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Intertemporal Demand Allocation for Inventory Control in Online Marketplaces
Rene Caldentey, Tong Xie
Intelligence
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Summary
The paper investigates how online marketplaces can influence third-party sellers' inventory decisions and fulfillment mode choices (FBM vs. FBP) by strategically allocating aggregate demand over time. By controlling the predictability of seller-level demand streams, the platform manipulates safety-stock requirements, creating a trade-off between inventory-related revenue and platform fulfillment adoption, even under neutrality constraints.
Entities (5)
Relation Signals (3)
Online Marketplace → allocatesdemandto → Third-party Seller
confidence 95% · the platform observes aggregate demand, allocates orders across sellers over time
Third-party Seller → choosesfulfillmentmode → Fulfill-by-Platform
confidence 92% · sellers choose between two fulfillment options, fulfill-by-merchant (FBM) and fulfill-by-platform (FBP)
Online Marketplace → influencesinventoryof → Third-party Seller
confidence 90% · platform control over demand allocation affects not only a seller’s average demand, but also the information revealed
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Abstract
Abstract:Online marketplaces increasingly do more than simply match buyers and sellers: they route orders across competing sellers and, in many categories, offer ancillary fulfillment services that make seller inventory a source of platform revenue. We investigate how a platform can use intertemporal demand allocation to influence sellers' inventory choices without directly controlling stock. We develop a model in which the platform observes aggregate demand, allocates orders across sellers over time, and sellers choose between two fulfillment options, fulfill-by-merchant (FBM) and fulfill-by-platform (FBP), while replenishing inventory under state-dependent base-stock policies. The key mechanism we study is informational: by changing the predictability of each seller's sales stream, the platform changes sellers' safety-stock needs even when average demand shares remain unchanged. We focus on nondiscriminatory allocation policies that give sellers the same demand share and forecast risk. Within this class, uniform splitting minimizes forecast uncertainty, whereas any higher level of uncertainty can be implemented using simple low-memory allocation rules. Moreover, increasing uncertainty above the uniform benchmark requires routing rules that prevent sellers from inferring aggregate demand from their own sales histories. These results reduce the platform's problem to choosing a level of forecast uncertainty that trades off adoption of platform fulfillment against the inventory held by adopters. Our analysis identifies demand allocation as a powerful operational and informational design lever in digital marketplaces.
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- Source: https://arxiv.org/abs/2604.07312v1
- Canonical: https://arxiv.org/abs/2604.07312v1
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Caldentey, Xie Demand Allocation for Inventory Control Demand Allocation for Inventory Control in Online Marketplaces é Caldentey and Tong Xie School of Business, The University of Chicago, Chicago, IL 60637 .caldentey@chicagobooth.edu, txie0@chicagobooth.edu Online marketplaces increasingly do more than simply match buyers and sellers: they route orders across competing sellers and, in many categories, offer additional ancillary fulfillment services that make seller inventory a source of platform revenue. We investigate how a platform can use intertemporal demand allocation to influence sellers’ inventory choices without directly controlling stock. We develop a model in which the platform observes aggregate demand, allocates orders across sellers over time, and sellers choose between two fulfillment options, fulfill-by-merchant (FBM) and fulfill-by-platform (FBP), while replenishing inventory under state-dependent base-stock policies. The key mechanism we study is informational: by changing the predictability of each seller’s sales stream, the platform changes sellers’ safety-stock needs even when average demand shares remain unchanged. We focus on nondiscriminatory allocation policies that give sellers the same demand share and forecast risk. Within this class, uniform splitting minimizes forecast uncertainty, whereas any higher level of uncertainty can be implemented using simple low-memory allocation rules. Moreover, increasing uncertainty above the uniform benchmark requires routing rules that prevent sellers from inferring aggregate demand from their own sales histories. These results reduce the platform’s problem to choosing a level of forecast uncertainty that trades off adoption of platform fulfillment against the inventory held by adopters. Our analysis identifies demand allocation as a powerful operational and informational design lever in digital marketplaces. Inventory control; Dynamic demand allocation; Online marketplace operations; Time-series forecasting. 1 Introduction Digital marketplaces increasingly operate not only as matching intermediaries, but also as market makers that control how demand is allocated across third-party sellers over time. In many categories, multiple sellers offer identical or nearly identical products, and platform algorithms determine which offer is made salient, and hence which seller is most likely to receive the next order. For example, on Amazon, the “buy box” (“featured offer”) selects the default merchant for one-click purchasing, thereby routing marginal order flow among competing sellers (Lee and Musolff, 2025; Raval, 2023; Jeffries and Yin, 2021).***This control is economically consequential because third-party sellers account for a large share of marketplace activity; in recent quarters, they represented more than 60% of Amazon’s worldwide paid units (Amazon.com, Inc., 2026). Motivated by this reality, the central question we investigate in this paper is: “How can a platform shape sellers’ inventory decisions without directly controlling their stock levels?” Platforms often have several levers at their disposal to influence inventory choices, including restock limits, capacity reservations, minimum performance thresholds, and automated replenishment recommendations. In this paper, however, we focus on a different and more indirect mechanism that, to the best of our knowledge, has not been studied in the literature. The mechanism we exploit builds on the informational asymmetry between the platform and sellers: because sellers replenish inventory based on their own realized sales histories, platform control over demand allocation affects not only a seller’s average demand, but also the information revealed through its order stream and, hence, the forecastability of its future demand. Because forecastability determines safety-stock requirements under standard inventory policies, it becomes a key operational lever through which the platform can influence decentralized inventory decisions. This mechanism is especially salient when the platform also provides fulfillment services. In programs such as Fulfillment by Amazon (FBA) and Walmart Fulfillment Services (WFS), sellers retain ownership of inventory while placing stock inside the platform’s logistics network, where the platform provides storage together with downstream fulfillment services such as pick–pack–ship, returns, and customer service (Amazon.com, Inc., 2025; Lai et al., 2022; Li et al., 2024a; Walmart Corporate, 2020; Walmart Inc., 2024). We use FBP as a generic label for such arrangements and FBM for seller-managed fulfillment.†Comparable programs include Walmart Fulfillment Services (WFS) for Walmart Marketplace sellers (Walmart Corporate, 2020; Supply Chain Dive, 2020; Walmart Inc., 2024; Walmart Marketplace, 2024), Mercado Libre’s managed fulfillment offering (Reuters, 2024; Process Excellence Network, 2025), and Zalando Fulfillment Solutions (ZFS) in Europe (Zalando SE, 2021). These programs are often framed as service-quality upgrades for consumers, but they also create a second operational relationship between the platform and sellers: the platform becomes an inventory carrier that can monetize storage and fulfillment activity (Shi et al., 2021; Amazon.com, Inc., 2025; Toogood, 2024). This dimension has become economically important in its own right. For Amazon, “third-party seller services” net sales—which include commissions and related fulfillment and shipping fees—grew from $117.7B in 2022 to $156.1B in 2024 (Amazon.com, Inc., 2025). Similar growth in fulfillment services has been observed for Walmart and Alibaba‡Walmart reports continued rapid marketplace growth and notes that more than 60% of its top sellers are enrolled in WFS (Walmart Inc., 2024; Walmart Marketplace, 2024). In Alibaba’s ecosystem, Cainiao’s segment revenue reached RMB 99.0B in fiscal year 2024, up 28% from the previous year (Alibaba Group Holding Limited, 2024).. Although inventory remains a decentralized seller decision, FBP gives the platform a direct economic stake in inventory through storage and fulfillment fees.§Fulfillment services can account for a substantial share of the platform’s effective take rate. For example, while Amazon’s referral fees are around 15%, FBA fees for storage, packing, and delivery often consume 20–35% of seller revenue, making fulfillment one of the largest cost components for many third-party sellers (Kaziukėnas, 2023). The platform therefore influences inventory not by dictating stock levels, but by shaping the realized demand process on which sellers base their replenishment decisions. We formalize this idea in a stylized model where aggregate demand is allocated across sellers who manage inventory using base-stock policies. Because safety stock depends on forecast errors, the platform’s allocation policy affects inventory by altering the predictability of seller-level demand, even when long-run demand shares remain unchanged. This mechanism generates an extensive–intensive trade-off. Increasing forecast uncertainty raises sellers’ safety stock and inventory levels but can reduce participation in the platform’s fulfillment program if higher inventory costs make it less attractive. The platform therefore trades off higher inventory among participating sellers against lower adoption. Our results characterize both the power and the limits of this lever. The main takeaway is that the platform can materially influence on-platform inventory, and thus potentially service levels and inventory-related revenues, using simple history-dependent allocation mechanisms. First, we characterize the levels of seller-level short-horizon forecast uncertainty (root MSFE) that are implementable when the platform acts in a neutral manner, meaning that sellers are treated symmetrically in a sense we make precise later, and subject to seller participation constraints. Second, we show that any implementable increase relative to the uniform-split benchmark, under which demand is allocated evenly across sellers, can be generated by low-order moving-average modifications of the uniform allocation rule. Third, we identify a structural role for informational asymmetry in the form of non-invertibility: to induce strictly higher safety stocks under neutrality, the platform must exploit its informational advantage in how aggregate demand innovations are revealed through demand allocation. Taken together, our results show that platform allocation is not only a demand-routing tool but also an inventory-design tool. A platform that cannot directly control seller stock levels can still shape on-platform inventory by shaping what sellers are able to forecast. Roadmap. The rest of the paper is organized as follows. Section 2 reviews the related literature and positions our work within it. Section 3 introduces the model, including the platform’s demand-allocation problem, sellers’ inventory decisions, the FBP/FBM choice, and the neutrality requirement. Section 4 develops the forecasting framework that links allocation dynamics to seller-level forecastability. Section 5 characterizes optimal neutral allocation policies and clarifies the role of non-invertibility. Section 6 presents an illustrative numerical example. Section 7 discusses extensions, and Section 8 provides concluding remarks. 2 Related Literature This paper lies at the intersection of three research streams. The first studies how digital marketplaces operating as two-sided platforms shape market outcomes by controlling the allocation of demand (order flow) across competing sellers, and how these allocation choices interact with seller operations—especially inventory—when the platform also offers fulfillment and inventory-carrying services. The second concerns neutrality and non-discrimination constraints in platform design: beyond operational objectives, platforms may face reputational, legal, and political pressures that limit overt discrimination among comparable sellers. The third stream is methodological and concerns forecasting stationary time series and how forecast uncertainty propagates into inventory decisions through base-stock policies, safety-stock levels, and information-sharing (or information-design) frictions. In what follows, we review these streams in tandem and position our contribution relative to them. 2.1 Demand allocation and inventory in online marketplaces A growing literature in operations and economics emphasizes that many digital marketplaces do not merely match buyers and sellers; they actively shape market outcomes by controlling ranking and order-routing rules. When multiple sellers offer close substitutes, the platform effectively implements a demand-allocation mechanism that shifts marginal orders and therefore affects competition, entry incentives, and surplus. Recent work formalizes how platform-guided search, algorithmic order routing, and default-merchant selection (e.g., Amazon’s “buy box”) reshape two-sided market outcomes, seller competition, and inventory incentives (Lee and Musolff, 2025; Cao and Hu, 2024; Hagiu and Jullien, 2011; Raval, 2023). A complementary literature in operations management examines platforms that bundle marketplace services with fulfillment and inventory management. Amazon’s FBA is the canonical example: sellers pre-position inventory in the platform’s logistics network, and the platform handles pick-pack-ship and customer service. Researchers study why a platform would offer such services even when third-party sellers compete with its own retail arm (Lai et al., 2022), how seller-side tying of fulfillment services intensifies price competition (de Cornière et al., 2025), and how fulfillment design interacts with fee structure, asymmetric information, and format choice (Li et al., 2024a, b; Sun et al., 2020, 2021). A common theme is that fulfillment programs transform the platform’s revenue model: inventory held within the platform ecosystem becomes a direct source of monetization through storage fees and fulfillment activity (Shi et al., 2021; Li et al., 2024a). Our contribution is complementary: rather than focusing on fee design or contracting, we study how a platform can influence sellers’ endogenous inventory decisions through temporal engineering of demand allocation under neutrality-style constraints. 2.2 Neutrality and Platform Accountability Platforms face growing reputational, legal, and political pressures to treat comparable sellers even-handedly. Recent policy scrutiny of recommendation fairness (Phillips, 2024), regulatory discussions under the Digital Markets Act (Amazon, 2024), and investigative work on platform-controlled defaults such as Amazon’s “buy box” (Jeffries and Yin, 2021; Raval, 2023) all highlight these accountability concerns. In response, a growing operations literature models explicit neutrality or fairness constraints that restrict the platform’s feasible set of strategies. In the context of search neutrality and online retail platforms, recent work studies how imposing such constraints changes equilibrium outcomes, platform incentives, and welfare (Zou and Zhou, 2025). Related constraint-based approaches appear in fairness-aware assortment and recommendation design such as market-share balancing (Housni et al., 2025), pairwise fairness (Chen et al., 2024), and minimum exposure or share constraints (Lu et al., 2024). Our modeling choice follows this tradition: we impose neutrality on demand allocation by requiring that comparable sellers be treated symmetrically in expectation—each seller receives the same long-run mean demand share and faces the same one-step-ahead forecast uncertainty. This notion captures a practical accountability constraint while still allowing the platform to retain design flexibility through the temporal structure of allocation. 2.3 Forecasting stationary time series and its role in inventory and supply chains Standard inventory theory suggests that sellers replenish inventory using base-stock policies, under which the order-up-to level decomposes into a forecast component plus a safety-stock component, thereby creating a direct link between inventory holdings and forecast uncertainty. A modern synthesis of the forecasting–inventory interface is provided by Goltsos et al. (2022), which highlights both the historical separation between forecasting and inventory research and recent efforts to integrate them. Our paper contributes from a platform-design perspective: the platform sits upstream of sellers’ forecasting problems and can shape the time-series properties of the demand data sellers observe. A related literature studies how forecast-error estimation, autocorrelation, and estimation uncertainty affect safety-stock calculations for lead-time demand (Prak et al., 2017; Babai et al., 2022; Beutel and Minner, 2012). These results reinforce a core premise behind our mechanism-design problem: inventory depends on the structure of forecast uncertainty at the relevant horizons, which is shaped not only by the mean and variance of demand but also by its autocovariance structure. Finally, a long-standing operations literature studies the value of information and information sharing in supply chains, and time-series perspectives integrate forecasting and inventory control across stages (Candogan and Gurkan, 2025; Gavirneni et al., 1999; Cachon and Fisher, 2000). More recently, Caldentey et al. (2025) develop an information-design view of inventory systems and show how the structure of an ordering process and non-invertibility can determine the value of information. Our paper is closely aligned with this perspective but shifts the setting to a marketplace with multiple symmetric sellers: the platform’s demand allocation mechanism governs how aggregate innovations are revealed to sellers over time and therefore how forecastability propagates into base-stock choices under neutrality-style constraints. Positioning and contributions. Relative to the platform and fulfillment literatures, we study a distinct mechanism: dynamic demand allocation that engineers the forecastability of seller-level demand, thereby shaping inventory and FBP adoption. Relative to the inventory-forecasting and information-sharing literatures, we endogenize each seller’s demand process through a platform design choice under neutrality constraints. Unlike other information-design mechanisms, such as the burgeoning literature on Bayesian persuasion (Kamenica and Gentzkow, 2011; Bergemann and Morris, 2019), the platform’s instrument here is not a signal structure but an allocation rule constrained by the requirement that seller-level demands sum to market demand period by period. As a result, our proposed mechanism operates not through belief manipulation, but through the manipulation of forecastability. We view our contributions as having both a conceptual and a concrete component. At a conceptual level, we identify a novel operational mechanism through which a platform can influence sellers’ inventory-management decisions. By controlling how realized aggregate demand is allocated across sellers over time, the platform effectively shapes the demand histories—and the information they carry—that sellers observe and use for forecasting, thereby influencing both their fulfillment-mode choices (FBP versus FBM) and the inventory level. More concretely, we characterize the implementable range of seller-level one-step-ahead forecast uncertainty under neutral demand allocation, showing that uniform splitting attains a sharp lower bound, while simple low-order moving-average perturbations—with low memory and based only on recent demand realizations—can generate higher uncertainty; we show that achieving strict improvements beyond the uniform benchmark necessarily requires non-invertibility, thereby identifying a precise sense in which the platform’s informational advantage can matter even under symmetric treatment; and we reduce the platform’s design problem to a tractable one-dimensional optimization problem, which makes transparent the trade-off between fulfillment adoption and inventory level. 3 Model Setup This section introduces the model of a platform that allocates demand across sellers who manage inventory and choose between FBM and FBP, as schematically depicted in Figure 1. Figure 1: Overview of the platform marketplace. Aggregate demand DtD_t is allocated across sellers, who choose between FBM and FBP and manage their inventory stocks using state-dependent base-stock policies. The platform observes aggregate demand DtD_t each period and allocates it across N sellers, generating seller-level demand streams Dntn∈[N]\D_nt\_n∈[N] satisfying ∑nDnt=Dt _nD_nt=D_t. The allocation may depend on past demand. Sellers choose between self-fulfillment (FBM) and platform fulfillment (FBP). Each seller manages inventory using a base-stock policy with base-stock level SnS_n based on its allocated demand process, so the platform’s allocation affects both inventory levels and fulfillment-mode choices. The remainder of this section formalizes these elements. ⋄ Platform intermediation and fulfillment modes. We consider a platform that intermediates transactions between consumers and a pool of N third-party sellers, indexed by n∈[N]:=1,…,Nn∈[N]:=\1,…,N\. Time is discrete, t=0,1,2,…t=0,1,2,…. In each period t, aggregate market demand for a representative product (or product class) is DtD_t. The platform acts as a market maker: it determines how realized market demand is split across sellers by choosing an allocation vector (Dnt)n∈[N](D_nt)_n∈[N] satisfying ∑n∈[N]Dnt=Dt,t=0,1,2,…. _n∈[N]D_nt\;=\;D_t, t=0,1,2,…. (1) The allocation may be history-dependent and may reflect the platform’s information at time t. The platform charges a fixed intermediation fee ρ per transaction (per unit of fulfilled demand), independently of how the order is fulfilled. Each seller chooses between two fulfillment modes: fulfill-by-merchant (FBM) or fulfill-by-platform (FBP). Under FBM, the seller manages fulfillment independently; under FBP, the platform provides storage and fulfillment services. ⋄ Seller heterogeneity and operating costs. Sellers differ in their operating costs. If seller n fulfills orders itself (FBM), its per-unit, per-period costs are (hn,bn,fn)(h_n,b_n,f_n), where hnh_n is the holding cost, bnb_n the stockout/backorder penalty, and fnf_n the fulfillment cost. Under platform fulfillment (FBP), the relevant triplet is (H,bn,F)(H,b_n,F), where H and F are the effective holding and fulfillment costs under FBP. Let ℳn∈FBM,FBP M_n∈\ FBM, FBP\ denote seller n’s mode and define h¯n:=hn,ℳn=FBM,H,ℳn=FBP,f¯n:=fn,ℳn=FBM,F,ℳn=FBP. h_n:= casesh_n,& M_n= FBM,\\ H,& M_n= FBP, cases f_n:= casesf_n,& M_n= FBM,\\ F,& M_n= FBP. cases Thus, seller n always faces backorder cost bnb_n, while holding and fulfillment costs depend on ℳn M_n. In what follows, we impose the following assumption to focus on instances that most clearly capture the canonical trade-off between the FBM and FBP operating modes: Assumption 1 For each seller n∈[N]n∈[N], F≤fnF≤ f_n and H≥hnH≥ h_n. That is, platform fulfillment weakly reduces per-order fulfillment cost but weakly increases per-unit inventory carrying cost. This canonical trade-off is empirically grounded: FBP pools shipments across sellers and positions inventory in distributed warehouses, lowering per-order costs (F≤fnF≤ f_n), but premium urban warehouse space and storage fees raise carrying costs (H≥hnH≥ h_n) (Toogood, 2024; Houde et al., 2017; Li et al., 2024a). The assumption is not essential for the theory or the analysis we develop, but it ensures that neither mode trivially dominates the other along both cost dimensions. ⋄ Market demand. We focus on stationary demand to isolate the effects of demand allocation from other forces such as seasonality, life-cycle trends, or macroeconomic shocks. This assumption still allows for a rich class of demand processes, including ARMA models, while ensuring that the long-run statistical properties of demand remain stable. As a result, sellers face constant one-step-ahead forecast uncertainty and can therefore manage inventory using time-invariant base-stock levels. Assumption 2 (Stationary Demand) The demand process Dtt≥0\D_t\_t≥ 0 is a weakly stationary, purely non-deterministic Gaussian process that admits the MA(∞)MA(∞)†See Appendix C for discussion on MA(∞)MA(∞) representation and Gaussian demand. representation Dt=μ+∑k=0∞ψkϵt−k,D_t=μ+ _k=0^∞ _k\, _t-k, (2) for some sequence ψ=ψkk≥0∈ℓ2ψ=\ _k\_k≥ 0∈ ^2, where μ=[Dt]μ=E[D_t] and ϵt\ _t\ is a Gaussian white-noise sequence with ar(ϵt)=σϵ2V ar( _t)= _ε^2. We further assume that Dt\D_t\ is invertible‡A formal definition and discussion of invertibility is provided in Section 4. with respect to ϵt\ _t\. In what follows, we normalize σϵ=1 _ε=1. This is without loss of generality because any scale factor in the innovations can be absorbed into ψ (equivalently, we measure demand in units of the innovation standard deviation). Letting ℬ B denote the Backshift operator, i.e., ℬϵt=ϵt−1 B _t= _t-1, the MA(∞) representation of the market demand in (2) can be written compactly as Dt=μ+ψ(ℬ)ϵtD_t=μ+ψ( B)\, _t, where ψ(z)=∑k=0∞ψkzkψ(z)= _k=0^∞ _k\,z^k is the z-transform representation of centered demand process Dt−μD_t-μ. ⋄ Admissible demand allocation policies. In each period t, the platform observes DtD_t and chooses an allocation (Dnt)n∈[N](D_nt)_n∈[N] satisfying (1), possibly via a history-dependent rule measurable with respect to the platform’s information set ℱtF_t, the filtration generated by Dss≤t\D_s\_s≤ t. We model allocation policies as linear filters applied to aggregate demand, which preserves stationarity and enables tractable characterization of forecastability. Definition 1 (Admissible Policy) An admissible demand allocation policy§Under invertibility, representations based on Dt\D_t\ and ϵt\ _t\ are equivalent for demand processes whose z-transform ψ(z)ψ(z) satisfies mild technical conditions, see Appendix C for details. π=(μn,n(z):n∈[N])π= ( _n,T_n(z):n∈[N] ), consists, for each seller n∈[N]n∈[N], of a mean allocation μn _n and a transfer function n(z)T_n(z), where n(z)=∑k=0∞τnkzkT_n(z)= _k=0^∞ _nkz^k for some sequence τnkk≥0∈ℓ2\ _nk\_k≥ 0∈ ^2. Under such a policy, seller n’s demand process is given by Dnt=μn+∑k=0∞τnk(Dt−k−μ),or equivalently,Dnt=μn+n(ℬ)(Dt−μ).D_nt= _n+ _k=0^∞ _nk\,(D_t-k-μ), or equivalently, D_nt= _n+T_n(B)\,(D_t-μ). The feasibility condition Dt=∑n∈[N]DntD_t= _n∈[N]D_nt then requires both ∑n∈[N]μn=μand∑n∈[N]n(z)=1. _n∈[N] _n=μ and _n∈[N]T_n(z)=1. We let Π(ψ) (ψ) denote the class of admissible allocation policies for market demand ψ.¶One could also consider allocation rules that do not satisfy Dt=∑n∈[N]DntD_t= _n∈[N]D_nt period-by-period, which would require the platform to absorb the mismatch, for example by holding its own inventory. We focus on contemporaneous splits to isolate the role of order allocation across sellers. One particular allocation that serves as a natural benchmark is the uniform allocation, under which Dnt=1NDt,that is,μn=μN,andn(z)=1Nfor all n∈[N].D_nt= 1N\,D_t, that is, _n= μN, and _n(z)= 1N for all n∈[N]. (Uniform Allocation) ⋄ Inventory management, seller payoffs, and participation. Each seller n manages inventory to satisfy its allocated demand stream Dnt\D_nt\. For expositional simplicity, our baseline model assumes zero replenishment lead time; in Appendix F we extend the model to allow for arbitrary lead times that may vary across sellers and may also depend on the seller’s chosen fulfillment mode. Let ℱntF_nt denote seller n’s information set at time t, defined as the filtration generated by Dnss≤t\D_ns\_s≤ t. Just before demand in period t+1t+1 is realized, seller n chooses a state-dependent order-up-to level SntS_nt measurable with respect to ℱntF_nt and places an order Qnt=(Snt−Int)+,Q_nt= (S_nt-I_nt )^+, where IntI_nt denotes net inventory after serving demand in period t and (x)+=maxx,0(x)^+= \x,0\. Following the standard base-stock logic, seller n selects SntS_nt to minimize expected holding and backorder costs for period t+1t+1 given ℱntF_nt: Snt= [h¯n(S−Dn,t+1)++bn(Dn,t+1−S)+|ℱnt].S_nt= _S\;E\! [\, h_n\,(S-D_n,t+1)^++b_n\,(D_n,t+1-S)^+\; |\;F_nt ]. Under 2, Dn,t+1D_n,t+1 is Gaussian conditional on ℱntF_nt, and the optimal base-stock policy takes the familiar form Snt=mnt+ζnσn,ζn:=Φ−1(bnh¯n+bn),S_nt=m_nt+ _n\, _n, _n:= ^-1\! ( b_n h_n+b_n ), (3) where mnt:=[Dn,t+1∣ℱnt]m_nt:=E[D_n,t+1 _nt] is the one-step-ahead mean forecast and the one-step-ahead MSFE is given by σn2:=ar[Dn,t+1−mnt∣ℱnt]. _n^2:=V ar\! [D_n,t+1-m_nt _nt ]. (4) Thus, safety stock is proportional to the one-step-ahead MSFE, so inventory depends directly on forecastability. Our model assumes that sellers act as optimal forecasters. This assumption allows us to isolate the impact of the platform’s allocation policy from suboptimal forecasting heuristics on the sellers’ part. In Section 7.1, we discuss the case in which sellers use suboptimal forecasting methods. Under an admissible allocation policy π=(μn,n(z):n∈[N])π= ( _n,T_n(z):n∈[N] ), seller n receives a stationary demand stream with mean μn _n and one-step-ahead root MSFE σnπ _n^π, as defined in (4). An explicit representation of σnπ _n^π in terms of ψ(z)ψ(z) and n(z)T_n(z) is provided in Section 4. Substituting the optimal base-stock level into the seller’s expected single-period profit yields Unm(π)=(r−ρ−fnm)μn−Knmσnπ,m∈FBM,FBP,U_n^m(π)=(r-ρ-f_n^m)\, _n-K_n^m\, _n^π, m∈\ FBM,FBP\, (5) where (fnFBM,fnFBP)=(fn,F)(f_n 0.45 FBM,f_n 0.45 FBP)=(f_n,F) and KnmK_n^m represents the seller’s expected inventory cost coefficient. It aggregates the expected holding and backorder costs associated with the optimal safety-stock level ζnm _n^m, and is given by Knm=hnmζnm+(hnm+bn)ℒ(ζnm),ζnm=Φ−1(bnhnm+bn),K_n^m=h_n^m\, _n^m+(h_n^m+b_n)\,L( _n^m), _n^m= ^-1\! ( b_nh_n^m+b_n ), with (hnFBM,hnFBP)=(hn,H)(h_n 0.45 FBM,h_n 0.45 FBP)=(h_n,H) and ℒ(z)=ϕ(z)−z(1−Φ(z))L(z)=φ(z)-z(1- (z)) denoting the standard Normal loss function. Stationarity implies that σnπ _n^π is independent of t, so Unm(π)U_n^m(π) is time-independent as well.∥Equation (5) follows from substituting the optimal base-stock level into the expected single-period holding/backorder cost under Gaussian demand. Therefore, seller n chooses FBP over FBM whenever UnFBP(π)≥UnFBM(π).U_n 0.45 FBP(π)≥ U_n 0.45 FBM(π). (6) ⋄ Neutral (non-discriminatory) allocation policies. When a platform steers order flow, its allocation algorithm may be scrutinized for favoritism toward particular sellers or the platform’s own retail arm (Jeffries and Yin, 2021; Phillips, 2024; Amazon, 2024). To capture a baseline notion of non-discrimination, we impose a neutrality constraint requiring symmetric treatment of sellers in expectation: each seller receives the same mean demand share and faces the same one-step-ahead forecast uncertainty (MSFE). This is the notion of non-discrimination most relevant for inventory decisions. We measure the demand risk induced by policy π using the one-step-ahead root MSFE σnπ _n^π, which governs sellers’ inventory and safety-stock decisions. This leads to the following definition. Definition 2 (Neutral admissible policies) Let π=(μn,n(z):n∈[N])π= ( _n,T_n(z):n∈[N] ) be an admissible demand-allocation policy. We say that π is neutral if there exists a constant σπ>0σ^π>0 such that, for all n∈[N]n∈[N], μn=μN _n= μN, and σnπ=σπ _n^π=σ^π, where σnπ _n^π denotes seller n’s one-step-ahead root MSFE under policy π as defined in (4). We denote by ΠN(ψ) _ 0.45 N(ψ) the class of neutral admissible policies. Since every neutral policy assigns the same mean demand μn=μN _n= μN to each seller, we will abuse notation slightly and write a neutral admissible policy simply as π=(n(z):n∈[N])π= (T_n(z):n∈[N] ), suppressing the dependence on the common mean demand allocation. Under neutrality, seller n prefers FBP if and only if ΔFn≥NσπμΔKn,whereΔFn:=fn−F,ΔKn:=KnFBP−KnFBM. F_n\;≥\; N\,σ^πμ\, K_n, F_n:=f_n-F, K_n:=K_n 0.45 FBP-K_n 0.45 FBM. (7) Under Assumption 1, we restrict attention to instances in which ΔKn≥0 K_n≥ 0 and ΔFn≥0 F_n≥ 0 for all n∈[N]n∈[N]. Thus, higher forecast uncertainty raises inventory costs and reduces FBP adoption. Here, ΔFn F_n is the seller’s per-order fulfillment cost savings from using FBP, while ΔKn K_n is the corresponding increase in the inventory holding cost. Under a neutral policy, the boundary ΔF=NσπμΔK F= N\,σ^πμ\, K separates FBP adopters from FBM sellers; see Figure 2. As σπσ^π increases, this boundary rotates upward, reducing adoption while increasing safety stock for remaining FBP sellers. Hence the platform faces an extensive–intensive trade-off: higher σπσ^π lowers participation but raises inventory among adopters. Because both margins affect profits, neither extreme is generally optimal. ΔK KΔF F0ΔF=NσπμΔK F= N\,σ^πμ\, KFBPFBM Figure 2: Partition of sellers (dots) in the (ΔK,ΔF)( K, F) plane under a neutral policy π. Sellers above ΔF=NσπμΔK F= Nσ^πμ K adopt FBP (shaded region), while sellers below prefer FBM. ⋄ Platform payoff and objective. Under an allocation policy π, the platform collects a fixed intermediation fee ρ for each fulfilled unit. Because total transaction volume equals aggregate demand,†In our baseline model demand is backordered rather than lost, so every unit of demand eventually becomes a transaction. Allowing for lost sales would make total transactions (and therefore intermediation revenue) depend on service level. the platform’s per-period expected intermediation revenue equals [ρDt]=ρμE[ρ\,D_t]=ρ\,μ, which is independent of π. In addition, the platform monetizes participation in its FBP service. Specifically, we assume that for each unit transacted by a seller using FBP, the platform collects a fulfillment payoff (net of delivery cost) Δf f, and for each unit of inventory stored in the platform’s warehouse network it collects a rental fee Δh h per unit per period. Thus, to compute the platform’s total payoff under π, we must first determine which sellers choose FBP vs. FBM. As illustrated in Figure 2, the set of sellers who adopt the FBP mode is given by NFBP(π):=n∈[N]:ΔFn≥NσπμΔKn.N 0.45 FBP(π):= \\,n∈[N]: F_n≥ N\,σ^πμ\, K_n \. Let S¯nπ:=[Snt] S_n^π:=E[S_nt] denote the average inventory level held by seller n under policy π. By (3), S¯nπ=μn+σπζnFBP S_n^π= _n+σ^π\,ζ 0.45 FBP_n, and the platform’s expected gross per-period payoff can be written as Vπ V^π := := ρμ+∑n∈NFBP(π)[Δfμn+ΔhS¯nπ]=ρμ+∑n∈NFBP(π)[ΔfμN+Δh(μN+σπζnFBP)] ρ\,μ+ _n∈ N 0.45 FBP(π) [ f\, _n+ h\, S_n^π ]=ρ\,μ+ _n∈ N 0.45 FBP(π) [ f\, μN+ h ( μN+σ^π\,ζ 0.45 FBP_n ) ] (8) = = ρμ+(Δf+Δh)μN|NFBP(π)|+ΔhΓFBP(π). ρ\,μ+( f+ h)\, μN\,|N 0.45 FBP(π)|+ h\, 0.45 FBP(π). where |NFBP(π)||N 0.45 FBP(π)| denotes the number of sellers adopting FBP and ΓFBP(π) 0.45 FBP(π) their aggregate safety stock under policy π. Equation (8) shows that the platform’s payoff depends on π through (i) which sellers adopt FBP and (i) the inventory levels induced by the demand allocation rule. Putting all the pieces together, the platform’s optimization problem is given by Platform’s Problem: supπ∈ΠN(ψ)Vπ=ρμ+(Δf+Δh)μN|NFBP(π)|+ΔhΓFBP(π) _π∈ _ 0.45 N(ψ)V^π=ρ\,μ+( f+ h)\,\,μ N\,|N 0.45 FBP(π)|+ h\, 0.45 FBP(π) (9) where NFBP(π)=n∈[N]:ΔFn≥NσπμΔKnandΓFBP(π)=σπ∑n∈NFBP(π)ζnFBP. N 0.45 FBP(π)= \\,n∈[N]: F_n≥ N\,σ^πμ\, K_n \ and 0.45 FBP(π)=σ^π\, _n∈ N 0.45 FBP(π)ζ 0.45 FBP_n. This problem lacks a simple closed-form solution because varying σπσ^π rotates the adoption boundary, causing discrete changes in the adopter set. Thus, both |NFBP(π)||N 0.45 FBP(π)| and ΓFBP(π) 0.45 FBP(π) are non-smooth in σπσ^π, see Figure 3. 00.50.5111.51.5222.52.533055101015152020252530303535σπσ^πNumber of FBP adopters |NFBP(π)||N 0.45 FBP(π)|00.50.5111.51.5222.52.53301010202030304040σπσ^πCumulative safety stock of FBP adopters ΓFBP(π) 0.45 FBP(π) Figure 3: Left: number of FBP adopters |NFBP(π)||N 0.45 FBP(π)|. Right: cumulative safety stock of FBP adopters, ΓFBP(π) 0.45 FBP(π). Nevertheless, the problem can be solved efficiently by exploiting the step-function nature of |NFBP(π)||N 0.45 FBP(π)|. Under Assumption 1, seller n adopts FBP under a neutral policy if and only if σπ≤μΔFn/(NΔKn)σ^π≤μ\, F_n/(N\, K_n)‡Note that under this convention, the right-hand side is +∞+∞ whenΔKn=0 K_n=0. Hence, the number of adopters |NFBP(π)||N 0.45 FBP(π)| is a piecewise-constant, non-increasing function of σπσ^π, with possible jumps only at the finitely many threshold values μΔFnNΔKn:n∈[N],ΔKn>0. \\; μ\, F_nN\, K_n\;:\;n∈[N],\ K_n>0 \. Between two consecutive thresholds, the adopter set is fixed, so ΓFBP(π) 0.45 FBP(π) is linear in σπσ^π. As a result, the platform’s payoff is piecewise linear in σπσ^π, and an optimal solution can be obtained by evaluating VπV^π at the finitely many critical thresholds. This procedure, however, requires knowing the range of feasible values of σπσ^π that the platform can implement under a neutral allocation policy. Although an arbitrarily large σπσ^π is theoretically feasible, we impose a participation constraint requiring nonnegative seller payoffs, which yields an upper bound σU _ U. Based on (5), define Un(σ):=maxUnFBP(σ),UnFBM(σ)U_n(σ):= \U_n 0.45 FBP(σ),\,U_n 0.45 FBM(σ)\, and let σU:=supσ≥0:Un(σ)≥0for all n∈[N]. _ U:= \σ≥ 0 U_n(σ)≥ 0\ for all n∈[N] \. (10) On the other hand, in Section 4.3 we show that there exists a lower bound σL _ L on the set of values of σπσ^π that are attainable under neutral demand-allocation policies. This bound is derived after introducing the mathematical framework that links an allocation policy π to the sellers’ root MSFE σπσ^π in Section 4. ⋄ Discussion of Neutrality. We close this section by clarifying two central and interconnected modeling choices: our use of neutral demand-allocation policies (see 2) and our use of forecastability (MSFE), rather than an alternative measure such as unconditional variance, as the relevant notion of demand risk. At a high level, the neutrality assumption reflects growing expectations of non-discrimination in platforms that control order flow (Jeffries and Yin, 2021; Phillips, 2024; Amazon, 2024). In our formulation, neutrality requires symmetry in expectation along two dimensions: mean exposure (equal long-run demand shares) and demand risk (equal root MSFE). Root MSFE is the natural measure here because sellers’ safety-stock decisions depend on short-horizon predictability, and it can in principle be audited from realized order histories. This notion implicitly assumes sellers form optimal one-step-ahead forecasts; in Section 7.1 we examine how the analysis changes under suboptimal forecasting methods. A natural alternative is to measure demand risk, and hence neutrality, in terms of unconditional variance rather than MSFE. The next example shows that these two measures are not necessarily aligned. Example 1 (Forecastability and Unconditional Variance) Consider a system with two sellers. Case 1: Same MSFE, Different Variances. Let demand follow the MA(1) process Dt=μ+ϵt+0.8ϵt−1D_t=μ+ _t+0.8 _t-1. Suppose the platform allocates demand so that D1t=μ2+0.5ϵt−0.2ϵt−1−0.48ϵt−2,D2t=μ2+0.5ϵt+1.0ϵt−1+0.48ϵt−2.D_1t= μ2+0.5 _t-0.2 _t-1-0.48 _t-2, D_2t= μ2+0.5 _t+1.0 _t-1+0.48 _t-2. Then ar(D1t)=0.5204V ar(D_1t)=0.5204 and ar(D2t)=1.4804V ar(D_2t)=1.4804, but the two allocations have the same one-step-ahead forecast uncertainty, with MSFE1=MSFE2=0.36MSFE_1=MSFE_2=0.36. Thus, if sellers 1 and 2 forecast optimally, they should be indifferent between these two allocations, whereas naïve sellers who forecast demand as if it were i.i.d. would rank allocation 1 as less risky, and hence preferable, than allocation 2. Case 2: Same Variance, Different MSFEs. Now consider aggregate demand Dt=μ+ϵt+0.5ϵt−1D_t=μ+ _t+0.5 _t-1 and D1t=μ2+0.2ϵt+0.85ϵt−1,D2t=μ2+0.8ϵt−0.35ϵt−1.D_1t= μ2+0.2 _t+0.85 _t-1, D_2t= μ2+0.8 _t-0.35 _t-1. In this case, the two allocations have the same variance, ar(D1t)=ar(D2t)=0.7625,V ar(D_1t)=V ar(D_2t)=0.7625, but different forecast errors: MSFE1=0.7225,MSFE2=0.64.MSFE_1=0.7225,\,MSFE_2=0.64. In this case, optimal forecasters prefer allocation 2 over allocation 1, whereas naïve forecasters would be indifferent. Consequently, a neutrality criterion based solely on unconditional variance may fail to detect economically relevant disparities in the actual inventory risk borne by sellers. ■ −2-2022D1tD_1t Seller 1 Demand: Var=0.5204∣MSFE=0.36Var=0.5204 =0.36 05510101515202025253030353540404545−2-2022Time (t)D2tD_2t Seller 2 Demand: Var=1.4804∣MSFE=0.36Var=1.4804 =0.36 −2-2022D1tD_1t Seller 1 Demand: Var=0.7625∣MSFE=0.7225Var=0.7625 =0.7225 05510101515202025253030353540404545−2-2022Time (t)D2tD_2t Seller 2 Demand: Var=0.7625∣MSFE=0.64Var=0.7625 =0.64 Figure 4: Demand sample paths: Left: equal MSFE, different variances; Right: equal variances, different MSFE. 4 Preliminaries The platform’s allocation rule reshapes the demand signal observed by each seller and, therefore, its forecastability. To analyze this mechanism, we work in the z-domain rather than the time domain and use inner–outer factorization to separate two key features of demand: forecastability, governed by the outer factor, and phase structure, captured by the inner factor. This representation allows us to characterize the range of forecast uncertainty that can be implemented under neutrality. 4.1 Foundations: Inner–outer factorization and invertibility We briefly review the essential mathematical tools that underlie our analysis. These tools are mainly drawn from the theory of the ℍ2H^2 Hardy space, with particular emphasis on inner–outer factorization and its time-series interpretation. The interested reader can consult Appendix B for a more detailed presentation, including additional technical background and references. Our analysis is facilitated by representing stationary demand processes in the z-domain. In particular, under an admissible demand allocation policy π=(μn,n(z):n∈[N])π= ( _n,T_n(z):n∈[N] ), seller n’s demand process satisfies Dnt=μn+n(ℬ)(Dt−μ).D_nt= _n+T_n(B)\,(D_t-μ). Moreover, by 2, the centered market demand can be represented by Dt−μ=ψ(ℬ)ϵtD_t-μ=ψ(B)\, _t, where ψ(z)=∑k=0∞ψkzkψ(z)= _k=0^∞ _kz^k is the z-transform of Dt−μ\D_t-μ\. Thus, seller n’s demand can equivalently be written as Dnt=μn+ψn(ℬ)ϵt,whereψn(z)=n(z)ψ(z).D_nt= _n+ _n(B)\, _t, where _n(z)=T_n(z)ψ(z). Thus, under policy π, the forecastability of seller n’s demand—and in particular its root MSFE σnπ _n^π—is determined by the transfer function ψn(z) _n(z). Note also that the feasibility condition Dt=∑n∈[N]DntD_t= _n∈[N]D_nt implies ∑n∈[N]ψn(z)=ψ(z) _n∈[N] _n(z)=ψ(z). A key tool is the inner–outer factorization of ψn(z) _n(z). A function ℐ∈ℍ2I ^2 is inner if it is unimodular on the unit circle, i.e., |ℐ(z)|=1|I(z)|=1 a.e. on |z|=1|z|=1. A function ∈ℍ2O ^2 is outer if it has no zeros in the open unit disk, i.e., (z)≠0O(z)≠ 0 for |z|<1|z|<1. We write I and O for the classes of inner and outer functions in ℍ2H^2. Lemma 1 (Inner–outer factorization) Every f(z)∈ℍ2f(z) ^2 admits a factorization f(z)=(z)ℐ(z),f(z)=O(z)\,I(z), where ℐ∈I is inner and ∈O is outer. Inner and outer functions admit a natural forecasting interpretation. Inner factors act as all-pass filters: multiplying a transfer function by an inner factor preserves its spectral density and, hence, its entire autocovariance structure. Thus, these inner factors are “invisible” to an optimal forecaster. By contrast, outer factors are causal, minimum-phase, and invertible, and therefore determine the fundamental one-step-ahead forecastability of the process. Thus, by 1, any transfer function can be decomposed into an inner component, which captures phase distortions, and an outer component, which governs forecast uncertainty. The following lemma summarizes the implication most relevant for our analysis. Lemma 2 (Root MSFE) Let Dnt=μn+n(ℬ)(Dt−μ)D_nt= _n+T_n(B)\,(D_t-μ) denote seller n’s allocated demand process under an admissible allocation policy π=(μn,n(z):n∈[N])π= ( _n,T_n(z):n∈[N] ), and define ψn(z):=n(z)ψ(z) _n(z):=T_n(z)\,ψ(z). Then ψn∈ℍ2 _n ^2 and, by 1, admits an inner–outer factorization of the form ψn(z)=n(z)ℐn(z), _n(z)=O_n(z)\,I_n(z), for some n∈O_n and ℐn∈I_n . Moreover, seller n’s one-step-ahead root mean squared forecast error under policy π is given by σnπ=|n(0)|. _n^π=|O_n(0)|. This lemma identifies the channel through which the platform can influence seller n’s root MSFE, and hence its inventory decisions, by shaping the outer component of the allocated demand process. By contrast, inner factors do not affect forecastability, but they provide the platform with additional flexibility to satisfy the feasibility condition ∑n∈[N]ψn(z)=ψ(z) _n∈[N] _n(z)=ψ(z). Another important concept in our analysis is invertibility. Given seller n’s allocated demand process Dnt=μn+ψn(ℬ)ϵtD_nt= _n+ _n(B)\, _t, we ask whether seller n can recover the underlying demand shocks ϵt\ _t\ using only its own observed demand stream Dnt\D_nt\. If so, we say that Dnt\D_nt\ is invertible with respect to ϵt\ _t\. The next result gives a necessary and sufficient condition. Lemma 3 (Invertibility) Seller n’s demand process Dnt\D_nt\ is invertible with respect to the demand shocks ϵt\ _t\ if and only if ψn(z) _n(z) is outer. As we show later, the platform’s optimal allocation policy generally induces a nontrivial inner factor in ψn(z) _n(z), so seller-level demand is not invertible. In that case, seller n cannot fully recover the underlying demand shocks and therefore has incomplete information about aggregate market demand. We conclude with a simple example that illustrates the main ideas in this section. Example 2 (MA(q)(q) demand allocation) Suppose the z-transform ψn(z) _n(z) of seller n’s demand admits an MA(q)(q) representation ψn(z)=∑k=0qφkzk. _n(z)= _k=0^q _k\,z^k. By fundamental theorem of algebra, we can factor ψn(z) _n(z) in terms of its q (possibly complex) roots ajj=1q\a_j\_j=1^q as ψn(z)=c∏j=1q(z−aj). _n(z)=c\, _j=1^q(z-a_j). To obtain the inner–outer factorization of ψn(z) _n(z), split the roots according to whether they lie inside the unit disk or not. Let =aj:|aj|<1A_I=\a_j: a_j<1\ and =aj:|aj|≥1A_O=\a_j: a_j≥ 1\. It follows that ψn(z) _n(z) =c∏j∈(z−aj)∏j∈(z−aj)=c∏j∈(z−aj)∏j∈(1−a¯jz)⏟n(z)∏j∈z−aj1−a¯jz⏟ℐn(z). =c\, _j _O(z-a_j)\, _j _I(z-a_j)= c\, _j _O(z-a_j)\, _j _I(1- a_jz)_O_n(z) _j _I z-a_j1- a_jz_I_n(z). The roots of n(z)O_n(z) are given by the set ∪a¯j−1:aj∈A_O∪\ a_j^-1:a_j _I\, all of which lie outside the unit disk; hence n(z)O_n(z) is outer. On the other hand, by construction, one can check that |ℐn(z)|=1 I_n(z)=1 for all |z|=1 z=1, so ℐn(z)I_n(z) is inner; alternatively, it is called a Blaschke product. Therefore, ψn(z) _n(z) admits the inner–outer factorization ψn(z)=n(z)ℐn(z) _n(z)=O_n(z)\,I_n(z) given above. It then follows from 2 that the root MSFE of ψn(z) _n(z) is σn=|n(0)|=|c|∏j∈|aj|. _n=|O_n(0)|=|c|\, _j _O a_j. Finally, by 3, seller n’s demand is invertible with respect to the demand shocks if and only if =∅A_I= , that is, if and only if |aj|≥1 a_j≥ 1 for all j=1,…,qj=1,…,q. ■ 4.2 Demand allocation in the z-domain We now apply the inner-outer representation to the platform’s demand-allocation problem in (9). Let ψ(z)=∑k=0∞ψkzkψ(z)= _k=0^∞ _kz^k denote the z-transform of the market demand process. Because (2) is the Wold representation of market demand DtD_t (see Appendix C for details), the corresponding transfer function ψ is outer, i.e., ψ∈ψ . In the z-transform domain, an admissible allocation policy π is specified by a collection ψn(z)=n(z)ψ(z)n∈[N]\ _n(z)=T_n(z)\,ψ(z)\_n∈[N] of transfer functions representing the seller-level allocated demand streams, subject to the admissibility constraint ∑n∈[N]ψn(z)=ψ(z). _n∈[N] _n(z)=ψ(z). Equivalently, writing the inner–outer factorization ψn(z)=n(z)ℐn(z) _n(z)=O_n(z)\,I_n(z) with ℐn∈I_n and n∈O_n , we may represent an admissible policy as π=(ℐn,n)n∈[N]π=\(I_n,O_n)\_n∈[N] satisfying ∑n∈[N]n(z)ℐn(z)=ψ(z) _n∈[N]O_n(z)\,I_n(z)=ψ(z). 2 makes explicit the link between outer factors and sellers’ forecastability. In particular, seller n’s one-step-ahead root MSFE is given by σn=|n(0)| _n=|O_n(0)|. We use this relation to express the notion of a neutral allocation policy as the algebraic condition |n(0)|=σ|O_n(0)|=σ for all n∈[N]n∈[N], where σ is a platform-chosen constant independent of n. This allows us to rewrite the platform’s optimization problem (9) directly in terms of inner and outer functions: V∗=supπ∈ΠN(ψ)ρμ+(Δf+Δh)μN|NFBP(σ)|+ΔhΓFBP(σ) V^*= _π∈ _ 0.45 N(ψ)ρ\,μ+( f+ h)\, μN\,|N 0.45 FBP(σ)|+ h\, 0.45 FBP(σ) (11) subject to ∑n∈[N]ψn(z)=ψ(z), _n∈[N] _n(z)=ψ(z), ψn(z)=n(z)ℐn(z),n∈,ℐn∈,∀n∈[N], _n(z)=O_n(z)\,I_n(z), _n ,\ I_n , ∀ n∈[N], |n(0)|=σ,∀n∈[N], |O_n(0)|=σ, ∀ n∈[N], where NFBP(σ)=n∈[N]:ΔFn≥NσμΔKnandΓFBP(σ)=σ∑n∈NFBP(σ)ζnFBP. N 0.45 FBP(σ)= \\,n∈[N]: F_n≥ N\,σμ\, K_n \ and 0.45 FBP(σ)=σ\, _n∈ N 0.45 FBP(σ)ζ 0.45 FBP_n. As previously noted, we can solve (11) by conducting an exhaustive search over the finitely many values of σ at which the piecewise-constant function |NFBP(σ)||N 0.45 FBP(σ)| jumps. This search, however, must be carried out over the range of values of σ that are feasible under a neutral allocation policy π∈ΠN(ψ)π∈ _ 0.45 N(ψ). We investigate this implementability question next by deriving a lower bound on the feasible set σπ:π∈ΠN(ψ)\σ^π:π∈ _ 0.45 N(ψ)\. 4.3 Uniform allocation as a lower bound on root MSFE In this section we derive a lower bound on the minimum possible value of σπσ^π under any neutral demand allocation policy. As we will see, the bound is attained by the simple uniform allocation in which each seller receives a fixed fraction of market demand each period. Intuitively, uniform splitting minimizes volatility by removing unnecessary dispersion across sellers: when total demand DtD_t is divided equally, each seller receives the same deterministic share 1/N1/N, so only aggregate uncertainty remains. Any deviation from this equal split reintroduces dispersion and increases the mean squared forecast error (MSFE). We first state a lower-bound result for neutral allocation policies. Proposition 1 (Lower bound) Let market demand DtD_t be given as in (2), and consider an admissible neutral allocation policy π=(ℐn,n)n∈[N]π=\(I_n,O_n)\_n∈[N]. Let σnπσ^π_n denote the root MSFE of seller n under π. Then ∑n∈[N]σnπ≥|ψ(0)|. _n∈[N]σ^π_n\;≥\; ψ(0). It follows that for any neutral demand allocation policy π, σnπ≥|ψ(0)|Nσ^π_n≥ ψ(0)N. For future reference, let σL:=|ψ0|N _ L:= _0N denote the lower bound on a seller’s root MSFE that the platform can implement using an admissible policy. The following corollary shows that this lower bound is attained by uniform splitting. Corollary 1 (Uniform allocation) Let π be the uniform demand allocation policy under which each seller n faces a demand process with z-transform ψn(z)=ψ(z)/N _n(z)=ψ(z)/N. Then σnπ=σL _n^π= _ L for all n∈[N]n∈[N]. The lower bound in 1 has an immediate implication for the platform’s ability to induce FBP adoption. Recall from Equation 7 that under a neutral allocation policy π, seller n adopts FBP if and only if ΔFn≥NσπμΔKn. F_n≥ N\,σ^πμ\, K_n. By 1, any neutral policy must satisfy σπ≥σLσ^π≥ _ L. As a result, the platform cannot necessarily induce every seller to join the FBP system. In particular, sellers with ΔFn<NσLμΔKn=|ψ(0)|μΔKn F_n< N\, _ Lμ\, K_n= ψ(0)μ\, K_n strictly prefer FBM under every implementable neutral policy π∈ΠN(ψ)π∈ _ 0.45 N(ψ). In words, the existence of the lower bound σL _ L implies that some sellers may never find it profitable to adopt the platform’s fulfillment service, regardless of how the platform allocates demand while maintaining neutrality. The lower-bound result above reveals that uniform allocation provides the best forecasting environment the platform can create under neutrality, since it minimizes each seller’s root MSFE and therefore serves as a natural benchmark. A complementary question is whether the platform can move in the opposite direction and deliberately make seller-level demand less forecastable. As noted above, doing so may be attractive because higher forecast uncertainty raises safety stocks and, through that channel, affects inventory held inside the platform’s logistics system. The next result shows, however, that increasing seller-level root MSFE relative to the uniform benchmark is not costless. In particular, any deviation from uniform allocation that preserves neutrality must also increase the unconditional variance of the allocated demand, thereby imposing a volatility tax on sellers. Proposition 2 Suppose market demand DtD_t is given as in (2). Let arunif=1N2‖ψ‖2V ar 0.45 unif= 1N^2 ψ^2 denote the variance of each seller’s demand under the uniform demand allocation policy. If an admissible policy π∈Π(ψ)π∈ (ψ) satisfies ar(Dnt)=arunifV ar(D_nt)=V ar 0.45 unif for all n∈[N]n∈[N], then π must be the uniform demand allocation policy, meaning ψn=1Nψ _n= 1Nψ for all n∈[N]n∈[N]. This proposition shows that the platform cannot raise sellers’ forecast uncertainty above the uniform-allocation benchmark while keeping their unconditional demand variance unchanged. Any non-uniform allocation therefore comes with strictly higher overall demand volatility for the sellers. 5 Optimal Neutral Demand Allocation Policies Building on the inner–outer factorization of demand allocation introduced in the previous section, we now solve (11) over the class ΠN(ψ) _ 0.45 N(ψ) of neutral demand-allocation policies, as defined in 2, and characterize the structure and implementability of optimal solutions. Our analysis shows that, absent participation constraints, the platform can induce any target root MSFE σπ≥σLσ^π≥ _ L using remarkably simple low-order moving-average perturbations of the uniform split: when the number of sellers N is even, an alternating MA(1)MA(1) filter suffices, whereas when N is odd, a minimal MA(2)MA(2) modification applied to two sellers restores admissibility while preserving neutrality. We derive these constructions through an inner–outer factorization of seller transfer functions, which clarifies how introducing a nontrivial inner component reallocates phase across sellers and raises each seller’s forecast error by changing the outer factor at the origin. A key implication is that achieving σπ>σLσ^π> _ L under neutrality requires non-invertibility for at least one seller, so that no seller can recover the innovation sequence ϵt\ _t\, and hence aggregate demand, from its own demand stream alone. Finally, we translate the resulting transfer-function policies into explicit time-domain allocation rules and complement the ex post benchmark with an order-level routing mechanism that implements the allocation dynamically as orders arrive, tracking the benchmark up to a small rounding error. Our construction of optimal solutions proceeds in three steps. First, for expositional convenience, we reparametrize the platform’s demand allocation to the class of transfer functions that express each seller’s allocation relative to the uniform baseline introduced in 1. Definition 3 (Transfer Function Representation) Using the uniform allocation ψ(z)/Nψ(z)/N as a baseline, we represent the z-transform for seller n as ψn(z)=ψ(z)Nn(z),with ∑n=1Nψn(z)=ψ(z). _n(z)= ψ(z)N\,T_n(z), \; _n=1^N _n(z)=ψ(z). (12) Because ψ(z)/Nψ(z)/N represents a uniform allocation, n(z)T_n(z) captures deviations from this baseline while preserving admissibility. Second, we decompose nT_n via its inner–outer factorization n(z)=n(z)ℐn(z)T_n(z)=O_n(z)\,I_n(z), where n∈O_n and ℐnI_n is a finite Blaschke product whose order is 11 when N is even, and at most 22 when N is odd. This decomposition isolates variance scaling (the outer factor) from phase reallocation (the inner factor). Third, for a given target value σ≥σLσ≥ _ L of a seller’s root MSFE, we choose the zeros of the Blaschke factors so that (i) neutrality holds and (i) each seller’s target root MSFE binds with equality. This pins down the constants n(0)O_n(0) and, using σn=|ψ(0)||n(0)|/N _n= ψ(0)\, O_n(0)/N, attains the desired target σ. Our derivation also implies that the optimization problem in (11) admits multiple solutions, allowing the platform to impose additional criteria beyond neutrality to break ties. We discuss this non-uniqueness in Section 5.2. Before turning to the formal details of this program, we first illustrate the approach with an example. Example 3 (IID Demand) Suppose there are two sellers, N=2N=2, and market demand is i.i.d., given by Dt=μ+ψ0ϵtD_t=μ+ _0\, _t, for some scalar ψ0>0 _0>0. In this case, ψ(z)=ψ0ψ(z)= _0 and σL=ψ0/2 _ L= _0/2. Consider the following class of admissible allocation policies based on MA(1)MA(1) processes: ψn(z)=ψ02n(z),where n(z)=1−αnz,n=1,2, _n(z)= _02\,T_n(z), where T_n(z)=1- _n\,z, n=1,2, (13) for scalars α1 _1 and α2 _2 such that α1=−α2 _1=- _2. Here, ψ0(z)/2 _0(z)/2 is the uniform allocation. This policy is admissible since ψ1(z)+ψ2(z)=ψ(z) _1(z)+ _2(z)=ψ(z). Seller n’s mean squared forecast error depends on whether its demand process is invertible (see Brockwell and Davis, 2006 for details). Invertibility is determined by the location of the root znz_n of the transfer function ψn(z) _n(z), which here equals zn:=1/αnz_n:=1/ _n. We distinguish two cases: 1. If |zn|>1 z_n>1, i.e., |αn|<1 _n<1, then Dnt\D_nt\ is invertible with ψn(z)∈ _n(z) and σn=|ψn(0)|=ψ0/2=σL _n= _n(0)= _0/2= _ L. 2. If |zn|≤1 z_n≤ 1, i.e., |αn|≥1 _n≥ 1, then Dnt\D_nt\ is noninvertible. In this case, ψn(z) _n(z) admits the inner–outer factorization ψn(z)=ψ02(1−αnz)=n(z)ℐn(z),withn(z)=ψ02(αn−z)andℐn(z)=1−αnzαn−z, _n(z)= _02\,(1- _nz)=O_n(z)\,I_n(z), with _n(z)= _02\,( _n-z) and _n(z)= 1- _nz _n-z, where n(z)∈O_n(z) and ℐn(z)∈I_n(z) is a Blaschke factor. Hence, 2 implies σn=|n(0)|=ψ02|αn|=σL|αn|. _n= O_n(0)= _02\, _n= _ L\, _n. Therefore, combining cases 1 and 2, under the MA(1)MA(1) allocation in (13), each seller’s root MSFE equals σn=σLmax1,|αn| _n= _ L\, \1, _n\. The platform can thus implement any target root MSFE σ≥σLσ≥ _ L by choosing α1 _1 and α2 _2 so that |αn|σL=σ _n\, _ L=σ. Finally, note that when σ>σLσ> _ L under the MA(1)MA(1) allocation in (13), each seller’s demand process DntD_nt is non-invertible in the demand shocks ϵt\ _t\. In the i.i.d. demand case, this implies that no single seller can infer aggregate market demand DtD_t solely from observing its own demand process DntD_nt. By contrast, under the uniform allocation in 1, each seller can recover DtD_t exactly from DntD_nt, provided the number of sellers N operating on the platform is common knowledge. ■ As we show in the following result, the method from the previous example extends to a general demand process when the number of sellers is even. The case of an odd number of sellers requires a slightly different approach, which we discuss next. Proposition 3 Consider a target root MSFE σ>σLσ> _ L and suppose the number of sellers N is even. Consider a policy π∈ΠN(ψ)π∈ _ 0.45 N(ψ) such that seller n’s demand allocation admits the z-transform representation ψn(z)=1Nψ(z)n(z) _n(z)= 1N\,ψ(z)\,T_n(z) with transfer function n(z)=1+(−1)nz,whereNσ|ψ(0)|=σL.T_n(z)=1+(-1)^n\,\,z, where N\,σ ψ(0)=σ _ L. Then π is neutral and satisfies σπ=σ^π=σ. It is worth noting that the allocation for seller n in 3 is obtained by taking the uniform split ψ(z)/Nψ(z)/N and multiplying it by the deliberately non-invertible transfer n(z)=1−αnzT_n(z)=1- _nz. This introduces an inner factor—specifically, a Blaschke product—into the representation of seller n’s demand, which in turn modifies the outer component and increases the root MSFE relative to the uniform split. The value of αn _n is carefully selected so as to match the target root MSFE σ. When N is odd, the previous construction, that is, multiplying the uniform allocation by the monomial factor (1−αnz) (1- _nz ), cannot satisfy admissibility (∑nαn=0 _n _n=0) and neutrality (the quantities |αn| _n are equal to the same positive constant for every n). Consequently, for odd N the optimal design requires a different (e.g., higher-order or mixed) inner–outer modification rather than the simple monomial factor. Proposition 4 Consider a target root MSFE σ>σLσ> _ L and suppose the number of sellers N≥3N≥ 3 is odd. Let π∈ΠN(ψ)π∈ _ 0.45 N(ψ) be a demand allocation policy such that the z-transform for seller n’s demand admits the representation ψn(z)=1Nψ(z)n(z) _n(z)= 1N\,ψ(z)\,T_n(z), with transfer function 1(z) _1(z) =1+z+z2, =1+z+\,z^2, 2(z) _2(z) =1−z2, =1-z^2, n(z) _n(z) =1+(−1)nz,for n≥3, =1+(-1)^n\,\,z, for n≥ 3, where Nσ|ψ(0)| N\,σ ψ(0). Then π is neutral and implements σπ=σ^π=σ. A distinctive feature of the optimal allocation policies identified in Propositions 3 and 4 is that the transfer functions n(z)T_n(z) used to perturb the uniform allocation possess a nontrivial inner component. As a result, the induced demand process Dnt\D_nt\ faced by any seller n is non-invertible with respect to the demand-shock sequence ϵt\ _t\. Indeed, as the following proposition shows, non-invertibility is typically necessary for a neutral allocation policy to achieve σπ>σLσ^π> _ L. Proposition 5 Suppose the platform uses a neutral policy π∈ΠNπ∈ _ 0.45 N whose associated root MSFE σπσ^π is strictly larger than the lower bound σL _ L. Then at least one of the demand processes Dnt\D_nt\ must be non-invertible with respect to ϵt\ _t\. 5 establishes that, under the neutrality requirement, the platform can increase the system’s safety stock beyond the normal level achieved under uniform allocation only by adopting allocation policies that render the demand process non-invertible for some sellers. 5.1 Implementable order-level routing The transfer-function policies characterized in Propositions 3 or 4 describe seller-level demand processes in the z-transform domain. To connect these prescriptions to marketplace operations, this subsection (i) translates the optimal policies into explicit time-domain allocation rules, and (i) provides a simple order-by-order routing mechanism that implements the resulting benchmark allocations as orders arrive. The benchmark should be interpreted as an idealized target that is computed ex post from aggregate demand, while the routing mechanism is online and uses only information available at the time each order arrives. Corollary 2 Consider the demand allocation with z-transform representation ψn(z)=1Nψ(z)n(z) _n(z)= 1N\,ψ(z)\,T_n(z), where the transfer function n(z)T_n(z) is given in Propositions 3 or 4 depending on whether N is even or odd. Then, the time-domain representation of seller n’s demand DntD_nt is given by (a) For N even: Dnt=DtN+(−1)nσNσL(Dt−1−μ),n∈[N].D_nt= D_tN+(-1)^n\, σN\, _ L\,(D_t-1-μ), n∈[N]. (b) For N odd: D1t D_1t =DtN+σNσL[(Dt−1−μ)+(Dt−2−μ)], = D_tN+ σN\, _ L [(D_t-1-μ)+(D_t-2-μ) ], D2t D_2t =DtN−σNσL(Dt−2−μ), = D_tN- σN\, _ L\,(D_t-2-μ), Dnt D_nt =DtN+(−1)nσNσL(Dt−1−μ),n≥3. = D_tN+(-1)^n\, σN\, _ L\,(D_t-1-μ), n≥ 3. The time-domain representations in Corollary 2 can be interpreted as an ex post allocation rule: at the end of period t the platform observes the realized aggregate demand DtD_t and then assigns seller-level demands Dntn∈[N]\D_nt\_n∈[N] according to a simple linear function of (Dt,Dt−1,Dt−2)(D_t,D_t-1,D_t-2) (and μ). Under this interpretation, implementation is immediate: the platform maintains a running record of past aggregate demand realizations, computes the adjustments prescribed by the corollary, and posts the corresponding seller-level quantities for period t. In particular, the rule is lightweight—it requires only μ, σ/σLσ/ _ L, and one or two lags of DtD_t—and it is transparent, since each seller’s allocation is a uniform share Dt/ND_t/N plus a signed correction that depends on recent deviations of market demand from its mean. In most operational settings, however, demand is not allocated in a single end-of-period “batch”. Rather, orders arrive sequentially and must be routed in real time to a specific seller (or fulfillment mode) subject to service levels, capacity, and latency constraints. When demand allocations are executed unit-by-unit, the platform typically cannot wait until the end of the period to learn DtD_t before deciding Dnt\D_nt\, and even if some form of batching is possible, deferring assignment would delay fulfillment and degrade customer experience. Moreover, practical marketplaces often impose additional frictions—partial cancellations, heterogeneous shipping promises, and seller availability—that make a pure after-the-fact redistribution of realized demand infeasible. For these reasons, Corollary 2 is best viewed as a theoretical benchmark describing the target seller-level demand processes implied by the transfer-function design. We therefore complement it with a dynamic allocation mechanism that routes each arriving order immediately, using only information available up to that moment, while ensuring that the induced seller-level demand streams track (as closely as possible) the benchmark allocations prescribed by the corollary. Corollary 2 implies that the benchmark allocation can be written as Dnt=DtN+bn,t,D_nt= D_tN+b_n,t, (14) where the offset bn,tb_n,t depends only on lagged aggregate demand (and parameters), hence is known at the beginning of period t even though DtD_t is not yet realized. Concretely, bn,t=σNσL(−1)n(Dt−1−μ),if N is even or N is odd and n≥3,(Dt−1−μ)+(Dt−2−μ),if N is odd and n=1,−(Dt−2−μ),if N is odd and n=2.b_n,t= σN\, _ L\, cases(-1)^n\,(D_t-1-μ),&if $N$ is even or $N$ is odd and $n≥ 3$,\\[5.16663pt] (D_t-1-μ)+(D_t-2-μ),&if $N$ is odd and $n=1$,\\[5.16663pt] -(D_t-2-μ),&if $N$ is odd and $n=2$. cases (15) Within period t, index arriving orders by k=1,2,…k=1,2,…. Define An,t(k)A_n,t(k) as the number of orders assigned to seller n among the first k−1k-1 arrivals in period t, with An,t(1)=0A_n,t(1)=0. The platform routes orders using the offset-tracking policy in Algorithm 1. Algorithm 1 Offset-Tracking 1:Number of sellers N, parameters (μ,σ,σL)(μ,σ, _ L), lagged aggregate demand (Dt−1,Dt−2)(D_t-1,D_t-2). 2:Compute offsets bn,tn∈[N]\b_n,t\_n∈[N] from (15). 3:Initialize counters An←0A_n← 0 for all n∈[N]n∈[N]. 4:for k=1,2,…k=1,2,… (as orders arrive during period t) do 5: Choose nk∈argminn∈[N]An−bn,tn_k∈ _n∈[N] \A_n-b_n,t \ (break ties uniformly at random). 6: Assign order k to seller nkn_k and update Ank←Ank+1A_n_k← A_n_k+1. 7:end for 8:return Implemented seller demands D^nt←An D_nt← A_n for all n∈[N]n∈[N]. Because the rule always assigns the next order to the seller with the smallest “offset-adjusted” count An−bn,tA_n-b_n,t, it equalizes these adjusted counts as tightly as possible as orders arrive. Consequently, at the end of period t the implemented allocation matches the benchmark up to indivisibility: that is, |D^nt−Dnt|≤1 | D_nt-D_nt |≤ 1 for all n∈[N]n∈[N]. Proposition 6 (Offset-tracking matches the benchmark up to indivisibility) Fix a period t and suppose that Dt∈ℤ+D_t _+ orders arrive during the period. Let bn,tn∈[N]\b_n,t\_n∈[N] be the offsets in (15) and define the benchmark target shares xn,t:=DtN+bn,t,n∈[N].x_n,t\;:=\; D_tN+b_n,t, n∈[N]. Assume that xn,t≥0x_n,t≥ 0 for all n.†This feasibility condition is automatically satisfied whenever the benchmark allocation in (14) is interpreted as a (nonnegative) order allocation for the realized DtD_t. Let D^nt D_nt denote the implemented seller demands returned by Algorithm 1. Then for every n∈[N]n∈[N], |D^nt−xn,t|=|D^nt−(DtN+bn,t)|≤1. | D_nt-x_n,t |= | D_nt- ( D_tN+b_n,t ) |≤ 1. Remark 1 (Additional operational constraints) In practice, order assignments may be subject to additional operational constraints (e.g., inventory availability, delivery promises, or capacity limits). The same policy can be modified by restricting the minimization in Algorithm 1 to a feasible set t(k)⊆[N]S_t(k) [N] of sellers available to serve the kthk th order arriving during period t. 5.2 Non-Uniqueness Propositions 3 and 4 are existence results: for any admissible target σ, typically infinitely many distinct allocation rules generate the same seller-level root MSFE. For instance, when N is even, the demand allocation policy ψn(z)=ψ(z)Nn(z) _n(z)= ψ(z)N\,T_n(z) with n(z)=1+(−1)nα¯zk,α¯=σL,T_n(z)=1+(-1)^n\, α\,z^k, α= σ _ L, (16) implements σnπ=σ _n^π=σ for every n∈[N]n∈[N] and every integer k∈ℕk , so varying the lag length k yields a distinct policy with identical forecasting implications.‡Indeed, for each n, the transfer function n(z)T_n(z) admits an inner–outer factorization of the form n(z)=n(z)ℐn(z)T_n(z)=O_n(z)\,I_n(z), with n(z)=(−1)nα¯+zkO_n(z)=(-1)^n α+z^k and ℐn(z)=(1+(−1)nα¯zk)/((−1)nα¯+zk)I_n(z)=(1+(-1)^n α\,z^k)/((-1)^n α+z^k). Since |n(0)|=α¯=σ/σL|O_n(0)|= α=σ/ _ L, it follows from 2 that the induced root MSFE is exactly σ. Our preference for the constructions in Propositions 3 and 4 is driven by parsimony: they deliver minimal-memory policies with transparent time-domain interpretations. By 2, the even-N policy depends only on current and one-period-lagged demand, whereas (16) with k>1k>1 requires demand k periods in the past. Thus, although both policies induce exactly the same seller-level root MSFE, the former is operationally simpler, requires less historical information, and is arguably a more natural benchmark for implementation. 6 Illustrative Example Consider ten sellers serving demand concentrated in a single urban market, but located at different distances from that market, as depicted in the left panel of Figure 5. Geography creates a fundamental cost tension. Sellers closer to the city can ship cheaply because they are nearer to customers, but they pay more for urban warehouse space. Sellers farther away face the opposite problem: storage is cheap, but every shipment travels far and costs more (see Fried and Goodchild, 2023 for a detailed discussion).§Last-mile facilities command substantially higher rents than more remote warehouses; see, for example, Dablanc et al. (2014), Kirk (2022). Likewise, parcel shipping costs rise with delivery distance and shipping zone; see United Parcel Service (2026). The right panel of Figure 5 reports the cost parameters (hn,bn,fn)(h_n,b_n,f_n), which reflect this geographical trade-off, along with the implied safety-stock coefficients KnFBMK_n 0.45 FBM and KnFBPK_n 0.45 FBP for each seller. This cost tension shapes each seller’s choice between FBP and FBM. Distant sellers, facing high fulfillment costs, gain the most from outsourcing logistics to the platform. Urban sellers, who already ship cheaply on their own, have less benefit. The question is then how the platform’s demand-allocation policy interacts with this geographic heterogeneity. Urban Demand Center(h10,f10)(h_10,f_10)(h9,f9)(h_9,f_9)(h6,f6)(h_6,f_6)(h7,f7)(h_7,f_7)(h4,f4)(h_4,f_4)(h8,f8)(h_8,f_8)(h2,f2)(h_2,f_2)(h3,f3)(h_3,f_3)(h5,f5)(h_5,f_5)(h1,f1)(h_1,f_1)Urban Demand Center n hnh_n bnb_n fnf_n KnFBMK^FBM_n KnFBPK^FBP_n 1 0.60 12.00 24.50 1.250 3.703 2 0.80 9.00 24.40 1.479 3.382 3 0.90 13.00 23.10 1.757 3.791 4 1.10 8.00 22.30 1.830 3.250 5 1.20 11.00 21.40 2.115 3.606 6 1.50 10.00 20.00 2.438 3.500 7 1.70 9.00 18.80 2.591 3.382 8 2.00 12.00 12.50 3.159 3.703 9 2.00 13.00 11.90 3.229 3.791 10 2.10 11.00 10.48 3.191 3.606 All costs as % of r=$100r= 100. Figure 5: Geographic trade-off between holding and fulfillment costs (left) and seller cost parameters with safety-stock coefficients (right). Each seller n is located at holding cost hnh_n and fulfillment cost fnf_n; sellers nearer the city face higher hnh_n but lower fnf_n. The shaded region highlights the urban area containing sellers 8, 9, and 10, who switch from FBP to FBM in the optimal policy. To make this concrete, we normalize the gross per-unit margin to r=$100r= 100 and express all operating costs as percentages of r (and hence, numerically, in dollars). The platform charges an intermediation fee ρ=15ρ=15 per unit, and its effective FBP seller-side costs are (F,H)=(10,2.5).(F,H)=(10,2.5). On the platform side, the net FBP fulfillment payoff and storage payoff are Δf=2,Δh=2. f=2, h=2. These values imply that F≤fnF≤ f_n and H≥hnH≥ h_n for all sellers: the platform fulfills more cheaply than any seller can on its own, but its storage costs exceed every seller’s holding cost.¶For example, Amazon’s FBA storage and fulfillment fees provide a useful benchmark for the values of H and F used here; see Amazon Seller Central (2026); Sellerapp (2026). For simplicity, we assume i.i.d. market demand, Dt=μ+5ϵt,μ=15,ar(ϵt)=1,D_t=μ+5\, _t, μ=15, ar( _t)=1, so that the market-demand z-transform is ψ(z)=5ψ(z)=5. Hence, by 1, σL=|ψ(0)|N=5N. _ L= |ψ(0)|N= 5N. With N=10N=10, this yields σL=0.5. _ L=0.5. Because N is even, 3 implies that the platform can implement any σ≥σLσ≥ _ L using an MA(1) modification of the uniform split. The platform’s design space is therefore the participation-truncated interval [σL,σU][ _ L, _ U], where σU _ U is defined in (10). In this example, σU=minn∈[N]max(r−ρ−fn)(μ/N)KnFBM,(r−ρ−F)(μ/N)KnFBP≈33. _ U= _n∈[N] \! \ (r-ρ-f_n)(μ/N)K_n 0.45 FBM, (r-ρ-F)(μ/N)K_n 0.45 FBP \≈ 33. The platform thus faces a wide range of implementable forecast-error levels. To see how this lever interacts with the geographic trade-off, Figure 6 maps each seller into the (ΔK,ΔF)( K, F) plane. The shaded cone corresponds to feasible values of σ∈[σL,σU]σ∈[ _ L, _ U], bounded by the two lines ΔF=NσLμΔKandΔF=NσUμΔK. F= N _ Lμ K F= N _ Uμ K. Sellers inside this region may choose either FBP or FBM, depending on the root MSFE induced by the platform’s allocation rule. 00.50.5111.51.5222.52.50551010151512345678910ΔF=NσLμΔK F= N _ Lμ\, KΔF=NσUμΔK F= N _ Uμ\, KΔF=Nσ⋆μΔK F= Nσ μ\, KΔK KΔF FSellers in (ΔK,ΔF)( K, F)-space Figure 6: Sellers in the (ΔK,ΔF)( K, F) plane. The shaded cone corresponds to feasible values σ∈[σL,σU]σ∈[ _ L, _ U], defined by the boundaries ΔF=(NσL/μ)ΔK F=(N _ L/μ) K and ΔF=(NσU/μ)ΔK F=(N _ U/μ) K. For each σ∈[σL,σU]σ∈[ _ L, _ U], the platform payoff under neutrality is V(σ)=ρμ+(Δf+Δh)μN|NFBP(σ)|+Δhσ∑n∈NFBP(σ)ζnFBP,whereV(σ)=ρμ+( f+ h) μN\, |N 0.45 FBP(σ) |+ h\,σ _n∈ N 0.45 FBP(σ)ζ 0.45 FBP_n,\;where NFBP(σ)=n∈[N]:ΔFn≥(Nσ/μ)ΔKn.N 0.45 FBP(σ)= \n∈[N]: F_n≥(Nσ/μ) K_n \. As σ rises, each seller n reaches a breakpoint σn=μΔFnNΔKn _n= μ\, F_nN\, K_n at which it switches from FBP to FBM, causing a jump in V(σ)V(σ). The sellers who switch first—at low values of σn _n—are the urban sellers, for whom the fulfillment advantage ΔFn F_n is small relative to the safety-stock disadvantage ΔKn K_n. Remote sellers, with large ΔFn F_n, remain in FBP even at high MSFE. Figure 7 plots the number of FBP adopters and the platform payoff as functions of σ. The shaded band marks the feasible interval σ∈[σL,σU]σ∈[ _ L, _ U]. By the participation constraint, we set V(σ)=0V(σ)=0 for σ>σUσ> _ U. σL _ Lσ⋆σ σU _ U40400224466881010σ\,σ|NFBP(σ)||N^FBP(σ)|Number of Sellers using FBPσL _ Lσ⋆σ σU _ U40400100100200200300300σ\, (σ)V(σ)Platform’s Payoff Figure 7: Number of FBP adopters (left) and platform payoff V(σ)V(σ) (right) as functions of σ. The shaded area corresponds to the feasible region σ∈[σL,σU]σ∈[ _ L, _ U], with σL=0.5 _ L=0.5 and σU=33 _ U=33. The platform-optimal choice is σ⋆≈8.87σ ≈ 8.87, far above the minimum σL=0.5 _ L=0.5 under a uniform allocation. At this level of forecast error, the most urban sellers, 88, 99, and 1010, switch to FBM, while the remaining seven, more remote sellers stay in FBP. This partition is captured by the shaded region in the left panel of Figure 5. Table 1 contrasts the platform’s optimal strategy with the uniform benchmark. Demand Allocation Uniform Optimal Root MSFE: σ σL=0.50 _ L=0.50 σ⋆=8.87σ =8.87 FBP adopters: NFBP(σ)N 0.45 FBP(σ) 1,2,3,4,5,6,7,8,9,10\1,2,3,4,5,6,7,8,9,10\ 1,2,3,4,5,6,7\1,2,3,4,5,6,7\ Number of FBP adopters: |NFBP(σ)| |N 0.45 FBP(σ) | 1010 77 Platform payoff: V(σ)V(σ) 293.78293.78 372.45372.45 Cumulative FBP safety stock: ΓFBP(σ) 0.45 FBP(σ) 4.394.39 52.7352.73 Cumulative FBM safety stock: ΓFBM(σ) 0.45 FBM(σ) 0 28.1228.12 Cumulative seller utility: ∑nUn(σ) _nU_n(σ) 1107.141107.14 814.47814.47 Table 1: Outcomes under the uniform allocation σ=σLσ= _ L and the platform-optimal choice σ=σ⋆σ=σ . Under uniform allocation, all ten sellers adopt FBP and carry minimal safety stock but generates a platform payoff of only V(σL)=293.78V( _ L)=293.78. Under the optimal policy, the platform raises σ nearly eighteen-fold. The urban sellers leave FBP because their fulfillment savings were modest—their proximity to the demand center already kept shipping costs low, so the extra inventory burden is too costly. The remote sellers stay, because the platform’s fulfillment advantage is large enough to offset the higher safety stock cost. The platform’s payoff rises by 26.78%26.78\% to V(σ⋆)=372.45V(σ )=372.45. This improvement is accompanied by a sharp increase in aggregate safety stock: ΓFBP(σL)+ΓFBM(σL)=4.39versusΓFBP(σ⋆)+ΓFBM(σ⋆)=80.85. 0.45 FBP( _ L)+ 0.45 FBM( _ L)=4.39 0.45 FBP(σ )+ 0.45 FBM(σ )=80.85. Total buffer inventory rises by a factor of approximately 1818, while cumulative seller utility falls from 1107.141107.14 to 814.47814.47, a decline of 26.44%26.44\%. The geographical heterogeneity is what makes this work: the platform can afford to lose the urban sellers because the remote sellers, who benefit most from FBP, are also the ones most willing to carry more inventory. In short, the platform exploits the holding–fulfillment trade-off created by geography to simultaneously raise service reliability and extract more surplus from the sellers who rely the most on the platform’s logistics network. 7 Discussion This section discusses one extension of the baseline model that helps clarify its scope and robustness. In particular, we examine how our analysis changes when sellers rely on simple forecasting heuristics rather than optimal one-step-ahead predictors. A second extension, which considers heterogeneous replenishment lead times across sellers and fulfillment modes, is relegated to Appendix F. Together, these extensions illustrate how the platform’s allocation policy continues to shape FBP adoption and inventory levels once we move beyond the benchmark assumptions. 7.1 Suboptimal forecasts Our baseline assumes sellers form the optimal one-step-ahead forecast, so the uncertainty in the base-stock rule (3) is the minimal MSFE, pinned down by the outer factor of the seller-level demand filter (2). In practice, forecasting is often done with low-touch heuristics—moving averages, exponential smoothing, or Holt–Winters—rather than the optimal predictor (Fildes et al., 2009). Suboptimal forecasts typically raise one-step-ahead MSFE and hence require additional safety stock, which could benefit the platform by increasing inventory intensity. But higher perceived uncertainty can also deter FBP adoption by worsening sellers’ adoption incentives. Thus, suboptimal forecasting can increase safety stock conditional on adoption while reducing the set of adopters, making the net ex ante effect on the platform’s payoffs ambiguous. Suppose seller n uses some ad-hoc point-forecast method that estimates the demand Dn,t+1D_n,t+1 in period t+1t+1 by means of a stable linear filter of the seller’s own demand history, m^nt=∑k=0∞ψ^nkDn,t−k≡ψ^n(ℬ)Dnt, m_nt= _k=0^∞\, ψ_nk\,D_n,t-k≡ ψ_n(B)\,D_nt, where ℬB denotes the backshift operator and ψ^n(z) ψ_n(z) satisfies ψ^n(1)=1 ψ_n(1)=1 so that [m^nt]=[Dnt]=μnE[ m_nt]=E[D_nt]= _n. For example, if seller n forecasts using simple exponential smoothing (SES) with a smoothing parameter λ∈(0,1]λ∈(0,1], as in ChenRyanSimchiLevi2000ESBullwhip (see also Gardner1990ForecastPerfInventoryControl; SnyderKoehlerHyndmanOrd2004MeansVariancesLTD; EppenMartin1988SafetyStockStochasticLeadTime), ψ^nSES(z)=λ∑k=0∞(1−λ)kzk=λ1−(1−λ)z. ψ 0.45 SES_n(z)=λ\, _k=0^∞(1-λ)^k\,z^k=λ 1-(1-λ)\,z. In the steady state, seller n’s forecast error can be written as en,t+1:=Dn,t+1−m^nt=Dn,t+1−ψ^n(ℬ)Dnt=(1−ψ^n(ℬ)ℬ)Dn,t+1.e_n,t+1:=D_n,t+1- m_nt=D_n,t+1- ψ_n(B)\,D_nt= (1- ψ_n(B)\,B )\,D_n,t+1. We measure perceived uncertainty by the stationary root MSFE, σ~nπ(λ):=([en,t+12])1/2, σ_n^π(λ):= (E[e_n,t+1^2] )^1/2, which replaces σnπ _n^π in the seller’s base-stock calculation when forecasting is performed via the linear-filter ψ^n(z) ψ_n(z). As in (12), write ψn(z)=1Nψ(z)n(z) _n(z)= 1Nψ(z)T_n(z) for a transfer function n(z)T_n(z) with an inner–outer factorization n(z)=n(z)ℐn(z)T_n(z)=O_n(z)I_n(z). It follows that (σ~nπ(λ))2=12πN∫−π|1−ψ^n(e−iθ)e−iθ|2|ψ(e−iθ)|2|n(e−iθ)|2dθ≥1N|ψ(0)|2|n(0)|2=(σnπ)2,( σ_n^π(λ))^2= 12π N _-π^π |1- ψ_n(e^-iθ)\,e^-iθ |^2 |ψ(e^-iθ) |^2\, |O_n(e^-iθ) |^2\, dθ\;≥\; 1N ψ(0)^2 O_n(0)^2=( _n^π)^2, (17) where the inequality follows from Parseval/Cauchy–Schwarz in ℍ2H^2∥For f(z)=∑k≥0fkzk∈ℍ2f(z)= _k≥ 0f_kz^k ^2, 12π∫−π|f(e−iθ)|2dθ=∑k=0∞|fk|2≥|f0|2=|f(0)|2. 12π _-π^π f(e^-iθ)^2\, dθ= _k=0^∞ f_k^2\;≥\; f_0^2= f(0)^2. . Equation (17) formalizes that (for a fixed allocation policy) suboptimal forecasts weakly inflate a seller’s one-step-ahead MSFE, and therefore inflate the safety stock required to maintain a given service level. When sellers forecast optimally, their demand uncertainty is driven solely by the baseline, unpredictable “noise” in their demand stream (|ψ(0)||n(0)| ψ(0)\, O_n(0)). They can perfectly filter out all other predictable patterns. In contrast, when sellers rely on simpler, ad-hoc forecasting rules, they cannot perfectly filter the signal. As a result, their perceived uncertainty σ~nπ(λ) σ_n^π(λ) is inflated by the entire pattern of the demand stream they receive. This means that if sellers use simple heuristics, the platform cannot simply tune a single, overall level of volatility; it must also carefully consider the specific shape of the fluctuations it routes to sellers (for example, whether the allocated demand bounces up and down rapidly day-by-day, or drifts in slow, extended waves). To illustrate this point, let us return to the i.i.d. demand example in Section 6 and assume seller n uses SES to forecast demand with smoothing parameter λn∈[0,1) _n∈[0,1). Suppose the platform chooses its allocation as if sellers were using optimal forecasts, implementing the MA(1) perturbation ψn(z)=|ψ(0)|N(1+(−1)nα⋆z),α⋆=σ⋆σL. _n(z)= ψ(0)N (1+(-1)^nα z ), α = σ _ L. Then the induced SES MSFE admits the closed form (Appendix E for a derivation), (σ~nπ(λn))2=|ψ(0)|2N2[1+|(−1)nα⋆−λn|2+λn2−λn|1−λn+(−1)nα⋆|2].( σ_n^π( _n))^2= ψ(0)^2N^2 [1+ (-1)^nα - _n^2+ _n2- _n 1- _n+(-1)^nα ^2 ]. (18) Furthermore, minimizing over λn∈[0,1] _n∈[0,1] yields λn⋆=0 _n =0. Consequently, (σ~nπ(λn⋆))2=|ψ(0)|2N2+(σ⋆)2=(σL)2+(σ⋆)2,( σ_n^π( _n ))^2= ψ(0)^2N^2+(σ )^2=( _ L)^2+(σ )^2, (19) so restricting sellers to SES inflates the MSFE relative to the optimal-forecast benchmark (σ⋆)2(σ )^2 by the additive term (σL)2( _ L)^2. Interestingly, the corner solution λn⋆=0 _n =0 corresponds to an infinitely sluggish SES rule: the update m^nt=λDnt+(1−λ)m^n,t−1 m_nt=λ D_nt+(1-λ) m_n,t-1 stops reacting to new observations, so the forecast m^nt m_nt becomes essentially constant. In effect, the seller behaves as if demand were unpredictable from its own history (as if i.i.d. around a fixed mean), so the one-step-ahead error is realized demand relative to that baseline. Let us apply these results to the numerical instance in Section 6 with N=10N=10 and σL=0.5 _ L=0.5. If the platform keeps the benchmark design σ⋆≈8.87σ ≈ 8.87, then, according to (19), sellers perceive σ~≈8.88 σ≈ 8.88. As a result, the set of FBP adopters shrinks from 1,2,3,4,5,6,7\1,2,3,4,5,6,7\ to 2,3,4,5,6,7\2,3,4,5,6,7\ (seller 11 drops out). Total FBP safety stock falls from ΓFBP(σ⋆)≈52.73 0.45 FBP(σ )≈ 52.73 to ΓSESFBP(σ⋆)≈44.35 0.45 FBP_ 0.45 SES(σ )≈ 44.35 (a 15.9%15.9\% reduction), and the platform payoff drops from V(σ⋆)≈372.45V(σ )≈ 372.45 to VSES(σ⋆)≈349.70V_ 0.45 SES(σ )≈ 349.70 (a 6.1%6.1\% reduction). To summarize, when sellers rely on low-touch forecasting heuristics, allocation policy affects not only realized demand risk but also perceived risk, and the latter feeds directly into both safety-stock levels and platform adoption. The key implication is that “turning up volatility” to stimulate inventory intensity can backfire by pushing marginal sellers out of platform fulfillment. Platforms should therefore (i) anticipate the forecasting rules that sellers actually use when optimizing allocation, and (i) design policies that are robust to forecast suboptimality—either by moderating induced volatility near adoption breakpoints or by reducing the perception gap via forecasting support, defaults, or information that makes demand more predictable from the sellers’ perspective. 8 Conclusion Digital marketplaces increasingly act as market makers: beyond matching, they algorithmically route order flow among competing sellers and, in many categories, provide managed fulfillment and inventory-carrying services. This paper studies how a platform can use intertemporal demand allocation to influence sellers’ inventory decisions when sellers replenish via base-stock policies and choose between fulfill-by-merchant (FBM) and fulfill-by-platform (FBP). Motivated by accountability and fairness concerns, we focus on neutral allocation mechanisms that treat sellers symmetrically in expectation by equalizing both long-run mean demand shares and one-step-ahead forecast uncertainty (root MSFE). Our main results show that neutrality still leaves substantial scope for operational design. Uniform splitting attains a sharp lower bound on sellers’ root MSFE, and any higher level can be implemented via simple low-order moving-average perturbations of the uniform split—MA(1) when N is even, MA(2) when N is odd—without changing mean demand shares. This characterization yields a tractable platform problem: under neutrality, the platform effectively chooses a scalar uncertainty level that determines (i) which sellers adopt FBP (the extensive margin) and (i) the safety stock carried by adopters (the intensive margin). A distinctive structural implication is that any strict improvement beyond the uniform benchmark requires non-invertibility from the sellers’ perspective, highlighting the platform’s informational advantage. Managerially, the model clarifies a fundamental extensive–intensive trade-off for platforms that monetize both fulfillment activity and on-platform storage. Increasing forecast uncertainty raises safety stocks (and therefore on-platform inventory level) among remaining FBP adopters, but it can also rotate marginal sellers toward FBM by making FBP more costly. Because adoption changes only at seller-specific thresholds while induced safety stock scales linearly between thresholds, the platform’s payoff is piecewise linear in the induced root MSFE, making optimal neutral policies transparent and implementable. We also provide a demand-allocation interpretation of our transfer-function allocations, and we show that when sellers forecast using common heuristics (e.g., simple exponential smoothing), the platform’s design lever shifts from controlling a scalar MSFE to shaping the frequency content of allocated demand. Several directions for future research would sharpen these conclusions and address natural limitations of our baseline model. First, neutrality can be defined in richer ways. Beyond equal mean shares and equal one-step-ahead MSFE, one could require neutrality over longer horizons (multi-step forecast errors), preserve the entire second-order structure (e.g., identical autocovariances or spectral densities), or impose exposure/share floors motivated by fairness regulation; characterizing the implementable set and the platform’s optimal design under such alternative constraints is an important next step. Second, we restrict attention to stationary demand and stationary allocation rules. Extending the analysis to nonstationary demand (seasonality, trends, regime shifts) and time-varying allocations would connect the mechanism to realistic planning environments. Third, incorporating lost sales and fulfillment-capacity congestion would link allocation design more directly to service levels and operational bottlenecks and would endogenize additional participation and feasibility constraints. 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Note: Company news release Cited by: §1, footnote †, footnote ‡. Walmart Marketplace (2024) WFS strategies for growth every walmart seller should know. Note: Walmart Marketplace Cited by: footnote †, footnote ‡. Zalando SE (2021) Building a platform that is relevant for all our partners. Note: https://corporate.zalando.com/en/investor-relations/capital-markets-day-2021 Cited by: footnote †. T. Zou and B. Zhou (2025) Self-preferencing and search neutrality in online retail platforms. Management Science 71 (5), p. 4087–4107. Cited by: §2.2. Appendix Appendix A Proofs Proof of 1: Let ψ(z)ψ(z) and ψn(z) _n(z) denote the z-transforms of the market demand process Dt\D_t\ and seller n’s demand process Dnt\D_nt\, respectively. Because the policy π is admissible, we have ψ(z)=∑n∈[N]ψn(z),z∈.ψ(z)= _n∈[N] _n(z), z . For each n∈[N]n∈[N], let ψn(z)=n(z)ℐn(z) _n(z)=O_n(z)\,I_n(z) denote the Nevanlinna inner–outer factorization of ψn(z) _n(z). It follows that σnπ=|n(0)|σ^π_n= O_n(0), and hence ∑n∈[N]σnπ=∑n∈[N]|n(0)|. _n∈[N]σ^π_n\;=\; _n∈[N] O_n(0). Also, from the admissibility of π, ψ(0)=∑n∈[N]ψn(0)=∑n∈[N]n(0)ℐn(0).ψ(0)= _n∈[N] _n(0)= _n∈[N]O_n(0)\,I_n(0). As a result, |ψ(0)|=|∑n∈[N]n(0)ℐn(0)|≤∑n∈[N]|n(0)||ℐn(0)|≤∑n∈[N]|n(0)|=∑n∈[N]σnπ, ψ(0)= | _n∈[N]O_n(0)\,I_n(0) |≤ _n∈[N] O_n(0)\, I_n(0)≤ _n∈[N] O_n(0)= _n∈[N]σ^π_n, where the second inequality follows from the fact that each ℐn(z)I_n(z) is inner. Indeed, since ℐnI_n is analytic on the unit disc =z∈ℂ:|z|<1D=\z |z|<1\, the maximum modulus principle implies |ℐn(0)|≤maxz∈|ℐn(z)|=maxz∈|ℐn(z)|=1, I_n(0)≤ _z I_n(z)= _z I_n(z)=1, and the last equality holds because ℐnI_n is unimodular on the unit circle =z∈ℂ:|z|=1T=\z |z|=1\. We therefore have that ∑n∈[N]σnπ≥|ψ(0)| _n∈[N]σ^π_n\;≥\; ψ(0). Finally, if π is a neutral demand allocation policy then σnπ=σ^π_n=σ, independent of n, and the previous inequality implies σnπ≥|ψ(0)|Nσ^π_n≥ ψ(0)N. ∎ Proof of 1: Since the market demand in (2) is invertible by construction, it follows that ψ(z)ψ(z) is outer. As a result, under the uniform allocation policy ψn(z)=ψ(z)/N _n(z)=ψ(z)/N, seller n’s allocated demand filter is also outer. Hence, by 2, σnπ=|ψn(0)|=|ψ(0)|N=σL. _n^π= _n(0)= ψ(0)N= _ L. ∎ Proof of 2: In this context, with a slight abuse of notation, we let ψ∈ℓ2ψ∈ ^2 and ψn∈ℓ2 _n∈ ^2 denote the coefficient sequences of the MA(∞)MA(∞) representations of the aggregate market demand and seller n’s allocated demand, respectively. By definition, the unconditional variance of a weakly stationary MA(∞)MA(∞) process driven by standard white noise (σϵ=1 _ε=1) is the squared ℓ2 ^2-norm of this coefficient sequence. We are given the variance constraints for all n∈[N]n∈[N]: ‖ψn‖2=1N2‖ψ‖2. _n^2= 1N^2 ψ^2. Admissibility (1) requires that the allocation perfectly clears the market: ∑n=1Nψn=ψ. _n=1^N _n=ψ. We expand the squared norm of the aggregate demand using the inner product space properties of ℓ2 ^2: ‖ψ‖2 ψ^2 =‖∑n=1Nψn‖2 = _n=1^N _n^2 =∑n=1N‖ψn‖2+∑i≠j⟨ψi,ψj⟩. = _n=1^N _n^2+ _i≠ j _i, _j . Substituting the given variance constraints into the expansion yields: ‖ψ‖2=N(1N2‖ψ‖2)+∑i≠j⟨ψi,ψj⟩=1N‖ψ‖2+∑i≠j⟨ψi,ψj⟩. ψ^2=N ( 1N^2 ψ^2 )+ _i≠ j _i, _j = 1N ψ^2+ _i≠ j _i, _j . Rearranging this equation to isolate the sum of the inner products gives: ∑i≠j⟨ψi,ψj⟩=N−1N‖ψ‖2. _i≠ j _i, _j = N-1N ψ^2. (A1) We then apply the Cauchy-Schwarz inequality, which states that ⟨x,y⟩≤‖x‖‖y‖ x,y ≤ x y. Applying this to each pair of our allocation sequences: ⟨ψi,ψj⟩≤‖ψi‖‖ψj‖=(1N‖ψ‖)(1N‖ψ‖)=1N2‖ψ‖2. _i, _j ≤ _i _j= ( 1N ψ ) ( 1N ψ )= 1N^2 ψ^2. Because there are exactly N(N−1)N(N-1) terms in the summation over i≠ji≠ j, the maximum for the sum of the inner products is: ∑i≠j⟨ψi,ψj⟩≤N(N−1)(1N2‖ψ‖2)=N−1N‖ψ‖2. _i≠ j _i, _j ≤ N(N-1) ( 1N^2 ψ^2 )= N-1N ψ^2. Comparing this upper bound to our derived exact value in (A1), we see that the Cauchy-Schwarz inequality must hold with strict equality for every pair (i,j)(i,j). Therefore, we have ψ1=ψ2=⋯=ψN _1= _2=…= _N. Substituting this back into the admissibility constraint, we obtain ψn=1Nψ _n= 1Nψ for all n∈[N]n∈[N]. ∎ Proof of 3. Define αn=(−1)n _n=(-1)^n\,. First, since N is even, we have ∑nαn=0 _n _n=0, which implies ∑nψn(z)=ψ(z) _n _n(z)=ψ(z); hence π is an admissible demand allocation. Moreover, since σ≥σLσ≥ _ L, we have |αn|≥1 _n≥ 1 for all n. Thus, as in 3, transfer function n(z)=1+αnzT_n(z)=1+ _nz admits the inner–outer factorization n(z)=n(z)ℐn(z),with n(z)=αn+zandℐn(z)=1+αnzαn+z,T_n(z)=O_n(z)\,I_n(z), 10000\ 10000\ O_n(z)= _n+z _n(z)= 1+ _nz _n+z, where n(z)∈O_n(z) and ℐn(z)∈I_n(z) is a Blaschke factor. It follows from 2 that σnπ=1N|ψ(0)||n(0)|=|ψ(0)||αn|N=σ.σ^π_n= 1N\, ψ(0)\, O_n(0)= ψ(0)\, _nN=σ. Thus, π is neutral and implements the target root MSFE σ as required. ∎ Proof of 4. To show that π is admissible, note that since N is odd we have ∑n=13n(z) _n=1^3T_n(z) =1+z+z2+1−z2+1−z=3, =1+z+\,z^2+1-z^2+1-\,z=3, ∑n=4Nn(z) _n=4^NT_n(z) =∑n=4N(1+(−1)nz)=N−3. = _n=4^N (1+(-1)^n\,z )=N-3. It follows that ∑n=1Nψn(z)=ψ(z) _n=1^N _n(z)=ψ(z). From the proof of 3, we know that the root MSFE for each seller n≥3n≥ 3 is σnπ=σ^π_n=σ. For seller 11, the transfer 1(z)=1+z+z2T_1(z)=1+z+\,z^2 admits the inner–outer factorization 1(z)=1(z)ℐ1(z),where1(z)=z+z2,ℐ1(z)=1+z+z2z+z2.T_1(z)=O_1(z)\,I_1(z), where _1(z)=\,z+z^2, _1(z)=1+z+\,z^2 \,z+z^2. The fact that >1>1 implies that the two roots of the polynomial 1+z+z21+z+\,z^2 lie in D, and therefore ℐ1(z)I_1(z) is a Blaschke product and hence an inner factor (see GarciaMashreghiRoss2018). It follows that σ1π=1N|ψ(0)||1(0)|=|ψ(0)|N=σ.σ^π_1= 1N\, ψ(0)\, O_1(0)= \, ψ(0)N=σ. Similarly, for seller 2, we have that 2(z)=1−z2=2(z)ℐ2(z),where2(z)=z2,ℐ2(z)=1−z2z2T_2(z)=1-\,z^2=O_2(z)\,I_2(z), where _2(z)=z^2, _2(z)=1-\,z^2 z^2 where again the fact that >1>1 implies that ℐ2(z)I_2(z) is a Blaschke product and σ2π=1N|ψ(0)||2(0)|=|ψ(0)|N=σ.σ^π_2= 1N\, ψ(0)\, O_2(0)= \, ψ(0)N=σ. In conclusion, for all sellers n∈[N]n∈[N], we have σnπ=σ^π_n=σ under policy π, thereby proving that π is neutral and implements σπ=σ^π=σ. ∎ Proof of 5: Let ψ(z)ψ(z) and ψn(z) _n(z) denote the z-transforms of the market demand DtD_t and seller n’s demand DntD_nt, respectively. Let ψn(z)=1Nψ(z)n(z),with n(0)=1,for all n∈[N]and∑n=1Nn(z)=Nfor z∈, _n(z)= 1N\,ψ(z)\,T_n(z), \;T_n(0)=1,\; for all n∈[N] and _n=1^NT_n(z)=N\ for z , Suppose, by contradiction, that all the demand processes DntD_nt are invertible. Thus, by 2 we have σnπ=|ψn(0)|=|ψ(0)n(0)|/N=|ψ(0)|/N=σLσ^π_n= _n(0)= ψ(0)\,T_n(0)/N= ψ(0)/N= _ L, which violates the assumption σπ>σLσ^π> _ L. ∎ Proof of 2: Write the (mean-corrected) market demand in its Wold form as Dt−μ=ψ(ℬ)εt,D_t-μ=ψ(B)\, _t, where εt\ _t\ is white noise. Under an allocation with seller-n transfer function ψn(z)=1Nψ(z)n(z), _n(z)= 1N\,ψ(z)\,T_n(z), the corresponding time-domain representation is Dnt−μN=ψn(ℬ)εt=1Nψ(ℬ)n(ℬ)εt.D_nt- μN= _n(B)\, _t= 1N\,ψ(B)\,T_n(B)\, _t. Since ψ(ℬ)ψ(B) and n(ℬ)T_n(B) are both functions of the backshift operator, they commute, hence Dnt−μN=1Nn(ℬ)ψ(ℬ)εt=1Nn(ℬ)(Dt−μ).D_nt- μN= 1N\,T_n(B)\,ψ(B)\, _t= 1N\,T_n(B)\,(D_t-μ). Therefore, Dnt=μN+1Nn(ℬ)(Dt−μ).D_nt= μN+ 1N\,T_n(B)\,(D_t-μ). Now plug in the explicit polynomial forms of nT_n from Propositions 3–4 and expand. (a) N even. From Proposition 3, n(z)=1+(−1)nσLz.T_n(z)=1+(-1)^n\, σ _ L\,z. Hence n(ℬ)(Dt−μ)=(Dt−μ)+(−1)nσL(Dt−1−μ),T_n(B)(D_t-μ)=(D_t-μ)+(-1)^n\, σ _ L\,(D_t-1-μ), and substituting into Dnt=μN+1Nn(ℬ)(Dt−μ)D_nt= μN+ 1NT_n(B)(D_t-μ) gives Dnt=DtN+(−1)nσNσL(Dt−1−μ).D_nt= D_tN+(-1)^n\, σN\, _ L\,(D_t-1-μ). (b) N odd. From Proposition 4, the transfer functions are (with s:=σ/σLs:=σ/ _ L) 1(z)=1+s(z+z2),2(z)=1−sz2,n(z)=1+(−1)nsz(n≥3).T_1(z)=1+s(z+z^2), _2(z)=1-sz^2, _n(z)=1+(-1)^nsz\ \ (n≥ 3). Applying these polynomials to (Dt−μ)(D_t-μ) yields 1(ℬ)(Dt−μ)=(Dt−μ)+s[(Dt−1−μ)+(Dt−2−μ)],T_1(B)(D_t-μ)=(D_t-μ)+s [(D_t-1-μ)+(D_t-2-μ) ], 2(ℬ)(Dt−μ)=(Dt−μ)−s(Dt−2−μ),T_2(B)(D_t-μ)=(D_t-μ)-s(D_t-2-μ), n(ℬ)(Dt−μ)=(Dt−μ)+(−1)ns(Dt−1−μ),n≥3.T_n(B)(D_t-μ)=(D_t-μ)+(-1)^ns(D_t-1-μ), n≥ 3. Substituting into Dnt=μN+1Nn(ℬ)(Dt−μ)D_nt= μN+ 1NT_n(B)(D_t-μ) and using μN+1N(Dt−μ)=DtN μN+ 1N(D_t-μ)= D_tN gives exactly the stated time-domain formulas: D1t=DtN+σNσL[(Dt−1−μ)+(Dt−2−μ)],D2t=DtN−σNσL(Dt−2−μ),D_1t= D_tN+ σN _ L [(D_t-1-μ)+(D_t-2-μ) ], D_2t= D_tN- σN _ L(D_t-2-μ), Dnt=DtN+(−1)nσNσL(Dt−1−μ),n≥3.D_nt= D_tN+(-1)^n\, σN _ L(D_t-1-μ), n≥ 3. ∎ Proof of 6: Summing (14) over n yields Dt=∑n=1NDnt=Dt+∑n=1Nbn,tD_t= _n=1^ND_nt=D_t+ _n=1^Nb_n,t, hence ∑n=1Nbn,t=0. _n=1^Nb_n,t=0. (A2) Therefore ∑n=1Nxn,t=Dt _n=1^Nx_n,t=D_t, so xn,t\x_n,t\ is a fractional allocation of the DtD_t orders. Let An(k)A_n(k) denote the counter for seller n after the first k orders of period t have been assigned (by definition An(0)=0A_n(0)=0 and D^nt=An(Dt) D_nt=A_n(D_t)). Define the discrepancy process δn(k):=An(k)−xn,t,k=0,1,…,Dt. _n(k)\;:=\;A_n(k)-x_n,t, k=0,1,…,D_t. Because xn,t≥0x_n,t≥ 0, we have δn(0)=−xn,t≤0 _n(0)=-x_n,t≤ 0 for all n. At any arrival k, Algorithm 1 chooses nk∈argminnAn(k−1)−bn,tn_k∈ _n\A_n(k-1)-b_n,t\. Since xn,t=Dt/N+bn,tx_n,t=D_t/N+b_n,t, we have An(k−1)−bn,t=(An(k−1)−xn,t)+DtN=δn(k−1)+DtN,A_n(k-1)-b_n,t= (A_n(k-1)-x_n,t )+ D_tN= _n(k-1)+ D_tN, so the platform equivalently chooses nk∈argminnδn(k−1)n_k∈ _n _n(k-1). After assigning order k to nkn_k, δnk(k)=δnk(k−1)+1,δn(k)=δn(k−1) for n≠nk. _n_k(k)= _n_k(k-1)+1, _n(k)= _n(k-1)\ for n≠ n_k. In particular, each δn(k) _n(k) is nondecreasing in k. Let δn:=δn(Dt) _n:= _n(D_t) be the terminal discrepancies and pick an index i∈argmaxnδni∈ _n _n. If δi≤0 _i≤ 0, then δn≤0 _n≤ 0 for all n. Since ∑n=1Nδn=∑n=1NAn(Dt)−∑n=1Nxn,t=Dt−Dt=0, _n=1^N _n= _n=1^NA_n(D_t)- _n=1^Nx_n,t=D_t-D_t=0, it follows that δn=0 _n=0 for all n and the claim holds trivially. Suppose instead that δi>0 _i>0. Since δi(0)≤0 _i(0)≤ 0 and δi(Dt)>0 _i(D_t)>0, seller i must have been chosen at least once. Let τ be the (arrival) index of the last order assigned to seller i. Immediately before that assignment, seller i is among the minimizers of δ(τ−1)δ(τ-1), hence δi(τ−1)≤δj(τ−1)for all j∈[N]. _i(τ-1)≤ _j(τ-1) all j∈[N]. Because τ is the last time i is chosen, we have δi(τ−1)=δi−1 _i(τ-1)= _i-1. Moreover, since each δj(k) _j(k) is nondecreasing in k, δj(τ−1)≤δj _j(τ-1)≤ _j for all j. Therefore, δi−1≤δjfor all j∈[N], _i-1≤ _j all j∈[N], or equivalently δi≤δj+1 _i≤ _j+1 for all j. Taking the minimum over j yields maxnδn=δi≤minnδn+1. _n _n= _i\;≤\; _n _n+1. Hence maxnδn−minnδn≤1 _n _n- _n _n≤ 1. In conclusion, since ∑nδn=0 _n _n=0, we have minnδn≤0≤maxnδn _n _n≤ 0≤ _n _n. Combined with maxnδn≤minnδn+1 _n _n≤ _n _n+1, this implies minnδn≥−1 _n _n≥-1 and maxnδn≤1 _n _n≤ 1. Thus |δn|≤1| _n|≤ 1 for every n, i.e., |An(Dt)−xn,t|≤1for all n∈[N]. |A_n(D_t)-x_n,t |≤ 1 all n∈[N]. Recalling that D^nt=An(Dt) D_nt=A_n(D_t) and xn,t=Dt/N+bn,tx_n,t=D_t/N+b_n,t completes the proof. ∎ Appendix Appendix B Hardy-space foundations: ℍ2H^2 This appendix formalizes the z-domain tools used in Section 4.1. Our goal is to make precise how demand-allocation filters affect seller-level forecastability, why inner factors preserve second moments while altering invertibility, and why the one-step-ahead root MSFE is determined by the outer component of the transfer function. Detailed background on Hardy spaces can be found in Rudin1987RealComplex; MartinezAvendanoRosenthal2007; Nikolski2019HardySpaces. Transforming a stationary demand process into the z-domain is useful because time-domain convolutions become multiplications. In our setting, this means that the platform’s allocation rule acts as an algebraic filter on market demand. The resulting representation separates the magnitude and phase components of demand, which in turn allows us to distinguish the part of seller-level demand that governs forecastability from the part that merely distorts phase while leaving second moments unchanged. Throughout, let =z∈ℂ:|z|<1D=\z :|z|<1\ denote the open unit disk, let =e−iθ:θ∈[−π,π)T=\e^-iθ:θ∈[-π,π)\ denote its boundary, and let ℍ2H^2 denote the Hardy–Hilbert space of analytic functions on D with square-summable Taylor coefficients (equivalently, square-integrable boundary values on T). A convenient starting point is the Wold decomposition theorem. It implies that every weakly stationary purely nondeterministic process admits a unique one-sided moving-average representation in terms of its innovation sequence. In particular, if Xt\X_t\ is such a process, then Xt=X¯+∑k=0∞xkϵt−k,xk∈ℓ2,X_t= X+ _k=0^∞x_k\, _t-k, \x_k\∈ ^2, where ϵt\ _t\ is a white-noise innovation sequence orthogonal to the closed linear span of the past Xt−1,Xt−2,…\X_t-1,X_t-2,…\. The sequence ϵt\ _t\ coincides with the one-step-ahead forecast errors of the process, and the coefficients xk\x_k\ are uniquely determined. This representation provides the time-series foundation for the z-transform used below. Definition 4 (z-transform representation) Let Xt\X_t\ be a weakly stationary, purely nondeterministic Gaussian process with Wold MA(∞)(∞) representation Xt=X¯+∑k=0∞xkϵt−k,xk∈ℓ2,X_t= X+ _k=0^∞x_k\, _t-k, \x_k\∈ ^2, where ϵt\ _t\ is a white-noise innovation sequence, normalized to have unit variance unless stated otherwise. The z-transform of Xt\X_t\ is the function ∈ℍ2X ^2 defined by (z)=∑k=0∞xkzk,z∈.X(z)= _k=0^∞x_kz^k, z . Under an admissible allocation policy π=(μn,n(z):n∈[N])π= ( _n,T_n(z):n∈[N] ), seller n’s demand process satisfies Dnt=μn+n(ℬ)(Dt−μ).D_nt= _n+T_n(B)(D_t-μ). Since market demand admits the representation Dt−μ=ψ(ℬ)ϵtD_t-μ=ψ(B) _t, seller n’s allocated demand can equivalently be written as Dnt=μn+ψn(ℬ)ϵt,ψn(z):=n(z)ψ(z).D_nt= _n+ _n(B) _t, _n(z):=T_n(z)ψ(z). Hence seller n’s forecastability is governed by the transfer function ψn(z) _n(z). The feasibility condition Dt=∑n∈[N]DntD_t= _n∈[N]D_nt implies ∑n∈[N]ψn(z)=ψ(z) _n∈[N] _n(z)=ψ(z). Invertibility. A stationary process is said to be invertible if its driving innovation sequence can be recovered from the history of the observed process. In the notation above, the representation Xt=X¯+(ℬ)ϵtX_t= X+X(B) _t is invertible if there exists a square-summable sequence ϖk∈ℓ2\ _k\∈ ^2 such that ϵt=∑k=0∞ϖk(Xt−k−X¯). _t= _k=0^∞ _k\,(X_t-k- X). Equivalently, the shocks can be recovered by applying the inverse filter (ℬ)−1X(B)^-1 to the centered process. Thus, invertibility means that past observations contain enough information to reconstruct the underlying shocks. As we explain below, this is exactly the property captured by the outer component of the transfer function. Definition 5 (Inner and outer functions) A function ℐ∈ℍ2I ^2 is inner if its boundary values are unimodular almost everywhere on T, that is, |ℐ(e−iθ)|=1for a.e. θ∈[−π,π).|I(e^-iθ)|=1 a.e.\ θ∈[-π,π). A function ∈ℍ2O ^2 is outer if it it has no zeros in the open unit disk, i.e., (z)≠0O(z)≠ 0 for |z|<1|z|<1†Equivalently, O is determined by its boundary modulus through the Poisson integral: (z)=cexp12π∫−πe−iθ+ze−iθ−zlog|(e−iθ)|dθ,z∈,O(z)=c \! \ 12π _-π^π e^-iθ+ze^-iθ-z\, |O(e^-iθ)|\,dθ \, z , for some unimodular constant c∈c , with log|(e−iθ)|∈L1([−π,π)) |O(e^-iθ)|∈ L^1([-π,π)). In particular, outer functions are zero-free on D.. We write I and O for the classes of inner and outer functions in ℍ2H^2, respectively. The forecasting interpretation of these two classes is central to our analysis. Inner factors are all-pass filters: they preserve the boundary magnitude of a transfer function and therefore leave its spectral density unchanged. Outer factors, by contrast, are the minimum-phase, invertible components of a transfer function. Accordingly, they encode the part of the process that is relevant for one-step-ahead forecasting. Lemma 4 (All-pass property of inner factors) If ℐ∈I and f∈ℍ2f ^2, then |(ℐf)(e−iθ)|2=|f(e−iθ)|2for a.e. θ∈[−π,π).|(If)(e^-iθ)|^2=|f(e^-iθ)|^2 a.e.\ θ∈[-π,π). Consequently, multiplying a transfer function by an inner factor preserves the spectral density and hence the entire autocovariance structure of the associated stationary process. The next result is the basic structural theorem from Hardy-space theory that we use throughout. Lemma 5 (Nevanlinna inner–outer factorization) Every ∈ℍ2X ^2 admits a factorization (z)=(z)ℐ(z),X(z)=O(z)\,I(z), where ∈O is outer and ℐ∈I is inner. This factorization is unique up to multiplication by a unimodular constant. Because inner factors preserve second moments while outer factors govern invertibility, 5 gives a canonical decomposition of any stationary demand stream into a forecast-relevant component and a pure phase distortion. Lemma 6 (Root MSFE and the outer factor) Let Xt=X¯+(ℬ)ϵtX_t= X+X(B) _t be a weakly stationary, purely nondeterministic Gaussian process with ∈ℍ2X ^2, and let =ℐX=OI be its inner–outer factorization. Then the one-step-ahead root mean squared forecast error of Xt\X_t\ is σX=|(0)|. _X=|O(0)|. Equivalently, only the outer component of the transfer function affects one-step-ahead forecastability. Proof of 6: In ℍ2H^2, the one-step predictor is the orthogonal projection of X onto zℍ2zH^2, and the prediction error is the orthogonal complement. Multiplication by an inner function is an isometry on ℍ2H^2, hence dist(,zℍ2)=dist(,zℍ2).dist (X,\,zH^2 )=dist (O,\,zH^2 ). For any g(z)=∑k≥0gkzk∈ℍ2g(z)= _k≥ 0g_kz^k ^2, the orthogonal complement of zℍ2zH^2 is spanned by the constant function, so dist(g,zℍ2)=|g0|=|g(0)|dist(g,zH^2)=|g_0|=|g(0)|. Applying this to g=g=O yields σX=|(0)| _X=|O(0)|.∎ The intuition is simple. Since ℐI is inner, the filtered sequence ηt:=ℐ(ℬ)ϵt _t:=I(B) _t is again white noise with unit variance by 4. Hence Xt=X¯+(ℬ)ηt,X_t= X+O(B) _t, so the process admits an equivalent outer representation whose innovation variance is unchanged. The one-step-ahead forecast error is therefore the contemporaneous coefficient in the outer filter, namely |(0)||O(0)|. Applying this result to seller-level demand gives the main text formula. Corollary 3 (Seller-level root MSFE) Under an admissible allocation policy π=(μn,n(z):n∈[N])π= ( _n,T_n(z):n∈[N] ), let seller n’s transfer function be ψn(z)=n(z)ψ(z) _n(z)=T_n(z)ψ(z), and let ψn=nℐn _n=O_nI_n be its inner–outer factorization. Then seller n’s one-step-ahead root MSFE under policy π is σnπ=|n(0)|. _n^π=|O_n(0)|. The next result formalizes the connection between outer functions and invertibility. Lemma 7 (Invertibility and outer functions) Let Xt=X¯+(ℬ)ϵtX_t= X+X(B) _t with ∈ℍ2X ^2. Then the process Xt\X_t\ is invertible with respect to ϵt\ _t\ if and only if X is outer. Thus, nontrivial inner factors represent precisely the part of the filter that is invisible from second moments yet prevents the underlying shocks from being recovered from the observed demand history. Corollary 4 (Seller-level invertibility) Under an admissible allocation policy π=(μn,n(z):n∈[N])π= ( _n,T_n(z):n∈[N] ), seller n’s demand process Dnt\D_nt\ is invertible with respect to the aggregate demand shocks ϵt\ _t\ if and only if ψn(z) _n(z) is outer. The inner part can itself be decomposed more finely. Although this is not needed for the main argument, it is useful in examples and in finite-order constructions. Remark 2 (Structure of inner functions) Any ℐ∈I can be written as the product of a unimodular constant, a Blaschke product, and a singular inner factor: ℐ(z)=cℬ(z)(z),c∈.I(z)=c\,B(z)\,S(z), c . The Blaschke part collects zeros ak⊂\a_k\ through ℬ(z)=∏kz−ak1−ak¯z,B(z)= _k z-a_k1- a_kz, while the singular inner factor is generated by a finite positive singular measure ν on T: (z)=exp−12π∫−πe−iθ+ze−iθ−zν(θ).S(z)= \! \- 12π _-π^π e^-iθ+ze^-iθ-z\,dν(θ) \. For more details, see Nikolski2019HardySpaces. The appendix results above provide the formal underpinning for the main-text discussion: the platform affects seller-level forecastability by shaping the outer components of the transfer functions ψn _n, while inner factors supply additional degrees of freedom that preserve second moments and help satisfy the feasibility constraint ∑n∈[N]ψn(z)=ψ(z) _n∈[N] _n(z)=ψ(z). Appendix Appendix C MA(∞)MA(∞) Representations of Gaussian Demand and the Equivalence of Demand Representations This appendix provides additional details for the demand assumption. In particular, we discuss about the MA(∞)MA(∞) representation of demand, the assumption of Gaussian shocks, and clarify the equivalence between formulations based on observed demand Dt\D_t\ and underlying demand shocks ϵt\ _t\ under standard invertibility conditions. The MA(∞)MA(∞) representation in (2) is sufficiently general to capture any weakly stationary ARMA process, though it excludes nonstationary models like ARIMA. By Wold’s representation theorem (BrockwellDavis2006), there exists a valid square-summable sequence ψk\ _k\ for this formulation. We also assume invertibility with respect to the demand shocks ϵt _t; in our setting, this entails no real loss of generality, since demand is exogenous to the seller and only DtD_t is observed. Because the demand shocks ϵt\ _t\ are Gaussian, the model can in principle generate negative realized demand and, under base-stock control, negative orders.‡This is a standard issue in Gaussian inventory models; see, for example, JohnsonThompson, LST2000, Aviv2003, Chen_Lee_2010, and CGH_OR_Letter. We therefore interpret the Gaussian-shock formulation as a convenient approximation of non-negative demand. This approximation works well for products with moderate demand volatility, where the probability of negative demand is negligible (see Table 4 in CGH_OR_Letter). Additional discussion and probability estimates are provided in Appendix D. Finally, we comment on the equivalence between representations based on Dt\D_t\ and ϵt\ _t\. Our analysis focuses on allocation policies defined in terms of observed demand histories Dt,Dt−1,…\D_t,D_t-1,…\. This formulation is equivalent to the one based on the shock history ϵt,ϵt−1,…\ _t, _t-1,…\ when the demand is invertiable and its z-transform ψ(z)ψ(z) satisfies mild technical conditions, e.g., when ψ(z)ψ(z) belongs to the class of causal and invertible ARMA processes. However, the demand-based representation is both more natural and more transparent, as the shocks must otherwise be recovered indirectly (e.g., via spectral factorization). Appendix Appendix D On the Likelihood of Negative Demand The classes of Gaussian demand models considered in 2 allow, in principle, for negative realized market demand and allocated demand. In what follows, we quantify the probability that market demand is negative and compare it to the probability that an individual seller observes negative allocated demand. Recall that the market-demand process Dt\D_t\ admits the one-sided MA(∞)MA(∞) representation Dt=μ+∑k=0∞ψkϵt−k,D_t=μ+ _k=0^∞ _k\, _t-k, where μ>0μ>0 is the mean demand and ϵt\ _t\ is a Gaussian white-noise sequence with [ϵt]=0E[ _t]=0 and ar(ϵt)=1V ar( _t)=1. Hence Dt∼(μ,σD2)D_t N(μ, _D^2) with σD2=∑k=0∞ψk2 _D^2= _k=0^∞ _k^2, and therefore ℙ(Dt≤0)=Φ(−μσD)=Φ(−1CV),CV:=σDμ=1μ∑k=0∞ψk2,P(D_t≤ 0)= \! (- μ _D )= \! (- 1CV ), := _Dμ= 1μ _k=0^∞ _k^2, where Φ(⋅) (·) is the standard normal cdf and CVCV is the coefficient of variation. In particular, ℙ(Dt≤0)→0P(D_t≤ 0)→ 0 as CV→0CV→ 0, so the Gaussian approximation is most appropriate for product categories with mean demand large relative to its standard deviation. To illustrate how the allocation rule affects this probability at the seller level, suppose for simplicity that N is even and the platform uses the allocation in 3, i.e., ψn(z)=1Nψ(z)n(z) _n(z)= 1Nψ(z)T_n(z) with transfer function n(z)=1+(−1)nα¯z,α¯:=σL,T_n(z)=1+(-1)^n αz, α:= σ _ L, for some σ≥σLσ≥ _ L. By 2, Dnt D_nt =DtN+(−1)nσNσL(Dt−1−μ) = D_tN+(-1)^n σN _ L(D_t-1-μ) =μN+1N∑k=0∞[ψkϵt−k+(−1)nα¯ψkϵt−1−k] = μN+ 1N _k=0^∞ [ _k\, _t-k+(-1)^n α\, _k\, _t-1-k ] =μN+1Nψ0ϵt+1N∑k=1∞[ψk+(−1)nα¯ψk−1]ϵt−k. = μN+ 1N _0\, _t+ 1N _k=1^∞ [ _k+(-1)^n α\, _k-1 ] _t-k. Thus DntD_nt is Gaussian with mean μ/Nμ/N and variance σn2=1N2(ψ02+∑k=1∞[ψk+(−1)nα¯ψk−1]2), _n^2= 1N^2 ( _0^2+ _k=1^∞ [ _k+(-1)^n α\, _k-1 ]^2 ), so ℙ(Dnt≤0)=Φ(−μψ02+∑k=1∞[ψk+(−1)nα¯ψk−1]2)=Φ(−1CVn),P(D_nt≤ 0)= \! (- μ _0^2+ _k=1^∞[ _k+(-1)^n α\, _k-1]^2 )= \! (- 1CV_n ), where CVn:=1μψ02+∑k=1∞[ψk+(−1)nα¯ψk−1]2.CV_n:= 1μ _0^2+ _k=1^∞ [ _k+(-1)^n α\, _k-1 ]^2. By the inequality (x+y)2≤2(x2+y2)(x+y)^2≤ 2(x^2+y^2), ∑k=1∞[ψk+(−1)nα¯ψk−1]2≤2∑k=1∞(ψk2+α¯2ψk−12), _k=1^∞ [ _k+(-1)^n α\, _k-1 ]^2≤ 2 _k=1^∞ ( _k^2+ α^2 _k-1^2 ), which yields the crude bound CVn≤CV2(1+α¯2).CV_n \, 2(1+ α^2). Although conservative, this bound provides a simple screening criterion for when negative seller-level demand is negligible. Using U.S. Census Bureau data, Table 4 in CGH_OR_Letter reports coefficients of variation of monthly sales across retail sectors. For example, for Furniture and Home Furnishings (FHF), CVCV ranges from 0.0810.081 to 0.1080.108 over the last 30 years, while for Electronics and Appliance (EA) it ranges from 0.1660.166 to 0.2440.244. Using midpoints of these ranges, ℙ(Dt≤0)P(D_t≤ 0) is on the order of 10−2610^-26 for FHF and 10−710^-7 for EA. In turn, the bound above implies that for α¯ α below roughly 55 (FHF) and 22 (EA), ℙ(Dnt≤0)P(D_nt≤ 0) remains below 5%5\%. If market demand is i.i.d., these thresholds increase to about 77 and 33, respectively. Appendix Appendix E Derivation of Equation 18 In this appendix, we derive the value of the MSFE when sellers use simple exponential smoothing to forecast demand. In particular, we consider the case in which market demand is i.i.d. (i.e., ψ(z)ψ(z) is a constant) and seller n’s demand allocation admits the z-transform representation ψn(z)=1Nψ(z)n(z) _n(z)= 1N\,ψ(z)\,T_n(z). For simplicity we consider the case with an even number of sellers. According to 3, in this case the platform sets ψn(z)=|ψ(0)|N(1+(−1)nαz),whereα=Nσ|ψ(0)|. _n(z)= ψ(0)N\, (1+(-1)^n\,α z ), α= N\,σ ψ(0). It follows that (σ~nπ(λ))2 ( σ_n^π(λ))^2 =12π|ψ(0)|2N2∫−π|1−e−iθ1−(1−λ)e−iθ|2|1+(−1)nαe−iθ|2dθ = 12π\, ψ(0)^2N^2 _-π^π | 1-e^-iθ1-(1-λ)e^-iθ |^2\, |1+(-1)^nα e^-iθ |^2\, dθ =|ψ(0)|2N2⋅12π∫−π|F(e−iθ)|2dθ,whereF(z):=1−z1−(1−λ)z(1+(−1)nαz). = ψ(0)^2N^2· 12π _-π^π |F(e^-iθ) |^2\, dθ, F(z):= 1-z1-(1-λ)z\, (1+(-1)^n\,α z ). Using a geometric-series expansion, F(z) F(z) =(1−λ∑k=1∞(1−λ)k−1zk)(1+(−1)nαz) = (1-λ _k=1^∞(1-λ)^k-1z^k ) (1+(-1)^n\,α z ) =1+(−1)nαz−λ∑k=1∞(1−λ)k−1zk−λ(−1)nα∑k=1∞(1−λ)k−1zk+1. =1+(-1)^n\,α z-λ _k=1^∞(1-λ)^k-1z^k-λ(-1)^nα _k=1^∞(1-λ)^k-1z^k+1. Therefore, writing F(z)=∑k≥0fkzkF(z)= _k≥ 0f_kz^k, the coefficients are f0=1,f1=(−1)nα−λ,fk=−λ(1−λ)k−2(1−λ+(−1)nα),k≥2.f_0=1, f_1=(-1)^nα-λ, f_k=-λ(1-λ)^k-2 (1-λ+(-1)^nα ),\ \ k≥ 2. Since F∈ℍ2F ^2, Parseval’s identity yields 12π∫−π|F(e−iθ)|2dθ=∑k=0∞|fk|2=1+|(−1)nα−λ|2+∑k=2∞λ2(1−λ)2(k−2)|1−λ+(−1)nα|2. 12π _-π^π |F(e^-iθ) |^2\, dθ= _k=0^∞ f_k^2=1+ (-1)^n\,α-λ^2+ _k=2^∞λ^2(1-λ)^2(k-2) 1-λ+(-1)^n\,α^2. Summing the geometric series, we obtain (σ~nπ(λ))2=|ψ(0)|2N2[1+|(−1)nα−λ|2+λ2−λ|1−λ+(−1)nα|2].( σ_n^π(λ))^2= ψ(0)^2N^2 [1+ (-1)^nα-λ^2+ λ2-λ 1-λ+(-1)^nα^2 ]. Appendix Appendix F Heterogeneous Replenishment Lead Times Our baseline model assumes that all sellers face identical and zero replenishment lead times, regardless of the fulfillment mode they select. We now extend the framework to allow for heterogeneous lead times that may vary both across sellers and across fulfillment modes. In particular, for each seller n∈[N]n∈[N] let LFBP≥0L 0.45 FBP≥ 0 and LFBM≥0L 0.45 FBM≥ 0 denote, respectively, the replenishment lead time when seller n operates under FBP and under FBM. Throughout the discussion, we use L¯n L_n to denote the relevant (mode-dependent) lead time faced by seller n under the prevailing fulfillment mode. Define seller n’s lead-time demand as n,t:=∑ℓ=0L¯nDn,t+ℓ,D_n,t:= _ =0 L_nD_n,t+ , that is, the cumulative demand realized during the replenishment delay. Just before demand in period t+1t+1 is realized, seller n chooses an order-up-to level SntS_nt measurable with respect to ℱntF_nt and orders Qnt=(Snt−Int)+,Q_nt= (S_nt-I_nt )^+, where IntI_nt denotes net inventory after serving demand in period t. As in the baseline model, seller n selects SntS_nt to minimize expected holding and backorder costs associated with lead-time demand: Snt= [h¯n(S−n,t+1)++bn(S−n,t+1)−|ℱnt].S_nt= _S\;E\! [\, h_n\,(S-D_n,t+1)^++b_n\,(S-D_n,t+1)^-\; |\;F_nt ]. Under 2, n,t+1D_n,t+1 is Gaussian conditional on ℱntF_nt, and the optimal base-stock policy takes the familiar form Snt=m¯nt+ζnσ¯n,ζn:=Φ−1(bnh¯n+bn),S_nt\;=\; m_nt+ _n\, σ_n, _n:= ^-1\! ( b_n h_n+b_n ), (A3) where m¯nt:=[n,t+1∣ℱnt] m_nt:=E\! [D_n,t+1 _nt ] is the conditional mean forecast of lead-time demand and σ¯n2:=ar[n,t+1−m¯nt∣ℱnt] σ_n^2:=V ar\! [D_n,t+1- m_nt _nt ] is the corresponding MSFE of the lead-time demand. Thus, seller n’s safety stock equals ζnσ¯n _n\, σ_n, implying that inventory holdings depend on the predictability of cumulative lead-time demand rather than one-step-ahead demand alone. Just as in (5), seller n’s expected per-period operating profit can be written as Un=(r−ρ−f¯n)μn−Knσ¯n,Kn:=h¯nζn+(h¯n+bn)ℒ(ζn).U_n\;=\;(r-ρ- f_n)\, _n\;-\;K_n\, σ_n, K_n:= h_n\, _n+( h_n+b_n)\,L( _n). (A4) A key difference between (5) and (A4) is that under lead times the operating mode affects not only the effective cost parameters (h¯n,f¯n)( h_n, f_n) but also the relevant lead time: FBP and FBM may entail different replenishment delays, so that L¯n∈LFBP,LFBM L_n∈\L 0.45 FBP,L 0.45 FBM\ and, consequently, the corresponding root MSFE σ¯n σ_n (computed for lead-time demand) may differ across modes even under the same demand allocation policy. Let us now extend the result in 2 to account for a general leadtime. Proposition 7 Suppose seller n faces a demand process with z-transform ψn(z)=n(z)ℐn(z) _n(z)=O_n(z)\,I_n(z), where n∈O_n and ℐn∈I_n . Then, the MSFE σ¯n2 σ^2_n of the lead-time demand is given by σ¯n2=∑ℓ=0L¯n(∑k=0L¯n−ℓθnk)2,whereθnk:=12π∫−πeikxn(e−ix)dx. σ^2_n= _ =0 L_n ( _k=0 L_n- _nk )^2, where _nk:=1 2π _-π^πe^i\,k\,x\,O_n(e^-i\,x)\, dx. (A5) The following corollary directly adapts the previous result to our proposed class of demand allocation policies in Propositions 3 and 4. Corollary 5 Consider the demand allocation ψn(z)=1Nψ(z)n(z) _n(z)= 1N\,ψ(z)\,T_n(z), where the transfer function n(z)T_n(z) is given in Propositions 3 or 4. Then, the MSFE σ¯n2 σ^2_n of seller n’s lead-time demand is given by (A5), where the sequence θnk\ _nk\ satisfies If N is even, or if N is odd and n≥3n≥ 3: θnk=1N(αnψk−ψk−1), _nk= 1N\, ( _n\, _k- _k-1 ), If N is odd and n=1n=1: θnk=1N(αnψk+αnψk−1+ψk−2), _nk= 1N\, ( _n\, _k+ _n\, _k-1+ _k-2 ), If N is odd and n=2n=2: θnk=1N(αnψk−ψk−2), _nk= 1N\, ( _n\, _k- _k-2 ), where αn:=(−1)nσ/σL _n:=(-1)^n\,σ/ _ L and we adopt the convention that ψk=0 _k=0 for k<0k<0. Equipped with this corollary, for any given value of σ each seller can compute the root MSFE of its lead-time demand under each fulfillment mode by applying 7 using the corresponding mode-specific replenishment leadtime. The seller then selects its preferred mode by maximizing its expected operating payoff as characterized in (A4). This mode choice also pins down the seller’s average safety stock through the base-stock rule (A3). Anticipating these endogenous mode choices, the platform subsequently determines the optimal value of σ by maximizing the appropriate lead-time version of (11), in which both the set of sellers operating under FBP, NFBP(σ)N 0.45 FBP(σ), and the associated cumulative safety stock ΓFBP(σ) 0.45 FBP(σ) are adjusted to reflect the mode-dependent lead-time root MSFE and safety-stock calculations described above. Proof of 7: Fix seller n and write the mean–zero demand as D~nt:=Dnt−μn=ψn(ℬ)ϵt, D_nt:=D_nt- _n= _n(B)\, _t, where ℬB is the backshift operator and ϵt\ _t\ is the Gaussian sequence of shocks appearing in the market-demand Wold representation in (2). Since ψn(z)=n(z)ℐn(z) _n(z)=O_n(z)I_n(z) with n∈O_n and ℐn∈I_n , we can write D~nt=n(ℬ)ℐn(ℬ)ϵt=n(ℬ)ϵ^nt,ϵ^nt:=ℐn(ℬ)ϵt. D_nt=O_n(B)I_n(B)\, _t=O_n(B)\, ε_nt, ε_nt:=I_n(B)\, _t. Because ℐnI_n is an inner function (i.e., an all-pass filter), ϵ^nt\ ε_nt\ is again a Gaussian white-noise sequence. Thus, D~nt=n(ℬ)ϵ^nt D_nt=O_n(B)\, ε_nt is the Wold representation of D~nt D_nt, i.e., D~nt=∑k=0∞θkϵ^n,t−k, D_nt= _k=0^∞ _k\, ε_n,t-k, (A6) where θkk≥0\ _k\_k≥ 0 are the Taylor coefficients of the outer factor nO_n, namely n(z)=∑k=0∞θkzkO_n(z)= _k=0^∞ _kz^k. Since n∈H2O_n∈ H^2, these coefficients can be recovered via the Fourier formula θk=12π∫−πeikxn(e−ix)x,k≥0. _k= 12π _-π^πe^ikx\,O_n(e^-ix)\,dx, k≥ 0. Now consider the lead-time demand n,t+1=∑ℓ=0L¯nDn,t+1+ℓ=(L¯n+1)μn+∑ℓ=0L¯nD~n,t+1+ℓ.D_n,t+1= _ =0 L_nD_n,t+1+ \;=\;( L_n+1) _n+ _ =0 L_n D_n,t+1+ . Because the constant mean term is ℱntF_nt-measurable, it does not affect the MSFE, so we work with ∑ℓ=0L¯nD~n,t+1+ℓ _ =0 L_n D_n,t+1+ . Substituting (A6) and re-indexing gives ∑ℓ=0L¯nD~n,t+1+ℓ _ =0 L_n D_n,t+1+ =∑ℓ=0L¯n∑k=0∞θkϵ^n,t+1+ℓ−k = _ =0 L_n _k=0^∞ _k\, ε_n,t+1+ -k =∑ℓ=0L¯n∑k=ℓ+1∞θkϵ^n,t+1+ℓ−k⏟ℱnt-measurable+∑ℓ=0L¯n∑k=0ℓθkϵ^n,t+1+ℓ−k. = _ =0 L_n _k= +1^∞ _k\, ε_n,t+1+ -k_$ F_nt$-measurable\;+\; _ =0 L_n _k=0 _k\, ε_n,t+1+ -k. The first term only involves innovation shocks ϵ^n,s ε_n,s with s≤ts≤ t and is therefore measurable with respect to ℱnt=σ(Dnss≤t)F_nt=σ(\D_ns\_s≤ t). Hence, the forecast error n,t+1−[n,t+1∣ℱnt]D_n,t+1-E[D_n,t+1 _nt] equals the orthogonal (innovation) part ∑ℓ=0L¯n∑k=0ℓθkϵ^n,t+1+ℓ−k. _ =0 L_n _k=0 _k\, ε_n,t+1+ -k. Collecting terms by innovation time t+1+ℓt+1+ (set u=ℓ−ku= -k) yields n,t+1−[n,t+1∣ℱnt]=∑ℓ=0L¯n(∑k=0L¯n−ℓθk)ϵ^n,t+1+ℓ.D_n,t+1-E[D_n,t+1 _nt]= _ =0 L_n ( _k=0 L_n- _k )\, ε_n,t+1+ . Since ϵ^nt\ ε_nt\ is white noise with unit variance, the conditional variance (and hence the MSFE) is σ¯n2=ar(n,t+1−[n,t+1∣ℱnt]|ℱnt)=∑ℓ=0L¯n(∑k=0L¯n−ℓθk)2, σ_n^2=V ar\! (D_n,t+1-E[D_n,t+1 _nt]\, |\,F_nt )= _ =0 L_n ( _k=0 L_n- _k )^2, which proves the claim. ∎