arXiv:2605.20036v1 [cs.LG] 19 May 2026
D3-Subsidy: Online and Sequential Driver Subsidy Decision-Making for Large-Scale Ride-Hailing Market Taijie Chen
Rui Su
Siyuan Feng∗
[email protected] University of Hong Kong Hong Kong, China
[email protected] Harbin Institute of Technology Harbin, Heilongjiang, China
[email protected] Hong Kong Polytechnic University Hong Kong, China
Laoming Zhang
Hongyang Zhang
Haijiao Wang
[email protected] Didi International Business Group Beijing, China
[email protected] Hong Kong Polytechnic University Hong Kong, China
[email protected] Didi International Business Group Beijing, China
Zhaofeng Ma
Jintao Ke∗
[email protected] Didi International Business Group Beijing, China
[email protected] University of Hong Kong Hong Kong, China
Abstract
1
Ride-hailing platforms like DiDi Chuxing operate in highly dynamic environments where balancing driver supply and passenger demand is critical. Although driver-side subsidies serve as a primary lever to align these forces and improve key KPIs like completed rides (Rides) and gross merchandise value (GMV), optimizing them in production requires simultaneously meeting three constraints: (i) responsiveness to stochastic shocks, (ii) strict subsidy-rate caps, and (iii) low-latency execution at city scale. These requirements rule out expensive per-order optimization, calling for a forwardlooking, constraint-aware city-level controller for online sequential decision making. To meet these requirements, we introduce D3 Subsidy (Dynamic Driver-side Diffusion-based Subsidy), a hierarchical diffusion-based framework for deployable city-wide subsidy control. To bridge the train-inference gap, D3 -Subsidy employs a prefix-conditioned diffusion model that samples plausible future trajectories from immutable historical observations, ensuring the training protocol aligns with the fixed-history nature of online deployment. These generated plans are then decoded by a contextconditioned inverse module into low-dimensional city-level control signals. For scalable execution, we bridge the gap between city-level planning and fine-grained dispatch via a Lagrangian-dual-derived mapping, which embeds subsidy-rate caps directly into order-driver incentives without iterative optimization. Additionally, a multi-city pretraining strategy with parameter-efficient fine-tuning enables robust transfer across heterogeneous cities. Extensive offline evaluations demonstrate that D3 -Subsidy improves Rides and GMV while enhancing cap compliance, and a real-world A/B test confirms significant uplift while keeping budget-related violation metrics within operational thresholds.
Ride-hailing platforms such as Uber and DiDi Chuxing are a core part of urban mobility, matching millions of passengers and drivers in rapidly changing spatiotemporal environments [19]. A persistent operational challenge is ensuring sufficient driver participation: due to heterogeneous preferences and opportunity costs, many drivers are reluctant to accept low-attractiveness requests, which can trigger severe supply–demand imbalance and degraded service quality [6]. Driver-side subsidies are therefore widely used to stimulate acceptance and completion, improving platform-level objectives such as completed rides (Rides) and gross merchandise value (GMV) [24]. Subsidies, however, must be allocated under strict profitability constraints. In practice, platforms impose a global budget over a long horizon (e.g., one day), often expressed as a cap on the subsidy rate relative to GMV. This makes subsidy allocation inherently online and sequential: a decision at the current time window affects not only immediate driver supply but also the remaining budget, thereby constraining future decisions. The goal is to learn a policy that maximizes cumulative platform utility over the day while respecting the budget cap. A further complication is scalability. Optimizing a subsidy for each individual order–driver pair in real time is computationally prohibitive at city scale, and overly personalized incentives can raise fairness and consistency concerns among drivers facing similar conditions. As a result, industrial systems commonly deploy a twostage mechanism: the platform first decides a city-level subsidy intensity 𝜆𝑡 for each time window 𝑡, and then deterministically maps 𝜆𝑡 to pair-level incentives (e.g., as a function of order value and pickup distance) via a Lagrangian-dual-derived rule. This design offers low-latency execution and a transparent, consistent incentive structure. Even with this low-dimensional action space, learning an effective controller is challenging. The ride-hailing market is stochastic and non-stationary, and standard reinforcement learning (RL) approaches can be unstable and data-hungry in such settings [11].
Keywords Ride-hailing, Driver Subsidy, Diffusion, Reinforcement Learning
∗ Corresponding author
1
Introduction
KDD ’26, August 9–13, 2026, Jeju, Republic of Korea
Chen et al.
More importantly, the trial-and-error exploration required by online RL is misaligned with strict budget caps: aggressive exploration may cause irreversible loss, premature budget exhaustion, or operational violations. These constraints motivate offline learning, where policies are learned purely from historical logs. In this work, we draw on recent advances in generative modeling and propose D3 -Subsidy (Dynamic Driver-side Diffusion-based Subsidy), a diffusion-based framework for safe and deployable city-level subsidy control. To bridge the train–inference gap in sequential decision making, we introduce a prefix-conditional diffusion scheme that learns to generate plausible future city trajectories conditioned on an observed history prefix. To explicitly promote budget feasibility, we incorporate a constraint-aware score that penalizes infeasible trajectories. We further propose a context-conditioned inverse dynamics module to decode the control signal 𝜆𝑡 under specific operational contexts, and adopt multi-city pretraining with parameter-efficient fine-tuning for scalable transfer across heterogeneous cities. Our main contributions are summarized as follows: • We formulate city-level driver subsidy allocation as a budgetconstrained online sequential decision-making problem, capturing the coupling between current incentives, future feasibility, and platform-level KPIs under market stochasticity. • We propose D3 -Subsidy, a diffusion-based offline control framework featuring (i) prefix-conditional diffusion for deploymentconsistent trajectory generation, (ii) a constraint-aware score for budget-feasible planning, and (iii) a context-conditioned inverse dynamics decoder for accurate and controllable action inference. • We evaluate our method on real-world data from three cities and validate it in production A/B tests, achieving a 1.59% increase in Rides and a 2.06% improvement in GMV under the same budget constraints.
box constraints: ∑︁ max 𝑟𝑖 𝑗 𝑝𝑖 𝑗 (𝑏𝑖 𝑗 ), 𝑏𝑖 𝑗
s.t.
𝑖,𝑗
∑︁
𝑏𝑖 𝑗 𝑝𝑖 𝑗 (𝑏𝑖 𝑗 ) − (𝐶 + 𝛿)
𝑖,𝑗
∑︁
𝑟𝑖 𝑗 𝑝𝑖 𝑗 (𝑏𝑖 𝑗 ) ≤ 0,
(3)
𝑖,𝑗
0 ≤ 𝑏𝑖 𝑗 ≤ 𝑏 max(𝑖 ) ,
∀𝑖, 𝑗 .
Dual relaxation and closed-form solution. To obtain a deployable mapping with analytical transparency, we adopt a locally linear approximation of the completion probability with respect to incentives. Consistent with empirical evidence that completion likelihood increases with incentives [6], we use 𝑝𝑖 𝑗 (𝑏𝑖 𝑗 ) ≈ 𝑎𝑖 𝑗 𝑏𝑖 𝑗 ,
𝑎𝑖 𝑗 > 0,
0 ≤ 𝑏𝑖 𝑗 ≤ 𝑏 max(𝑖 ) .
(4)
Here 𝑎𝑖 𝑗 is a function of observable order- and context-level characteristics (e.g., pickup distance and order value) and does not depend on driver-identity features. Therefore, any two driver–order pairs with the same observable characteristics share the same 𝑎𝑖 𝑗 , yielding a uniform, non-discriminatory subsidy rule (i.e., no driver-specific differentiated incentives). In practice, 𝑝𝑖 𝑗 (𝑏) is bounded in [0, 1] and can be modeled by a monotone response such as a logit specification. Since any smooth monotone function on [0, 𝑏 max ] can be uniformly approximated by a piecewise-linear function, the dualbased mapping derived below extends to such nonlinear response models by replacing Equation (4) with its piecewise-linear approximation, leading to a piecewise closed-form policy with the same clipping structure. Introducing a Lagrange multiplier 𝜆 ≥ 0 for the subsidy-rate constraint, the Lagrangian of Equation (3) becomes i ∑︁ h L (𝑏, 𝜆) = (1 + 𝜆(𝐶 + 𝛿)) 𝑟𝑖 𝑗 𝑎𝑖 𝑗 𝑏𝑖 𝑗 − 𝜆𝑎𝑖 𝑗 𝑏𝑖2𝑗 . (5) 𝑖,𝑗
Because L (𝑏, 𝜆) is separable across pairs (𝑖, 𝑗), the inner maximization admits a closed-form solution. Define 𝜅 = (𝐶 + 𝛿 + 1/𝜆)/2. The optimal subsidy is n o 𝑏𝑖∗𝑗 (𝜆) = min max 0, 𝜅𝑟𝑖 𝑗 , 𝑏 max(𝑖 ) . (6)
2 Preliminary 2.1 Problem Formulation We consider a broadcast-based order dispatching system [7]. When an order 𝑖 is broadcast to a candidate driver 𝑗, the platform may offer a monetary subsidy 𝑏𝑖 𝑗 to incentivize acceptance and completion. Each subsidy is bounded by an order-specific cap: 0 ≤ 𝑏𝑖 𝑗 ≤ 𝑏 max(𝑖 ) ,
∀ 𝑖, 𝑗 .
Equation (6) shows that the pair-level subsidy is proportional to reward 𝑟𝑖 𝑗 , scaled by a global dual variable that reflects the tightness of the daily budget.
(1)
2.2 Let 𝑝𝑖 𝑗 (𝑏𝑖 𝑗 ) denote the probability that driver 𝑗 completes order 𝑖 under subsidy 𝑏𝑖 𝑗 , and let 𝑟𝑖 𝑗 be the platform revenue obtained upon completion. To ensure sustainable operations, the platform enforces a global daily subsidy-rate constraint: total subsidy spending should not exceed a fraction 𝐶 ∈ (0, 1) of total revenue. We allow a tolerance 𝛿 ≥ 0 to accommodate demand volatility and measurement noise. This tolerance is specified by the platform and kept fixed across all experiments. In expectation, the constraint can be written as Í 𝑖,𝑗 𝑏𝑖 𝑗 𝑝𝑖 𝑗 (𝑏𝑖 𝑗 ) Í ≤ 𝐶 + 𝛿. (2) 𝑖,𝑗 𝑟 𝑖 𝑗 𝑝𝑖 𝑗 (𝑏𝑖 𝑗 )
Sequential Subsidy Decision-Making
Equation (6) specifies a tractable pair-level rule given a dual variable, but directly optimizing 𝑏𝑖 𝑗 for millions of order–driver pairs in real time is still impractical. In production, platforms therefore update a single city-level control signal 𝜆𝑡 at a coarse cadence (e.g., every 2 minutes) and apply the induced incentives within each time window. We model this online control process as a Markov Decision Process (MDP). State and action. Let 𝑠𝑡 ∈ S denote the observable market state at time window 𝑡 (e.g., supply–demand gap and spatial distribution). To track constraint satisfaction over the day, we augment the state with the realized subsidy rate w.r.t. GMV: Í Í 𝑡 ′ ≤𝑡 (𝑖,𝑗 ) ∈ E𝑡 ′ 𝑏𝑖 𝑗,𝑡 ′ 𝑦𝑖 𝑗,𝑡 ′ Í 𝜌𝑡 = Í , (7) ′ 𝑡 ≤𝑡 (𝑖,𝑗 ) ∈ E𝑡 ′ 𝑔𝑖 𝑗,𝑡 ′ 𝑦𝑖 𝑗,𝑡 ′
The platform’s objective is to decide subsidies {𝑏𝑖 𝑗 } to maximize expected total revenue subject to the subsidy-rate constraint and 2
D3 -Subsidy: Online and Sequential Driver Subsidy Decision-Making for Large-Scale Ride-Hailing Market
(1) Inputs
(2) Multi-City Pretraining A. Prefix-Conditional Diffusion Prior
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Diffusion Mechanics
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KDD ’26, August 9–13, 2026, Jeju, Republic of Korea
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Figure 1: Overview of the proposed D3 -Subsidy framework. where E𝑡 ′ is the set of broadcasted order–driver pairs in period 𝑡 ′ , 𝑦𝑖 𝑗,𝑡 ′ ∈ {0, 1} indicates whether order 𝑖 is completed by driver 𝑗, and 𝑔𝑖 𝑗,𝑡 ′ denotes the GMV of pair (𝑖, 𝑗) if completed. The augmented state is 𝑥𝑡 = (𝑠𝑡 , 𝜌𝑡 ), and the action is the scalar city-level control 𝜆𝑡 .
control variable 𝑎𝑡 = 𝜆𝑡 at a coarse cadence, and then induces finegrained pair-level subsidies through the dual-based mapping in Equation (6). Given offline trajectories collected from multiple cities, D3 -Subsidy learns a transferable and controllable decision prior that can be deployed under heterogeneous operating regimes (e.g., different subsidy-rate tightness and KPI preferences). The framework consists of two coupled modules: (i) a prefix-conditional diffusion model that generates plausible future city trajectories conditioned on an observed history prefix, and (ii) a context-conditioned inverse dynamics model that decodes the city-level control 𝜆𝑡 from short state windows under the same context. To improve feasibility under strict subsidy-rate caps, we further introduce a constraint-aware scoring objective, and adopt a two-stage training pipeline (multi-city pretraining followed by single-city adaptation) to balance generalization and local calibration.
From city-level control to pair-level subsidies. Given 𝜆𝑡 , the platform deterministically computes incentives using Equation (6): n o 𝑏𝑖 𝑗,𝑡 = min max 0, 𝜅𝑡 𝑟𝑖 𝑗,𝑡 , 𝑏 max(𝑖 ) . (8) This mapping serves as a low-dimensional interface, converting one scalar decision into fine-grained subsidies at scale. Reward and dynamics. The period reward 𝑟𝑡 aggregates the target KPI (e.g., Rides or GMV) over the time window. After executing 𝜆𝑡 and observing outcomes {𝑦𝑖 𝑗,𝑡 }, the realized subsidy rate 𝜌𝑡 is updated accordingly, and the system transitions to 𝑥𝑡 +1 under stochastic market dynamics P (· | 𝑥𝑡 , 𝜆𝑡 ). The platform seeks a policy 𝜋 (𝜆𝑡 | 𝑥𝑡 ) that maximizes long-term performance while keeping 𝜌𝑡 within the subsidy-rate cap.
3.2
Prefix-Conditional Diffusion Model
To avoid ambiguity, we use the subscript 𝑡 ∈ {0, . . . ,𝑇 −1} to denote the index of the time window, and the superscript 𝜏 ∈ {0, 1, . . . , 𝐿} to denote the diffusion step. In particular, 𝜏 = 0 denotes a clean (undiffused) trajectory, while larger 𝜏 indicates higher noise. We represent each city-day as a length-𝑇 clean trajectory 𝜉 = x00:𝑇 −1 = (𝑥 00, . . . , 𝑥𝑇0 −1 ), where each state is
3 D3 -Subsidy 3.1 Overview Building on the sequential decision formulation in Section 2.2, we propose D3 -Subsidy, a diffusion-based offline decision framework for platform-controlled, city-level driver-side subsidy control. In line with industrial practice, the platform updates a single city-level
𝑥𝑡0 = [ 𝑠𝑡 ; 𝜌𝑡 ] ∈ R𝑑 . 3
(9)
KDD ’26, August 9–13, 2026, Jeju, Republic of Korea
Chen et al.
Standard Diffusion
Here 𝑠𝑡 summarizes the observable market condition and 𝜌𝑡 is the realized subsidy rate defined in Section 2.2. Each trajectory is paired with a context vector c that encodes spatial information, temporal semantics, the budget regime, and the target Rides. Given an observed history prefix x00:𝐾 −1 , our goal is to learn a conditional generative prior over the future suffix: 0 0 𝑝𝜃 x𝐾:𝑇 (10) −1 | x0:𝐾 −1 , c .
Sampled / Denoised Full Sequence (Past + Future)
𝑝𝜃 (z𝜏 −1 | z𝜏 , 𝜏, c, 𝐾) =
𝑇−1
Sampled / Denoised Suffix (Generated)
−1 𝑇Ö 𝜏 𝛿 𝑧𝑡𝜏 −1 − 𝑥𝑡0 · N 𝑧𝑡𝜏 −1 ; 𝜇𝜃,𝑡 , 𝛽˜𝜏 𝐼𝑑 . 𝑡 =𝐾
(17) 0 The final generated future is given by the suffix of z0 , i.e., x𝐾:𝑇 −1 = 0 z𝐾:𝑇 −1 . For completeness, pseudocode for the prefix-clamped forward noising and reverse sampling procedures is provided in Appendix B. 3.2.3 Mask-Normalized Denoising Loss (MNDL). A practical challenge in prefix-conditioned training is that the number of supervised positions varies across examples and minibatches due to (i) different prefix lengths 𝐾 and (ii) padding. Without accounting for this variation, the denoising loss scale becomes coupled to the effective suffix length, causing large loss/gradient fluctuations and destabilizing optimization. To address this, we propose the MNDL, which normalizes the denoising error by the number of valid (unmasked) suffix positions. 𝑁 , let 𝑚 For a minibatch of 𝑁 trajectories {𝜉 𝑛 }𝑛=1 𝑛,𝑡 ∈ {0, 1} be a binary mask indicating whether timestep 𝑡 in trajectory 𝜉 𝑛 is a valid suffix position (i.e., 𝑡 ≥ 𝐾 and not padded). We set 𝑚𝑛,𝑡 = 0 for the fixed prefix 𝑡 < 𝐾 and for any padded timesteps. Noise variables 𝜀𝑛,𝑡 are sampled only for suffix positions (𝑡 ≥ 𝐾); the deterministic prefix contributes neither noise nor gradients. We optimize 2 Í𝑁 Í𝑇 −1 𝑛=1 𝑡 =0 𝑚𝑛,𝑡 𝑓𝜃 (z𝜏𝑛 , 𝜏, c𝑛 ) 𝑡 − 𝜀𝑛,𝑡 2 . (18) Ldiff (𝜃 ) = E Í𝑁 Í𝑇 −1 𝑛=1 𝑡 =0 𝑚𝑛,𝑡 MNDL decouples the loss magnitude from the effective horizon, substantially reducing minibatch-to-minibatch variance when 𝐾 varies or when trajectories contain padding.
−1 √ 𝑇Ö 𝛿 𝑧𝑡𝜏 − 𝑥𝑡0 · N 𝑧𝑡𝜏 ; 𝛼¯𝜏 𝑥𝑡0, (1 − 𝛼¯𝜏 )𝐼𝑑 , 𝑡 =𝐾
(12) where 𝛿 (·) denotes the Dirac delta distribution, which places all its mass at 𝑧𝑡𝜏 = 𝑥𝑡0 and thus leaves the prefix tokens 𝑡 ∈ [0, 𝐾 −1] unchanged (i.e., no noise is added to the prefix), and 𝐼𝑑 is the 𝑑 × 𝑑 identity matrix. Equivalently, for each 𝑡 ≥ 𝐾 we can sample √ √ 𝑧𝑡𝜏 = 𝛼¯𝜏 𝑥𝑡0 + 1 − 𝛼¯𝜏 𝜀𝑡 , 𝜀𝑡 ∼ N (0, 𝐼𝑑 ), (13) and set 𝑧𝑡𝜏 = 𝑥𝑡0 for all 𝑡 < 𝐾 deterministically. 3.2.2 Reverse process. At inference time, we initialize the suffix as −1 ∼ N (0, 𝐼 ) and clamp the prefix by setting 𝑧 𝐿 = 𝑥 0 for all {𝑧𝑡𝐿 }𝑇𝑡 =𝐾 𝑑 𝑡 𝑡 𝑡 < 𝐾. We then sample z𝜏 −1 from z𝜏 for 𝜏 = 𝐿, . . . , 1. Let 𝑓𝜃 be a Temporal U-Net denoiser that takes (z𝜏 , 𝜏, c) and outputs a sequence of noise predictions 𝜺ˆ𝜏 = 𝑓𝜃 (z𝜏 , 𝜏, c) ∈ R𝑇 ×𝑑 . We denote by 𝜀ˆ𝑡𝜏 = [ˆ𝜺 𝜏 ] 𝑡 the 𝑡-th slice (time window) of this output and only use it on suffix positions: 𝜀ˆ𝑡𝜏 = 𝑓𝜃 (z𝜏 , 𝜏, c) 𝑡 , 𝑡 ≥ 𝐾. (14)
and use the standard DDPM reverse mean/variance: 𝛽𝜏 1 1 − 𝛼¯𝜏 −1 𝜏 𝑧𝑡𝜏 − √ 𝜀ˆ𝑡𝜏 , 𝛽˜𝜏 = 𝛽𝜏 · . 𝜇𝜃,𝑡 =√ 1 − 𝛼¯𝜏 𝛼𝜏 1 − 𝛼¯𝜏
𝐾 −1 Ö 𝑡 =0
3.2.1 Forward process. We adopt the DDPM parameterization [13] with a cosine noise schedule [18], but restrict noising to the suffix. Formally, the forward distribution over the entire z𝜏 factorizes as
For 𝑡 ≥ 𝐾, we form the implied clean estimate 1 𝜏 √ 𝑥ˆ𝑡0 = √ 𝑧𝑡 − 1 − 𝛼¯𝜏 𝜀ˆ𝑡𝜏 , 𝛼¯𝜏
𝐾−1 𝐾
Crucially, the reverse transition is defined over the entire z with hard prefix constraints:
where x𝜏𝐾:𝑇 −1 denotes the noised suffix at diffusion step 𝜏. Thus, the prefix is enforced for all 𝜏, while only the suffix evolves along the Î diffusion chain. We write 𝛼𝜏 = 1 − 𝛽𝜏 and 𝛼¯𝜏 = 𝜏𝑖=1 𝛼𝑖 .
𝑡 =0
0
Fixed History / Prefix (Real Data)
Figure 2: Comparison of standard trajectory diffusion and prefix-conditional diffusion.
z𝜏 ≜ (x00:𝐾 −1, x𝜏𝐾:𝑇 −1 ).
𝐾 −1 Ö
𝑇−1
0
Crucially, the prefix corresponds to realized history and must remain immutable. Therefore, in both training and inference, we treat the prefix as a hard condition and diffuse/denoise only the suffix. As illustrated in Figure 2, we clamp the realized history prefix and apply diffusion-based denoising/sampling only to the future suffix. −1 over the entire We define a single diffusion variable z𝜏 = {𝑧𝑡𝜏 }𝑇𝑡 =0 horizon: ( 𝑥 0, 𝑡 < 𝐾, 𝜏 𝑧𝑡 ≜ 𝑡𝜏 𝑥𝑡 , 𝑡 ≥ 𝐾, (11)
𝑞(z𝜏 | x0, 𝐾) =
Prefix-Conditional (Ours)
3.3
Context-Conditioned Inverse Dynamics
Rather than directly diffusing actions, we infer the city-level control via inverse dynamics. This design is motivated by city-scale subsidy control, where the same state transition can correspond to different actions under different operating targets and constraint regimes. For example, a transition that is desirable under a growth-oriented target may be infeasible (or suboptimal) under a tighter subsidy cap. As a result, an unconditional inverse dynamics model 𝑔(𝑧𝑡0−2:𝑡 +1 ) tends to average across heterogeneous objectives and produces actions that are poorly calibrated for deployment-time steering. We therefore propose a context-conditioned inverse dynamics decoder that treats the operating target and constraint regime as an
(15)
(16) 4
D3 -Subsidy: Online and Sequential Driver Subsidy Decision-Making for Large-Scale Ride-Hailing Market
4 Experiments 4.1 Experimental Setup
explicit task specification. Let c encode city-level embedding, temporal covariates (e.g., time-of-day and day-of-week), budget regime, and target Rides preference. We decode the city-level control as 𝜆ˆ𝑡 = 𝑔𝜙 (𝑧𝑡0−2, 𝑧𝑡0−1, 𝑧𝑡0, 𝑧𝑡0+1, c).
4.1.1 Dataset. We compile an offline dataset from DiDi broadcast logs to evaluate city-level subsidy control. It contains three aligned versions that use the same cities/days, state–action setup, and evaluation protocol, and differ only in the aggregation window (2/5/10 minutes). The data are stored as city-day trajectories. At each step, we aggregate city-level supply–demand signals and context, log the average online control 𝜆, and compute key KPIs including Rides, GMV, and driver revenue (DRV). The dataset covers 133 Brazilian cities over 28 days; full statistics are in Table 7 in Appendix C. To evaluate cold-start transfer, we reserve 3 cities and exclude all their city-days from training.
(19)
Here 𝑧𝑡0 denotes the clean (final) prefix-clamped trajectory at diffusion step 𝜏 = 0: during training, 𝑧𝑡0 is constructed from the groundtruth trajectory, whereas at inference it is obtained from the sampled trajectory z0 produced by the diffusion prior. This conditioning makes the decoder controllable: varying c at deployment enables the same diffusion prior to yield different valid controls consistent with the desired objective–constraint trade-off. We train 𝑔𝜙 with a mask-normalized mean-squared error "Í # inv ˆ 2 𝑛,𝑡 𝑚𝑛,𝑡 ∥ 𝜆𝑛,𝑡 − 𝜆𝑛,𝑡 ∥ 2 Linv (𝜙) = E , (20) Í inv 𝑛,𝑡 𝑚𝑛,𝑡 + 𝜖
4.1.2
Constraint-Aware Score
To balance the primary KPI with strict subsidy-rate feasibility, we introduce a constraint-aware score defined on a trajectory 𝜉. Let Rides(𝜉) be the number of Rides and let 𝐶 real (𝜉) be the realized subsidy rate over 𝜉. Given a subsidy-rate cap 𝐶 ∈ (0, 1) and a penalty exponent 𝛽 > 0, we define 𝛽 𝐶 Rides(𝜉), if 𝐶 real (𝜉) > 𝐶 + 𝛿, Score(𝜉) = 𝐶 real (𝜉) (21) Rides(𝜉), otherwise. When the subsidy-rate constraint is satisfied, the score equals the KPI. When the cap is violated, the score applies a smooth multiplicative penalty that increases with the degree of violation. The exponent 𝛽 controls the penalty strength and provides a simple knob to trade off Rides improvement against subsidy-rate compliance.
3.5
All offline RL baselines are trained to maximize the Score(𝜉). Therefore, the compared offline RL methods already correspond to soft-constrained policy learning under the subsidy-rate cap. 4.1.3 Evaluation. For offline evaluation, we hold out 7 consecutive days from 3 cities (City A, City B, and City C), yielding 21 cityday trajectories for testing; all remaining trajectories are used for training. We report Score(𝜉) as the primary metric and also report Rides, GMV, and DRV. All policies are evaluated via closed-loop rollouts in DiDi’s high-fidelity production simulator. Simulator fidelity is verified by replaying the production Online policy, achieving 7.49% MAPE on daily Rides over the held-out trajectories. At each time window, the policy maps the current city-level state to an action 𝜆𝑡 . The simulator applies 𝜆𝑡 together with the broadcast context within the time window, and uses its internal predictive model to generate the resulting KPIs and the next state. Rolling out the full day yields a simulated city-day trajectory 𝜉, from which we compute Score(𝜉) via Equation (21). For each city, we report the mean Score(𝜉) over its 7 test days for each method and compare algorithms using these city-level averages. In production, we allow a small upper tolerance band and count a violation only if 𝐶 real (𝜉) > 𝐶 + 𝛿. We additionally report under-utilization as UnderGap(𝜉) = max(0, 𝐶 − 𝐶 real (𝜉)), where smaller is better.
Two-Stage Training
We first pretrain (𝑓𝜃 , 𝑔𝜙 ) on pooled multi-city trajectories and condition both modules on context c. This stage learns (i) a transferable diffusion prior over city dynamics and (ii) a generic inversedynamics decoder for the city-level control 𝑎𝑡 = 𝜆𝑡 . To improve deployment calibration on a target city 𝑐 ∗ using offline-only data, we apply parameter-efficient fine-tuning (PEFT): we freeze the diffusion model parameters and fine-tune only the inverse dynamics decoder. Let (𝜃 0, 𝜙 0 ) denote pretrained parameters. We optimize min
Linv (𝜙) + 𝜆anc ∥𝜙 − 𝜙 0 ∥ 22,
Baselines. We compare against the following baselines:
• Online: DiDi’s production subsidy policy (predict-then-optimize). • BC: behavior cloning—supervised policy learning from logged actions. • BCQ [10]: offline RL that restricts actions to the dataset support (and nearby) to curb out-of-distribution (OOD) overestimation. • CQL [16]: conservative offline RL that penalizes the 𝑄-function to reduce overestimation. • IQL [15]: offline RL that improves BC via advantage-weighted regression without explicit OOD action maximization. • TD3+BC [9]: TD3 updates with a BC regularizer to limit deviation from logged behavior. • DT (Decision Transformer) [4]: an autoregressive Transformer policy that generates actions from trajectory history conditioned on a target return-to-go. • DD (Decision Diffuser) [1]: a diffusion-based generative policy that produces actions via iterative denoising, conditioned on the current context and a target return.
inv masks padded timesteps in the 𝑛-th trajectory. where 𝑚𝑛,𝑡 Finally, the decoded city-level control 𝜆ˆ𝑡 is converted into deployable pair-level subsidies by applying the closed-form dual mapping in Equation (6) (with 𝑟𝑖 𝑗 replaced by 𝑟𝑖 𝑗,𝑡 for window 𝑡), i.e., 𝑏𝑖 𝑗,𝑡 = 𝑏𝑖∗𝑗 (𝜆ˆ𝑡 ).
3.4
KDD ’26, August 9–13, 2026, Jeju, Republic of Korea
(22)
𝜙
where the anchor term acts as a trust-region surrogate to prevent excessive drift from the pretrained decoder. In practice, we use a smaller learning rate than in pretraining and update only 𝜙 while keeping 𝜃 = 𝜃 0 fixed. 5
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Chen et al.
Table 1: Offline evaluation score. Scores are averaged over the 7 hold-out days. Best in bold, second-best underlined. City
Time Window
D3 -Subsidy (Ours)
Online
BC
BCQ
CQL
IQL
TD3+BC
DT
DD
City A
2-min 5-min 10-min Average
10802.72 10824.43 10712.05 10779.73
10368.08 10359.22 10341.50 10356.27
10441.40 10324.27 10423.46 10396.38
10494.38 10445.51 10260.77 10400.22
10382.75 10513.69 10659.68 10518.71
10685.53 10612.73 10262.76 10520.34
10638.74 10452.29 10696.46 10595.83
10784.55 10618.92 10658.45 10687.31
10730.26 10488.95 10332.50 10517.24
City B
2-min 5-min 10-min Average
2008.16 2000.00 2018.06 2008.74
1940.09 1938.57 1936.16 1938.27
1910.12 1911.20 1938.49 1919.94
1992.22 1883.80 1908.35 1928.12
1981.83 1958.27 1967.94 1969.35
1963.18 1953.19 1973.27 1963.21
1981.78 1958.20 1988.49 1976.16
1957.06 1954.61 1984.76 1965.48
1992.83 1987.89 1983.93 1988.22
City C
2-min 5-min 10-min Average
1736.80 1792.48 1718.31 1749.20
1650.70 1643.23 1631.43 1641.79
1628.70 1613.38 1670.16 1637.41
1694.30 1657.01 1597.11 1649.47
1669.96 1661.04 1689.27 1673.42
1656.00 1642.01 1679.28 1659.10
1686.09 1650.17 1649.69 1661.98
1722.55 1674.31 1672.04 1689.63
1709.86 1737.39 1717.02 1721.42
Overall Average
4845.89
4645.44
4651.24
4659.27
4720.49
4714.22
4744.66
4780.81
4742.29
Note: “Average” rows for each city represent the mean value over time windows; “Overall Average” is computed by averaging over all city samples.
4.2
Table 2: Ablation results in City C (5-min).
Main Results
Table 1 summarizes the city-level results on the test split. D3 Subsidy achieves the best performance across all cities and settings, delivering the highest overall average score and consistently outperforming the online strategy. Compared with representative offline RL baselines, our gains remain consistent. In particular, DT is the strongest competitor among baselines on average, yet still falls behind D3 -Subsidy. The gain holds across all windows, suggesting robust city-level subsidy control under the same subsidy-rate constraint. Since the performance trends are consistent across temporal granularities, we use the 5-minute variant in the remaining experiments for ablations and in-depth analysis.
4.3
Score
Compare
D3 -Subsidy
1792.48 1737.58 1704.91 1715.78
– −3.1% −4.9% −4.3%
D3 -Subsidy-C D3 -Subsidy-M D3 -Subsidy-P
Scoreours (𝜉) − Scorebaseline (𝜉) and perform a one-sided paired 𝑡test with 𝐻 0 : E[Δ] ≤ 0 and 𝐻 1 : E[Δ] > 0. Table 3 reports the mean gain, 95% confidence interval, and 𝑡-statistic. D3 -Subsidy yields significant improvements over all baselines (all 𝑝 < 0.05), indicating that the advantage is robust and reproducible.
Ablation Studies
We elucidate the contribution of key design components in D3 Subsidy through an ablation study by evaluating the following variants:
Table 3: Statistical significance of D3 -Subsidy improvements.
• D3 -Subsidy-C: removes the Condition in the inverse dynamics module, predicts the control signal without trajectory-conditioned decoding. • D3 -Subsidy-M: removes multi-city pretraining, training only on the target city’s trajectories. • D3 -Subsidy-P: removes the PEFT/fine-tuning stage and directly applies the pretrained backbone for inference. The results are reported in Table 2. As shown, removing any key component leads to a clear degradation in score, validating that the performance gain of D3 -Subsidy is not driven by a single component. Removing conditioning (C), multi-city pretraining (M), and PEFT (P) reduces the score by 3.1%, 4.9%, and 4.3%, respectively, with the largest drop from w/o multi-city pretraining.
4.4
Model
Baseline
Mean Diff.
95% Confidence Interval
𝑡 20
𝑝-value
Online BC BCQ CQL IQL TD3+BC DT DD
+225.28 +256.01 +210.18 +161.29 +136.31 +185.40 +123.01 +134.21
[120.79, 329.78] [155.05, 356.96] [108.35, 312.02] [78.98, 243.60] [69.55, 203.08] [102.08, 268.73] [53.81, 192.21] [17.30, 251.12]
4.50 5.29 4.31 4.09 4.26 4.64 3.71 2.39
< 0.001 < 0.001 < 0.001 < 0.001 < 0.001 < 0.001 < 0.001 0.013
4.4.2 Temporal Dynamics of KPI under Subsidy Control. Beyond average Score(𝜉), we further inspect how different algorithms shape intra-day operational dynamics under the same subsidy rate control. Using City C as a representative case, we visualize the time evolution of key KPIs at the 5-minute time window. As shown in Figure 3a, our method delivers steady daily improvements over the online strategy in Rides, GMV, and DRV, suggesting the overall gain does not rely on trade-offs among these indicators.
In-depth Analysis
4.4.1 Statistical Comparison. To verify that the Score(𝜉) gains are consistent rather than driven by a few trajectories, we perform paired significance tests against each baseline on the held-out set (3 cities × 7 days, 𝑁 =21). For each trajectory 𝜉, we compute Δ(𝜉) = 6
D3 -Subsidy: Online and Sequential Driver Subsidy Decision-Making for Large-Scale Ride-Hailing Market
KDD ’26, August 9–13, 2026, Jeju, Republic of Korea
Normalized Value
1.0
GMV
0.8
Rides
0.6
Score
0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
Score Factor
Figure 5: KPI-conditional policy steering.
0.53
Diffusion Loss Inv Loss
0.52 0.51
180
160
0.50 0.49
140 200
Diffusion Loss Inv Loss
1.08 1.06
200 180
1.04 160
1.02 1.00
Inv Loss
Diffusion Loss
Figure 3: Intra-day Rides, GMV and DRV dynamics in City C.
Inv Loss
(b) Per-window
Diffusion Loss
(a) Cumulative
140
400
200
400
Epoch
Epoch
(a) w/o MNDL
(b) w/ MNDL
Figure 4: Daily Subsidy Rate in City C.
Figure 6: Training loss comparison under different settings.
Notably, the advantage accumulates steadily over time within each day, suggesting a stable and sustained improvement rather than a chance lead. To understand where the cumulative gain comes from, Figure 3b reports per-window KPI trajectories, showing that our method preserves a similar daily pattern to the online strategy while remaining consistently higher across most time windows. Figure 4 visualizes the daily subsidy rate as deviation from the cap 𝐶 (reported in this form for confidentiality), where the shaded area indicates the allowed upper tolerance band. Compared with the online strategy, D3 -Subsidy ends near 𝐶 (or slightly above) with no overshoot beyond 𝐶 + 𝛿 on the held-out test days, while maintaining a consistently small UnderGap. Results for the other held-out test cities are provided in Appendix F (Figures 9–14).
preventing the loss scale from drifting when the effective prediction horizon changes due to padding or varying prefix lengths. We also note that the diffusion loss may appear numerically larger after introducing MNDL. This increase is expected because the normalization denominator changes, so the absolute magnitudes across the two losses are not on the same scale. In practice, we focus on the improved stability and consistent convergence behavior rather than the raw loss values. 4.4.5 Cold-Start Performance. Cold-start deployment is common in practice because a new city may have no city-specific offline data at launch. We evaluate cold-start transfer on 3 held-out cities and exclude all their city-days from training. PEFT is not applicable in this setting since it requires target-city offline data to fine-tune the inverse-dynamics decoder. Table 4 reports results on the 5-min setting, averaged over 7 consecutive days. D3 -Subsidy achieves the highest average score in the cold-start setting, outperforming both the online strategy and DT, which indicates stronger cross-city generalization.
4.4.3 Target-KPI Controllability Analysis. A practical requirement for subsidy control is deployment-time steerability: the platform should be able to shift operating targets by adjusting the targetKPI inputs without retraining. To test this, we scale the target-KPI signals in the context by 𝛾 ∈ 0.2, 0.4, . . . , 2.0 while keeping the subsidy-rate constraint fixed, and roll out the resulting policies aggregated over the three held-out test cities. As shown in Figure 5, varying 𝛾 induces the expected shifts in realized Score, Rides, and GMV, indicating that the model converts target-KPI conditioning into controllable subsidy decisions and outcomes.
Table 4: Offline evaluation score on cold-start cities. Best in bold, second-best underlined.
4.4.4 Stabilizing Diffusion Training via MNDL. As shown in Figure 6, replacing the vanilla denoising loss with MNDL yields more stable optimization, and the loss curve accordingly becomes smoother with large fluctuations significantly reduced. MNDL normalizes the denoising error by the number of valid (unmasked) suffix steps, 7
City
D3 -Subsidy (Ours)
Online
DT
City D City E City F
532.20 285.54 1152.65
532.06 277.44 1142.31
531.74 266.66 1132.58
Average
656.80
650.60
643.66
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Chen et al.
Table 6: Online A/B Test Result.
Sensitivity Analysis
4.5.1 Sensitivity to Reverse Diffusion Steps. We sweep reverse diffusion steps 𝜏 ∈ {10, 50, 100, 150} in City C while keeping other settings fixed (Figure 7). Performance peaks at 50 steps: fewer steps degrade quality and more steps do not consistently help. Overall, this sweep confirms that our default choice of 50 steps is both effective and reasonably robust to step variations.
5
4.5.2 Sensitivity to the Penalty Exponent 𝛽. Penalty exponent 𝛽 controls how strongly trajectories are penalized when the realized daily subsidy rate exceeds the target cap 𝐶. Table 5 reports results under 𝛽 ∈ {0.5, 1.0, 2.0} in City C. Notably, the online strategy stays below the cap in this city, so its score remains unchanged as 𝛽 varies. In contrast, D3 -Subsidy aims to closely track the target budget and may finish slightly above the soft cap on some days; therefore, increasing 𝛽 leads to a gradual decrease in score as expected. Importantly, D3 -Subsidy consistently outperforms the online strategy across all tested 𝛽 values, demonstrating that our improvement is robust and not driven by a particular choice of 𝛽. Table 5: Score sensitivity to penalty exponent 𝛽 in City C.
4.6
D3 -Subsidy (Ours)
Online
0.5 1.0 2.0
1792.48 1747.28 1662.15
1643.23 1643.23 1643.23
Rides
GMV
DRV
Inference Time
compare
+1.59%
+2.06%
+2.31%
+20 ms
Related Work
Offline RL. Offline RL is widely adopted in real-world systems where online exploration is prohibitively expensive or risky [8]. It learns policies from static logged data but suffers from distribution shift when generalizing beyond observed actions. Classical offline RL methods, such as BCQ [10] and IQL [15], mitigate this issue through behavior constraints or conservative value estimation. More recently, generative decision models have emerged as an alternative paradigm. Chen et al. [4] formulate offline RL as conditional sequence modeling using Transformers, enabling long-horizon decision making without explicit value functions. Diffusion models have also been introduced into offline RL; for example, HansenEstruch et al. [12] combine diffusion-based generation with implicit Q-learning to improve robustness. Diffusion Models for Decision Making. Diffusion models provide strong generative priors for complex, multi-modal data and support conditional generation with diverse samples [2, 5], making them attractive for decision modeling from offline data. Prior work has connected diffusion models to offline RL and policy learning [12, 21], as well as to behavior and trajectory modeling with temporal conditioning [3, 14]. Beyond RL, diffusion-based models have been applied to constrained decision-making and industrial planning, including autonomous driving and large-scale bidding systems [11, 17, 20, 26]. In contrast, diffusion-based frameworks have not been explored for ride-hailing automated driver-subsidy control, where decisions must be made under non-stationary dynamics and subsidy-rate constraints. Automated Driver-Subsidy Control in Ride-Hailing. Existing automated driver-subsidy control methods in ride-hailing primarily differ in subsidy granularity and decision scope. Zhang et al. [25] study acquisition subsidies and show that they are most effective when one side of the two-sided market remains thin. Zhu et al. [27] formulate spatiotemporal subsidies as sequential control using a mean-field MDP and advocate zone-based surge subsidies over uniform payments. Moving to finer granularity, Xie et al. [22] propose personalized incentives by estimating heterogeneous treatment effects. Yang et al. [23] consider city-level budget allocation and propose a decision-focused learning framework that optimizes predictive models for subsidy allocation objectives. However, their formulation treats budget allocation as a static or single-stage decision problem and does not explicitly model the sequential MDP structure induced by budget consumption and evolving system dynamics.
Figure 7: Score under different diffusion steps in City C.
Penalty exponent 𝛽
Metrics
Online A/B Test
We further validate D3 -Subsidy via a 7-day online A/B test on DiDi’s production system (Feb 1–7, 2026) in a city, using the online strategy as the baseline. Orders are randomly assigned to the online strategy and D3 -Subsidy, with 50% of orders allocated to each variant (202,279 broadcasts in total), and no other concurrent strategies or operational campaigns were running in the test scope. Both variants share the same subsidy-rate cap, traffic scope, and serving constraints. Across all 7 days, the subsidy-rate cap was never violated. Table 6 reports the relative lift on Rides, GMV, and DRV: D3 -Subsidy delivers consistent gains, confirming that offline improvements translate to real-market benefits while meeting the cap. Inference is operationally feasible: compared with the baseline, it adds around 20 ms per decision.
6
Conclusion
In this paper, we propose D3 -Subsidy for automated driver-subsidy control decisions in the ride-hailing market, which can optimize the KPI and satisfy the subsidy-rate constraint at the same time. Extensive experiments on offline simulation and the online A/B testing demonstrate the effectiveness of D3 -Subsidy. In the future, 8
D3 -Subsidy: Online and Sequential Driver Subsidy Decision-Making for Large-Scale Ride-Hailing Market
KDD ’26, August 9–13, 2026, Jeju, Republic of Korea
References
scholars can develop new methods to improve the efficiency and controllability of D3 -Subsidy.
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Algorithm 2 Prefix-Conditional Trajectory Diffusion Sampling
[24] Enpeng Yuan and Pascal Van Hentenryck. 2021. Real-time pricing optimization for ride-hailing quality of service. In 30th International Joint Conference on Artificial Intelligence (IJCAI-21. [25] Qi Zhang, Yang Liu, and Zhi-Ping Fan. 2023. Short-term subsidy strategy for new users of ride-hailing platform with user base. Computers & Industrial Engineering 179 (2023), 109177. [26] Yinan Zheng, Ruiming Liang, Kexin ZHENG, Jinliang Zheng, Liyuan Mao, Jianxiong Li, Weihao Gu, Rui Ai, Shengbo Eben Li, Xianyuan Zhan, et al. [n. d.]. Diffusion-Based Planning for Autonomous Driving with Flexible Guidance. In The Thirteenth International Conference on Learning Representations. [27] Zheng Zhu, Jintao Ke, and Hai Wang. 2021. A mean-field Markov decision process model for spatial-temporal subsidies in ride-sourcing markets. Transportation Research Part B: Methodological 150 (2021), 540–565.
A
Require: Observed prefix x00:𝐾 −1 , context c, diffusion steps 𝐿, 𝐿 , denoiser 𝑓 schedule {𝛼𝜏 , 𝛽𝜏 , 𝛼¯𝜏 , 𝛽˜𝜏 }𝜏=1 𝜃 0 Ensure: Sampled future suffix x𝐾:𝑇 −1 Initialize (clamp prefix, randomize suffix). 1: for 𝑡 ← 0 to 𝐾 − 1 do 2: 𝑧𝑡𝐿 ← 𝑥𝑡0 ⊲ hard prefix 3: end for 4: for 𝑡 ← 𝐾 to 𝑇 − 1 do 5: 𝑧𝑡𝐿 ∼ N (0, 𝐼𝑑 ) ⊲ noisy suffix 6: end for Reverse diffusion (denoise suffix with hard prefix). 7: for 𝜏 ← 𝐿, 𝐿 − 1, . . . , 1 do 8: z𝜏 ← Clamp𝐾 (z𝜏 ; x00:𝐾 −1 ) 9: 𝜺ˆ𝜏 ← 𝑓𝜃 (z𝜏 , 𝜏, c) ⊲ 𝜺ˆ𝜏 ∈ R𝑇 ×𝑑 10: for 𝑡 ← 𝐾 to 𝑇 − 1 do 11: 𝜀ˆ𝑡𝜏 ← [ˆ𝜺 𝜏 ] 𝑡 ⊲ 𝑡-th slice (time window)
Operations in Ride-hailing Platforms
In a ride-hailing platform, operational decisions arise from the continuous interactions among passengers, drivers, and the platform. When a passenger submits a trip request, the platform first assesses the local supply-demand conditions (e.g., nearby driver availability and anticipated demand) and then determines a driver-side subsidy to influence driver participation. This subsidy is offered to nearby drivers as additional compensation for accepting and completing the order, thereby shaping drivers’ acceptance decisions based on perceived profitability and travel distance [6]. Given drivers’ responses to the subsidy offer, the platform proceeds to match the order to one or more candidate drivers and updates system states accordingly (e.g., driver availability, spatial distribution of demand). This closed-loop process repeats over time and across regions, forming a large-scale, dynamic marketplace in which subsidy interventions serve as a key lever for steering driver behavior and improving driver-passenger matching outcomes.
𝛽
𝜏 ← √1 𝑧𝜏 − √ 𝜏 𝜀ˆ𝜏 𝜇𝜃,𝑡 𝑡 𝛼𝜏 1−𝛼¯𝜏 𝑡 13: if 𝜏 > 1 then 𝜏 , 𝛽˜ 𝐼 ) 14: 𝑧𝑡𝜏 −1 ∼ N (𝜇𝜃,𝑡 𝜏 𝑑 15: else 1 16: 𝑧𝑡0 ← 𝜇𝜃,𝑡 17: end if 18: end for 19: z𝜏 −1 ← Clamp𝐾 (z𝜏 −1 ; x00:𝐾 −1 ) 20: end for 0 0 21: return x𝐾:𝑇 −1 ← z𝐾:𝑇 −1
12:
C
B Prefix-Conditional Diffusion: Pseudocode B.1 Forward Noising with a Clamped Prefix
⊲ maintain hard prefix
Data Description
This appendix reports summary statistics of our offline dataset under three temporal granularities (2/5/10 minutes). All three versions use the same cities/days and the same state–action setup and evaluation protocol.
Given a clean trajectory x00:𝑇 −1 , we keep the prefix deterministic for all diffusion steps and add Gaussian noise only to the suffix. Here 𝐼𝑑 denotes the 𝑑 × 𝑑 identity matrix.
Table 7: Data statistics under different time windows.
Algorithm 1 Forward Noising for Prefix-Conditional Diffusion Require: Clean trajectory x00:𝑇 −1 , prefix length 𝐾, diffusion step 𝜏 ∈ {1, . . . , 𝐿}, 𝛼¯𝜏 , state dimension 𝑑 Ensure: Noised variable z𝜏 = (x00:𝐾 −1, x𝜏𝐾:𝑇 −1 ) 1: z𝜏0:𝐾 −1 ← x00:𝐾 −1 ⊲ clamp prefix (no noise) 2: 𝜺 𝐾:𝑇 −1 ∼ N (0, 𝐼𝑑 ) ⊲ (𝑇 −𝐾) × 𝑑 suffix noise √ √ 0 ¯ 3: z𝜏𝐾:𝑇 −1 ← 𝛼¯𝜏 x𝐾:𝑇 1 − 𝛼 𝜺 + 𝜏 𝐾:𝑇 −1 −1 4: return z𝜏
B.2
⊲ final step: no noise
D
Parameters
2-min
5-min
10-min
Trajectories (city×day) Cities / Days Trajectory length State dimension Action dimension Return-To-Go dimension Action range
3,724 133 / 28 720 20 1 1 (0, 30]
3,724 133 / 28 288 20 1 1 (0, 30]
3,724 133 / 28 144 20 1 1 (0, 30]
Closed-Form Dual-Based Subsidy under a Linear Completion Model
We now present the complete derivation of the closed-form optimal subsidy under the linear completion model introduced in the main text. The result is stated as a lemma followed by its proof.
Reverse Sampling with a Clamped Prefix
Algorithm 2 summarizes the reverse sampling procedure. We define a clamping operator Clamp𝐾 (·; x00:𝐾 −1 ) that overwrites the first 𝐾 states by the observed prefix, ensuring exact feasibility of the hard constraint at every step.
Lemma D.1 (Closed-Form Optimal Subsidy under Linear Completion). Suppose the completion probability is linear in the subsidy: 𝑝𝑖 𝑗 (𝑏𝑖 𝑗 ) = 𝑎𝑖 𝑗 𝑏𝑖 𝑗 , 10
𝑎𝑖 𝑗 > 0.
D3 -Subsidy: Online and Sequential Driver Subsidy Decision-Making for Large-Scale Ride-Hailing Market
KDD ’26, August 9–13, 2026, Jeju, Republic of Korea
Intra-Window Execution with FIXED GLOBAL 𝝀𝒕 (during Window 𝑡)
Inter-Window Transition (at start of window 𝒕) Observe state (From Window 𝒕-1)
Sequence broadcast loop
Broadcast aggregates (supply/demand, value) + evaluated outcomes
Assign available driver 𝒋 Order 𝒊 arrives
Real-time subsidizing
···
∗ Calculate subsidy 𝑏"# 𝜆% using 𝝀𝒕
Update global action (For Window 𝒕)
𝑏!",$ = min max 0, 𝜅$ 𝑟!",$ , 𝑏%&' 𝑖
Next driver 𝑗~𝐽!
Compute new dual variable 𝝀𝒕 based on a specific algorithm.
START
NO
Outcome Reward 𝑟"#,% realized. Stop broadcast for order 𝑖 .
Time Window 𝑡+1
Time Window 𝒕
Time Window 𝑡-1
···
YES
Driver 𝑗 accepts?
···
END
Figure 8: Problem Formulation Let 𝐶 ∈ (0, 1) be the global subsidy-rate cap. Consider the primal problem ∑︁ max 𝑟 𝑖 𝑗 𝑎𝑖 𝑗 𝑏 𝑖 𝑗 , 𝑏𝑖 𝑗
Introduce a Lagrange multiplier 𝜆 ≥ 0. Using the standard form L (𝑏, 𝜆) = 𝑅(𝑏) − 𝜆 𝐶 (𝑏) − (𝐶 + 𝛿)𝑅(𝑏) , the Lagrangian is ∑︁ L (𝑏, 𝜆) = (1 + 𝜆(𝐶 + 𝛿)) 𝑟𝑖 𝑗 𝑎𝑖 𝑗 𝑏𝑖 𝑗 − 𝜆𝑎𝑖 𝑗 𝑏𝑖2𝑗 .
𝑖,𝑗 𝑖,𝑗
s.t.
∑︁
𝑎𝑖 𝑗 𝑏𝑖2𝑗 − (𝐶 + 𝛿)
𝑖,𝑗
∑︁
𝑟𝑖 𝑗 𝑎𝑖 𝑗 𝑏𝑖 𝑗 ≤ 0,
Because the Lagrangian is separable across (𝑖, 𝑗), we solve the inner maximization independently for each pair: max (1 + 𝜆(𝐶 + 𝛿)) 𝑟𝑖 𝑗 𝑎𝑖 𝑗 𝑏𝑖 𝑗 − 𝜆𝑎𝑖 𝑗 𝑏𝑖2𝑗 .
𝑖,𝑗
0 ≤ 𝑏𝑖 𝑗 ≤ 𝑏 max(𝑖 ) , ∀𝑖, 𝑗 .
0≤𝑏𝑖 𝑗 ≤𝑏 max(𝑖 )
Let 𝜆 ≥ 0 be the Lagrange multiplier associated with the subsidyrate constraint. Then the optimal subsidy for each (𝑖, 𝑗) under dual parameter 𝜆 (with 𝜆 > 0) is 𝑏𝑖∗𝑗 (𝜆) = min max{0, 𝜅𝑟𝑖 𝑗 }, 𝑏 max(𝑖 ) ,
Ignoring the box constraint temporarily, the first-order condition is 𝜕L = (1 + 𝜆(𝐶 + 𝛿)) 𝑟𝑖 𝑗 𝑎𝑖 𝑗 − 2𝜆𝑎𝑖 𝑗 𝑏𝑖 𝑗 = 0. 𝜕𝑏𝑖 𝑗
where
Solving for 𝑏𝑖 𝑗 yields the stationary point
𝐶 + 𝛿 + 1/𝜆 . 𝜅= 2 Letting 𝜆 ∗ denote the optimal dual multiplier, the final dual-based optimal subsidy is 𝑏𝑖∗𝑗 = min max{0, 𝜅𝑟𝑖 𝑗 }, 𝑏 max(𝑖 ) .
𝑏𝑖◦𝑗 (𝜆) =
𝐶 + 𝛿 + 1/𝜆 (1 + 𝜆(𝐶 + 𝛿)) 𝑟𝑖 𝑗 = 𝑟𝑖 𝑗 , 2𝜆 2
and defining 𝜅=
𝐶 + 𝛿 + 1/𝜆 2
gives 𝑏𝑖◦𝑗 (𝜆) = 𝜅𝑟𝑖 𝑗 .
Proof. Under the linear completion model, the expected reward and total subsidy are ∑︁ ∑︁ 𝑅(𝑏) = 𝑟 𝑖 𝑗 𝑎𝑖 𝑗 𝑏 𝑖 𝑗 , 𝐶 (𝑏) = 𝑎𝑖 𝑗 𝑏𝑖2𝑗 . 𝑖,𝑗
The feasible region is the interval [0, 𝑏 max(𝑖 ) ]. Thus the optimal solution is the projection of the stationary point onto this interval: 𝑏𝑖∗𝑗 (𝜆) = min max{0, 𝜅𝑟𝑖 𝑗 }, 𝑏 max(𝑖 ) .
𝑖,𝑗
The subsidy-rate constraint 𝐶 (𝑏)/𝑅(𝑏) ≤ 𝐶 is equivalent to ∑︁ ∑︁ 𝑎𝑖 𝑗 𝑏𝑖2𝑗 − (𝐶 + 𝛿) 𝑟𝑖 𝑗 𝑎𝑖 𝑗 𝑏𝑖 𝑗 ≤ 0. 𝑖,𝑗
The dual function is ∑︁ 𝑔(𝜆) = max
𝑖,𝑗
𝑖,𝑗
11
0≤𝑏𝑖 𝑗 ≤𝑏 max(𝑖 )
(1 + 𝜆(𝐶 + 𝛿)) 𝑟𝑖 𝑗 𝑎𝑖 𝑗 𝑏𝑖 𝑗 − 𝜆𝑎𝑖 𝑗 𝑏𝑖2𝑗 ,
KDD ’26, August 9–13, 2026, Jeju, Republic of Korea
Chen et al.
and the dual problem is min𝜆≥0 𝑔(𝜆). Letting 𝜆 ∗ denote the optimal multiplier and substituting it into the expression above yields 𝑏𝑖∗𝑗 = min max{0, 𝜅𝑟𝑖 𝑗 }, 𝑏 max(𝑖 ) , which completes the proof.
E
exactly the clipped form in (23). This establishes statement (2) and completes the proof. □
F
□
Temporal Dynamics of KPIs in Additional Cities
This appendix provides the temporal KPI trajectories for the two additional held-out test cities (City A and City B), complementing the City C case in the main text. For each city, we show (i) cumulative Rides/GMV/DRV, (ii) per-window Rides/GMV/DRV curves (5-minute), and (iii) the day-to-date subsidy rate as deviation from the cap 𝐶. Figures 9–11 correspond to City A and exhibit the same pattern as in City C: D3 -Subsidy delivers consistent gains across KPIs while keeping the realized subsidy rate tightly aligned with the cap under the same tolerance-band criterion. Figures 12–14 correspond to City B and confirm similar behavior in another held-out market, with steady KPI uplift and stable subsidy rate control throughout the day.
General Completion Models: Optimal Subsidy Characterization
We now provide a general structural result for the dual-based subsidy when the completion function 𝑝𝑖 𝑗 (𝑏) is nonlinear (e.g., logistic or saturating). The result shows that the optimal subsidy retains the same clipped form as in the linear case, except that the interior optimum is determined implicitly by a one-dimensional equation. This lemma serves as the theoretical foundation for the general expression in the main text. Lemma E.1 (Structure of Dual-Based Optimal Subsidy for General Completion Models). Consider the primal subsidy optimization problem in the main text and assume that the completion function 𝑝𝑖 𝑗 (𝑏) satisfies: (A1) 𝑝𝑖 𝑗 is continuously differentiable on [0, 𝑏 max(𝑖 ) ], (A2) 𝑝𝑖′𝑗 (𝑏) ≥ 0 (monotonicity), (A3) 𝑝𝑖′′𝑗 (𝑏) ≤ 0 (concavity), (A4) 𝑝𝑖 𝑗 (0) = 0 and 𝑝𝑖 𝑗 (𝑏 max(𝑖 ) ) ≤ 1. Let 𝜆 ≥ 0 be the Lagrange multiplier associated with the subsidy-rate constraint, and define, for each (𝑖, 𝑗), 𝐹𝑖 𝑗 (𝑏; 𝜆) ≜ (1 + 𝜆(𝐶 + 𝛿)) 𝑟𝑖 𝑗 − 𝜆𝑏 𝑝𝑖′𝑗 (𝑏) − 𝜆 𝑝𝑖 𝑗 (𝑏). Then: (1) There exists at most one 𝑏𝑖◦𝑗 (𝜆) ∈ (0, 𝑏 max(𝑖 ) ) satisfying 𝐹𝑖 𝑗 (𝑏𝑖◦𝑗 (𝜆); 𝜆) = 0. (2) The dual-based optimal subsidy for each pair (𝑖, 𝑗) is given by 𝑏𝑖∗𝑗 (𝜆) = min max{0, 𝑏b𝑖 𝑗 (𝜆)}, 𝑏 max(𝑖 ) , (23) where 𝑏b𝑖 𝑗 (𝜆) = 𝑏𝑖◦𝑗 (𝜆) if the root lies in (0, 𝑏 max(𝑖 ) ), and any value outside this interval otherwise. Proof. Fix (𝑖, 𝑗) and 𝜆 ≥ 0. Using the standard Lagrangian form L = 𝑓 − 𝜆𝑔 with 𝑓 = 𝑟𝑖 𝑗 𝑝𝑖 𝑗 (𝑏) and 𝑔 = 𝑏 𝑝𝑖 𝑗 (𝑏) − (𝐶 + 𝛿) 𝑟𝑖 𝑗 𝑝𝑖 𝑗 (𝑏), the inner maximization problem decouples across (𝑖, 𝑗) and reduces to max (1 + 𝜆(𝐶 + 𝛿)) 𝑟𝑖 𝑗 − 𝜆𝑏 𝑝𝑖 𝑗 (𝑏) . 0≤𝑏 ≤𝑏 max(𝑖 )
For an interior maximizer 𝑏 ∈ (0, 𝑏 max(𝑖 ) ), the first-order condition is 𝑑L = (1 + 𝜆(𝐶 + 𝛿)) 𝑟𝑖 𝑗 − 𝜆𝑏 𝑝𝑖′𝑗 (𝑏) − 𝜆 𝑝𝑖 𝑗 (𝑏) = 0, 𝑑𝑏 which is exactly 𝐹𝑖 𝑗 (𝑏; 𝜆) = 0. Under (A1)–(A3), 𝑝𝑖 𝑗 is increasing and concave. For common completion curves (e.g., logistic or saturating), the equation 𝐹𝑖 𝑗 (𝑏; 𝜆) = 0 admits at most one root in (0, 𝑏 max(𝑖 ) ), establishing statement (1). To characterize the maximizer, observe that the sign of 𝑑𝑑𝑏L at the endpoints determines whether an endpoint is optimal; when neither endpoint is optimal, the (unique) root gives the interior optimum. Therefore the optimal solution is obtained by projecting 𝑏𝑖◦𝑗 (𝜆) (if it exists) onto the feasible interval [0, 𝑏 max(𝑖 ) ], which yields 12
D3 -Subsidy: Online and Sequential Driver Subsidy Decision-Making for Large-Scale Ride-Hailing Market
KDD ’26, August 9–13, 2026, Jeju, Republic of Korea
Figure 11: Daily Subsidy Rate in City A.
Figure 9: Cumulative Rides, GMV and DRV in City A.
Figure 12: Cumulative Rides, GMV and DRV in City B.
Figure 10: Per-Window Rides, GMV and DRV in City A.
Figure 13: Per-Window Rides, GMV and DRV in City B.
13
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Chen et al.
Figure 14: Daily Subsidy Rate in City B.
14