Analysis of an on-demand food delivery platform: Participatory equilibrium and two-sided pricing strategy

Abstract The on-demand food delivery platform assigns orders from customers to independent couriers, known as agents. An agent’s decision to provide services depends on the wage and the order assignment. The platform can adjust the actual supply and demand through two-sided pricing, such that an equilibrium between customers and agents that maximizes the platform’s profit can be formed. We develop a stylized model to investigate the optimal price and wage of a platform facing delay-sensitive customers and income-sensitive agents. We find three possibilities for participatory equilibrium in the platform as the pair of wage and price changes. Meanwhile, all participatory equilibrium regions have two asymptotically stable equilibria, one of which is a nonparticipatory equilibrium. The platform can form a participatory equilibrium only when it attracts enough customers and agents in the initial phase. Our analysis shows that the optimal pricing strategy and the maximum revenue for a platform depend on the valuation and delay sensitivity of the target customers. Furthermore, contradicting the intuition, our results show that the optimal wage is non-increasing in the total demand rate and that the optimal price is non-decreasing in the service capacity of the platform.


Introduction
With the advances in computer technology and mobile Internet, the sharing economy impacts service industries in the form of on-demand service platforms that coordinate resources from two sides in real-time, i.e., by recruiting independent service providers to meet the demands of customers (Zhong et al., 2020).Ondemand food delivery platforms have boomed in recent years, especially during the COVID-19 crisis.In 2019, the worldwide revenue from online food delivery reached 94.385 billion dollars, of which China accounted for more than 42% (Mao et al., 2019).Meituan, the largest online food delivery platform in China, currently has more than 50 million registered customers, 4.7 million couriers, and 7 million restaurants.On-demand food delivery platforms implement a more advanced system for ordering and delivering food via computer technology and mobile Internet.Specifically, customers, restaurants, and couriers are connected through a platform.Restaurants display their food on the platform, and customers pick and place their orders.Afterward, the platform assigns the delivery of orders to independent couriers registered on the platform, which we refer to as agents.The current practice in the industry is that the platform shares a portion of the restaurant's food revenue and charges a flat delivery fee per order from customers (Chen et al., 2022).At the same time, the agent will receive wages from the platform after completing each food delivery.
This study aims to investigate the optimal twosided pricing strategy of the food delivery platform when facing delay-sensitive customers and incomesensitive agents.Given the fierce competition among platforms, customers have multiple platforms to choose from and switching from one platform to another is trivial (Williams et al., 2020).Moreover, they can order from the same restaurant at similar prices on different platforms.As a result, compared with the price of food, the fee and efficiency of delivery have a greater effect on the customers' platform choice.The delivery fee is directly determined by the platform, whereas the delivery efficiency is influenced by the number of agents involved.Agents within on-demand service platforms are usually independent, i.e., they can decide whether and when to work (Taylor, 2018).On-demand food delivery platforms must attract enough agents to provide delivery services and avoid losing customer demand.However, agents only choose to provide services when the expected income is sufficient.Factors that affect agents' expected income include the wage paid by the platform for each delivery service and the probability of being matched to a customer's request.Therefore, through two-sided pricing, the platform can directly or indirectly regulate the decisions of customers and agents and thus balance supply and demand.
We construct a model framework that characterizes the cross-and within-group network effects of customers and agents.Customers' and agents' choices influence each other, which refers to the cross-group network effect.Meanwhile, the withingroup network effect among customers is reflected by the fact that an increased customer demand reduces customer's utility by increasing their waiting time (Bai et al., 2019).Similarly, when a large number of agents choose to provide services at the same time, not all of them will be assigned to delivery work, such that the agent's utility decreases in the number of agents providing the service.As another key feature, our model captures the equilibrium state of the food delivery platform, which is determined by customers' and agents' choices after they are informed about the price (i.e., delivery fee) and wage.
This study establishes a stylized model of the food delivery platform that enriches the related research from the perspective of operations management.Our contributions are fourfold.First, our paper examines the relationship between customers and agents on the food delivery platform, in which they are affected by both the cross-and withingroup network effects.The platform, as a medium of transaction, can regulate the actual supply and demand and improve its profitability by two-sided pricing after understanding these two effects.Second, for some wage-price region, there exist two asymptotically stable equilibria for the platform, demonstrating that the platform should guide the choice between customers and agents, to induce the service system to reach the more profitable equilibrium.Furthermore, our analysis shows that when the platform cannot attract enough customers and agents in the initial phase, a nonparticipatory equilibrium is formed.It is worth noting that setting a low price and high wage does not completely free the platform from this dilemma.It is also essential that customers and agents have high enough expectations of service and demand rates.Third, our study shows that the optimal pricing strategy and the maximum revenue for a platform depend on the valuation and delay sensitivity of the target customers.When the customers' valuation is high enough, the higher their delay sensitivity is, the higher the optimal wage and price of the platform are.Finally, in contrast to the intuition that the platform should raise the wage for agents when the total demand rate increases and lower the price of service when the platform's service capacity expands, our result shows that the optimal wage is non-increasing in the total demand rate and that the optimal price is non-decreasing in the service capacity of the platform.This finding reflects the influence of the cross-group network effect on the platform's optimal decisions.
The rest of this paper is organized as follows.Section Literature review reviews the related literature.By analyzing the model described in Section Model, Section Optimal strategy and revenue derives the optimal strategy and revenue for the platform.Section Discussion further discusses the effective service capacity of the platform.Section Conclusion concludes the paper.All proofs are relegated to the Appendix, which is provided as supplementary material.

Literature review
Our work relates to the literature on on-demand service platforms in a sharing economy environment.On-demand service platforms cover a variety of industries, such as ride-sharing, food delivery, healthcare, household services, and grocery delivery service (Xu et al., 2021).Three main streams of platform literature relate to our study: (i) food delivery platforms, (ii) ride-sharing platforms, and (iii) grocery delivery platforms.
Our study relates to the literature on food delivery platforms.On-demand food delivery platform is a three-sided market with network interaction among three parties: customers, restaurants, and couriers.Due to the complexity of its modeling and analysis, most of the literature focuses on a certain aspect of the food delivery platform.For example, Chen et al. (2022) and Feldman et al. (2023) investigate the impact of food delivery services on the restaurant industry.See a brief review of food delivery platforms provided by Seghezzi et al. (2021), the existing research on on-demand food delivery services can be summarized as follows: (i) associated pickup and delivery issues in the logistics and operational research domains (Tu et al., 2020); (ii) discussions of this business model from the finance, strategy, and microeconomics perspectives (Seghezzi & Mangiaracina, 2020); (iii) research on increasing consumer purchase intentions by scholars in marketing; see e.g., Mao et al. (2019) and Williams et al. (2020); and (iv) discussions of related labor policy and legal issues; see e.g., Belanche et al. (2021) and Li and Wang (2021).In practice, the customer's willingness to pay is affected by the fee and efficiency of delivery.However, previous studies have ignored this fact.Hence, differing from the mentioned works, we restrict attention to the relationship among customers, couriers, and the food delivery platform.
Our study also relates to the previous literature on ride-sharing platforms highlighting the impact of price and wage on the operation of platforms; see e.g., Bai et al. (2019), Lin and Zhou (2019), Guda and Subramanian (2019), Krishnaprasad and Tripathi (2020) and Zhong et al. (2020).Previous studies focused on finding a more effective pricing strategy by comparing dynamic and static pricing.Riquelme et al. (2015) reveal that static pricing performs well when customers are heterogeneous and the payout ratio is exogenous.By contrast, Guda and Subramanian (2019) state that surge pricing can play a good scheduling role for free service providers, especially when the platform requires them to provide services across regions.Different from ride-sharing platforms that need to consider customers' across region service requirements, food delivery platforms have a smaller order delivery range.Thus, our model focuses on how a static two-sided pricing strategy affects the platform equilibrium evolved by customers' and agents' decisions.Furthermore, our model takes into account the case where multiple orders are matched to one agent at the same time, as orders on the delivery platform are more concentrated at meal times.
Another stream of relevant literature examines the mechanism of on-demand grocery delivery platforms.Couriers for grocery delivery services typically have more time and route flexibility than for food delivery services (Tao et al., 2023).The relevant works mainly focus on routing operations and matching drivers with delivery tasks, such as Arslan et al. (2019), Bahrami et al. (2021), andArslan et al. (2021).The paper most relevant to our work on delivery services is Kung and Zhong (2017), which also studies the optimal pricing strategy of the twosided platform composed of shoppers (i.e., couriers) and consumers.They explore the optimal equilibrium of the platform with cross-side network effect under three pricing strategies.In our work, we focus on the effect of the delay time on the customer's utility, rather than assuming that it is affected by the quality of delivery service.Furthermore, our model considers both within-and cross-group network effects, which affect the customer's and courier's decision-making during meal times.

Model
The on-demand food delivery platform is a three-sided platform composed of customers, restaurants, and couriers (called agents hereafter).To explore the equilibrium formed by the evolution of customers' and agents' decisions, we focus on the interaction between customers and agents in a food delivery platform and do not concentrate on the role of the restaurant.In our model, restaurants, as third-party food suppliers, are assumed to have sufficient variety and quantity.Customers order according to their location and taste, without the congestion caused by the insufficient capacity of restaurants.
Assume that the potential market demand rate is K (called total demand rate hereafter).Meanwhile, suppose that the upper limit of service rate for agents registered within the platform is l (called service capacity hereafter).When all agents participate in the service, the platform's service rate reaches its service capacity.In our model, both K and l are assumed to be constants.Customers' arrival rate, k, varies in the price and efficiency of service, which is upper bounded by K. Similarly, the platform's actual service rate l is no higher than its service capacity l: For each delivered service, the platform charges the customer a price p and pays a wage w to the agent who provides the service.
We assume that each customer has a valuation v of receiving the service and is delay sensitive.Therefore, the customer's expected utility function is given by U c ðk, lÞ ¼ v À p À t Á Wðk, lÞ, where t is the delay cost and Wðk, lÞ is the expected delay.Assume that Wðk, lÞ strictly increases in k and decreases in l with Wð0, lÞ ¼ 0: The customer chooses to wait for service if and only if U c is nonnegative, i.e., v !t Á Wðk, lÞ þ p: On the other hand, the income of agents will be influenced by wage and cost.Suppose they will incur an opportunity cost c when providing services.Note that not all idle agents can be matched immediately with the customers' demand if too many participating agents are available.Accordingly, we define qðk, lÞ as the matching ratio, which is strictly increasing (resp., decreasing) in k (resp., l).If an agent participates in the service, then his/her expected utility is given by U a ðk, lÞ ¼ w Á qðk, lÞ À c: Consequently, the agent chooses to provide services if and only if qðk, lÞ > c=w: Considering the interaction between the customers' and agents' decisions, the actual arrival and service rates are determined based on their expectations for the other party.Hence, we use the notation k a ðlÞ to indicate that the actual arrival rate is a function of the service rate l and use l a ðkÞ to indicate that the actual service rate is also a function of arrival rate k.Given that Wðk, lÞ increases in k and decreases in l strictly, for a given service rate l, we observe three possibilities for the actual arrival rate.If the expected utility is nonnegative for a customer even if all customers join the platform (i.e., v À p À t Á WðK, lÞ !0), then every customer will choose to wait for a service, which implies that k a ðlÞ ¼ K: Another extreme situation is that the expected utility is negative even if no customer joins the queue (i.e., v À p < 0).Therefore, all customers will choose to leave, which yields k a ðlÞ ¼ 0: In other cases, when 0 v À p < t Á WðK, lÞ, each customer plays a mixed strategy in equilibrium, which indicates that each customer has a certain probability of joining or leaving the platform, with the equilibrium arrival rate k a ðlÞ 2 ð0, KÞ satisfying v À p À t Á Wðk a ðlÞ, lÞ ¼ 0: Similarly, for a given arrival rate k, we observe three possibilities for the actual service rate.The above discussion leads to the following result: Lemma 3.1.a.For a given service rate l, the actual arrival rate is given by which by convention takes a value of 0 if the set is empty.b.For a given arrival rate k, the actual service rate is given by which by convention takes a value of 0 if the set is empty.
Lemma 3.1 shows the decisions of customers and agents, respectively.We assume that neither customers nor agents have full information.Customers' and agents' decisions will be based on their expectations of service and arrival rates.Moreover, they constantly revise their expectations of each other until their expectations are in line with reality.In other words, the equilibrium state of the platform is formed through continuous learning and adjustment by the customers and the agents, which is consistent with evolutionary game theory.Suppose that the proportion of customers choosing to stay on the platform to request services is x, and thus the proportion choosing to leave is 1 À x: Hence, the platform's actual arrival rate is k a ðlÞ ¼ xK: Similarly, supposing that the proportion of agents choosing to participate in services on the platform is y, the platform's actual service rate satisfies l a ðkÞ ¼ y l: According to the evolutionary game theory, the replicator dynamic equations are given by _ x ¼ xð1 À xÞU c ðxK, y lÞ and _ y ¼ yð1 À yÞU a ðxK, y lÞ, where both x and y are time varying, and _ x and _ y denote their rates of change over time.By solving the above two replicator dynamic equations, we can obtain the equilibrium state of the platform, which consists of the effective arrival rate and the effective service rate.Introducing the concept of evolutionary game to study innovation platforms has been applied in the literature, such as Wang (2020) and Mai et al. (2023).
In our model, the equilibrium state of the platform is composed of the arrival and the service rates, which we call the equilibrium rate pair.
Definition 3.2.The equilibrium rate pair is ðk e , l e Þ, where l e ¼ l a ðk e Þ and k e ¼ k a ðl e Þ: Note that there is always a nonparticipatory equilibrium, i.e., ðk e , l e Þ ¼ ð0, 0Þ: First, for a high price or a low wage such as p > v and w < c=qðk, lÞ, customers and agents are bound to leave.However, a nonparticipatory equilibrium is still possible even if the platform sets the appropriate price and wage.The reason is that the platform does not attract enough customers and agents in the initial phase.When actual service and demand arrival rates consistently fall below customers' and agents' expectations, they continue to lower their expectations.Eventually, no customer will request a service or no agents will participate in the service, forming a nonparticipatory equilibrium.After analyzing the equilibrium rate pairs, we arrive at the following theorem.
Theorem 3.3.There are four possibilities of participatory equilibrium: ðK, lÞ, ðK, l 1 Þ, ðk 1 , lÞ, and lÞ, c=qðk 2 , 0Þg, c=qðk 2 , lÞ and p 2 ðv À t Á WðK, lÞ, v, then the participatory equilibrium rate pairs are ðk 1 , lÞ and ðk 2 , l 2 Þ: Theorem 3.3 demonstrates that, given the wage and price, the equilibrium may not be unique.In the face of multiple equilibria, it is crucial to determine which equilibrium is the most preferable for the platform before determining its optimal pricing strategy.Next, we need to introduce expressions for Wðk, lÞ and qðk, lÞ to derive the specific multiple equilibrium rate pairs and discuss their stability.

Optimal strategy and revenue
In this section, we explore the optimal strategy and revenue for the platform.For analytical simplicity, we take the following assumption for the remaining of this paper.
Assumption 1. a.The expected delay of the customer Wðk, lÞ is given by k=ðlðl À kÞÞ for k < l and is 1 for k !l: b.The matching ratio of the agent qðk, lÞ is given by ck=ðal þ kÞ, where both a and c are positive constants.
We make the following remarks about Assumption 1. First, both the expected delay of the customer and the matching ratio of the agent are consistent with the basic assumptions of our model.Second, Assumption 1(a) shows that the expected delay of the customer increases with the utilization (i.e., k=l) of the agents and approaches infinity when the utilization is greater than 1.Finally, Assumption 1(b) demonstrates that agents' matching ratio is also affected by external factors.For example, unexpected events, such as customer order change, restaurant order cancellation due to insufficient food preparation capacity, and poor communication network connection, will have a negative impact on the agents' working status.Therefore, we adopt a parameter a to measure the platform's matching inefficiency, such that a large value of a corresponds to a low agents' matching ratio.Meanwhile, we use parameter c to capture this feature of the food delivery platform considering that an agent may match multiple orders in one trip.We find that the matching ratio of the agent is greater than one when c > 1 þ al=k occurs.

Equilibrium rate pairs
Under Assumption 1, the equilibrium of the platform under different wage and price conditions can be characterized as follows.
Proposition 4.1.a.When K < l, there are four cases: i.If w > ðac l þ cKÞ=cK and p v À tK= lð l À KÞ, the asymptotically stable equilibria are (0, 0) and ðK, lÞ: ii.If w 2 ðð1 þ aÞc=c, ðac l þ cKÞ=cK and p v À a 2 c 2 t=Kðcw À cÞðcw À c À acÞ, the asymptotically stable equilibria are (0, 0) and ðK, Kðcw À cÞ=acÞ: iii.If w !ðac l þ cKÞ=cK and p 2 ðv À tK= lð l ÀKÞ, v À act= lðcw À c À acÞ, the asymptotically stable equilibria are (0, 0) and ð l 2 ðv À pÞ=ð lðv À pÞ þ tÞ, lÞ: iv.Otherwise, the asymptotically stable equilibrium is (0, 0).b.When K !l, there are two cases: i.If w > ð1 þ aÞc=c and p v À act= lðcw À c À acÞ, the asymptotically stable equilibria are (0, 0) and ð l 2 ðv À pÞ= ð lðv À pÞ þ tÞ, lÞ: ii.Otherwise, the asymptotically stable equilibrium is (0, 0).Proposition 4.1 demonstrates three possibilities of a participatory equilibrium within a certain wage-price region.Once the service capacity of the platform is sufficient, i.e., K < l, all three participatory equilibria are likely to occur.When the wage is high (w > ðac l þ cKÞ=cK) and the price is low (p v À tK= lð l À KÞ), the platform achieves a participatory equilibrium in which all customers choose to request the service and all agents participate in providing the service.The supply and demand sides receive positive utility due to the high wage and low price, respectively.By contrast, the low wage (w 2 ðð1 þ aÞc=c, ðac l þ cKÞ=cK) triggers such a participatory equilibrium in which part of the agents serves all customers, whereas the high price (p 2 ðv À tK= lð l À KÞ, v À act= lðcw À c À acÞ) forms another participatory equilibrium in which part of the demand is lost when all agents participate in the service.If the service capacity of the platform is insufficient, i.e., K !l, it is only possible to form the participatory equilibrium in which all agents serve part of the customers.Moreover, the necessary condition for this participatory equilibrium is that the wage and price satisfy w > ð1 þ aÞc=c and p v À act= lðcw À c À acÞ: To clarify this result, we use a concrete example to examine the effect of both wage and price in the participatory equilibrium.Figure 1(a) shows the situation in which the platform's service capacity exceeds the total demand, whereas Figure 1(b) depicts the opposite situation.For K < l, we define wage-price regions A 1 , A 2 , and A 3 as follows: Meanwhile, for K !l, we define the wage-price region A 4 by These regions with different-colored shadows in Figure 1 correspond to different equilibrium rate pairs.The blue-colored region corresponds to ðK, lÞ covering A 1 , the red-colored region represents ðK, Kðcw À cÞ=acÞ covering A 2 , and the green-colored region corresponds to ð l 2 ðv À pÞ=ð lðv À pÞ þ tÞ, lÞ covering A 3 and A 4 .As mentioned in Section Model, there is always a nonparticipatory equilibrium.Therefore, as shown in Figure 1, wage-price regions can be divided into two categories: 1) the only-nonparticipatory equilibrium region, which is the blank area in the figure; and 2) the participatory equilibrium regions, which are denoted by A 1 , A 2 , A 3 , and A 4 marked with colors.
Recall from Theorem 3.3 that there exist multiple participatory equilibria in the participatory equilibrium regions.To avoid trivial discussions, as in Wang and Fang (2022), we focus only on the stable equilibrium of the platform.Proposition 4.1 illustrates that all participatory equilibrium regions have two asymptotically stable equilibria, and one of them is a nonparticipatory equilibrium.According to Li et al. (2020), these equilibrium rate pairs are considered evolutionarily stable strategies.If there are two evolutionarily stable strategies under the same wage-price condition, which one is formed will depend on the initial state of the platform.To avoid nonparticipatory equilibrium, the platform must ensure that it attracts enough customers and agents in the initial phase.Note that there are also two evolutionarily stable strategies in region A 1 .It implies that setting a low price and high wage does not completely make the platform attract enough customers and agents to get rid of nonparticipatory equilibrium.It is essential that customers and agents have sufficiently high expectations of service and demand rates.

Optimal strategy of the platform
Let R(p, w) denote the platform's expected revenue, which can be written as Rðp, wÞ ¼ ðp À wÞk e : The platform's objective is to find the optimal wage and price to maximize its expected revenue R(p, w).For the only-nonparticipatory equilibrium region, the unique equilibrium arrival rate k e ¼ 0 leads to Rðp, wÞ ¼ 0: If the wage and price are located in the participatory equilibrium regions in which two stable equilibria exist, then the equilibrium that brings the higher expected revenue is treated as the optimal equilibrium rate pair.To investigate the optimal strategy for the platform, we need to solve the optimal equilibrium rate pair and corresponding revenue for participatory equilibrium regions A 1 -A 4 .
For wage-price regions A 1 and A 2 , the arrival rate of participatory equilibrium is K. Therefore, the expected revenue of the platform is R K ðp, wÞ :¼ ðp À wÞK: By contrast, in regions A 3 and A 4 , the arrival rate of participatory equilibrium is l 2 ðv À pÞ=ð lðv À pÞ þ tÞ: Therefore, the expected revenue of the platform is R k ðp, wÞ :¼ ð l 2 ðp À wÞðv À pÞÞ=ð lðv À pÞ þ tÞ: As a result, we formulate the following three optimization problems for regions A 1 , A 2 , and A 3 when K < l : For K !l, we formulate the following optimization problem for region A 4 : , and R Ã 4 by solving these optimization problems and then deduce the optimal strategy for the platform.If R Ã i > 0 (i ¼ 1, 2, 3, 4), the participatory equilibrium rate pair is optimal for region A i .Otherwise, the nonparticipatory equilibrium, which generates zero profit, will be optimal for region A i .
First, we consider the case where the platform's service capacity is sufficient, i.e., K < l: Theorem 4.2 below characterizes the optimal decisions for a platform in each wage-price region with participatory equilibrium, and Theorem 4.3 determines the optimal price and wage by comparing each region's maximum expected revenue obtained from Theorem 4.2.For brevity, we introduce the following notations: where w 2 is the unique root of FðwÞ :¼ a 2 c 2 tcðcw À c À ac þ cw À cÞ À Kðcw À cÞ 2 ðcw À c À acÞ 2 ¼ 0 on ðð1 þ aÞc=c, ðac l þ cKÞ=cKÞ, and p 3 is the unique root of GðpÞ :¼ c lðv À pÞ 2 þ ðcv À 2cp þ cÞt ¼ 0 on ðv À tK=ð lð l À KÞÞ, vÞ: Theorem 4.2.When K < l, the platform's optimal decisions for each participatory equilibrium region as shown in Table 1.
Theorem 4.2 shows the optimal decisions of the platform in each participatory equilibrium region when the service capacity is sufficient.To show the results (including Theorems 4.2 and 4.3) clearly, we plot Figure 2, where the first three subgraphs show the maximum expected revenue of wage-price regions A 1 À A 3 : Similar to Figure 1, we use different color shades to indicate the optimal equilibrium rate pair for each participatory equilibrium region.We derive several findings from Theorem 4.2 and Figure 2. First, the platform's expected revenue must be nonpositive among all wage-price regions when customers have a low valuation or significantly high delay sensitivity.These customers do not accept the high price or have the patience to wait.To attract more customers and agents, the platform should decrease the price to offset the negative effect of low valuation and increase the wage to reduce the customers' expected delay.However, the platform is hardly profitable in such a case.Second, (w 1 , p 1 ) is likely to be optimal among regions A 1 , A 2 , and A 3 when K < l (i.e., R Ã 1 in panels (a), (b), and (c) of Figure 2).This wage and price vector is located at the intersection of these three regions, and at this point, Kðcw À cÞ=ac ¼ l and l 2 ðv À pÞ=ð lðv À pÞ þ tÞ ¼ K: In other words, at this time, the optimal equilibrium rate pair in regions A 2 and A 3 is ðK, lÞ: Third, when customers' valuation is sufficiently high, the optimal equilibrium rate pair will shift as customers' delay sensitivity increases.For example, when v > v 1 in region A 2 (Figure 2(b)), the optimal equilibrium rate pair changes from ðK, Kðcw 2 À cÞ=acÞ to ðK, lÞ and then to (0, 0) as the delay sensitivity t rises because customers tend to leave rather than wait when they are highly impatient (increased delay sensitivity).Therefore, the platform should attract all agents by increasing the wage to reduce service delays.However, when customers' delay sensitivity continues to increase, the platform's service capacity can no longer meet the customers' demands in time, and the platform also cannot benefit by paying too high wages to attract agents.As a result, the optimal equilibrium in this region becomes a nonparticipatory equilibrium.Similarly, when v > v 2 in region A 3 (Figure 2(c)), the optimal equilibrium rate pair shifts from ðK, lÞ to ð l 2 ðv À p 3 Þ=ð lðv À p 3 Þ þ t, lÞ and then to (0, 0) as t rises.
Theorem 4.3.For K < l, we have: w and t < t, then the optimal wage and price vector, denoted by ðw Ã , p Ã Þ, is (w 2 , p 2 ).Thus, the maximum expected revenue of the platform is R Ã ¼ R Ã 2 ; b.If v > v 1 and t 2 ½ t, minft Ã , tgÞ, then ðw Ã , p Ã Þ is (w 1 , p 1 ).Thus, the maximum expected revenue of the platform is R lðcv À c À acÞ 2 =ð4accÞÞ, then ðw Ã , p Ã Þ is (w 3 , p 3 ).Thus, the maximum expected revenue of the platform is R Ã ¼ R Ã 3 ; d.Otherwise, the maximum expected revenue of the platform is R Ã ¼ 0: Theorem 4.3 demonstrates the impact of customers' valuation and delay sensitivity on the platform's optimal decisions when its service capacity is sufficient.If the customers' delay sensitivity is low, then all customers' demands can be met even if only some agents provide the service.For moderately delay-sensitive customers, the platform should ensure that all agents provide services to reduce delay.When the customers' delay sensitivity is sufficiently high, some customers still leave the platform even if all agents provide services.Besides, the higher the customers' delay sensitivity, the higher the bottom of the valuation can make the platform earn positive revenues.Figure 2(d) shows the maximum expected revenue of the platform for different types of customers when the service capacity is sufficient, which is equivalent to overlapping panels (a), (b), and (c) of Figure 2 and keeping the largest revenue in these panels.This panel again shows the results of Theorem 4.3 intuitively.It also reveals that the lower the customers' valuation, especially when v < v 2 , the equilibrium rate pair ðK, Kðcw 2 À cÞ=acÞ (R Ã 2 corresponding region) is more likely to become the optimal participatory equilibrium rate pair of the platform.This result implies that when its service capacity is sufficient, the platform tends to set a moderate wage (ð1 þ aÞc=c < w ðac l þ cKÞ=cK) even though doing so would reduce the effective service rate.The R Ã 1 region in Figure 2(d) shows that the high wage (w > ðac l þ cKÞ=cK) and low price (p v À tK= lð l KÞ) pair can also bring maximum revenue to the platform under certain conditions.In such a case, both the arrival and service rates reach their peak, i.e., ðK, lÞ: We then consider the case where the platform's service capacity is insufficient.
Theorem 4.4 illustrates the optimal decisions of the platform when its service capacity is insufficient.The platform's expected revenue is positive if and only if the customers' valuation is high (i.e., v > ð1 þ aÞc=c) and their delay sensitivity is low (i.e., t < lðcv À c À acÞ 2 =ð4accÞ).Figure 3 clearly illustrates this result.In this case, the platform should set w 3 and p 3 to achieve the equilibrium rate pair ð l 2 ðv À p 3 Þ=ð lðv À p 3 Þ þ t, lÞ and obtain the maximum expected revenue R Ã 3 : When customers have low valuation or high delay sensitivity, the platform cannot obtain positive returns regardless of the wage and price in or outside region A 4 .
For the platform, the optimal decisions can be determined when the type of customers (i.e., (v, t)) is clear.If the platform wants to obtain revenue R Ã i (i ¼ 1, 2, 3), it should set the wage to w i and the price to p i , and pay attention to the attitudes of customers and agents toward each other, especially in the initial phase.To avoid the formation of other non-optimal equilibria, the platform can take some measures to guide customers and agents, such as the real-time announcement of the actual demand/supply information.Once it is realized that the equilibrium is moving towards a non-optimal equilibrium, the platform can prevent this trend and guide the evolution of the optimal equilibrium by the realtime announcement of information.

Discussion
In this section, we discuss how the total demand rate and service capacity affect the optimal decision and revenue of the platform.The impact of K and l on p Ã , w Ã , and R Ã will be analyzed in the following proposition: Proposition 5.1.The optimal wage w Ã is nonincreasing in K, and the optimal price p Ã is nondecreasing in l.Thus, the platform's maximum revenue R Ã is always non-decreasing in K and l.Specifically, the impact of K and l on the optimal decisions of the platform is shown in Table 2.
At first glance, the wage pay for agents should increase along with the total demand rate, and the price of the service should decrease along with an increasing service capacity of the platform.The former can be ascribed to the fact that a higher wage attracts a sufficient number of agents to provide services to meet the increased customer demand, whereas the latter can be explained by the sufficient number of agents who can provide services despite lower prices when the total service rate increases.However, Proposition 5.1 shows that the optimal wage is non-increasing in K and that the optimal price is non-decreasing in l, which seems to be counterintuitive.Two opposite drivers explain the impact of the increasing total demand rate on the optimal wage.On the one hand, the platform should increase the wage to attract more agents to provide services so as to reduce the customers' expected delay.On the other hand, an increase in the matching ratio makes agents more willing to participate in the service, and the platform can maintain the Table 2. Monotonicity of the optimal solution and revenue with respect to system parameters.
Parameter " optimal equilibrium service rate even if the wage decreases.These two drives demonstrate the influence of within-and cross-group network effects of customers and agents on platform decisions.Proposition 5.1 states that the cross-group network effect exerts a stronger influence on the optimal wage when the total demand rate increases.
We now analyze why the optimal price is nondecreasing as the service capacity of the platform increases.The reason is that the expansion of the platform's service capacity will result in a shorter expected delay for customers when all agents provide services.At this time, price increases can still preserve the equilibrium demand rate expected by the platform.Therefore, Proposition 5.1 demonstrates that only when the optimal equilibrium rate pair is ðK, Kðcw 2 À cÞ=acÞ, due to the limited total demand, the increased service capacity does not affect the equilibrium service rate and thus does not affect the optimal decisions of the platform.In two other cases, the optimal price increases as the platform's service capacity expands.
Note that the valuation and delay sensitivity thresholds mentioned in Theorem 4.3 also change along with the total demand rate or service capacity.For notational brevity hereinafter, we keep in mind that the thresholds (i.e., t, t, t Ã , v 1 , v 2 , and w 2 þ w) are all functions of K and l even if not explicitly stated.We further analyze below the impact of K and l on these thresholds.
Proposition 5.2.a.Both t and t are strictly decreasing in K and increasing in l: t Ã is strictly increasing in l if and only if v > v 1 and decreasing in K if and only if v > v 2 : b.Both v 1 and v 2 are strictly increasing in l and decreasing in K.Moreover, w 2 þ w is strictly decreasing in K.
Proposition 5.2 shows that the delay sensitivity and valuation thresholds are strictly increasing in the service capacity and decreasing in the total demand rate.By observing Figure 2 (d), we find that the increase of the total demand rate K expands the area of region R Ã ¼ R Ã 3 , while the area of region R Ã ¼ R Ã 2 shrinks.Note that p 2 < p 1 < p 3 and w 2 < w 1 < w 3 : Thus, R Ã ¼ R Ã 2 indicates that the platform's optimal strategy is to set a low price to attract all customers and set a low wage to retain part of the agents to participate in the service.On the other hand, R Ã ¼ R Ã 3 implies that the optimal strategy is to set a high wage to attract all agents to participate in the service and set a high price to retain part of customers' demands.Therefore, as the total demand rate increases, the platform's optimal strategy should gradually shift to one with both a higher price and wage.The increase in service capacity l has the opposite effect on the platform decision.Specifically, in Figure 2(d), the area of region R Ã ¼ R Ã 2 expands while the area of region R Ã ¼ R Ã 3 shrinks as l increases.In other words, the platform should gradually shift to a strategy with both a lower wage and price as the service capacity rises.
The above discussions can be formally summarized in the following corollary.
Panels (a) and (b) in Figure 4 show the optimal strategy of the platform when the total demand rate K and the service capacity l change, respectively.The results in Figure 4 and Table 2 are consistent with Corollary 5.3.The increase in the total demand rate gradually shifts the optimal strategy of the platform towards directing all agents to participate in the service, while the increase in service capacity makes the platform more inclined to retain all demand.Overall, increases in both the total demand rate and service capacity positively impact the platform's maximum revenue.Note that if the total demand rate is large enough, further increases will no longer change the platform's optimal decision, and the same is true for service capacity.As shown in Corollary 5.3, once the service capacity of the platform exceeds l 3 , the optimal equilibrium rate pair will remain at ðK, Kðcw 2 À cÞ=acÞ, which means that always only part of the agents choose to provide the services.In our model, a proportion strictly between 0 and 1 of agents will provide the service, such that the utilities of providing and not providing the service are both zero.In practice, when the offered service capacity is larger than the demand size, part of the agents involved in the service can be screened out in two scenarios.In the first scenario, the model takes into account the effect of random factors on the agent's decision.For example, we can set the utility function of agents as U a ðk, lÞ þ , where is a random parameter.Parameter represents the agent's heterogeneous factors that are not captured by the current model.In the second scenario, the platform prioritizes the allocation of orders to agents with higher ratings through the carefully designed matching mechanism, and thus agents with low ratings leave because they cannot be assigned orders.The rating here can depend on the agent's history of order fulfillment, customer reviews received, etc.

Conclusion
On-demand food delivery platforms have become popular in recent years, providing a convenient and safe way for customers to enjoy their food, especially during the COVID-19 crisis.Although some scholars are interested in food delivery platforms, only a few have explored the mechanism of these on-demand platforms.To fill this gap, this paper explores the optimal two-sided pricing strategy of the on-demand food delivery platform and analyzes the equilibrium results from the operations management perspective.We initially build a model that takes into account the within-and cross-group network effects of customers and agents.After a preliminary calculation, we summarize the wage-price regions into two categories, the participatory equilibrium, and the only-nonparticipatory equilibrium regions.We derive the optimal strategy in each participatory equilibrium region and then obtain the platform's optimal two-sided pricing strategy through further comparison.After that, we discuss the impact of the total demand rate and service capacity on the platform decision, respectively.Our theoretical analysis provides insights into the operation and management of the on-demand food delivery platform.
Our modeling framework has several limitations.First, for practical reasons and tractability, we assume that both customers and agents are homogeneous.If the assumption of customer heterogeneity and agent heterogeneity is introduced, the equilibrium state of the platform will become more diversified.In addition to two-sided pricing, platform decisions also include the design of a matching mechanism.Incorporating matching priorities and equilibrium analysis is an interesting future direction.Second, we focus on food delivery services and ignore the connection between the platform and restaurants.However, in practice, cooperation with restaurants is an important part of food delivery platform operations.Moreover, some restaurants tend to arrange for their employees to deliver the orders they receive.This topic deserves further exploration in future research by establishing a new model.Third, in real life, the customer flow of food delivery platforms is also affected by external factors, such as weather and transportation.

Figure 2 .
Figure 2. The maximum expected revenue of the platform when the service capacity is sufficient, where c ¼ 3, a ¼ 0:4, c ¼ 1, K ¼ 11, and l ¼ 15:

Table 1 .
Conditions and outcomes for each participatory equilibrium region.