M_{a}^{2} \left( {\sqrt {3V_{q} } + \sqrt {3V_{t} } } \right)^{2} }}{{9\left( {V_{a} + M_{a}^{2} } \right)}}.$ Thus, we are able to simplify Eq. (17) as follows,

$$\begin{aligned} \Delta \pi & = \frac{{\left[ {\beta x_{q} + \left( {1 - \beta } \right)x_{t} } \right]^{2} \left( {M_{q} + M_{t} + \sqrt {3V_{q} } + \sqrt {3V_{t} } } \right)^{3} }}{{54\beta \left( {1 - \beta } \right)\left( {\sqrt {3V_{q} } + \sqrt {3V_{t} } } \right)^{2} }} - \frac{{M_{a}^{5} \left[ {\beta x_{q} + \left( {1 - \beta } \right)x_{t} } \right]^{2} \left[ {V_{a} + M_{a}^{2} + \sqrt {2V_{acc} \left( {M_{a}^{2} + V_{a} } \right)} } \right]^{3} }}{{243\beta \left( {1 - \beta } \right)V_{a} V_{acc} \left( {M_{a}^{2} + V_{a} } \right)^{3} }} \\ & = \frac{{\left[ {\beta x_{q} + \left( {1 - \beta } \right)x_{t} } \right]^{2} \left[ {V_{a} + M_{a}^{2} + \sqrt {2V_{adt} \left( {M_{a}^{2} + V_{a} } \right)} } \right]^{3} \left[ {V_{a} \left( {M_{a}^{2} + V_{a} } \right)^{2} - M_{a}^{6} } \right]}}{{54\beta \left( {1 - \beta } \right)V_{a} \left( {M_{a}^{2} + V_{a} } \right)^{2} M_{a}^{3} \left( {\sqrt {3V_{q} } + \sqrt {3V_{t} } } \right)^{2} }}. \\ \end{aligned}$$

It indicates that the profit difference depends on $V_{a} \left( {M_{a}^{2} + V_{a} } \right)^{2} - M_{a}^{6}$, that is, $f\left( {1 - f} \right)^{3} .$ When the likelihood of a dissatisfied delivery time f^* > 0.317, the subdivided review system arouses much more profits than the aggregated review system does.

Let $A = \left[ {V_{a} + M_{a}^{2} + \sqrt {2V_{acc} \left( {M_{a}^{2} + V_{a} } \right)} } \right]^{3} \left[ {V_{a} \left( {M_{a}^{2} + V_{a} } \right)^{2} - M_{a}^{6} } \right]$ and $B = 54V_{a} \left( {M_{a}^{2} + V_{a} } \right)^{2} M_{a}^{3} \left( {\sqrt {3V_{q} } + \sqrt {3V_{t} } } \right)^{2}$, substituting these terms back to $\Delta \pi_{2} = \pi_{2}^{subdivided*} - \pi_{2}^{aggregated*}$ yields $\Delta \pi_{2} = \frac{{A\left[ {\beta x_{t} + \left( {1 - \beta } \right)x_{u} } \right]^{2} }}{{\beta \left( {1 - \beta } \right)B}}$. Thus, we derive $\frac{{\partial \Delta \pi_{2} }}{\partial \beta } = \frac{{\beta x_{t} + \beta x_{u} - x_{u} }}{{\left[ {\beta \left( {1 - \beta } \right)B} \right]^{2} }}.$ Since $\frac{{\partial^{2} \Delta \pi_{2} }}{{\partial \beta^{2} }} = \frac{{x_{u}^{2} x_{t} + x_{t}^{2} x_{u} }}{{\left[ {\beta \left( {1 - \beta } \right)B} \right]^{4} }} > 0$, we know that if customers’ sensitivity to mismatched attributes $\beta^{*} = \frac{{x_{u} }}{{x_{t} + x_{u} }}$, online vendors have the minimum profit in the second period. Similarly, let $C = 2\hat{v}^{3} \left[ {f - \left( {1 - f} \right)^{3} } \right]$ and $D = 27\tau^{2} f$, substituting these terms back to $\Delta \pi_{ 1} = \pi_{ 1}^{subdivided*} - \pi_{ 1}^{aggregated*}$ yields $\Delta \pi_{ 1} = \frac{C}{{\beta \left( {1 - \beta } \right)D}}$. Thus, we derive $\frac{{\partial \Delta \pi_{1} }}{\partial \beta } = \frac{{ - \left( {1 - 2\beta } \right)D}}{{\left[ {\beta \left( {1 - \beta } \right)D} \right]^{ 2} }}$. Since $\frac{{\partial^{ 2} \Delta \pi_{1} }}{{\partial \beta^{ 2} }} = \frac{{ 2D^{2} \left[ {\beta \left( {1 - \beta } \right)} \right]^{ 3} + 4\left( {1 - 2\beta } \right)^{2} }}{{D\left[ {\beta \left( {1 - \beta } \right)} \right]^{ 3} }} > 0$, when $\beta^{*} = 0.5$, first-period profit reaches the lowest point. In terms of these results, we can deduce that online vendors’ profits for both online review systems are directly related to customers’ sensitivity to mismatched attributes in any periods.