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ERROR: type should be string, got "https://pubsonline.informs.org/journal/opre\nSubmitted to Operations Research\nOPERATIONS RESEARCH\nVol. 00, No. 0, Xxxxx 0000, pp. 000–000\nISSN 0030-364X, EISSN 1526-5463\nDual-Agent Deep Reinforcement Learning for\nDynamic Pricing and Replenishment\nYi Zheng\nGuanghua School of Management, Peking University, yizheng@stu.pku.edu.cn\nZehao Li\nGuanghua School of Management, Peking University, zehaoli@stu.pku.edu.cn\nPeng Jiang\nSchool of Management, Huazhong University of Science and Technology, pengjiang0731@outlook.com\nYijie Peng\nGuanghua School of Management, Peking University, pengyijie@pku.edu.cn\nAuthors are encouraged to submit new\npapers to INFORMS journals by means\nof a style file template, which includes\nthe journal title. However, use of a tem-\nplate does not certify that the paper\nhas been accepted for publication in the\nnamed journal. INFORMS journal tem-\nplates are for the exclusive purpose of\nsubmitting to an INFORMS journal and\nare not intended to be a true representa-\ntion of the article’s final published form.\nUse of this template to distribute papers\nin print or online or to submit papers\nto another non-INFORM publication is\nprohibited.\nAbstract. We study the dynamic pricing and replenishment problems under incon-\nsistent decision frequencies. Different from the traditional demand assumption, the\ndiscreteness of demand and the parameter within the Poisson distribution as a function\nof price introduce complexity into analyzing the problem property. We demonstrate\nthe concavity of the single-period profit function with respect to product price and\ninventory within their respective domains. The demand model is enhanced by inte-\ngrating a decision tree-based machine learning approach, trained on comprehensive\nmarket data. Employing a two-timescale stochastic approximation scheme, we address\nthe discrepancies in decision frequencies between pricing and replenishment, ensuring\nconvergence to local optimum. We further refine our methodology by incorporating\ndeep reinforcement learning (DRL) techniques and propose a fast-slow dual-agent\nDRL algorithm. In this approach, two agents handle pricing and inventory and are\nupdated on different scales. Numerical results from both single and multiple products\nscenarios validate the effectiveness of our methods.\nFunding: This research was supported by [grant number, funding agency].\nKey words: Dynamic pricing and replenishment, Stochastic approximation, Deep\nreinforcement learning\n1. Introduction\nDynamic pricing refers to the practice where sellers dynamically adjust prices based on factors\nsuch as sales period, historical demand information, and supply availability (Elmaghraby and\nKeskinocak 2003), which typically faces challenges of high costs or low feasibility in practice.\nDynamic pricing methods have been applied to industries where capacity (supply) is difficult to\n1\narXiv:2410.21109v1 [cs.LG] 28 Oct 2024\nX: Dual-Agent Deep Reinforcement Learning for Dynamic Pricing and Replenishment\n2\nArticle submitted to Operations Research\nchange in the short term, such as airlines, cruise lines, hotels, and power facilities (Gallego and\nVan Ryzin 1994, 1997). Sectors like retail, with more flexible replenishment or higher price-change\ncosts, have traditionally focused on enhancing inventory management. Appropriate coordination\nbetween pricing and inventory decisions can mitigate inventory-demand mismatches (Feng et al.\n2020) and optimize profits (Lei et al. 2018).\nThe rapid development of e-commerce and information systems has reduced the costs associated\nwith price adjustments and enhanced the ability for real-time inventory monitoring (Brynjolfsson\nand Smith 2000). The challenge lies in that retailers often need to coordinate the pricing and\ninventory of thousands of products simultaneously. The existing literature suffers the following\nlimitations: (i) The optimality of the base-stock list-price (BSLP) and (s,S,p) policies requires\nassuming restrictive functional forms of demand, such as additive or multiplicative demands. (ii)\nExisting literature often assumes that excess demand is backlogged. For retail firms selling to\nconsumers (i.e., B2C), it is more common for consumers to take their demand elsewhere during\nstock-outs, leading to lost-sales (Bijvank et al. 2023). (iii) Most previous works consider demand\nin a certain period that depends only on prices, without considering the impact of other market\nfactors, such as competitors. We take into consideration the impact of market competition (including\ncompetitors’ prices, market reference prices, and price rankings) while optimizing pricing and\nreplenishment strategies.\nIn the literature, strong assumptions about market demand have often been made. For example,\nCohen et al. (2018) assume that the demand is a linear function of the price plus a random noise\ncomponent, and Aviv and Pazgal (2005) assume that customer arrivals follow a Poisson distribution.\nIn reality, demand is not solely related to price, and the relationship is not as simple as a linear one.\nOur work is not constrained by the unrealistic assumptions mentioned above.\nA recent trend gaining steam is to introduce reinforcement learning (RL) to solve sequential\ninventory decision problems (Gijsbrechts et al. 2021, Oroojlooyjadid et al. 2022). Through con-\ntinuous trial-and-error interactions with an unknown demand, RL agents learn by themselves to\nachieve optimal strategies without imposing assumptions on the demand (Sutton and Barto 1999).\nDeep RL (DRL) is a subfield of RL, which employs deep neural networks to approximate the value\nor policy functions of Markov Decision Processes (MDPs). Rana and Oliveira (2014) investigate\nthe pricing problem for a given inventory within a fixed time period. Kastius and Schlosser (2021)\napply Deep Q-Networks (DQN) and Soft Actor Critic (SAC) RL algorithms to the pricing problem\nX: Dual-Agent Deep Reinforcement Learning for Dynamic Pricing and Replenishment\nArticle submitted to Operations Research\n3\nof a single product in duopoly and oligopoly markets where the customer behavior is described by\nthe logit model in Schlosser and Boissier (2018).\nWe first investigate a dynamic pricing and replenishment problem under competition, considering\ncompetitor strategies and market conditions. Different from the traditional demand assumption,\nthe discreteness of demand we consider, along with the parameter within the Poisson distribution\nbeing a function of price, introduces complexity into analyzing the property of the problem. We\nprove that the single-period profit function is concave with respect to product price and inventory\nwithin respective domains, but it is not jointly concave. Based on the respective concavity, a\ntwo-timescale algorithm is proposed. For multi-period problems, we propose a dual-agent DRL\nalgorithm combining the two-timescale approach with Multi-Agent DRL (MADRL). Two agents\nare introduced in the algorithm, responsible for pricing and inventory decisions respectively. This\nallows each agent to optimize its own decisions while fully considering the influence of the other\nagent. Furthermore, we allow the two agents for making decisions on pricing and inventory to\nbe updated in different scales. Finally, we explore a dynamic pricing and replenishment problem\nwhere product demand is directly learned from comprehensive market data, encompassing product\nattributes, customer behavior, and marketing information. The simulation results indicate that our\njoint pricing and replenishment strategy, which incorporates demand forecasting and further price\noptimization, results in lower inventory and higher profits.\nOur work consists of the following contributions.\n(i) We prove that the single-period profit function in dynamic pricing and replenishment problem\nis concave with respect to product price and inventory within respective domains, but it is not\njointly concave. The respective concavity ensures the convergence of our proposed two-timescale\nalgorithm, with solutions converging to the limit points of the ordinary differential equation (ODE).\n(ii) We formulate the dynamic pricing and replenishment problems as Markov Games (MGs) and\npropose a new solution integrating two-timescale algorithms with MADRL to construct intelligent\npricing and ordering policies. Additionally, we enhance our model’s capability to capture real\nmarket demand by combining a decision tree-based machine learning (ML) model with real market\ndata to train the ML model.\n(iii) We experimentally demonstrate that our algorithms achieve optimal performance on solv-\nable numerical examples and outperform baseline algorithms in cases where optimal strategies\nare not analytical, thus validating their effectiveness. Furthermore, our method exhibits linear\ncomputational complexity relative to problem scale, ensuring efficiency in practical pricing and\nreplenishment applications.\nX: Dual-Agent Deep Reinforcement Learning for Dynamic Pricing and Replenishment\n4\nArticle submitted to Operations Research\n2. Literature Review\nIn this section, a brief review on related literature is provided. We begin by reviewing the literature\non joint dynamic pricing and inventory control, and then proceed to review the problem of dynamic\npricing and replenishment under competition. Relevant computing approach and their applications\nin the field of dynamic pricing and inventory management are introduced.\n2.1. Joint Dynamic Pricing and Inventory Control\nThe joint pricing and inventory control has received considerable attention over the past decade,\noften formulated as a Markov Decision Process (MDP) for multi-period decision problems in finite\nor infinite horizons (Federgruen and Heching 1999). To solve this problem, strong assumptions\nhave been made in the literature. For instance, Jia and Hu (2011) assume the demand follows a\nrandom price-dependent function; Bisi and Dada (2007) study an additive and multiplicative price-\nsensitive demand; Raz and Porteus (2006) approximate a continuous demand model using a finite\nnumber of representative fractiles, which are piecewise linear functions of the price. Federgruen\nand Heching (1999) assume that replenishment, holding, and ordering costs are convex, and the\nseller has unlimited production capacity. Chen and Simchi-Levi (2004) and Chan et al. (2006)\nextend the previous models by considering fixed ordering costs and limited production capacity,\nrespectively.\nThowsen (1975) proves that a BSLP policy is optimal for the additive demand function by\nassuming the random noise follows a P´olya frequency function of order 2 (PF2) distribution.\nFedergruen and Heching (1999) further establish the optimality of a BSLP policy with a general\ndemand function for a backorder system without capacity constraints. Their sufficient conditions\nare as follows: (i) the demand is decreasing and concave in the list price; (ii) the single-period\nexpected inventory cost is jointly convex in the base-stock level and price, which can be guaranteed\nonly if the demand function is linear in the price (Feng et al. 2014). With fixed ordering costs, Chen\nand Simchi-Levi (2004) prove that the (𝑠, 𝑆, 𝑝) policy is optimal when the demand is additive in a\nfinite time horizon and the seller’s objective is to either maximize expected discounted profits or\nmaximize average long-run profits in the infinite-horizon model.\n2.2. Dynamic Pricing and Replenishment under Competition\nChen and Simchi-Levi (2012) note that research on jointly optimizing pricing and inventory deci-\nsions under competition remains limited due to model complexity. Competition typically drives\nX: Dual-Agent Deep Reinforcement Learning for Dynamic Pricing and Replenishment\nArticle submitted to Operations Research\n5\nprices lower, reducing firms’ profits compared to monopolistic scenarios (Mantin et al. 2011,\nMookherjee and Friesz 2008), and may lead to excessive inventory holdings (Xu and Hopp 2006).\nMahmoodi (2019) examines the pricing and replenishment problem of a single product with deter-\nministic linear demand in a duopoly market and also compares the case where two substitutable\nproducts are both sold by monopolists but does not consider products and competition simulta-\nneously. Schlosser and Boissier (2018) analyze stochastic dynamic pricing models in competitive\nmarkets with multiple offer dimensions (e.g., product qualities and seller ratings). They compare\ndifferent demand learning techniques, including least squares, gradient boosted trees, and logistic\nregression, to estimate sales probabilities from partially observable market data. Ultimately, they\nadopt a logistic regression approach for its interpretability and performance. In the follow-up study,\nthey further consider market reference prices, which are related to historical market prices, as well\nas competitor strategies and price rankings (Schlosser and Richly 2019).\n2.3. Two-timescale Stochastic Approximation and Deep Reinforcement Learning\nTwo-timescale algorithms aggregate data and update parameters at epochs with two timescales,\nwhich allows for discrete-valued observations and avoids the infrequent aggregation of data over\nregeneration epochs (Bhatnagar and Borkar 1998), and can handle the frequency discrepancy\nbetween different decision-making processes (Wernz 2013).\nMDPs provide a mathematical framework to model decision-making problems in sequential\nenvironments. RL learns optimal policies or value functions in MDPs through iterative interactions\nwith the environment, basing decisions on observed feedback. RL algorithms can be divided\ninto three categories: value-based, policy-based, and actor-critic algorithms (Arulkumaran et al.\n2017). Value-based algorithms focus on learning the value function, which estimates the expected\ncumulative reward for actions taken in specific states. Policy-based algorithms directly learn the\noptimal policy without explicitly estimating the value function. Actor-critic algorithms combine\nvalue-based and policy-based approaches, utilizing an actor to learn the policy and a critic to\nestimate the value function.\nWhile RL has shown promise in solving complex decision-making problems, it faces challenges\nwith high-dimensional inputs, scalability, and generalization, which could be addressed by DRL\nalgorithms, such as Asynchronous Advantage Actor-Critic (A3C) (Mnih et al. 2016) and Proxi-\nmal Policy Optimization (PPO) (Schulman et al. 2017) algorithms. A3C employs multiple agents\nX: Dual-Agent Deep Reinforcement Learning for Dynamic Pricing and Replenishment\n6\nArticle submitted to Operations Research\nrunning in parallel, enabling effective exploration and faster convergence, and PPO ensures mono-\ntonic policy improvement in each training iteration based on the trust region learning proposed by\nSchulman et al. (2015). These algorithms have demonstrated excellent performance in supply chain\nmanagement. For instance, Gijsbrechts et al. (2021) apply A3C to three classic inventory problems\nand their result shows that A3C beats a Base Stock policy by 9%-12%. Vanvuchelen et al. (2020)\napply the PPO algorithm to the joint replenishment problem for multiple items and they show that\nPPO outperforms two other heuristics.\nIn Multi-Agent Systems (MAS), multiple agents cooperate or compete within a shared environ-\nment to achieve common or individual goals. For any individual agent in a MAS, the transition\nof environmental states is no longer solely dependent on its own actions and the dynamics of the\nenvironment; it also depends on the actions taken by other agents, leading to non-stationarity in\nthe environment. This is particularly evident in the dynamic pricing and replenishment problem\ninvestigated in this study, where the local objectives of different decisions are conflicting: for exam-\nple, the pricing strategy aims to increase profits through high pricing, while inventory management\naims to quickly clear stock through low pricing. A single-agent DRL algorithm can not effectively\ncoordinate the competition between them, resulting in sub-optimal or unstable strategies. In Section\n4 of our experiments, we apply a single-agent DRL algorithm, but the results were unsatisfactory,\nwith a convergence speed merely 1/5th of the algorithms we subsequently adopted.\nTo solve MDPs in MAS, MADRL has been developed. The Centralized Training and Decentral-\nized Execution (CTDE) scheme, a mainstream MADRL architecture, addresses non-stationarity\nin MAS environments (Foerster et al. 2016). It trains agents with full environmental information\nbut executes actions based only on local information. This architecture enables global optimization\nduring training by sharing full information and decentralized execution during testing, ensuring\nboth agent cooperation and system efficiency. By applying CTDE to Deep Deterministic Policy\nGradient (DDPG), Lowe et al. (2017) show that Multi-Agent DDPG (MADDPG) has a desirable\nperformance under both cooperative and competitive multi-agent environments. Similarly, Yu et al.\n(2022) apply CTDE to PPO and introduce Multi-Agent PPO (MAPPO). Unlike the PPO algorithm,\nMAPPO does not guarantee monotonic improvement. To address these issues, Kuba et al. (2021)\npropose Heterogeneous-Agent Proximal Policy Optimization (HAPPO).\n3. Dynamic Pricing and Replenishment under Competition\nIn this section, we formulate the problem of dynamic pricing and replenishment under competition\nas an MDP and illustrate the dynamics of this problem. We first derive theoretical results for the\nX: Dual-Agent Deep Reinforcement Learning for Dynamic Pricing and Replenishment\nArticle submitted to Operations Research\n7\nsingle-period problem and propose a two-timescale approach based on these findings. Then, con-\nsidering the multi-period problem structure, we combine this approach with MADRL to develop\na novel fast-slow dual-agent DRL algorithm for solving multi-period dynamic pricing and replen-\nishment problems. Finally, we demonstrate the convergence and computational complexity of our\nalgorithm, which significantly outperforms other heuristic strategies in experimental results.\n3.1. Markov Decision Process Formulation\nAn MDP can be defined as a tuple < S, A, P, R, 𝛾>. S is the set of product states. The state 𝑠𝑡\nin period 𝑡encompasses: the previous period’s ending inventory level, lost-sales, demand, price,\nand ordering information within the lead time. A is the set of product actions 𝑎𝑡. Here, action\n𝑎𝑡in period 𝑡determines price 𝑝𝑡and replenishment quantity 𝑞𝑡. P : S × A × S →[0,1] is the\nstate transition function that maps the current state, action and next possible state to a transition\nprobability; R : S × A →R represents the reward function that maps current state and action to a\npayoff and 𝛾∈(0,1] is the discount factor. In each period 𝑡, the agent receives 𝑠𝑡∈S and takes an\naction 𝑎𝑡∼𝜋(·|𝑠𝑡), where 𝜋is a policy which maps states to action’s probability distribution. After\ntaking action 𝑎𝑡, a reward R(𝑎𝑡|𝑠𝑡) = 𝑟𝑡set as the sales profit in period 𝑡is observed. Meanwhile,\nthe system transfers to the next state 𝑠𝑡+1 ∼P(·|𝑎𝑡, 𝑠𝑡), where the state transition function P is\ndetermined by the dynamics of the pricing and replenishment model.\nAt every time period, the decision maker decides the price and replenishment quantity of a\nproduct for this period in order to maximize the discounted future profits, which are defined as\n𝑅𝑡=\n𝑇−𝑡\n∑︁\n𝑘=0\n𝛾𝑘𝑟𝑡+𝑘.\n(1)\nThe expected value of these discounted profits 𝑅𝑡corresponds to the value function 𝑉𝜋𝑡(𝑠𝑡),\nwhich indicates the value of being in state 𝑠𝑡following a certain policy 𝜋𝑡:\n𝑉𝜋𝑡(𝑠𝑡) =\n𝑇−𝑡\n∑︁\n𝑘=0\n𝛾𝑘E𝜋𝑡𝑟𝑡+𝑘.\n(2)\nThe value 𝑉(𝑠𝑡) is recursively dependent on the value of the next state 𝑉(𝑠𝑡+1). The optimal\npricing and replenishment action in period 𝑡can be determined by solving the well-known Bellman\nequations:\n𝑉𝜋∗(𝑠𝑡) = max\n𝑎𝑡∈A\nn\n𝑟𝑡+ 𝛾\n∑︁\n𝑠𝑡+1\nP(𝑠𝑡+1|𝑠𝑡, 𝑎𝑡)𝑉𝜋∗(𝑠𝑡+1)\no\n.\n(3)\nX: Dual-Agent Deep Reinforcement Learning for Dynamic Pricing and Replenishment\n8\nArticle submitted to Operations Research\nTraditional demand assumptions mainly follow the form:\n𝑑𝑡(𝑝𝑡, 𝜖𝑡) = 𝛾𝑡(𝑝𝑡)𝜖𝑡+ 𝛿𝑡(𝑝𝑡),\n(4)\nwith 𝛾(·) and 𝛿(·) being nonincreasing functions. The cases of 𝛾𝑡(𝑝) = 1 and 𝛿𝑡(𝑝) = 0 are often\nreferred to as the additive and multiplicative models, respectively. Continuous demand models are\noften preferred due to their mathematical tractability and computational efficiency (Babai et al.\n2011). However, real-world demands could be discrete. Our analysis of real market demand data,\nusing four common continuous demand models, results in poor fitness with R-squared values\nranging only between 0.106 and 0.146. Example 1 in Appendix B provides further illustration of\nthis inadequacy. Swaminathan and Shanthikumar (1999) point out that optimal ordering policies for\ndiscrete and continuous demand distributions may be totally different. In Section 3.2, we formulate\na problem with the discrete Poisson demand model.\n3.2. Dynamics of the Dynamic Pricing and Replenishment Model\nConsider a 𝑇-period problem, where Table 1 presents the notations. At the beginning of each\nperiod 𝑡(𝑡= 1, ...,𝑇), product prices and replenishment quantities need to be determined. Product\ndemand is uncertain, influenced not only by its own price but also by competitors’ prices and market\nconditions. We infer overall demand by modeling the probability of each individual consumer’s\nchoice. Specifically, customer behavior is captured through a logistic model, which is based on\nreal-world data (Schlosser and Richly 2019). Given our offer price 𝑝, the competitor’s price 𝑜, and\nthe current reference price 𝑗, the sales probability within a time span of length Δ ∈[0,1] is denoted\nby\n𝑃Δ(𝑝, 𝑜, 𝑗) := 𝑃𝑜𝑖𝑠\n\u0010\nΔ · 𝑒𝜅(𝑝,𝑜,𝑗)\n′ 𝛽/(1 + 𝑒𝜅(𝑝,𝑜,𝑗)\n′ 𝛽)\n\u0011\n,\n(5)\nwhere 𝜅(𝑝, 𝑜, 𝑗) represents regressors relating to sales, such as the competitor’s price and the refer-\nence price. 𝛽represents the corresponding parameter vector. This model calculates the probability\nof a sale to an individual customer in a given period based on these price-related features. The\ndemand for a product in period 𝑡follows\n𝑑𝑡∼𝑃𝑜𝑖𝑠\n\u0010\n𝜂Δ · 𝑒𝜅(𝑝,𝑜,𝑗)\n′ 𝛽/(1 + 𝑒𝜅(𝑝,𝑜,𝑗)\n′ 𝛽)\n\u0011\n.\n(6)\nThe scaling factor 𝜂is introduced to adjust the magnitude of the Poisson distribution parameter 𝜆.\nWe assume that excess demand is lost and unsold products have no salvage value. This lost-\nsales assumption complicates the model compared to backlog scenarios (Morton 1971). Arrow\nX: Dual-Agent Deep Reinforcement Learning for Dynamic Pricing and Replenishment\nArticle submitted to Operations Research\n9\nTable 1\nParameters and variables\nParameters\nℎ\nunit holding cost\n𝑏\nunit shortage cost\n𝑐\nunit ordering cost\n𝑧\nlead time\nVariables\n𝑞𝑡replenishment quantity offered by agents in period 𝑡\n𝑝𝑡price offered by the DRL agent in period 𝑡\n𝑜𝑡\nprice offered by the competitor in period 𝑡\n𝑑𝑡\ndemand in period 𝑡\n𝐼𝑡\ninventory level in period 𝑡\n𝐿𝑡lost-sales in period 𝑡\n𝑆𝑡\nactual sales in period 𝑡\net al. (1958) demonstrate that in a single sourcing model with backlogging and constant lead time,\ninventory and outstanding orders can be simplified to one inventory position. When excess demand\nis lost, this simplification does not apply, and the optimal policy relies on the full inventory pipeline\nvector, exponentially increasing complexity with lead time.\nAt the beginning of any period 𝑡, the replenishment quantity 𝑞𝑡and price 𝑝𝑡for a product must\nbe decided. Since the unmet demand is lost, the dynamics of the evolution of inventory can be\nexpressed by\n𝐼𝑡= [𝐼𝑡−1 + 𝑞𝑡−𝑧−𝑑𝑡]+.\n(7)\nThe lost-sales is\n𝐿𝑡= max{𝑑𝑡−(𝐼𝑡−1 + 𝑞𝑡−𝑧),0}.\n(8)\nThe actual sales is given by\n𝑆𝑡= min{𝑑𝑡, 𝐼𝑡−1 + 𝑞𝑡−𝑧}.\n(9)\nThe sales profit 𝑟𝑡is\n𝑟𝑡= 𝑝𝑡𝑆𝑡−ℎ𝐼𝑡−𝑏𝐿𝑡−𝑐𝑞𝑡.\n(10)\nX: Dual-Agent Deep Reinforcement Learning for Dynamic Pricing and Replenishment\n10\nArticle submitted to Operations Research\n3.3. Theoretical Analysis for Single-period Problem\nBefore studying multi-period decision problems, we first consider a single-period problem with\nzero lead time and ending inventory given by\n𝐼= [𝑥0 + 𝑞−𝑑]+,\n(11)\nwhere 𝑞represents the replenishment quantity, 𝑥0 is the initial inventory at the beginning of the\nperiod, which is known, and 𝑑is the demand for the current period.\nThe lost-sales is 𝐿= max{𝑑−(𝑥0 + 𝑞),0}. The actual sales is given by 𝑆= min{𝑑,𝑥0 + 𝑞}. The\nobjective is to find the optimal order quantity 𝑞∗and the optimal price 𝑝∗, in order to maximize the\nsingle-period expected profit depending on the demand model, as follows:\n𝐹(𝑝,𝑥) = E[𝑝𝑆(𝑝,𝑥) −ℎ𝐼(𝑝,𝑥) −𝑏𝐿(𝑝,𝑥) −𝑐(𝑥−𝑥0)],\nwhere 𝑥= 𝑥0 + 𝑞. Therefore, the problem comes down to deciding on 𝑝and 𝑥, given that 𝑥0 is\nknown.\nUnlike the traditional demand assumption in Eq.(4), our discrete Poisson demand model (Eq.(6))\nincorporates a price-dependent parameter 𝜆(𝑝). This dependency introduces complexity into the\nanalysis of related functions and deviates from conventional assumptions.\nProposition 1. The Poisson distribution 𝐷(𝑝) ∼𝑃𝑜𝑖𝑠(𝜆(𝑝)), where 𝜆(𝑝) is the parameter, does\nnot fall under the traditional demand assumption described in Eq.(4): 𝑑(𝑝, 𝜖) = 𝛾(𝑝)𝜖+ 𝛿(𝑝), with\n𝜖being a random variable that is independent of 𝑝.\nThe proof of Proposition 1, along with the following omitted proofs, can be found in Appendix A.\nThis proposition highlights the structural difference between our Poisson demand model 𝐷(𝑝) ∼\n𝑃𝑜𝑖𝑠(𝜆(𝑝)) and traditional models. The key distinction lies in the inseparability of the price-\ndependent function 𝜆(𝑝) from the distribution of the random variable in the Poisson model, unlike\nthe explicit separation of the random variable 𝜖and price-dependent functions in the traditional\nmodel 𝛾(𝑝)𝜖+ 𝛿(𝑝).\nFurthermore, previous studies on joint pricing and replenishment often focus on scenarios with\ndemand backlog, as in Federgruen and Heching (1999), where revenue depends on demand volume\nrather than sales, making the revenue function independent of inventory. In contrast, we analyze\nscenarios with lost-sales under discrete demand (Eq.(6)). We derive the concavity and convexity\nof the inventory function, revenue function, and lost-sales function under discrete demand Eq.(6).\nThe final comparison with traditional assumptions is presented in Proposition 4.\nX: Dual-Agent Deep Reinforcement Learning for Dynamic Pricing and Replenishment\nArticle submitted to Operations Research\n11\n3.3.1. Properties of Objective Function\nAssumption 1. We suppose 𝑙𝑝+ 𝑎> 0 in the domain of 𝑝and the domain is a bounded set.\nLemma 1. Under Assumption 1, define\n𝜆= 𝜂Δ · 𝑒𝑙𝑝+𝑎/(1 + 𝑒𝑙𝑝+𝑎),\nthen 𝜆(𝑝) is concave and monotonically decreasing with respect to 𝑝.\nProposition 2. Under Assumption 1, the inventory function 𝐼(𝑝,𝑥) = E[(𝑥−𝑑)+] is convex and\nmonotonically increasing with respect to price 𝑝.\nProof\nDenote ⌊𝑥⌋as the maximum integer no more than 𝑥. We rewrite 𝐼(𝑝,𝑥) as\nE[(𝑥−𝑑)+] =\n⌊𝑥⌋\n∑︁\n𝑘=0\n𝜆𝑘𝑒−𝜆\n𝑘!\n(𝑥−𝑘).\nThen we calculate the first order derivative and the second order derivative:\n𝜕𝐼(𝑝,𝑥)\n𝜕𝑝\n= 𝜕𝜆\n𝜕𝑝\n⌊𝑥⌋\n∑︁\n𝑘=0\n𝜆𝑘−1𝑒−𝜆\n𝑘!\n(𝑥−𝑘)(𝑘−𝜆),\n𝜕2𝐼(𝑝,𝑥)\n𝜕𝑝2\n= 𝜕2𝜆\n𝜕𝑝2\n⌊𝑥⌋\n∑︁\n𝑘=0\n𝜆𝑘−1𝑒−𝜆\n𝑘!\n(𝑥−𝑘)(𝑘−𝜆) + ( 𝜕𝜆\n𝜕𝑝)2\n⌊𝑥⌋\n∑︁\n𝑘=0\n𝜆𝑘−2𝑒−𝜆\n𝑘!\n(𝑥−𝑘)((𝑘−𝜆)2 −𝑘).\n(12)\nWe want to prove 𝜕2𝐼(𝑝,𝑥)\n𝜕𝑝2\n> 0. First, we consider the first term of Eq.(12). When 𝑥< 𝜆, for every\n𝑘≤⌊𝑥⌋, 𝑘−𝜆< 0 holds, we have Í⌊𝑥⌋\n𝑘=0\n𝜆𝑘−1𝑒−𝜆\n𝑘!\n(𝑥−𝑘)(𝑘−𝜆) < 0. When 𝑥≥𝜆, since (𝑥−𝑘)(𝑘−𝜆) ≤\n(𝑥−𝜆)(𝑘−𝜆), we have\n⌊𝑥⌋\n∑︁\n𝑘=0\n𝜆𝑘−1𝑒−𝜆\n𝑘!\n(𝑥−𝑘)(𝑘−𝜆) ≤(𝑥−𝜆)\n⌊𝑥⌋\n∑︁\n𝑘=0\n𝜆𝑘−1𝑒−𝜆\n𝑘!\n(𝑘−𝜆)\n= 𝑥−𝜆\n𝜆\n⌊𝑥⌋\n∑︁\n𝑘=0\nP(𝑋= 𝑘)(𝑘−𝜆) = 𝑥−𝜆\n𝜆\n(\n⌊𝑥⌋\n∑︁\n𝑘=0\n𝑘P(𝑋= 𝑘) −𝜆P(𝑋≤⌊𝑥⌋),\nwhere 𝑋is a random variable following a Poisson distribution with mean 𝜆.\nFigure 1\nPoisson distribution and its truncated version\nX: Dual-Agent Deep Reinforcement Learning for Dynamic Pricing and Replenishment\n12\nArticle submitted to Operations Research\nFigure 2\nEq.(14) and the second term of Eq.(12) without quadratic term with 𝜆= 5\n(a) Plot of Eq.(14).\n(b) Plot of the Second Term of Eq.(12) without Quadratic Term.\nConsidering a random variable 𝑋′ following Poisson distribution truncating at ⌊𝑥⌋, it is obvious\nthat 𝐸[𝑋′] < 𝐸[𝑋] due to the truncation, as shown in Figure 1. Therefore,\n𝐸[𝑋′] =\nÍ⌊𝑥⌋\n𝑘=0 𝑘P(𝑋= 𝑘)\nP(𝑋≤⌊𝑥⌋)\n< 𝐸[𝑋] = 𝜆,\nwhich implies Í⌊𝑥⌋\n𝑘=0\n𝜆𝑘−1𝑒−𝜆\n𝑘!\n(𝑥−𝑘)(𝑘−𝜆) < 0. Note that 𝜕2𝜆\n𝜕𝑝2 < 0, we prove that the first term of\nEq.(12) is positive. The conclusion 𝜕𝐼(𝑝,𝑥)\n𝜕𝑝\n> 0 holds because 𝜕𝜆\n𝜕𝑝< 0. Next, we claim that the second\nterm of Eq.(12) is also positive. Suppose that 𝑋is a random variable following a Poisson distribution\nwith mean 𝜆, then E[𝑋2] = 𝜆2 + 𝜆, E[𝑋3] = 𝜆3 + 3𝜆2 + 𝜆. We can find that\nE[(𝑥−𝑋)((𝑋−𝜆)2 −𝑋)] = −E[𝑋3] + (𝑥+ (2𝜆+ 1))E[𝑋2] + (−𝜆2 −(2𝜆+ 1)𝑥)E𝑋+ 𝜆2𝑥\n= −𝜆3 −3𝜆2 −𝜆+ (𝑥+ (2𝜆+ 1))(𝜆2 + 𝜆) + (−𝜆2 −(2𝜆+ 1)𝑥)𝜆+ 𝜆2𝑥= 0.\nTherefore, the equation below holds for every 𝑥:\n∞\n∑︁\n𝑘=0\n𝜆𝑘−2𝑒−𝜆\n𝑘!\n(𝑥−𝑘)((𝑘−𝜆)2 −𝑘) = 𝜆−2E[(𝑥−𝑋)((𝑋−𝜆)2 −𝑋)] = 0.\n(13)\nIt is obvious that the the second term of Eq.(12) is continuous and piecewise linear with respect\nto 𝑥with integer points as nodes, which implies that this function reaches its minimum at certain\ninteger nodes. In order to find its minimum with respect to 𝑥, we can calculate the difference by\ntaking 𝑥as the integers 𝑛and 𝑛+ 1, respectively, as follows:\n𝑛+1\n∑︁\n𝑘=0\n𝜆𝑘−2𝑒−𝜆\n𝑘!\n(𝑛+ 1 −𝑘)((𝑘−𝜆)2 −𝑘) −\n𝑛\n∑︁\n𝑘=0\n𝜆𝑘−2𝑒−𝜆\n𝑘!\n(𝑛−𝑘)((𝑘−𝜆)2 −𝑘) =\n𝑛\n∑︁\n𝑘=0\n𝜆𝑘−2𝑒−𝜆\n𝑘!\n((𝑘−𝜆)2 −𝑘).\n(14)\nGiven that ( 𝜕𝜆\n𝜕𝑝)2 > 0, the sign of Eq.(14) determines that of the derivative of the second term of\nEq.(12) with respect to 𝑥when 𝑥is not an integer. We illustrate Eq.(14) and the second term of\nEq.(12) without quadratic term ( 𝜕𝜆\n𝜕𝑝)2 with 𝜆= 5 in Figure 2. The quadratic equation (𝑘−𝜆)2 −𝑘\nwith respect to 𝑘has two roots, denoted as 𝑥1 and 𝑥2, where 0 < 𝑥1 < 𝑥2. These roots are indicated\nas two red points in Figure 2. As is shown in Figure 2a, when 0 < 𝑥≤𝑥1, the value of Eq.(14) is\nX: Dual-Agent Deep Reinforcement Learning for Dynamic Pricing and Replenishment\nArticle submitted to Operations Research\n13\nincreasing with respect to 𝑥. When 𝑥1 < 𝑥< 𝑥2, the value of Eq.(14) is decreasing for 𝑥1 < 𝑘< 𝑥2 as\neach added term containing (𝑘−𝜆)2 −𝑘is negative. When 𝑥> 𝑥2, the value of Eq.(14) is increasing\nfor 𝑘> 𝑥2 as each added term containing (𝑘−𝜆)2 −𝑘is positive. The minimum of the second term\nof Eq.(12) occurs at the point that Eq.(14) changes from negative to positive (if there exists) or at\nthe boundary. Therefore, the second term of Eq.(12) reaches its minimum when 𝑥= 𝑥∗≥⌊𝑥2⌋or\n𝑥= 0. Consequently, we have\n⌊𝑥⌋\n∑︁\n𝑘=0\n𝜆𝑘−2𝑒−𝜆\n𝑘!\n(𝑥−𝑘)((𝑘−𝜆)2 −𝑘) ≥min{0,\n𝑥∗\n∑︁\n𝑘=0\n𝜆𝑘−2𝑒−𝜆\n𝑘!\n(𝑥∗−𝑘)((𝑘−𝜆)2 −𝑘)}\n=min{0,0 −\n∞\n∑︁\n𝑘=𝑥∗+1\n𝜆𝑘−2𝑒−𝜆\n𝑘!\n(𝑥∗−𝑘)((𝑘−𝜆)2 −𝑘)} ≥0.\nAccording to Eq.(13), the second equality holds. The last inequality is based on the fact that when\n𝑘≥𝑥∗+ 1 ≥𝑥2, the expression (𝑥∗−𝑘)((𝑘−𝜆)2 −𝑘) is less than 0. Therefore, the second term of\nEq.(12) is positive as is shown in Figure 2b. In conclusion, the two terms of the second derivative\nof inventory function are positive and non-negative, respectively. This indicates that the inventory\nfunction is convex in 𝑝.\n□\nRemark 1. Under Assumption 1, the inventory function 𝐼(𝑝,𝑥) = E[(𝑥−𝑑)+] is neither jointly\nconvex nor concave with respect to the inventory 𝑥and price 𝑝at its differentiable points. This is\ndue to 𝐼(𝑝, 𝑥) being continuous and piecewise linear with respect to 𝑥, implying 𝜕2𝐼(𝑝,𝑥)\n𝜕𝑥2\n= 0 at its\ndifferentiable points. Besides,\n𝜕2𝐼(𝑝,𝑥)\n𝜕𝑥𝜕𝑝\n= 𝜕𝜆\n𝜕𝑝\n⌊𝑥⌋\n∑︁\n𝑘=0\n𝜆𝑘−1𝑒−𝜆\n𝑘!\n(𝑘−𝜆) > 0,\nwhich is implied by the proof of Proposition 2. Therefore, the Hessian matrix of 𝐼(𝑝,𝑥) is indeter-\nminate.\nProposition 3. Under\nAssumption\n1,\nthe\nrevenue\nfunction\n𝐸(𝑝,𝑥) = 𝑝E[𝑆(𝑝,𝑥)] =\n𝑝E[min(𝑑, 𝑥)] is concave with respect to price 𝑝.\nSince the random variable 𝑑follows the Poisson distribution and the parameter 𝜆is determined\nby price 𝑝through a complex function 𝜆(𝑝), we need more assumptions to guarantee the convexity\nof lost-sales function. We employ the Taylor expansion to approximate 𝜆and linearize it.\nAssumption 2. We suppose that ˆ𝜆= 𝜂Δ𝑒𝑎(1 + 𝑙𝑝) with 𝑙< 0 and 1 + 𝑙𝑝> 0 in the domain of 𝑝,\nwhere the domain is bounded.\nX: Dual-Agent Deep Reinforcement Learning for Dynamic Pricing and Replenishment\n14\nArticle submitted to Operations Research\nLemma 2. Under Assumption 2, the lost-sales function 𝐿(𝑝,𝑥) = E[(𝑑−𝑥)+] is convex with respect\nto the price 𝑝.\nRemark 2. Assumption 1 aims to guarantee the concavity of function 𝜆(𝑝), which can be replaced\nby Assumption 2 because linear function is concave. Henceforth, we use 𝜆to stand for ˆ𝜆in\nAssumption 2.\nHaving established the concavity or convexity of the expected inventory, revenue, and lost-sales\nfunctions, we can derive the concavity of the single-period expected profit function 𝐹(𝑝,𝑥) with\nrespect to 𝑝.\nTheorem 1. Under Assumption 2, 𝐹(𝑝,𝑥) is concave with respect to 𝑝and 𝑥within their respective\ndomains, though not jointly concave. Furthermore, 𝐹(𝑝,𝑥) takes its maximum either at the boundary\nor at an interior point that satisfies the following conditions:\n𝜕𝐹(𝑝,𝑥)\n𝜕𝑝\n= 𝜕𝐸(𝑝,𝑥)\n𝜕𝑝\n−ℎ𝜕𝐼(𝑝,𝑥)\n𝜕𝑝\n−𝑏𝜕𝐿(𝑝,𝑥)\n𝜕𝑝\n= 0,\n𝑏−𝑐+ 𝑝−(ℎ+ 𝑏+ 𝑝)\n𝑥∗−1\n∑︁\n𝑘=0\n𝜆𝑘(𝑝)𝑒−𝜆(𝑝)\n𝑘!\n≥0,\n𝑏−𝑐+ 𝑝−(ℎ+ 𝑏+ 𝑝)\n𝑥∗\n∑︁\n𝑘=0\n𝜆𝑘(𝑝)𝑒−𝜆(𝑝)\n𝑘!\n< 0,\n(15)\nwhere 𝑥∗is a positive integer.\nPrevious literature (Federgruen and Heching 1999) established the joint concavity of the profit\nfunction 𝐹(𝑝,𝑥) with respect to price and inventory under Assumptions 3(i) and 4(i). The Assump-\ntion 3(i) represents only a class of continuous distributions. The Assumption 4(i) means that the\nrevenue collected in each period depends on the demand volume instead of the sales amount. In\nthis paper, we cannot derive this conclusion primarily because we have revised these conditions to\nAssumption 3(ii) and Assumption 4(ii).\nAssumption 3. (i) 𝑑(𝑝, 𝜖) = 𝛾(𝑝)𝜖+ 𝛿(𝑝); (ii) 𝑑∼𝑃𝑜𝑖𝑠(𝜆(𝑝)).\nAssumption 4. (i) 𝐸(𝑝) = 𝑝E[𝑑] = 𝑝𝜆(𝑝); (ii) 𝐸(𝑝,𝑥) = 𝑝E[min(𝑑,𝑥)].\nNotably, altering Assumptions 3(i) or 4(i) to Assumptions 3(ii) or 4(ii) results in the loss of joint\nconcavity established in Federgruen and Heching (1999). We illustrate this point with the following\nproposition.\nX: Dual-Agent Deep Reinforcement Learning for Dynamic Pricing and Replenishment\nArticle submitted to Operations Research\n15\nProposition 4. If either Assumption 3(i) or Assumption 4(i) is changed to Assumption 3(ii) or\nAssumption 4(ii), respectively, the joint concavity of 𝐹(𝑝,𝑥) in Federgruen and Heching (1999)\nwill be lost.\nRemark 3. A one-to-one mapping from price 𝑝to expected demand 𝜆(𝑝) is constructed, thus\ntransforming the optimization variable from price to expected demand (Bernstein et al. 2016). Since\naffine transformations do not alter concavity or convexity, the conclusions in this paper remain\nunchanged due to the Poisson demand.\nProposition 4 illustrates the limitations of the traditional model. Even slight modifications to the\nassumptions can negate joint concavity, rendering the original model’s optimal strategies invalid.\nUnder new assumptions, there are still some conclusions consistent with the finding in previous\nliterature such as the following proposition (Federgruen and Heching 1999).\nProposition 5. Given the inventory 𝑥and by solving the first equation in condition (15), we can\ndetermine 𝑝(𝑥) to maximize 𝐹(𝑝,𝑥). The function 𝑝(𝑥) is a non-increasing function with respect\nto 𝑥.\nRemark 4. Figure 10 in Appendix C clearly shows that 𝐹(𝑝,𝑥) is not jointly concave with respect\nto 𝑝and 𝑥, because the value of 𝐹(𝑝,𝑥) at points between 𝐴and 𝐵lies below the corresponding\nlinear interpolation of the function values at 𝐴and 𝐵, indicating a violation of joint concavity\nwhich is crucial for deriving optimal policies like BSLP. Without it, optimal joint pricing and\nreplenishment decisions cannot be directly obtained from the profit function’s first-order conditions.\nFinding the global maximum requires identifying and comparing all local maxima, a process\nwhose computational complexity increases exponentially with the search space. Consequently,\nmore advanced optimization algorithms become necessary for efficient solutions.\n3.3.2. Two-timescale Stochastic Approximation\nIn commercial operations, the frequencies\nof product replenishment and pricing decisions often differ. The convergence of a two-timescale\nalgorithm can be guaranteed by the respective concavity of the profit function. This concavity can\nnow be utilized to guide the algorithm design. For the case where the frequency of pricing decisions\nexceeds that of replenishment decisions, two-timescale algorithms can adjust the relative updating\nrates of the price and inventory in the algorithm and use a faster search speed to update 𝑝in cases\nwhere prices are determined with a higher frequency than inventory. Conversely, for some products\nwhere replenishment decisions are made more frequently than pricing adjustments, two-timescale\nalgorithms use a faster search speed to update 𝑥in these cases.\nX: Dual-Agent Deep Reinforcement Learning for Dynamic Pricing and Replenishment\n16\nArticle submitted to Operations Research\nWe illustrate the algorithm when pricing decisions are more frequent than replenishment deci-\nsions. The single-period profit function is\n𝐹(𝑝,𝑥) = 𝑝E[min(𝑑,𝑥)] −(ℎ+ 𝑏)E[(𝑥−𝑑)+] −𝑏𝜆+ 𝑏𝑥−𝑐(𝑥−𝑥0),\nwhere inventory 𝑥changes relatively slower than price 𝑝in this case. We conduct simultaneous\ndescents for 𝑝and 𝑥in two coupled iterations, with the update rate of 𝑝being faster than that of 𝑥.\nWe express the partial derivatives of the profit function in Eq.(A.2) using the form of expectation,\nallowing us to derive their unbiased estimators, as outlined in the proposition below.\nProposition 6. Let 𝑔(𝑝,𝑥) and ℎ(𝑝,𝑥) denote the partial derivatives of the profit function in\nEq.(A.2) with respect to 𝑝and 𝑥, then\n𝑔(𝑝, 𝑥) \u0011 𝜕𝐹(𝑝,𝑥)\n𝜕𝑝\n= E[min(𝑑,𝑥)] + 𝑝E\n\u0014𝜕logP(𝑑)\n𝜕𝑝\nmin(𝑑,𝑥)\n\u0015\n−(ℎ+ 𝑏)E\n\u0014𝜕logP(𝑑)\n𝜕𝑝\n(𝑥−𝑑)+\n\u0015\n−𝑏𝜕𝜆\n𝜕𝑝,\nℎ(𝑝, 𝑥) \u0011 𝜕𝐹(𝑝,𝑥)\n𝜕𝑥\n= 𝑏−𝑐+ 𝑝−(ℎ+ 𝑏+ 𝑝)E[1𝑑≤𝑥],\nwhere P(𝑑) is the probability density function of random variable 𝑑.\nDue to Proposition 6, unbiased stochastic gradient estimators for 𝑔(𝑝,𝑥) and ℎ(𝑝,𝑥) can be\nobtained using the Monte Carlo method and are denoted as ˆ𝑔(𝑝,𝑥) and ˆℎ(𝑝,𝑥), respectively. The\niteration process of the two-timescale algorithm can be formulated as follows:\n𝑝𝑘+1 = 𝑝𝑘+ 𝛼𝑘ˆ𝑔(𝑝𝑘,𝑥𝑘)\n\u0011 𝑝𝑘+ 𝛼𝑘\n\u0012\nmin(𝑑𝑘,𝑥𝑘) + 𝑝𝑘\n𝜕logP(𝑑𝑘)\n𝜕𝑝\nmin(𝑑𝑘,𝑥𝑘) −(ℎ+ 𝑏) 𝜕logP(𝑑𝑘)\n𝜕𝑝\n(𝑥𝑘−𝑑𝑘)+ −𝑏𝜕𝜆\n𝜕𝑝\n\u0013\n,\n𝑥𝑘+1 = 𝑥𝑘+ 𝛽𝑘ˆℎ(𝑝𝑘,𝑥𝑘) \u0011 𝑥𝑘+ 𝛽𝑘\n\u0012\n(𝑏−𝑐+ 𝑝𝑘) −(ℎ+ 𝑏+ 𝑝𝑘)1𝑑𝑘≤𝑥𝑘\n\u0013\n,\n(16)\nwhere 𝑑𝑘is sampled from a Poisson distribution with parameter 𝜆(𝑝𝑘). Note that E[ ˆ𝑔(𝑝,𝑥)] =\n𝑔(𝑝, 𝑥) and E[ ˆℎ(𝑝,𝑥)] = ℎ(𝑝,𝑥). We could reduce the variance by taking more samples each time,\nwhile a single sample is sufficient for convergence to the optimum. Step size 𝛼𝑘represents the time\nscale for the first iteration, and 𝛽𝑘represents the time scale for the second iteration. The conditions\nÍ∞\n𝑘=1 𝛼𝑘= Í∞\n𝑘=1 𝛽𝑘= ∞, Í∞\n𝑘=1 𝛽2\n𝑘< Í∞\n𝑘=1 𝛼2\n𝑘< ∞, and 𝛽𝑘\n𝛼𝑘→0 as 𝑘tends to infinity are satisfied,\nindicating that the first time scale progresses faster than the second time scale.\nThe convergence of the two-timescale algorithm is guaranteed, analogous to the theoretical\nresults in Borkar (2009) and Harold et al. (1997). Specifically, we present the following lemma\nto analyze the convergence of 𝑝𝑘in the faster time scale, leveraging the concavity of the revenue\nfunction with respect to 𝑝.\nX: Dual-Agent Deep Reinforcement Learning for Dynamic Pricing and Replenishment\nArticle submitted to Operations Research\n17\nLemma 3. The iterative sequence {(𝑝𝑘,𝑥𝑘)} generated by the iteration Eq.(16) converges to a set\nwhich is defined as below:\n(𝑝𝑘,𝑥𝑘) →{(𝑝★(𝑥),𝑥)},\nwhere 𝑝★(𝑥) is a well defined function showing that when 𝑥is fixed, the value of 𝑝to make 𝑔(𝑝,𝑥)\nequal to zero. In other words, this lemma means |𝑝𝑘−𝑝★(𝑥𝑘)| →0 almost surely, that is, {𝑝𝑘}\nasymptotically tracks 𝑝★(𝑥𝑘).\nThen, we will combine the analysis of the second time scale and show that the sequence {(𝑝𝑘,𝑥𝑘)}\nconverges to a fixed point. Considering the second scale, the variable 𝑥, previously considered\nfixed, should now be treated as the time-varying 𝑥(𝑡). Based on Lemma 3, for sufficiently large 𝑡,\n𝑝(𝑡) closely tracks 𝑝★(𝑥(𝑡)). We then substitute it into the second scale and focus on the following\nODE:\n¤𝑥(𝑡) = ℎ(𝑝★(𝑥(𝑡)),𝑥(𝑡)).\n(17)\nTheorem 2. The iterative sequence {(𝑝𝑘,𝑥𝑘)} generated by the iteration Eq.(16) converges to\nsome limit point of the ODE (17), i.e.,\n(𝑝𝑘,𝑥𝑘) →(𝑝★(𝑥★),𝑥★).\nProof\nSince ℎ(𝑝,𝑥) = 𝑏−𝑐+ 𝑝−Í⌊𝑥⌋\n𝑘=0\n𝜆(𝑝)𝑘𝑒−𝜆( 𝑝)\n𝑘!\n(ℎ+ 𝑏+ 𝑝), note that ℎ(𝑝,𝑥) is a piecewise\nconstant function so the ODE (17) is equipped with discontinuous right hand side, for which\nstandard solution concepts are introduced in Filippov (1988). It is obvious that ℎ(𝑝,𝑥) satisfies a\nlinear growth condition: |ℎ(𝑝,𝑥)| ≤𝐾(1+ |𝑥|) for some 𝐾> 0. Define 𝐻(𝑝,𝑥) \u0011 ∩𝜖>0 ¯𝑐𝑜({ℎ(𝑝, 𝑦) :\n|𝑦−𝑥| < 𝜖}), where ¯𝑐𝑜is the notation of convex closure. Then the ODE limit gets replaced by a\ndifferential inclusion limit: ¤𝑥(𝑡) ∈𝐻(𝑝★(𝑥(𝑡)),𝑥(𝑡)). By Corollary 4 in Borkar (2009, Chap.5), the\nsequence {𝑥𝑘} generated by the iteration Eq.(16) converges to (possibly path dependent) the limit\npoint of Eq.(17) almost surely. Furthermore, the iterative sequences {𝑝𝑘} and {𝑥𝑘} generated by\niterations Eq.(16) converge to some limit point of the ODE below by Theorem 2 in Borkar (2009,\nChap.6).\n¤𝑝(𝑡) = 𝑔(𝑝(𝑡),𝑥(𝑡)),\n¤𝑥(𝑡) = ℎ(𝑝(𝑡),𝑥(𝑡)).\n□\nThe limit point of ODE in Lemma 3 and Theorem 2 satisfy the condition 𝑔(𝑝,𝑥) = 0 and ℎ(𝑝,𝑥) = 0,\nwhich shows 𝜕𝐹(𝑝,𝑥)\n𝜕𝑝\n= 𝜕𝐹(𝑝,𝑥)\n𝜕𝑥\n= 0 by definition of 𝑔(𝑝,𝑥) and ℎ(𝑝,𝑥). Therefore, the limit point\nof ODE is a local optimum in the optimization problem. Figure 11 in Appendix C shows that the\nX: Dual-Agent Deep Reinforcement Learning for Dynamic Pricing and Replenishment\n18\nArticle submitted to Operations Research\ntwo-timescale algorithm approximately converges to 𝑝= 55 and 𝑥= 5 under certain parameters,\nwhich fully satisfy the conditions in Theorem 1.\nFurthermore, we study the convergence rate of the two-timescale algorithm.\nTheorem 3. Assume that the function 𝑝★(𝑥) is a Lipschitz function, and then we have the order of\nthe mean squared error for the sequence {𝑥𝑘} generated by the iteration Eq.(16):\nE[|𝑝𝑘−𝑝★(𝑥𝑘)|] = 𝑂( 𝛽𝑘\n𝛼𝑘\n) + 𝑂(𝛼\n1\n2\n𝑘).\n3.4. Multi-period Problem Property\nWhen it comes to multi-period decision problems, the problem is formulated as MDP through\nBellman equation (3), with the goal of determining the optimal policy for the current state in each\nperiod.\nAt each step excluding the final period 𝑇, the optimization of the Bellman equation (3) includes\na future discounted expectation term with respect to the next state. To account for this, we need to\nfind the optimal policy 𝑎𝑡+1(𝑠𝑡+1) under every state 𝑠𝑡+1 in the 𝑡+ 1 period, calculate the 𝑉𝜋★(𝑠𝑡+1),\nand then take the expectation E𝑠𝑡+1[𝑉𝜋★(𝑠𝑡+1)]. This transforms the problem into a functional\noptimization task, as we need to find a function 𝑎𝑡+1(𝑠𝑡+1) that maximizes E𝑠𝑡+1[𝑉𝜋★(𝑠𝑡+1)]. The\ncomputational complexity involved in solving Eq.(3) by backward induction is detailed in Proposi-\ntion 7.\nProposition 7. Suppose the price of each product is discretized into P values and the replenish-\nment quantity is discretized into 𝑄values; there are 𝐾regressors in Eq.(5), the maximum dimension\nand the minimum dimension of the regressors are 𝑅𝑚𝑎𝑥and 𝑅𝑚𝑖𝑛. The lower and upper bounds of\nthe total computational complexity needed to recursively solve the optimality equation (3) via DP\nare 𝑂(𝑃𝑇𝑄𝑇(𝑅𝑚𝑖𝑛)𝐾𝑇) and 𝑂(𝑃𝑇𝑄𝑇(𝑅𝑚𝑎𝑥)𝐾𝑇), respectively.\nIt can be observed that the traditional dynamic programming technique suffers from the curse-\nof-dimensionality, rendering it computationally intractable for sizable problems. The curse-of-\ndimensionality arises from the high dimensions of the state space which grows in the number of\nproducts, the action space which includes all possible pricing and replenishment combinations\nand the transition space dependent on the number of possible demand realizations. To address\nsuch complex scenarios, particularly in high-dimensional, continuous spaces typical of functional\noptimization tasks, DRL can be applied. By interacting with the environment, DRL learns to make\ndecisions based on observed feedback while leveraging the power of deep neural networks.\nX: Dual-Agent Deep Reinforcement Learning for Dynamic Pricing and Replenishment\nArticle submitted to Operations Research\n19\n3.5. Two-timescale DRL Approach\nWe use two agents to make pricing and replenishment decisions, respectively, and integrate the\ntwo-timescale algorithm with HAPPO. To achieve optimal pricing and replenishment decisions,\neach agent should consider the influence of the action on not only its own reward but also the reward\nof other agents. Thus, the MDP constructed in Section 3.1 should be transformed into an MG.\n3.5.1. Markov Game Formulation\nWe formulate the dynamic pricing and replenishment\nproblem with 𝑇periods as an MG, which can be defined as a tuple < 𝑁,S, A, P, R, 𝛾>. Here\n𝑁∈N+ is the number of agents; S is the finite state space; A = Î𝑛\n𝑖=1 A𝑖is the product of finite\naction spaces of all agents, known as the joint action space, where A𝑖is the set of actions for\nagent 𝑖; P : S × A × S →[0,1] is the transition probability function that maps the current global\nstate, global action (including all agents’ actions) and next possible state to a transition probability;\nR : S × A →R is the reward function that maps current global state and global action to a payoff;\nand 𝛾∈(0,1] is the discount factor. The agents interact with the environment according to the\nfollowing protocol: in each period 𝑡, the agents are at state 𝑠𝑡∈S; every agent 𝑖takes an action\n𝑎𝑖\n𝑡∈A𝑖, drawn from its policy 𝜋𝑖(·|𝑠𝑡), which together with other agents’ actions gives a joint action\na𝑡∈A, drawn from the joint policy 𝜋(·|𝑠𝑡) = Î𝑛\n𝑖=1 𝜋𝑖(·|𝑠𝑡). After taking actions a𝑡, the agents receive\na joint reward 𝑟𝑡= R(𝑠𝑡|a𝑡), which is set as the sales profit in period 𝑡; see Eq.(10). Afterwards,\nthe system transfers to the next state 𝑠𝑡+1 with probability P(𝑠𝑡+1|𝑠𝑡,a𝑡), where the state transition\nfunction P is determined by the dynamics of the pricing and replenishment model, as detailed in\nSection 3.2. A trajectory is generated by repeating the interaction until period 𝑇is reached. All\nagents update their policies cooperatively, aiming to maximize the expected total reward:\n𝐽(𝜋) = E𝑎1\n1:𝑇∼𝜋1,...,𝑎𝑛\n1:𝑇∼𝜋𝑛,𝑠1:𝑇∼𝜌1:𝑇\n𝜋\n\" 𝑇\n∑︁\n𝑡=1\n𝛾𝑡𝑟𝑡\n#\n,\n(18)\nwhere 𝑎𝑖\n1:𝑇∼𝜋𝑖represents that each action in action sequence (𝑎𝑖\n1, ..., 𝑎𝑖\n𝑇) of actor 𝑖follows 𝜋𝑖, and\n𝜌𝑡\n𝜋is the marginal state distribution at time 𝑡with the joint policy 𝜋.\n3.5.2. Solution\nNow we show how to use our fast-slow dual-agent DRL algorithm (FSDA) to\nconstruct pricing and replenishment policies. FSDA applies an on-policy PPO for each agent and\nfollows a CTDE structure. We use the subscripts 𝜃and 𝜙to represent the parameters of policy and\nvalue functions, respectively. Each agent 𝑖has a policy function 𝜋𝑖\n𝜃. The value function 𝑣𝜙maps all\nagents’ state to state value 𝑣𝜙(𝑠𝑡). Both the policy function and the value function are constructed\nusing neural networks, known as the actor network and the critic network, respectively. To capture\nX: Dual-Agent Deep Reinforcement Learning for Dynamic Pricing and Replenishment\n20\nArticle submitted to Operations Research\nuseful historical information such as demand fluctuations and price changes during the lead time,\nwe choose a Recurrent Neural Network (RNN) with two stacked Gate Recurrent Units (GRUs) to\nconstruct our actor and critic networks (Liu et al. 2022). Thus, the agent can fully take historical\ninformation into account and make better decisions.\nFigure 3\nStructure of actor network\nWe illustrate the diagram for the actor network in Figure 3. For the actor network, we use a\nmulti-layer perceptron (MLP) with two hidden layers, namely MLP1, to encode the actor 𝑖’s state 𝑠𝑖\n𝑡\ninto high-dimensional features. The first GRU, namely GRU1, receives these features and combines\na hidden state ℎ1\n𝑡−1 as the input and outputs a hidden state ℎ1\n𝑡and some high-dimensional features\nwhich are served as the input of the second GRU (GRU2). GRU2 outputs an updated hidden state\nℎ2\n𝑡and new high-dimensional features. These features are then fed into the second multi-layer\nperceptron (MLP2) to match the size of the action space and mapped into the probability distribution\n𝜋𝑖\n𝜃(·|𝑠𝑖\n𝑡) of actions by a softmax layer. For the critic network, its only difference from the actor\nnetwork lies in that features produced by GRU2 are directly mapped into the state value.\nFor the actual training process, we use 𝜃𝑚and 𝜙𝑚to represent parameters for actor and critic\nnetworks after 𝑚updates. For each episode, we simulate the MG corresponding to the dynamic\npricing and replenishment problem for 𝑇periods with actors’ current policies (𝜋1\n𝜃𝑚,...,𝜋𝑛\n𝜃𝑚) and\ncollect all actors’ states, actions and rewards as the training data. Then, if the current episode is\ndivisible by the ratio 𝑘of the two time scales, each actor’s parameters denoted by 𝜃𝑚are updated\nto 𝜃𝑚+1 using the collected data, respectively. Otherwise, only the first actor’s parameters 𝜃𝑚are\nupdated to 𝜃𝑚+1. Note that the first actor can be either a pricing or a replenishment agent, depending\non which decision needs to be updated more quickly. After that, the critic’s parameters 𝜙𝑚are\nupdated to 𝜙𝑚+1. The ratio 𝑘of the two time scales tends to infinity as the algorithm iterates. That\nis to say, 𝑘is a function of 𝑚, for example, 𝑘= 𝑚/2. A pseudocode for the detailed simulation and\nparameter update process is presented in Appendix E.\nX: Dual-Agent Deep Reinforcement Learning for Dynamic Pricing and Replenishment\nArticle submitted to Operations Research\n21\nIn the application of the algorithm, we also incorporate several performance enhancement tech-\nniques to optimize its effectiveness, such as batch advantage normalization, reward scaling, and\northogonal initialization.\n3.6. Theoretical Analysis\nFor the FSDA in this section, we show its convergence below. Notice that (𝜃2\n𝑚, 𝜙𝑚) is updated at\nevery step, while 𝜃1\n𝑚is updated once every 𝑘steps. When using the Adam optimizer, with a step\nsize that satisfies the descent condition, we can consider (𝜃2\n𝑚, 𝜙𝑚) to be on a faster time scale and\n𝜃1\n𝑚on a slower time scale. The iterative formulas extracted from the algorithm are as follows:\n𝜃1\n𝑚+1 = 𝜃1\n𝑚+ 𝛽𝑚\n𝜕\n𝜕𝜃1𝑚\n(𝐿1\n𝑝+ 𝐿1\n𝑒) = 𝜃1\n𝑚+ 𝛽𝑚(ℎ1(𝜃1\n𝑚, 𝜙𝑚) + 𝑁(1)\n𝑚),\n𝜃2\n𝑚+1 = 𝜃2\n𝑚+ 𝛼𝑚\n𝜕\n𝜕𝜃2𝑚\n(𝐿2\n𝑝+ 𝐿2\n𝑒) = 𝜃2\n𝑚+ 𝛼𝑚(ℎ2(𝜃1\n𝑚, 𝜃2\n𝑚, 𝜙𝑚) + 𝑁(2)\n𝑚),\n𝜙𝑚+1 = 𝜙𝑚+ 𝛼𝑚\n𝜕\n𝜕𝜙𝑚\n𝐿𝑐= 𝜙𝑚+ 𝛼𝑚(ℎ3(𝜃1\n𝑚, 𝜃2\n𝑚, 𝜙𝑚) + 𝑁(3)\n𝑚),\n(19)\nwhere 𝑘𝑚= 𝛼𝑚\n𝛽𝑚→∞due to our setting and the deterministic functions ℎ1(𝜃1, 𝜙), ℎ2(𝜃1, 𝜃2, 𝜙) and\nℎ3(𝜃1, 𝜃2, 𝜙) are as follows:\nℎ1(𝜃1, 𝜙) = E𝑠∼𝜌𝜋𝜃𝑚,𝑎∼𝜋𝜃𝑚\n\"\nmin\n\u0010 𝜋1\n𝜃𝑚+1(𝑎1\n𝑡|𝑠1\n𝑡)\n𝜋1\n𝜃𝑚(𝑎1\n𝑡|𝑠1\n𝑡) 𝐴1\n𝑡, 𝑐𝑙𝑖𝑝\n\u0010 𝜋1\n𝜃𝑚(𝑎1\n𝑡|𝑠1\n𝑡)\n𝜋1\n𝜃𝑚−1 (𝑎1\n𝑡|𝑠1\n𝑡) ,1 −𝜖,1 + 𝜖\n\u0011\n𝐴1\n𝑡\n\u0011#\n,\nℎ2(𝜃1, 𝜃2, 𝜙) = E𝑠∼𝜌𝜋𝜃𝑚,𝑎∼𝜋𝜃𝑚\n\"\nmin\n\u0010 𝜋2\n𝜃𝑚+1(𝑎2\n𝑡|𝑠2\n𝑡)\n𝜋2\n𝜃𝑚(𝑎2\n𝑡|𝑠2\n𝑡) 𝐹2\n𝑡𝐴2\n𝑡, 𝑐𝑙𝑖𝑝\n\u0010 𝜋2\n𝜃𝑚(𝑎2\n𝑡|𝑠2\n𝑡)\n𝜋2\n𝜃𝑚−1(𝑎2\n𝑡|𝑠2\n𝑡) ,1 −𝜖,1 + 𝜖\n\u0011\n𝐹2\n𝑡𝐴2\n𝑡\n\u0011#\n,\nℎ3(𝜃1, 𝜃2, 𝜙) = E𝑠∼𝜌𝜋𝜃𝑚[(𝑣𝜙(𝑠𝑡) −𝑟𝑡)2].\n(20)\nwhere 𝑐𝑙𝑖𝑝(𝑥,min,max) confines 𝑥to the interval [min,max], returning min, max, or 𝑥accordingly\nas 𝑥falls below min, exceeds max, or lies between them.\nLemma 4. Suppose that the parameter space for (𝜃1\n𝑚, 𝜃2\n𝑚, 𝜙𝑚) is a convex bounded set. Also suppose\nthat ℎ1, ℎ2 and ℎ3 are Lipschitz functions and 𝑁(1)\n𝑚, 𝑁(2)\n𝑚\nand 𝑁(3)\n𝑚\nare martingales with finite\nconditional second order moment. The step sizes 𝛼𝑚and 𝛽𝑚satisfy the conditions Í∞\n𝑘=1 𝛼𝑘=\nÍ∞\n𝑘=1 𝛽𝑘= ∞, Í∞\n𝑘=1 𝛽2\n𝑘< Í∞\n𝑘=1 𝛼2\n𝑘< ∞, and 𝛽𝑘\n𝛼𝑘→0 as 𝑘tends to infinity. The iterative sequence\n{(𝜃1\n𝑚, 𝜃2\n𝑚, 𝜙𝑚)} generated by the iteration Eq.(19) converges to a set which is defined as below:\n(𝜃1\n𝑚, 𝜃2\n𝑚, 𝜙𝑚) →{(𝜃1\n𝑚, 𝜃2★\n𝑚(𝜃1\n𝑚), 𝜙★\n𝑚(𝜃1\n𝑚))},\nwhere 𝜃2★\n𝑚(𝜃1\n𝑚) and 𝜙★\n𝑚(𝜃1\n𝑚) are well-defined functions, similar to 𝑝∗(𝑥) defined in Lemma 3. In\nother words, this lemma means |𝜃2\n𝑚−𝜃2★\n𝑚(𝜃1\n𝑚)| →0 and |𝜙𝑚−𝜙★\n𝑚(𝜃1\n𝑚)| →0 almost surely, that is,\n{𝜃2\n𝑚} and {𝜙𝑚} asymptotically track 𝜃2★\n𝑚(𝜃1\n𝑚) and 𝜙★\n𝑚(𝜃1\n𝑚), respectively.\nX: Dual-Agent Deep Reinforcement Learning for Dynamic Pricing and Replenishment\n22\nArticle submitted to Operations Research\nTheorem 4. In the condition of Lemma 4, the iterative sequence {(𝜃1\n𝑚, 𝜃2\n𝑚, 𝜙𝑚)} generated by\nthe iteration Eq.(19) converges to some limit point of the ODE ¤𝜃1(𝑡) = ℎ1(𝜃1(𝑡), 𝜙★(𝜃1(𝑡))), i.e.,\n(𝜃1\n𝑚, 𝜃2\n𝑚, 𝜙𝑚) →(𝜃1★, 𝜃2★(𝜃1★), 𝜙★(𝜃1★)).\nThe proofs of Lemma 4 and Theorem 4 are similar to Lemma 3 and Theorem 2 and we omit them.\n3.7. Algorithmic Complexity Analysis\nIn this section, we analyze the computation complexity of the FSDA algorithm shown in Algorithm\n1. The computational complexity is associated with the scale of the pricing and replenishment\nproblem under consideration, including the length of the decision period and the dimensions of\nthe state and action. Additionally, the computational complexity is also influenced by the specific\ndesign of the neural networks in the FSDA, such as the input/output width and the number of hidden\nlayers of the MLP, the number of GRUs and the number of episodes for training. We provide the\ncomputation complexity of the model training and the model inference in Proposition 8.\nProposition 8. Suppose that the dimension of state 𝑠𝑖\n𝑡for each agent 𝑖∈{1, ..., 𝑁} in each period\n𝑡∈{1, ...,𝑇} is 𝐷𝐼, the dimension of the action 𝑎𝑖\n𝑡is 𝐷𝐴, the width of MLP1’s hidden layers are\nboth 𝑊1, and the input width of MLP2 is 𝑊2. Then the computational complexity of model training\nwith 𝑀episodes is\n𝑂\n\u0010"

ERROR: type should be string, got "https://pubsonline.informs.org/journal/opre Submitted to Operations Research OPERATIONS RESEARCH Vol. 00, No. 0, Xxxxx 0000, pp. 000–000 ISSN 0030-364X, EISSN 1526-5463 Dual-Agent Deep Reinforcement Learning for Dynamic Pricing and Replenishment Yi Zheng Guanghua School of Management, Peking University, yizheng@stu.pku.edu.cn Zehao Li Guanghua School of Management, Peking University, zehaoli@stu.pku.edu.cn Peng Jiang School of Management, Huazhong University of Science and Technology, pengjiang0731@outlook.com Yijie Peng Guanghua School of Management, Peking University, pengyijie@pku.edu.cn Authors are encouraged to submit new papers to INFORMS journals by means of a style file template, which includes the journal title. However, use of a tem- plate does not certify that the paper has been accepted for publication in the named journal. INFORMS journal tem- plates are for the exclusive purpose of submitting to an INFORMS journal and are not intended to be a true representa- tion of the article’s final published form. Use of this template to distribute papers in print or online or to submit papers to another non-INFORM publication is prohibited. Dynamic pricing refers to the practice where sellers dynamically adjust prices based on factors such as sales period, historical demand information, and supply availability (Elmaghraby and Keskinocak 2003), which typically faces challenges of high costs or low feasibility in practice. Dynamic pricing methods have been applied to industries where capacity (supply) is difficult to 1 arXiv:2410.21109v1 [cs.LG] 28 Oct 2024 X: Dual-Agent Deep Reinforcement Learning for Dynamic Pricing and Replenishment 2 Article submitted to Operations Research change in the short term, such as airlines, cruise lines, hotels, and power facilities (Gallego and Van Ryzin 1994, 1997). Sectors like retail, with more flexible replenishment or higher price-change costs, have traditionally focused on enhancing inventory management. Appropriate coordination between pricing and inventory decisions can mitigate inventory-demand mismatches (Feng et al. 2020) and optimize profits (Lei et al. 2018). The rapid development of e-commerce and information systems has reduced the costs associated with price adjustments and enhanced the ability for real-time inventory monitoring (Brynjolfsson and Smith 2000). The challenge lies in that retailers often need to coordinate the pricing and inventory of thousands of products simultaneously. The existing literature suffers the following limitations: (i) The optimality of the base-stock list-price (BSLP) and (s,S,p) policies requires assuming restrictive functional forms of demand, such as additive or multiplicative demands. (ii) Existing literature often assumes that excess demand is backlogged. For retail firms selling to consumers (i.e., B2C), it is more common for consumers to take their demand elsewhere during stock-outs, leading to lost-sales (Bijvank et al. 2023). (iii) Most previous works consider demand in a certain period that depends only on prices, without considering the impact of other market factors, such as competitors. We take into consideration the impact of market competition (including competitors’ prices, market reference prices, and price rankings) while optimizing pricing and replenishment strategies. In the literature, strong assumptions about market demand have often been made. For example, Cohen et al. (2018) assume that the demand is a linear function of the price plus a random noise component, and Aviv and Pazgal (2005) assume that customer arrivals follow a Poisson distribution. In reality, demand is not solely related to price, and the relationship is not as simple as a linear one. Our work is not constrained by the unrealistic assumptions mentioned above. A recent trend gaining steam is to introduce reinforcement learning (RL) to solve sequential inventory decision problems (Gijsbrechts et al. 2021, Oroojlooyjadid et al. 2022). Through con- tinuous trial-and-error interactions with an unknown demand, RL agents learn by themselves to achieve optimal strategies without imposing assumptions on the demand (Sutton and Barto 1999). Deep RL (DRL) is a subfield of RL, which employs deep neural networks to approximate the value or policy functions of Markov Decision Processes (MDPs). Rana and Oliveira (2014) investigate the pricing problem for a given inventory within a fixed time period. Kastius and Schlosser (2021) apply Deep Q-Networks (DQN) and Soft Actor Critic (SAC) RL algorithms to the pricing problem X: Dual-Agent Deep Reinforcement Learning for Dynamic Pricing and Replenishment Article submitted to Operations Research 3 of a single product in duopoly and oligopoly markets where the customer behavior is described by the logit model in Schlosser and Boissier (2018). We first investigate a dynamic pricing and replenishment problem under competition, considering competitor strategies and market conditions. Different from the traditional demand assumption, the discreteness of demand we consider, along with the parameter within the Poisson distribution being a function of price, introduces complexity into analyzing the property of the problem. We prove that the single-period profit function is concave with respect to product price and inventory within respective domains, but it is not jointly concave. Based on the respective concavity, a two-timescale algorithm is proposed. For multi-period problems, we propose a dual-agent DRL algorithm combining the two-timescale approach with Multi-Agent DRL (MADRL). Two agents are introduced in the algorithm, responsible for pricing and inventory decisions respectively. This allows each agent to optimize its own decisions while fully considering the influence of the other agent. Furthermore, we allow the two agents for making decisions on pricing and inventory to be updated in different scales. Finally, we explore a dynamic pricing and replenishment problem where product demand is directly learned from comprehensive market data, encompassing product attributes, customer behavior, and marketing information. The simulation results indicate that our joint pricing and replenishment strategy, which incorporates demand forecasting and further price optimization, results in lower inventory and higher profits. Our work consists of the following contributions. (i) We prove that the single-period profit function in dynamic pricing and replenishment problem is concave with respect to product price and inventory within respective domains, but it is not jointly concave. The respective concavity ensures the convergence of our proposed two-timescale algorithm, with solutions converging to the limit points of the ordinary differential equation (ODE). (ii) We formulate the dynamic pricing and replenishment problems as Markov Games (MGs) and propose a new solution integrating two-timescale algorithms with MADRL to construct intelligent pricing and ordering policies. Additionally, we enhance our model’s capability to capture real market demand by combining a decision tree-based machine learning (ML) model with real market data to train the ML model. (iii) We experimentally demonstrate that our algorithms achieve optimal performance on solv- able numerical examples and outperform baseline algorithms in cases where optimal strategies are not analytical, thus validating their effectiveness. Furthermore, our method exhibits linear computational complexity relative to problem scale, ensuring efficiency in practical pricing and replenishment applications. X: Dual-Agent Deep Reinforcement Learning for Dynamic Pricing and Replenishment 4 Article submitted to Operations Research 2. Literature Review In this section, a brief review on related literature is provided. We begin by reviewing the literature on joint dynamic pricing and inventory control, and then proceed to review the problem of dynamic pricing and replenishment under competition. Relevant computing approach and their applications in the field of dynamic pricing and inventory management are introduced. 2.1. Joint Dynamic Pricing and Inventory Control The joint pricing and inventory control has received considerable attention over the past decade, often formulated as a Markov Decision Process (MDP) for multi-period decision problems in finite or infinite horizons (Federgruen and Heching 1999). To solve this problem, strong assumptions have been made in the literature. For instance, Jia and Hu (2011) assume the demand follows a random price-dependent function; Bisi and Dada (2007) study an additive and multiplicative price- sensitive demand; Raz and Porteus (2006) approximate a continuous demand model using a finite number of representative fractiles, which are piecewise linear functions of the price. Federgruen and Heching (1999) assume that replenishment, holding, and ordering costs are convex, and the seller has unlimited production capacity. Chen and Simchi-Levi (2004) and Chan et al. (2006) extend the previous models by considering fixed ordering costs and limited production capacity, respectively. Thowsen (1975) proves that a BSLP policy is optimal for the additive demand function by assuming the random noise follows a P´olya frequency function of order 2 (PF2) distribution. Federgruen and Heching (1999) further establish the optimality of a BSLP policy with a general demand function for a backorder system without capacity constraints. Their sufficient conditions are as follows: (i) the demand is decreasing and concave in the list price; (ii) the single-period expected inventory cost is jointly convex in the base-stock level and price, which can be guaranteed only if the demand function is linear in the price (Feng et al. 2014). With fixed ordering costs, Chen and Simchi-Levi (2004) prove that the (𝑠, 𝑆, 𝑝) policy is optimal when the demand is additive in a finite time horizon and the seller’s objective is to either maximize expected discounted profits or maximize average long-run profits in the infinite-horizon model. 2.2. Dynamic Pricing and Replenishment under Competition Chen and Simchi-Levi (2012) note that research on jointly optimizing pricing and inventory deci- sions under competition remains limited due to model complexity. Competition typically drives X: Dual-Agent Deep Reinforcement Learning for Dynamic Pricing and Replenishment Article submitted to Operations Research 5 prices lower, reducing firms’ profits compared to monopolistic scenarios (Mantin et al. 2011, Mookherjee and Friesz 2008), and may lead to excessive inventory holdings (Xu and Hopp 2006). Mahmoodi (2019) examines the pricing and replenishment problem of a single product with deter- ministic linear demand in a duopoly market and also compares the case where two substitutable products are both sold by monopolists but does not consider products and competition simulta- neously. Schlosser and Boissier (2018) analyze stochastic dynamic pricing models in competitive markets with multiple offer dimensions (e.g., product qualities and seller ratings). They compare different demand learning techniques, including least squares, gradient boosted trees, and logistic regression, to estimate sales probabilities from partially observable market data. Ultimately, they adopt a logistic regression approach for its interpretability and performance. In the follow-up study, they further consider market reference prices, which are related to historical market prices, as well as competitor strategies and price rankings (Schlosser and Richly 2019). 2.3. Two-timescale Stochastic Approximation and Deep Reinforcement Learning Two-timescale algorithms aggregate data and update parameters at epochs with two timescales, which allows for discrete-valued observations and avoids the infrequent aggregation of data over regeneration epochs (Bhatnagar and Borkar 1998), and can handle the frequency discrepancy between different decision-making processes (Wernz 2013). MDPs provide a mathematical framework to model decision-making problems in sequential environments. RL learns optimal policies or value functions in MDPs through iterative interactions with the environment, basing decisions on observed feedback. RL algorithms can be divided into three categories: value-based, policy-based, and actor-critic algorithms (Arulkumaran et al. 2017). Value-based algorithms focus on learning the value function, which estimates the expected cumulative reward for actions taken in specific states. Policy-based algorithms directly learn the optimal policy without explicitly estimating the value function. Actor-critic algorithms combine value-based and policy-based approaches, utilizing an actor to learn the policy and a critic to estimate the value function. While RL has shown promise in solving complex decision-making problems, it faces challenges with high-dimensional inputs, scalability, and generalization, which could be addressed by DRL algorithms, such as Asynchronous Advantage Actor-Critic (A3C) (Mnih et al. 2016) and Proxi- mal Policy Optimization (PPO) (Schulman et al. 2017) algorithms. A3C employs multiple agents X: Dual-Agent Deep Reinforcement Learning for Dynamic Pricing and Replenishment 6 Article submitted to Operations Research running in parallel, enabling effective exploration and faster convergence, and PPO ensures mono- tonic policy improvement in each training iteration based on the trust region learning proposed by Schulman et al. (2015). These algorithms have demonstrated excellent performance in supply chain management. For instance, Gijsbrechts et al. (2021) apply A3C to three classic inventory problems and their result shows that A3C beats a Base Stock policy by 9%-12%. Vanvuchelen et al. (2020) apply the PPO algorithm to the joint replenishment problem for multiple items and they show that PPO outperforms two other heuristics. In Multi-Agent Systems (MAS), multiple agents cooperate or compete within a shared environ- ment to achieve common or individual goals. For any individual agent in a MAS, the transition of environmental states is no longer solely dependent on its own actions and the dynamics of the environment; it also depends on the actions taken by other agents, leading to non-stationarity in the environment. This is particularly evident in the dynamic pricing and replenishment problem investigated in this study, where the local objectives of different decisions are conflicting: for exam- ple, the pricing strategy aims to increase profits through high pricing, while inventory management aims to quickly clear stock through low pricing. A single-agent DRL algorithm can not effectively coordinate the competition between them, resulting in sub-optimal or unstable strategies. In Section 4 of our experiments, we apply a single-agent DRL algorithm, but the results were unsatisfactory, with a convergence speed merely 1/5th of the algorithms we subsequently adopted. To solve MDPs in MAS, MADRL has been developed. The Centralized Training and Decentral- ized Execution (CTDE) scheme, a mainstream MADRL architecture, addresses non-stationarity in MAS environments (Foerster et al. 2016). It trains agents with full environmental information but executes actions based only on local information. This architecture enables global optimization during training by sharing full information and decentralized execution during testing, ensuring both agent cooperation and system efficiency. By applying CTDE to Deep Deterministic Policy Gradient (DDPG), Lowe et al. (2017) show that Multi-Agent DDPG (MADDPG) has a desirable performance under both cooperative and competitive multi-agent environments. Similarly, Yu et al. (2022) apply CTDE to PPO and introduce Multi-Agent PPO (MAPPO). Unlike the PPO algorithm, MAPPO does not guarantee monotonic improvement. To address these issues, Kuba et al. (2021) propose Heterogeneous-Agent Proximal Policy Optimization (HAPPO). 3. Dynamic Pricing and Replenishment under Competition In this section, we formulate the problem of dynamic pricing and replenishment under competition as an MDP and illustrate the dynamics of this problem. We first derive theoretical results for the X: Dual-Agent Deep Reinforcement Learning for Dynamic Pricing and Replenishment Article submitted to Operations Research 7 single-period problem and propose a two-timescale approach based on these findings. Then, con- sidering the multi-period problem structure, we combine this approach with MADRL to develop a novel fast-slow dual-agent DRL algorithm for solving multi-period dynamic pricing and replen- ishment problems. Finally, we demonstrate the convergence and computational complexity of our algorithm, which significantly outperforms other heuristic strategies in experimental results. 3.1. Markov Decision Process Formulation An MDP can be defined as a tuple < S, A, P, R, 𝛾>. S is the set of product states. The state 𝑠𝑡 in period 𝑡encompasses: the previous period’s ending inventory level, lost-sales, demand, price, and ordering information within the lead time. A is the set of product actions 𝑎𝑡. Here, action 𝑎𝑡in period 𝑡determines price 𝑝𝑡and replenishment quantity 𝑞𝑡. P : S × A × S →[0,1] is the state transition function that maps the current state, action and next possible state to a transition probability; R : S × A →R represents the reward function that maps current state and action to a payoff and 𝛾∈(0,1] is the discount factor. In each period 𝑡, the agent receives 𝑠𝑡∈S and takes an action 𝑎𝑡∼𝜋(·|𝑠𝑡), where 𝜋is a policy which maps states to action’s probability distribution. After taking action 𝑎𝑡, a reward R(𝑎𝑡|𝑠𝑡) = 𝑟𝑡set as the sales profit in period 𝑡is observed. Meanwhile, the system transfers to the next state 𝑠𝑡+1 ∼P(·|𝑎𝑡, 𝑠𝑡), where the state transition function P is determined by the dynamics of the pricing and replenishment model. At every time period, the decision maker decides the price and replenishment quantity of a product for this period in order to maximize the discounted future profits, which are defined as 𝑅𝑡= 𝑇−𝑡 ∑︁ 𝑘=0 𝛾𝑘𝑟𝑡+𝑘. (1) The expected value of these discounted profits 𝑅𝑡corresponds to the value function 𝑉𝜋𝑡(𝑠𝑡), which indicates the value of being in state 𝑠𝑡following a certain policy 𝜋𝑡: 𝑉𝜋𝑡(𝑠𝑡) = 𝑇−𝑡 ∑︁ 𝑘=0 𝛾𝑘E𝜋𝑡𝑟𝑡+𝑘. (2) The value 𝑉(𝑠𝑡) is recursively dependent on the value of the next state 𝑉(𝑠𝑡+1). The optimal pricing and replenishment action in period 𝑡can be determined by solving the well-known Bellman equations: 𝑉𝜋∗(𝑠𝑡) = max 𝑎𝑡∈A n 𝑟𝑡+ 𝛾 ∑︁ 𝑠𝑡+1 P(𝑠𝑡+1|𝑠𝑡, 𝑎𝑡)𝑉𝜋∗(𝑠𝑡+1) o . (3) X: Dual-Agent Deep Reinforcement Learning for Dynamic Pricing and Replenishment 8 Article submitted to Operations Research Traditional demand assumptions mainly follow the form: 𝑑𝑡(𝑝𝑡, 𝜖𝑡) = 𝛾𝑡(𝑝𝑡)𝜖𝑡+ 𝛿𝑡(𝑝𝑡), (4) with 𝛾(·) and 𝛿(·) being nonincreasing functions. The cases of 𝛾𝑡(𝑝) = 1 and 𝛿𝑡(𝑝) = 0 are often referred to as the additive and multiplicative models, respectively. Continuous demand models are often preferred due to their mathematical tractability and computational efficiency (Babai et al. 2011). However, real-world demands could be discrete. Our analysis of real market demand data, using four common continuous demand models, results in poor fitness with R-squared values ranging only between 0.106 and 0.146. Example 1 in Appendix B provides further illustration of this inadequacy. Swaminathan and Shanthikumar (1999) point out that optimal ordering policies for discrete and continuous demand distributions may be totally different. In Section 3.2, we formulate a problem with the discrete Poisson demand model. 3.2. Dynamics of the Dynamic Pricing and Replenishment Model Consider a 𝑇-period problem, where Table 1 presents the notations. At the beginning of each period 𝑡(𝑡= 1, ...,𝑇), product prices and replenishment quantities need to be determined. Product demand is uncertain, influenced not only by its own price but also by competitors’ prices and market conditions. We infer overall demand by modeling the probability of each individual consumer’s choice. Specifically, customer behavior is captured through a logistic model, which is based on real-world data (Schlosser and Richly 2019). Given our offer price 𝑝, the competitor’s price 𝑜, and the current reference price 𝑗, the sales probability within a time span of length Δ ∈[0,1] is denoted by 𝑃Δ(𝑝, 𝑜, 𝑗) := 𝑃𝑜𝑖𝑠 \u0010 Δ · 𝑒𝜅(𝑝,𝑜,𝑗) ′ 𝛽/(1 + 𝑒𝜅(𝑝,𝑜,𝑗) ′ 𝛽) \u0011 , (5) where 𝜅(𝑝, 𝑜, 𝑗) represents regressors relating to sales, such as the competitor’s price and the refer- ence price. 𝛽represents the corresponding parameter vector. This model calculates the probability of a sale to an individual customer in a given period based on these price-related features. The demand for a product in period 𝑡follows 𝑑𝑡∼𝑃𝑜𝑖𝑠 \u0010 𝜂Δ · 𝑒𝜅(𝑝,𝑜,𝑗) ′ 𝛽/(1 + 𝑒𝜅(𝑝,𝑜,𝑗) ′ 𝛽) \u0011 . (6) The scaling factor 𝜂is introduced to adjust the magnitude of the Poisson distribution parameter 𝜆. We assume that excess demand is lost and unsold products have no salvage value. This lost- sales assumption complicates the model compared to backlog scenarios (Morton 1971). Arrow X: Dual-Agent Deep Reinforcement Learning for Dynamic Pricing and Replenishment Article submitted to Operations Research 9 Table 1 Parameters and variables Parameters ℎ unit holding cost 𝑏 unit shortage cost 𝑐 unit ordering cost 𝑧 lead time Variables 𝑞𝑡replenishment quantity offered by agents in period 𝑡 𝑝𝑡price offered by the DRL agent in period 𝑡 𝑜𝑡 price offered by the competitor in period 𝑡 𝑑𝑡 demand in period 𝑡 𝐼𝑡 inventory level in period 𝑡 𝐿𝑡lost-sales in period 𝑡 𝑆𝑡 actual sales in period 𝑡 et al. (1958) demonstrate that in a single sourcing model with backlogging and constant lead time, inventory and outstanding orders can be simplified to one inventory position. When excess demand is lost, this simplification does not apply, and the optimal policy relies on the full inventory pipeline vector, exponentially increasing complexity with lead time. At the beginning of any period 𝑡, the replenishment quantity 𝑞𝑡and price 𝑝𝑡for a product must be decided. Since the unmet demand is lost, the dynamics of the evolution of inventory can be expressed by 𝐼𝑡= [𝐼𝑡−1 + 𝑞𝑡−𝑧−𝑑𝑡]+. (7) The lost-sales is 𝐿𝑡= max{𝑑𝑡−(𝐼𝑡−1 + 𝑞𝑡−𝑧),0}. (8) The actual sales is given by 𝑆𝑡= min{𝑑𝑡, 𝐼𝑡−1 + 𝑞𝑡−𝑧}. (9) The sales profit 𝑟𝑡is 𝑟𝑡= 𝑝𝑡𝑆𝑡−ℎ𝐼𝑡−𝑏𝐿𝑡−𝑐𝑞𝑡. (10) X: Dual-Agent Deep Reinforcement Learning for Dynamic Pricing and Replenishment 10 Article submitted to Operations Research 3.3. Theoretical Analysis for Single-period Problem Before studying multi-period decision problems, we first consider a single-period problem with zero lead time and ending inventory given by 𝐼= [𝑥0 + 𝑞−𝑑]+, (11) where 𝑞represents the replenishment quantity, 𝑥0 is the initial inventory at the beginning of the period, which is known, and 𝑑is the demand for the current period. The lost-sales is 𝐿= max{𝑑−(𝑥0 + 𝑞),0}. The actual sales is given by 𝑆= min{𝑑,𝑥0 + 𝑞}. The objective is to find the optimal order quantity 𝑞∗and the optimal price 𝑝∗, in order to maximize the single-period expected profit depending on the demand model, as follows: 𝐹(𝑝,𝑥) = E[𝑝𝑆(𝑝,𝑥) −ℎ𝐼(𝑝,𝑥) −𝑏𝐿(𝑝,𝑥) −𝑐(𝑥−𝑥0)], where 𝑥= 𝑥0 + 𝑞. Therefore, the problem comes down to deciding on 𝑝and 𝑥, given that 𝑥0 is known. Unlike the traditional demand assumption in Eq.(4), our discrete Poisson demand model (Eq.(6)) incorporates a price-dependent parameter 𝜆(𝑝). This dependency introduces complexity into the analysis of related functions and deviates from conventional assumptions. Proposition 1. The Poisson distribution 𝐷(𝑝) ∼𝑃𝑜𝑖𝑠(𝜆(𝑝)), where 𝜆(𝑝) is the parameter, does not fall under the traditional demand assumption described in Eq.(4): 𝑑(𝑝, 𝜖) = 𝛾(𝑝)𝜖+ 𝛿(𝑝), with 𝜖being a random variable that is independent of 𝑝. The proof of Proposition 1, along with the following omitted proofs, can be found in Appendix A. This proposition highlights the structural difference between our Poisson demand model 𝐷(𝑝) ∼ 𝑃𝑜𝑖𝑠(𝜆(𝑝)) and traditional models. The key distinction lies in the inseparability of the price- dependent function 𝜆(𝑝) from the distribution of the random variable in the Poisson model, unlike the explicit separation of the random variable 𝜖and price-dependent functions in the traditional model 𝛾(𝑝)𝜖+ 𝛿(𝑝). Furthermore, previous studies on joint pricing and replenishment often focus on scenarios with demand backlog, as in Federgruen and Heching (1999), where revenue depends on demand volume rather than sales, making the revenue function independent of inventory. In contrast, we analyze scenarios with lost-sales under discrete demand (Eq.(6)). We derive the concavity and convexity of the inventory function, revenue function, and lost-sales function under discrete demand Eq.(6). The final comparison with traditional assumptions is presented in Proposition 4. X: Dual-Agent Deep Reinforcement Learning for Dynamic Pricing and Replenishment Article submitted to Operations Research 11 3.3.1. Properties of Objective Function Assumption 1. We suppose 𝑙𝑝+ 𝑎> 0 in the domain of 𝑝and the domain is a bounded set. Lemma 1. Under Assumption 1, define 𝜆= 𝜂Δ · 𝑒𝑙𝑝+𝑎/(1 + 𝑒𝑙𝑝+𝑎), then 𝜆(𝑝) is concave and monotonically decreasing with respect to 𝑝. Proposition 2. Under Assumption 1, the inventory function 𝐼(𝑝,𝑥) = E[(𝑥−𝑑)+] is convex and monotonically increasing with respect to price 𝑝. Proof Denote ⌊𝑥⌋as the maximum integer no more than 𝑥. We rewrite 𝐼(𝑝,𝑥) as E[(𝑥−𝑑)+] = ⌊𝑥⌋ ∑︁ 𝑘=0 𝜆𝑘𝑒−𝜆 𝑘! (𝑥−𝑘). Then we calculate the first order derivative and the second order derivative: 𝜕𝐼(𝑝,𝑥) 𝜕𝑝 = 𝜕𝜆 𝜕𝑝 ⌊𝑥⌋ ∑︁ 𝑘=0 𝜆𝑘−1𝑒−𝜆 𝑘! (𝑥−𝑘)(𝑘−𝜆), 𝜕2𝐼(𝑝,𝑥) 𝜕𝑝2 = 𝜕2𝜆 𝜕𝑝2 ⌊𝑥⌋ ∑︁ 𝑘=0 𝜆𝑘−1𝑒−𝜆 𝑘! (𝑥−𝑘)(𝑘−𝜆) + ( 𝜕𝜆 𝜕𝑝)2 ⌊𝑥⌋ ∑︁ 𝑘=0 𝜆𝑘−2𝑒−𝜆 𝑘! (𝑥−𝑘)((𝑘−𝜆)2 −𝑘). (12) We want to prove 𝜕2𝐼(𝑝,𝑥) 𝜕𝑝2 > 0. First, we consider the first term of Eq.(12). When 𝑥< 𝜆, for every 𝑘≤⌊𝑥⌋, 𝑘−𝜆< 0 holds, we have Í⌊𝑥⌋ 𝑘=0 𝜆𝑘−1𝑒−𝜆 𝑘! (𝑥−𝑘)(𝑘−𝜆) < 0. When 𝑥≥𝜆, since (𝑥−𝑘)(𝑘−𝜆) ≤ (𝑥−𝜆)(𝑘−𝜆), we have ⌊𝑥⌋ ∑︁ 𝑘=0 𝜆𝑘−1𝑒−𝜆 𝑘! (𝑥−𝑘)(𝑘−𝜆) ≤(𝑥−𝜆) ⌊𝑥⌋ ∑︁ 𝑘=0 𝜆𝑘−1𝑒−𝜆 𝑘! (𝑘−𝜆) = 𝑥−𝜆 𝜆 ⌊𝑥⌋ ∑︁ 𝑘=0 P(𝑋= 𝑘)(𝑘−𝜆) = 𝑥−𝜆 𝜆 ( ⌊𝑥⌋ ∑︁ 𝑘=0 𝑘P(𝑋= 𝑘) −𝜆P(𝑋≤⌊𝑥⌋), where 𝑋is a random variable following a Poisson distribution with mean 𝜆. Figure 1 Poisson distribution and its truncated version X: Dual-Agent Deep Reinforcement Learning for Dynamic Pricing and Replenishment 12 Article submitted to Operations Research Figure 2 Eq.(14) and the second term of Eq.(12) without quadratic term with 𝜆= 5 (a) Plot of Eq.(14). (b) Plot of the Second Term of Eq.(12) without Quadratic Term. Considering a random variable 𝑋′ following Poisson distribution truncating at ⌊𝑥⌋, it is obvious that 𝐸[𝑋′] < 𝐸[𝑋] due to the truncation, as shown in Figure 1. Therefore, 𝐸[𝑋′] = Í⌊𝑥⌋ 𝑘=0 𝑘P(𝑋= 𝑘) P(𝑋≤⌊𝑥⌋) < 𝐸[𝑋] = 𝜆, which implies Í⌊𝑥⌋ 𝑘=0 𝜆𝑘−1𝑒−𝜆 𝑘! (𝑥−𝑘)(𝑘−𝜆) < 0. Note that 𝜕2𝜆 𝜕𝑝2 < 0, we prove that the first term of Eq.(12) is positive. The conclusion 𝜕𝐼(𝑝,𝑥) 𝜕𝑝 > 0 holds because 𝜕𝜆 𝜕𝑝< 0. Next, we claim that the second term of Eq.(12) is also positive. Suppose that 𝑋is a random variable following a Poisson distribution with mean 𝜆, then E[𝑋2] = 𝜆2 + 𝜆, E[𝑋3] = 𝜆3 + 3𝜆2 + 𝜆. We can find that E[(𝑥−𝑋)((𝑋−𝜆)2 −𝑋)] = −E[𝑋3] + (𝑥+ (2𝜆+ 1))E[𝑋2] + (−𝜆2 −(2𝜆+ 1)𝑥)E𝑋+ 𝜆2𝑥 = −𝜆3 −3𝜆2 −𝜆+ (𝑥+ (2𝜆+ 1))(𝜆2 + 𝜆) + (−𝜆2 −(2𝜆+ 1)𝑥)𝜆+ 𝜆2𝑥= 0. Therefore, the equation below holds for every 𝑥: ∞ ∑︁ 𝑘=0 𝜆𝑘−2𝑒−𝜆 𝑘! (𝑥−𝑘)((𝑘−𝜆)2 −𝑘) = 𝜆−2E[(𝑥−𝑋)((𝑋−𝜆)2 −𝑋)] = 0. (13) It is obvious that the the second term of Eq.(12) is continuous and piecewise linear with respect to 𝑥with integer points as nodes, which implies that this function reaches its minimum at certain integer nodes. In order to find its minimum with respect to 𝑥, we can calculate the difference by taking 𝑥as the integers 𝑛and 𝑛+ 1, respectively, as follows: 𝑛+1 ∑︁ 𝑘=0 𝜆𝑘−2𝑒−𝜆 𝑘! (𝑛+ 1 −𝑘)((𝑘−𝜆)2 −𝑘) − 𝑛 ∑︁ 𝑘=0 𝜆𝑘−2𝑒−𝜆 𝑘! (𝑛−𝑘)((𝑘−𝜆)2 −𝑘) = 𝑛 ∑︁ 𝑘=0 𝜆𝑘−2𝑒−𝜆 𝑘! ((𝑘−𝜆)2 −𝑘). (14) Given that ( 𝜕𝜆 𝜕𝑝)2 > 0, the sign of Eq.(14) determines that of the derivative of the second term of Eq.(12) with respect to 𝑥when 𝑥is not an integer. We illustrate Eq.(14) and the second term of Eq.(12) without quadratic term ( 𝜕𝜆 𝜕𝑝)2 with 𝜆= 5 in Figure 2. The quadratic equation (𝑘−𝜆)2 −𝑘 with respect to 𝑘has two roots, denoted as 𝑥1 and 𝑥2, where 0 < 𝑥1 < 𝑥2. These roots are indicated as two red points in Figure 2. As is shown in Figure 2a, when 0 < 𝑥≤𝑥1, the value of Eq.(14) is X: Dual-Agent Deep Reinforcement Learning for Dynamic Pricing and Replenishment Article submitted to Operations Research 13 increasing with respect to 𝑥. When 𝑥1 < 𝑥< 𝑥2, the value of Eq.(14) is decreasing for 𝑥1 < 𝑘< 𝑥2 as each added term containing (𝑘−𝜆)2 −𝑘is negative. When 𝑥> 𝑥2, the value of Eq.(14) is increasing for 𝑘> 𝑥2 as each added term containing (𝑘−𝜆)2 −𝑘is positive. The minimum of the second term of Eq.(12) occurs at the point that Eq.(14) changes from negative to positive (if there exists) or at the boundary. Therefore, the second term of Eq.(12) reaches its minimum when 𝑥= 𝑥∗≥⌊𝑥2⌋or 𝑥= 0. Consequently, we have ⌊𝑥⌋ ∑︁ 𝑘=0 𝜆𝑘−2𝑒−𝜆 𝑘! (𝑥−𝑘)((𝑘−𝜆)2 −𝑘) ≥min{0, 𝑥∗ ∑︁ 𝑘=0 𝜆𝑘−2𝑒−𝜆 𝑘! (𝑥∗−𝑘)((𝑘−𝜆)2 −𝑘)} =min{0,0 − ∞ ∑︁ 𝑘=𝑥∗+1 𝜆𝑘−2𝑒−𝜆 𝑘! (𝑥∗−𝑘)((𝑘−𝜆)2 −𝑘)} ≥0. According to Eq.(13), the second equality holds. The last inequality is based on the fact that when 𝑘≥𝑥∗+ 1 ≥𝑥2, the expression (𝑥∗−𝑘)((𝑘−𝜆)2 −𝑘) is less than 0. Therefore, the second term of Eq.(12) is positive as is shown in Figure 2b. In conclusion, the two terms of the second derivative of inventory function are positive and non-negative, respectively. This indicates that the inventory function is convex in 𝑝. □ Remark 1. Under Assumption 1, the inventory function 𝐼(𝑝,𝑥) = E[(𝑥−𝑑)+] is neither jointly convex nor concave with respect to the inventory 𝑥and price 𝑝at its differentiable points. This is due to 𝐼(𝑝, 𝑥) being continuous and piecewise linear with respect to 𝑥, implying 𝜕2𝐼(𝑝,𝑥) 𝜕𝑥2 = 0 at its differentiable points. Besides, 𝜕2𝐼(𝑝,𝑥) 𝜕𝑥𝜕𝑝 = 𝜕𝜆 𝜕𝑝 ⌊𝑥⌋ ∑︁ 𝑘=0 𝜆𝑘−1𝑒−𝜆 𝑘! (𝑘−𝜆) > 0, which is implied by the proof of Proposition 2. Therefore, the Hessian matrix of 𝐼(𝑝,𝑥) is indeter- minate. Proposition 3. Under Assumption 1, the revenue function 𝐸(𝑝,𝑥) = 𝑝E[𝑆(𝑝,𝑥)] = 𝑝E[min(𝑑, 𝑥)] is concave with respect to price 𝑝. Since the random variable 𝑑follows the Poisson distribution and the parameter 𝜆is determined by price 𝑝through a complex function 𝜆(𝑝), we need more assumptions to guarantee the convexity of lost-sales function. We employ the Taylor expansion to approximate 𝜆and linearize it. Assumption 2. We suppose that ˆ𝜆= 𝜂Δ𝑒𝑎(1 + 𝑙𝑝) with 𝑙< 0 and 1 + 𝑙𝑝> 0 in the domain of 𝑝, where the domain is bounded. X: Dual-Agent Deep Reinforcement Learning for Dynamic Pricing and Replenishment 14 Article submitted to Operations Research Lemma 2. Under Assumption 2, the lost-sales function 𝐿(𝑝,𝑥) = E[(𝑑−𝑥)+] is convex with respect to the price 𝑝. Remark 2. Assumption 1 aims to guarantee the concavity of function 𝜆(𝑝), which can be replaced by Assumption 2 because linear function is concave. Henceforth, we use 𝜆to stand for ˆ𝜆in Assumption 2. Having established the concavity or convexity of the expected inventory, revenue, and lost-sales functions, we can derive the concavity of the single-period expected profit function 𝐹(𝑝,𝑥) with respect to 𝑝. Theorem 1. Under Assumption 2, 𝐹(𝑝,𝑥) is concave with respect to 𝑝and 𝑥within their respective domains, though not jointly concave. Furthermore, 𝐹(𝑝,𝑥) takes its maximum either at the boundary or at an interior point that satisfies the following conditions: 𝜕𝐹(𝑝,𝑥) 𝜕𝑝 = 𝜕𝐸(𝑝,𝑥) 𝜕𝑝 −ℎ𝜕𝐼(𝑝,𝑥) 𝜕𝑝 −𝑏𝜕𝐿(𝑝,𝑥) 𝜕𝑝 = 0, 𝑏−𝑐+ 𝑝−(ℎ+ 𝑏+ 𝑝) 𝑥∗−1 ∑︁ 𝑘=0 𝜆𝑘(𝑝)𝑒−𝜆(𝑝) 𝑘! ≥0, 𝑏−𝑐+ 𝑝−(ℎ+ 𝑏+ 𝑝) 𝑥∗ ∑︁ 𝑘=0 𝜆𝑘(𝑝)𝑒−𝜆(𝑝) 𝑘! < 0, (15) where 𝑥∗is a positive integer. Previous literature (Federgruen and Heching 1999) established the joint concavity of the profit function 𝐹(𝑝,𝑥) with respect to price and inventory under Assumptions 3(i) and 4(i). The Assump- tion 3(i) represents only a class of continuous distributions. The Assumption 4(i) means that the revenue collected in each period depends on the demand volume instead of the sales amount. In this paper, we cannot derive this conclusion primarily because we have revised these conditions to Assumption 3(ii) and Assumption 4(ii). Assumption 3. (i) 𝑑(𝑝, 𝜖) = 𝛾(𝑝)𝜖+ 𝛿(𝑝); (ii) 𝑑∼𝑃𝑜𝑖𝑠(𝜆(𝑝)). Assumption 4. (i) 𝐸(𝑝) = 𝑝E[𝑑] = 𝑝𝜆(𝑝); (ii) 𝐸(𝑝,𝑥) = 𝑝E[min(𝑑,𝑥)]. Notably, altering Assumptions 3(i) or 4(i) to Assumptions 3(ii) or 4(ii) results in the loss of joint concavity established in Federgruen and Heching (1999). We illustrate this point with the following proposition. X: Dual-Agent Deep Reinforcement Learning for Dynamic Pricing and Replenishment Article submitted to Operations Research 15 Proposition 4. If either Assumption 3(i) or Assumption 4(i) is changed to Assumption 3(ii) or Assumption 4(ii), respectively, the joint concavity of 𝐹(𝑝,𝑥) in Federgruen and Heching (1999) will be lost. Remark 3. A one-to-one mapping from price 𝑝to expected demand 𝜆(𝑝) is constructed, thus transforming the optimization variable from price to expected demand (Bernstein et al. 2016). Since affine transformations do not alter concavity or convexity, the conclusions in this paper remain unchanged due to the Poisson demand. Proposition 4 illustrates the limitations of the traditional model. Even slight modifications to the assumptions can negate joint concavity, rendering the original model’s optimal strategies invalid. Under new assumptions, there are still some conclusions consistent with the finding in previous literature such as the following proposition (Federgruen and Heching 1999). Proposition 5. Given the inventory 𝑥and by solving the first equation in condition (15), we can determine 𝑝(𝑥) to maximize 𝐹(𝑝,𝑥). The function 𝑝(𝑥) is a non-increasing function with respect to 𝑥. Remark 4. Figure 10 in Appendix C clearly shows that 𝐹(𝑝,𝑥) is not jointly concave with respect to 𝑝and 𝑥, because the value of 𝐹(𝑝,𝑥) at points between 𝐴and 𝐵lies below the corresponding linear interpolation of the function values at 𝐴and 𝐵, indicating a violation of joint concavity which is crucial for deriving optimal policies like BSLP. Without it, optimal joint pricing and replenishment decisions cannot be directly obtained from the profit function’s first-order conditions. Finding the global maximum requires identifying and comparing all local maxima, a process whose computational complexity increases exponentially with the search space. Consequently, more advanced optimization algorithms become necessary for efficient solutions. 3.3.2. Two-timescale Stochastic Approximation In commercial operations, the frequencies of product replenishment and pricing decisions often differ. The convergence of a two-timescale algorithm can be guaranteed by the respective concavity of the profit function. This concavity can now be utilized to guide the algorithm design. For the case where the frequency of pricing decisions exceeds that of replenishment decisions, two-timescale algorithms can adjust the relative updating rates of the price and inventory in the algorithm and use a faster search speed to update 𝑝in cases where prices are determined with a higher frequency than inventory. Conversely, for some products where replenishment decisions are made more frequently than pricing adjustments, two-timescale algorithms use a faster search speed to update 𝑥in these cases. X: Dual-Agent Deep Reinforcement Learning for Dynamic Pricing and Replenishment 16 Article submitted to Operations Research We illustrate the algorithm when pricing decisions are more frequent than replenishment deci- sions. The single-period profit function is 𝐹(𝑝,𝑥) = 𝑝E[min(𝑑,𝑥)] −(ℎ+ 𝑏)E[(𝑥−𝑑)+] −𝑏𝜆+ 𝑏𝑥−𝑐(𝑥−𝑥0), where inventory 𝑥changes relatively slower than price 𝑝in this case. We conduct simultaneous descents for 𝑝and 𝑥in two coupled iterations, with the update rate of 𝑝being faster than that of 𝑥. We express the partial derivatives of the profit function in Eq.(A.2) using the form of expectation, allowing us to derive their unbiased estimators, as outlined in the proposition below. Proposition 6. Let 𝑔(𝑝,𝑥) and ℎ(𝑝,𝑥) denote the partial derivatives of the profit function in Eq.(A.2) with respect to 𝑝and 𝑥, then 𝑔(𝑝, 𝑥) \u0011 𝜕𝐹(𝑝,𝑥) 𝜕𝑝 = E[min(𝑑,𝑥)] + 𝑝E \u0014𝜕logP(𝑑) 𝜕𝑝 min(𝑑,𝑥) \u0015 −(ℎ+ 𝑏)E \u0014𝜕logP(𝑑) 𝜕𝑝 (𝑥−𝑑)+ \u0015 −𝑏𝜕𝜆 𝜕𝑝, ℎ(𝑝, 𝑥) \u0011 𝜕𝐹(𝑝,𝑥) 𝜕𝑥 = 𝑏−𝑐+ 𝑝−(ℎ+ 𝑏+ 𝑝)E[1𝑑≤𝑥], where P(𝑑) is the probability density function of random variable 𝑑. Due to Proposition 6, unbiased stochastic gradient estimators for 𝑔(𝑝,𝑥) and ℎ(𝑝,𝑥) can be obtained using the Monte Carlo method and are denoted as ˆ𝑔(𝑝,𝑥) and ˆℎ(𝑝,𝑥), respectively. The iteration process of the two-timescale algorithm can be formulated as follows: 𝑝𝑘+1 = 𝑝𝑘+ 𝛼𝑘ˆ𝑔(𝑝𝑘,𝑥𝑘) \u0011 𝑝𝑘+ 𝛼𝑘 \u0012 min(𝑑𝑘,𝑥𝑘) + 𝑝𝑘 𝜕logP(𝑑𝑘) 𝜕𝑝 min(𝑑𝑘,𝑥𝑘) −(ℎ+ 𝑏) 𝜕logP(𝑑𝑘) 𝜕𝑝 (𝑥𝑘−𝑑𝑘)+ −𝑏𝜕𝜆 𝜕𝑝 \u0013 , 𝑥𝑘+1 = 𝑥𝑘+ 𝛽𝑘ˆℎ(𝑝𝑘,𝑥𝑘) \u0011 𝑥𝑘+ 𝛽𝑘 \u0012 (𝑏−𝑐+ 𝑝𝑘) −(ℎ+ 𝑏+ 𝑝𝑘)1𝑑𝑘≤𝑥𝑘 \u0013 , (16) where 𝑑𝑘is sampled from a Poisson distribution with parameter 𝜆(𝑝𝑘). Note that E[ ˆ𝑔(𝑝,𝑥)] = 𝑔(𝑝, 𝑥) and E[ ˆℎ(𝑝,𝑥)] = ℎ(𝑝,𝑥). We could reduce the variance by taking more samples each time, while a single sample is sufficient for convergence to the optimum. Step size 𝛼𝑘represents the time scale for the first iteration, and 𝛽𝑘represents the time scale for the second iteration. The conditions Í∞ 𝑘=1 𝛼𝑘= Í∞ 𝑘=1 𝛽𝑘= ∞, Í∞ 𝑘=1 𝛽2 𝑘< Í∞ 𝑘=1 𝛼2 𝑘< ∞, and 𝛽𝑘 𝛼𝑘→0 as 𝑘tends to infinity are satisfied, indicating that the first time scale progresses faster than the second time scale. The convergence of the two-timescale algorithm is guaranteed, analogous to the theoretical results in Borkar (2009) and Harold et al. (1997). Specifically, we present the following lemma to analyze the convergence of 𝑝𝑘in the faster time scale, leveraging the concavity of the revenue function with respect to 𝑝. X: Dual-Agent Deep Reinforcement Learning for Dynamic Pricing and Replenishment Article submitted to Operations Research 17 Lemma 3. The iterative sequence {(𝑝𝑘,𝑥𝑘)} generated by the iteration Eq.(16) converges to a set which is defined as below: (𝑝𝑘,𝑥𝑘) →{(𝑝★(𝑥),𝑥)}, where 𝑝★(𝑥) is a well defined function showing that when 𝑥is fixed, the value of 𝑝to make 𝑔(𝑝,𝑥) equal to zero. In other words, this lemma means |𝑝𝑘−𝑝★(𝑥𝑘)| →0 almost surely, that is, {𝑝𝑘} asymptotically tracks 𝑝★(𝑥𝑘). Then, we will combine the analysis of the second time scale and show that the sequence {(𝑝𝑘,𝑥𝑘)} converges to a fixed point. Considering the second scale, the variable 𝑥, previously considered fixed, should now be treated as the time-varying 𝑥(𝑡). Based on Lemma 3, for sufficiently large 𝑡, 𝑝(𝑡) closely tracks 𝑝★(𝑥(𝑡)). We then substitute it into the second scale and focus on the following ODE: ¤𝑥(𝑡) = ℎ(𝑝★(𝑥(𝑡)),𝑥(𝑡)). (17) Theorem 2. The iterative sequence {(𝑝𝑘,𝑥𝑘)} generated by the iteration Eq.(16) converges to some limit point of the ODE (17), i.e., (𝑝𝑘,𝑥𝑘) →(𝑝★(𝑥★),𝑥★). Proof Since ℎ(𝑝,𝑥) = 𝑏−𝑐+ 𝑝−Í⌊𝑥⌋ 𝑘=0 𝜆(𝑝)𝑘𝑒−𝜆( 𝑝) 𝑘! (ℎ+ 𝑏+ 𝑝), note that ℎ(𝑝,𝑥) is a piecewise constant function so the ODE (17) is equipped with discontinuous right hand side, for which standard solution concepts are introduced in Filippov (1988). It is obvious that ℎ(𝑝,𝑥) satisfies a linear growth condition: |ℎ(𝑝,𝑥)| ≤𝐾(1+ |𝑥|) for some 𝐾> 0. Define 𝐻(𝑝,𝑥) \u0011 ∩𝜖>0 ¯𝑐𝑜({ℎ(𝑝, 𝑦) : |𝑦−𝑥| < 𝜖}), where ¯𝑐𝑜is the notation of convex closure. Then the ODE limit gets replaced by a differential inclusion limit: ¤𝑥(𝑡) ∈𝐻(𝑝★(𝑥(𝑡)),𝑥(𝑡)). By Corollary 4 in Borkar (2009, Chap.5), the sequence {𝑥𝑘} generated by the iteration Eq.(16) converges to (possibly path dependent) the limit point of Eq.(17) almost surely. Furthermore, the iterative sequences {𝑝𝑘} and {𝑥𝑘} generated by iterations Eq.(16) converge to some limit point of the ODE below by Theorem 2 in Borkar (2009, Chap.6). ¤𝑝(𝑡) = 𝑔(𝑝(𝑡),𝑥(𝑡)), ¤𝑥(𝑡) = ℎ(𝑝(𝑡),𝑥(𝑡)). □ The limit point of ODE in Lemma 3 and Theorem 2 satisfy the condition 𝑔(𝑝,𝑥) = 0 and ℎ(𝑝,𝑥) = 0, which shows 𝜕𝐹(𝑝,𝑥) 𝜕𝑝 = 𝜕𝐹(𝑝,𝑥) 𝜕𝑥 = 0 by definition of 𝑔(𝑝,𝑥) and ℎ(𝑝,𝑥). Therefore, the limit point of ODE is a local optimum in the optimization problem. Figure 11 in Appendix C shows that the X: Dual-Agent Deep Reinforcement Learning for Dynamic Pricing and Replenishment 18 Article submitted to Operations Research two-timescale algorithm approximately converges to 𝑝= 55 and 𝑥= 5 under certain parameters, which fully satisfy the conditions in Theorem 1. Furthermore, we study the convergence rate of the two-timescale algorithm. Theorem 3. Assume that the function 𝑝★(𝑥) is a Lipschitz function, and then we have the order of the mean squared error for the sequence {𝑥𝑘} generated by the iteration Eq.(16): E[|𝑝𝑘−𝑝★(𝑥𝑘)|] = 𝑂( 𝛽𝑘 𝛼𝑘 ) + 𝑂(𝛼 1 2 𝑘). 3.4. Multi-period Problem Property When it comes to multi-period decision problems, the problem is formulated as MDP through Bellman equation (3), with the goal of determining the optimal policy for the current state in each period. At each step excluding the final period 𝑇, the optimization of the Bellman equation (3) includes a future discounted expectation term with respect to the next state. To account for this, we need to find the optimal policy 𝑎𝑡+1(𝑠𝑡+1) under every state 𝑠𝑡+1 in the 𝑡+ 1 period, calculate the 𝑉𝜋★(𝑠𝑡+1), and then take the expectation E𝑠𝑡+1[𝑉𝜋★(𝑠𝑡+1)]. This transforms the problem into a functional optimization task, as we need to find a function 𝑎𝑡+1(𝑠𝑡+1) that maximizes E𝑠𝑡+1[𝑉𝜋★(𝑠𝑡+1)]. The computational complexity involved in solving Eq.(3) by backward induction is detailed in Proposi- tion 7. Proposition 7. Suppose the price of each product is discretized into P values and the replenish- ment quantity is discretized into 𝑄values; there are 𝐾regressors in Eq.(5), the maximum dimension and the minimum dimension of the regressors are 𝑅𝑚𝑎𝑥and 𝑅𝑚𝑖𝑛. The lower and upper bounds of the total computational complexity needed to recursively solve the optimality equation (3) via DP are 𝑂(𝑃𝑇𝑄𝑇(𝑅𝑚𝑖𝑛)𝐾𝑇) and 𝑂(𝑃𝑇𝑄𝑇(𝑅𝑚𝑎𝑥)𝐾𝑇), respectively. It can be observed that the traditional dynamic programming technique suffers from the curse- of-dimensionality, rendering it computationally intractable for sizable problems. The curse-of- dimensionality arises from the high dimensions of the state space which grows in the number of products, the action space which includes all possible pricing and replenishment combinations and the transition space dependent on the number of possible demand realizations. To address such complex scenarios, particularly in high-dimensional, continuous spaces typical of functional optimization tasks, DRL can be applied. By interacting with the environment, DRL learns to make decisions based on observed feedback while leveraging the power of deep neural networks. X: Dual-Agent Deep Reinforcement Learning for Dynamic Pricing and Replenishment Article submitted to Operations Research 19 3.5. Two-timescale DRL Approach We use two agents to make pricing and replenishment decisions, respectively, and integrate the two-timescale algorithm with HAPPO. To achieve optimal pricing and replenishment decisions, each agent should consider the influence of the action on not only its own reward but also the reward of other agents. Thus, the MDP constructed in Section 3.1 should be transformed into an MG. 3.5.1. Markov Game Formulation We formulate the dynamic pricing and replenishment problem with 𝑇periods as an MG, which can be defined as a tuple < 𝑁,S, A, P, R, 𝛾>. Here 𝑁∈N+ is the number of agents; S is the finite state space; A = Î𝑛 𝑖=1 A𝑖is the product of finite action spaces of all agents, known as the joint action space, where A𝑖is the set of actions for agent 𝑖; P : S × A × S →[0,1] is the transition probability function that maps the current global state, global action (including all agents’ actions) and next possible state to a transition probability; R : S × A →R is the reward function that maps current global state and global action to a payoff; and 𝛾∈(0,1] is the discount factor. The agents interact with the environment according to the following protocol: in each period 𝑡, the agents are at state 𝑠𝑡∈S; every agent 𝑖takes an action 𝑎𝑖 𝑡∈A𝑖, drawn from its policy 𝜋𝑖(·|𝑠𝑡), which together with other agents’ actions gives a joint action a𝑡∈A, drawn from the joint policy 𝜋(·|𝑠𝑡) = Î𝑛 𝑖=1 𝜋𝑖(·|𝑠𝑡). After taking actions a𝑡, the agents receive a joint reward 𝑟𝑡= R(𝑠𝑡|a𝑡), which is set as the sales profit in period 𝑡; see Eq.(10). Afterwards, the system transfers to the next state 𝑠𝑡+1 with probability P(𝑠𝑡+1|𝑠𝑡,a𝑡), where the state transition function P is determined by the dynamics of the pricing and replenishment model, as detailed in Section 3.2. A trajectory is generated by repeating the interaction until period 𝑇is reached. All agents update their policies cooperatively, aiming to maximize the expected total reward: 𝐽(𝜋) = E𝑎1 1:𝑇∼𝜋1,...,𝑎𝑛 1:𝑇∼𝜋𝑛,𝑠1:𝑇∼𝜌1:𝑇 𝜋 \" 𝑇 ∑︁ 𝑡=1 𝛾𝑡𝑟𝑡 # , (18) where 𝑎𝑖 1:𝑇∼𝜋𝑖represents that each action in action sequence (𝑎𝑖 1, ..., 𝑎𝑖 𝑇) of actor 𝑖follows 𝜋𝑖, and 𝜌𝑡 𝜋is the marginal state distribution at time 𝑡with the joint policy 𝜋. 3.5.2. Solution Now we show how to use our fast-slow dual-agent DRL algorithm (FSDA) to construct pricing and replenishment policies. FSDA applies an on-policy PPO for each agent and follows a CTDE structure. We use the subscripts 𝜃and 𝜙to represent the parameters of policy and value functions, respectively. Each agent 𝑖has a policy function 𝜋𝑖 𝜃. The value function 𝑣𝜙maps all agents’ state to state value 𝑣𝜙(𝑠𝑡). Both the policy function and the value function are constructed using neural networks, known as the actor network and the critic network, respectively. To capture X: Dual-Agent Deep Reinforcement Learning for Dynamic Pricing and Replenishment 20 Article submitted to Operations Research useful historical information such as demand fluctuations and price changes during the lead time, we choose a Recurrent Neural Network (RNN) with two stacked Gate Recurrent Units (GRUs) to construct our actor and critic networks (Liu et al. 2022). Thus, the agent can fully take historical information into account and make better decisions. Figure 3 Structure of actor network We illustrate the diagram for the actor network in Figure 3. For the actor network, we use a multi-layer perceptron (MLP) with two hidden layers, namely MLP1, to encode the actor 𝑖’s state 𝑠𝑖 𝑡 into high-dimensional features. The first GRU, namely GRU1, receives these features and combines a hidden state ℎ1 𝑡−1 as the input and outputs a hidden state ℎ1 𝑡and some high-dimensional features which are served as the input of the second GRU (GRU2). GRU2 outputs an updated hidden state ℎ2 𝑡and new high-dimensional features. These features are then fed into the second multi-layer perceptron (MLP2) to match the size of the action space and mapped into the probability distribution 𝜋𝑖 𝜃(·|𝑠𝑖 𝑡) of actions by a softmax layer. For the critic network, its only difference from the actor network lies in that features produced by GRU2 are directly mapped into the state value. For the actual training process, we use 𝜃𝑚and 𝜙𝑚to represent parameters for actor and critic networks after 𝑚updates. For each episode, we simulate the MG corresponding to the dynamic pricing and replenishment problem for 𝑇periods with actors’ current policies (𝜋1 𝜃𝑚,...,𝜋𝑛 𝜃𝑚) and collect all actors’ states, actions and rewards as the training data. Then, if the current episode is divisible by the ratio 𝑘of the two time scales, each actor’s parameters denoted by 𝜃𝑚are updated to 𝜃𝑚+1 using the collected data, respectively. Otherwise, only the first actor’s parameters 𝜃𝑚are updated to 𝜃𝑚+1. Note that the first actor can be either a pricing or a replenishment agent, depending on which decision needs to be updated more quickly. After that, the critic’s parameters 𝜙𝑚are updated to 𝜙𝑚+1. The ratio 𝑘of the two time scales tends to infinity as the algorithm iterates. That is to say, 𝑘is a function of 𝑚, for example, 𝑘= 𝑚/2. A pseudocode for the detailed simulation and parameter update process is presented in Appendix E. X: Dual-Agent Deep Reinforcement Learning for Dynamic Pricing and Replenishment Article submitted to Operations Research 21 In the application of the algorithm, we also incorporate several performance enhancement tech- niques to optimize its effectiveness, such as batch advantage normalization, reward scaling, and orthogonal initialization. 3.6. Theoretical Analysis For the FSDA in this section, we show its convergence below. Notice that (𝜃2 𝑚, 𝜙𝑚) is updated at every step, while 𝜃1 𝑚is updated once every 𝑘steps. When using the Adam optimizer, with a step size that satisfies the descent condition, we can consider (𝜃2 𝑚, 𝜙𝑚) to be on a faster time scale and 𝜃1 𝑚on a slower time scale. The iterative formulas extracted from the algorithm are as follows: 𝜃1 𝑚+1 = 𝜃1 𝑚+ 𝛽𝑚 𝜕 𝜕𝜃1𝑚 (𝐿1 𝑝+ 𝐿1 𝑒) = 𝜃1 𝑚+ 𝛽𝑚(ℎ1(𝜃1 𝑚, 𝜙𝑚) + 𝑁(1) 𝑚), 𝜃2 𝑚+1 = 𝜃2 𝑚+ 𝛼𝑚 𝜕 𝜕𝜃2𝑚 (𝐿2 𝑝+ 𝐿2 𝑒) = 𝜃2 𝑚+ 𝛼𝑚(ℎ2(𝜃1 𝑚, 𝜃2 𝑚, 𝜙𝑚) + 𝑁(2) 𝑚), 𝜙𝑚+1 = 𝜙𝑚+ 𝛼𝑚 𝜕 𝜕𝜙𝑚 𝐿𝑐= 𝜙𝑚+ 𝛼𝑚(ℎ3(𝜃1 𝑚, 𝜃2 𝑚, 𝜙𝑚) + 𝑁(3) 𝑚), (19) where 𝑘𝑚= 𝛼𝑚 𝛽𝑚→∞due to our setting and the deterministic functions ℎ1(𝜃1, 𝜙), ℎ2(𝜃1, 𝜃2, 𝜙) and ℎ3(𝜃1, 𝜃2, 𝜙) are as follows: ℎ1(𝜃1, 𝜙) = E𝑠∼𝜌𝜋𝜃𝑚,𝑎∼𝜋𝜃𝑚 \" min \u0010 𝜋1 𝜃𝑚+1(𝑎1 𝑡|𝑠1 𝑡) 𝜋1 𝜃𝑚(𝑎1 𝑡|𝑠1 𝑡) 𝐴1 𝑡, 𝑐𝑙𝑖𝑝 \u0010 𝜋1 𝜃𝑚(𝑎1 𝑡|𝑠1 𝑡) 𝜋1 𝜃𝑚−1 (𝑎1 𝑡|𝑠1 𝑡) ,1 −𝜖,1 + 𝜖 \u0011 𝐴1 𝑡 \u0011# , ℎ2(𝜃1, 𝜃2, 𝜙) = E𝑠∼𝜌𝜋𝜃𝑚,𝑎∼𝜋𝜃𝑚 \" min \u0010 𝜋2 𝜃𝑚+1(𝑎2 𝑡|𝑠2 𝑡) 𝜋2 𝜃𝑚(𝑎2 𝑡|𝑠2 𝑡) 𝐹2 𝑡𝐴2 𝑡, 𝑐𝑙𝑖𝑝 \u0010 𝜋2 𝜃𝑚(𝑎2 𝑡|𝑠2 𝑡) 𝜋2 𝜃𝑚−1(𝑎2 𝑡|𝑠2 𝑡) ,1 −𝜖,1 + 𝜖 \u0011 𝐹2 𝑡𝐴2 𝑡 \u0011# , ℎ3(𝜃1, 𝜃2, 𝜙) = E𝑠∼𝜌𝜋𝜃𝑚[(𝑣𝜙(𝑠𝑡) −𝑟𝑡)2]. (20) where 𝑐𝑙𝑖𝑝(𝑥,min,max) confines 𝑥to the interval [min,max], returning min, max, or 𝑥accordingly as 𝑥falls below min, exceeds max, or lies between them. Lemma 4. Suppose that the parameter space for (𝜃1 𝑚, 𝜃2 𝑚, 𝜙𝑚) is a convex bounded set. Also suppose that ℎ1, ℎ2 and ℎ3 are Lipschitz functions and 𝑁(1) 𝑚, 𝑁(2) 𝑚 and 𝑁(3) 𝑚 are martingales with finite conditional second order moment. The step sizes 𝛼𝑚and 𝛽𝑚satisfy the conditions Í∞ 𝑘=1 𝛼𝑘= Í∞ 𝑘=1 𝛽𝑘= ∞, Í∞ 𝑘=1 𝛽2 𝑘< Í∞ 𝑘=1 𝛼2 𝑘< ∞, and 𝛽𝑘 𝛼𝑘→0 as 𝑘tends to infinity. The iterative sequence {(𝜃1 𝑚, 𝜃2 𝑚, 𝜙𝑚)} generated by the iteration Eq.(19) converges to a set which is defined as below: (𝜃1 𝑚, 𝜃2 𝑚, 𝜙𝑚) →{(𝜃1 𝑚, 𝜃2★ 𝑚(𝜃1 𝑚), 𝜙★ 𝑚(𝜃1 𝑚))}, where 𝜃2★ 𝑚(𝜃1 𝑚) and 𝜙★ 𝑚(𝜃1 𝑚) are well-defined functions, similar to 𝑝∗(𝑥) defined in Lemma 3. In other words, this lemma means |𝜃2 𝑚−𝜃2★ 𝑚(𝜃1 𝑚)| →0 and |𝜙𝑚−𝜙★ 𝑚(𝜃1 𝑚)| →0 almost surely, that is, {𝜃2 𝑚} and {𝜙𝑚} asymptotically track 𝜃2★ 𝑚(𝜃1 𝑚) and 𝜙★ 𝑚(𝜃1 𝑚), respectively. X: Dual-Agent Deep Reinforcement Learning for Dynamic Pricing and Replenishment 22 Article submitted to Operations Research Theorem 4. In the condition of Lemma 4, the iterative sequence {(𝜃1 𝑚, 𝜃2 𝑚, 𝜙𝑚)} generated by the iteration Eq.(19) converges to some limit point of the ODE ¤𝜃1(𝑡) = ℎ1(𝜃1(𝑡), 𝜙★(𝜃1(𝑡))), i.e., (𝜃1 𝑚, 𝜃2 𝑚, 𝜙𝑚) →(𝜃1★, 𝜃2★(𝜃1★), 𝜙★(𝜃1★)). The proofs of Lemma 4 and Theorem 4 are similar to Lemma 3 and Theorem 2 and we omit them. 3.7. Algorithmic Complexity Analysis In this section, we analyze the computation complexity of the FSDA algorithm shown in Algorithm 1. The computational complexity is associated with the scale of the pricing and replenishment problem under consideration, including the length of the decision period and the dimensions of the state and action. Additionally, the computational complexity is also influenced by the specific design of the neural networks in the FSDA, such as the input/output width and the number of hidden layers of the MLP, the number of GRUs and the number of episodes for training. We provide the computation complexity of the model training and the model inference in Proposition 8. Proposition 8. Suppose that the dimension of state 𝑠𝑖 𝑡for each agent 𝑖∈{1, ..., 𝑁} in each period 𝑡∈{1, ...,𝑇} is 𝐷𝐼, the dimension of the action 𝑎𝑖 𝑡is 𝐷𝐴, the width of MLP1’s hidden layers are both 𝑊1, and the input width of MLP2 is 𝑊2. Then the computational complexity of model training with 𝑀episodes is 𝑂 \u0010"