Pricing and Commission in Two-Sided Markets with Free Upgrades

Abstract

We address the problem of optimal pricing in two-sided markets with platforms that facilitate the exchange of services between freelance workers and customers. Often, such platforms offer multiple variants of the same service and in cases when the cheaper service variant witnesses a shortage of supply and the more expensive variant sees a surplus, the platform offers free upgrades to the customers of the cheaper service variant. In this work, we explore the impact of such free upgrades on the platform’s revenue and throughput. In addition, for the setting where the demand and supply are unknown to the platform and the platform has to perform the joint task of supply/demand estimation and pricing, we devise an algorithm based on a strategic division of the search space that enables the platform to efficiently determine throughput and revenue optimal prices. Further, we ascertain the optimal value of the commission retained by the platform per transaction to maximize its revenue.

Ankur’s work was supported in part by the grant SB/S3/EECE/0182/2014 of the Science and Engineering Research Board, Government of India.

7 Appendix

1.1 7.1 Proof of Theorem 1

We separately analyse the throughput function in subsets I, II, III, A and B of the feasible region defined in Sect. 3 (Fig. 2), first arriving at the optima within individual subsets and later finding the global optima. We additionally define region IV as ${A} \cup {B} \cup {C}$.

Region I. Here, $w_1(p_1) \le c_1(p_1),\ w_2(p_2) \le c_2(p_2)$. Thus, $N(p_1,p_2) = w_1(p_1)+w_2(p_2).$ Now, $\large \frac{\partial N}{\partial p_1}>0,\ \frac{\partial N}{\partial p_2}>0.$ From the sign of partial derivatives, the optimal price occurs at maximum feasible value of both $p_1$ and $p_2$ which is $(p_{1 \mathrm{bal}},p_{2 \mathrm{bal}})$.

Region II. Here, $c_1(p_1) \le w_1(p_1),\ w_2(p_2) \le c_2(p_2)$. Thus, $N(p_1,p_2) = c_1(p_1)+w_2(p_2).$ Now, $\large \frac{\partial N}{\partial p_1}<0,\ \frac{\partial N}{\partial p_2} > 0$. The optimal price occurs at minimum feasible value of $p_1$ and maximum feasible value of $p_2$ which is $(p_{1 \mathrm{bal}},p_{2 \mathrm{bal}})$.

Region III. Here, $c_1(p_1) \le w_1(p_1),\ c_2(p_2) \le w_2(p_2)$ and $N(p_1,p_2) = c_1(p_1)+c_2(p_2).$ Now, $\frac{\partial N}{\partial p_1}<0,\ \frac{\partial N}{\partial p_2} < 0$. The optimal price occurs at minimum feasible value of $p_1$ and $p_2$ which is again $(p_{1 \mathrm{bal}},p_{2 \mathrm{bal}})$.

Region IV. This is perhaps the most interesting region as there’s a possibility of free upgrades. Here, $w_1(p_1) \le c_1(p_1),\ c_2(p_2) \le w_2(p_2)$. For the purpose of analyzing the throughput, we separately analyze the following 2 regions,

– Region A: Here, $w_2(p_2)-c_2(p_2)\le c_1(p_1)-w_1(p_1),$ and $N(p_1,p_2) = w_1(p_1)+w_2(p_2).$ Moreover, $\frac{\partial N}{\partial p_1}>0,\ \frac{\partial N}{\partial p_2} > 0$. Thus, the optimal price occurs at maximum allowed feasible values of $p_1$ and $p_2$ which pushes the optimal price to curve C.

– Region B: Here, $c_1(p_1)-w_1(p_1) \le w_2(p_2)-c_2(p_2),$ and $N(p_1,p_2) =c_1(p_1)+c_2(p_2)$. Now, $\frac{\partial N}{\partial p_1}<0,\ \frac{\partial N}{\partial p_2} < 0$. As above, the optimal price occurs at minimum allowed feasible values of $p_1$ and $p_2$ which correspond to curve C.

1.2 7.2 Proof of Proposition 1

In Sect. 3, we have shown that throughput must lie on the curve C. The function $c_2(p_2)-w_2(p_2)$ is a strictly decreasing function of $p_2$ and therefore invertible. Hence, we can parameterize C as $p_1=(w_1-c_1)^{-1}(c_2(p_2)-w_2(p_2))$. Denote, $\eta (p_2):=w_1((w_1-c_1)^{-1}(c_2(p_2)-w_2(p_2)))+w_2(p_2).$ It can be verified that, $\dfrac{d\eta (p_2)}{dp_2} = \dfrac{w_1'( p_1 ) c_2'(p_2)-w_2'(p_2)c_1'(p_1)}{w_1'(p_1)-c_1'(p_1)}.$

Note that $w_1'(p_1)-c_1'(p_1)>0 \ \forall p_1$. Consequently, the sign of $w_1'( p_1 ) c_2'(p_2)-w_2'(p_2)c_1'(p_1)$ determines the sign of $\frac{d\eta }{dp_2}$. If $w_1'( p_1 ) c_2'(p_2)<w_2'(p_2)c_1'(p_1)$ along curve C, then throughput is optimal at the least feasible value $p_2$ which in fact corresponds to the equilibrium price $(p_{1 \mathrm{bal}},p_{2 \mathrm{bal}})$. On the other hand, if $w_1'( p_1 ) c_2'(p_2)>w_2'(p_2)c_1'(p_1)$, then the throughput is optimal at the least allowed value of $p_1$ or equivalently the greatest allowed $p_2$ along curve C. Noting that $|c_1'(p_1)|=-c_1'(p_1), |c_2'(p_2)|=-c_2'(p_2)$, and that $w'_1(p_1),w_2'(p_2)>0$ we get the required result.

1.3 7.3 Proof of Theorem 2

We separately analyse the revenue function in regions I, II, III, $\bar{A},\bar{B}$ and D of the feasible region as defined in Sect. 4 (Fig. 3). We additionally define region IV as $\bar{A} \cup \bar{B} \cup \bar{C} \cup D$.

Region I. Here, $w_1(p_1) \le c_1(p_1),\ w_2(p_2) \le c_2(p_2)$. Thus, $R(p_1,p_2) = \gamma p_1 w_1(p_1)+\gamma p_2 w_2(p_2)$ whereby, $\large \frac{\partial R}{\partial p_1}>0,\ \frac{\partial R}{\partial p_2}>0.$ From the sign of partial derivatives, the optimal price occurs at maximum feasible value of both $p_1$ and $p_2$ which is $(p_{1 \mathrm{bal}},p_{2 \mathrm{bal}})$.

Region II. Here, $c_1(p_1) \le w_1(p_1),\ w_2(p_2) \le c_2(p_2)$. Thus, $R(p_1,p_2) = \gamma p_1 c_1(p_1)+\gamma p_2 w_2(p_2).$ Now, $\frac{\partial R}{\partial p_2} > 0$. Thus, $\forall p_1, \ p_2$ takes the maximum feasible value. Hence, optimal price occurs on line $\{(p_1,p_2)|p_2= p_{2 \mathrm{bal}}\}$.

Region III. Here, $c_1(p_1) \le w_1(p_1),\ c_2(p_2) \le w_2(p_2)$ and consequently there are no free upgrades. As discussed in Sect. 4, the optimal price becomes $(\min \{p_{1 \mathrm{opt}},p_{2 \mathrm{opt}}\},p_{2 \mathrm{opt}})$.

Region IV. Here, $w_1(p_1) \le c_1(p_1),\ c_2(p_2) \le w_2(p_2)$ and there’s a possibility of free upgrades if the transaction price of an upgrade $p_1-(1-\gamma )p_2$ is positive. We separately analyze regions $\bar{A},\bar{B} \setminus \widehat{B}$ and D.

– Region D: Here, $w_1(p_1) \le c_1(p_1),\ c_2(p_2) \le w_2(p_2)$ but there are no free upgrades as $p_1<(1-\gamma )p_2$. Thus, $R(p_1,p_2) = \gamma p_1 w_1(p_1)+\gamma p_2 c_2(p_2).$ Now, $\large \frac{\partial R}{\partial p_1}>0$. Thus, $\forall p_2, \ p_1$ takes the maximum feasible value. Hence, the optimal prices cannot lie in interior of region D.

– Region $\bar{A}$: Here, $w_2(p_2)-c_2(p_2)\le c_1(p_1)-w_1(p_1), \ p_1 \ge (1-\gamma )p_2$. In this region, all unmatched Type 2 workers serve Type 1 customers, whereby,

$$\begin{aligned} R(p_1,p_2) = \gamma p_1 w_1(p_1)+\gamma p_2 c_2(p_2) +(p_1-(1-\gamma )p_2)(w_2(p_2)-c_2(p_2)). \end{aligned}$$

It is easy to see that $\frac{\partial R}{\partial p_1}>0.$ Thus, $\forall p_2, \ p_1$ takes the maximum feasible value. Hence, the optimal prices cannot lie in the interior of this region and are pushed to curve $\bar{C}$.

– Region$\bar{B}\setminus \widehat{B}$: Here, $p_2 \ge p_{2 \mathrm{opt}},\ p_1 \ge (1-\gamma )p_2, \ c_1(p_1)-w_1(p_1) \le w_2(p_2)-c_2(p_2).$ In this region, all unmatched Type 1 customers are provided a free upgrade to Type 2 service. Consequently, $R(p_1,p_2) = \gamma p_1 w_1(p_1) +\gamma p_2 c_2(p_2) +(p_1-(1-\gamma )p_2)(c_1(p_1)-w_1(p_1)).$ Observe that,

$$\begin{aligned} \frac{\partial R}{\partial p_2}&= \gamma c_2(p_2)+\gamma p_2 c_2'(p_2)-(1-\gamma )(c_1(p_1)-w_1(p_1)). \end{aligned}$$

Since the function $p_2 c_2(p_2)$ has its maxima at $p_{2 \mathrm{opt}}$ after which it decreases. Thus, $\forall p_2 > p_{2 \mathrm{opt}}, (p_2c_2(p_2))' = c_2(p_2)+p_2 c_2'(p_2)< 0.$ Hence, $\frac{\partial R}{\partial p_2}<0$. Thus $\forall p_1,\ p_2$ takes the minimum feasible value in this region. Thus, the optimal price is pushed to region $\widehat{B} \cup C$. Therefore, revenue optimal prices lie in $(\min \{p_{1 \mathrm{opt}},p_{2 \mathrm{opt}}\},p_{2 \mathrm{opt}}) \cup \bar{C} \cup \widehat{B}.$