cs.AISep 24, 2026

A General Framework for Budgeted Threshold Incentives on Request

Authors: Zhuolin Wu, Chengrui Zhu, Wenhua Nie, Kenny Ye Liang, Junming Lin, Haiyang Li, Zhilin Li, Wenjia Geng, +4 more

Organizations: Meituan · National Taiwan University · Tsinghua University · Tongji University

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Abstract

On-demand delivery platforms pay riders through incentive activities whose tiers are set from recent completions of riders with a similar history. Operators request such plans for changing periods, rider populations, payment rules and budgets, often for holidays or bad weather, where randomized trials are scarce and take months to collect. We present a request-driven framework that composes four stages (conditional prediction, population reduction, trajectory integration and budget allocation) through seven replaceable modules that exchange conditional trajectory laws, whose award probabilities and award-marked moments give payment and uplift for any activity rule. A response-correction step reweights trajectories from abundant no-offer history to match the moments of a short pilot. We prove that, on a fixed plan menu and given the stage errors, the end-to-end value loss is bounded by the sum of four stage terms, and that for every stage there are instances on which omitting it leaves an error floor the others cannot remove. On 3,000 riders over 45 weekly origins, all 127 windows of a week are answered 11.04x faster with identical scenarios and at most 0.92% value lost by the allocation. On 24 new controlled response laws, the response correction with a one-week pilot lowers regret by 51.2% relative to a trial with the same nominal randomized rider-weeks, and a four-week pilot with exact summation comes within +0.007 of an 18-week trial. In registered studies where windows, populations, rules and binding budgets change from request to request, the framework's regret is below that of a trial with the same nominal rider-weeks and below dose interpolation of the same pilot data, and reusing its one-off preparation answers 60 requests 14.1x and 2.70x faster with identical answers. Against a nine-offer trial fitted with the framework's own dose curve, one-week regret is 0.055 lower.

Figures & tables

Figure 1: Request-driven framework: four stages and seven modules. Dashed chips are inputs (request fields, candidate plans, activity rule, budget band, short pilot); arrows show data flow between modules. Predict (M1–M3) gives every eligible rider a conditional completion law; Reduce (M4) merges riders into activity layers with plan menus; Integrate (M5–M6) tilts the no-offer law to short-pilot moments and scores every plan by a weighted sum over stored samples of the corrected law, an exact sum when the trajectory types can be enumerated; Allocate (M7) chooses one plan per layer inside the two-sided budget band and returns the rewards that attain it. The bar names the four terms of the end-to-end bound (Proposition 1 ); the table marks what a new request recomputes.
Alternative per requestAccuracy the framework gives upTime: alternative →\to frameworkSpeed-up
Rider records: 3,000 riders, 45 weekly origins, five activity layers
Re-draw trajectories and solve exactlynone in the scenarios (1,431/1,431 requests identical); allocation loses value in 3/405 instances, at most 0.92%117.8 s →\to 3.07 s per week-long request38.6 ×\times [30.8, 43.0]; 11.0 ×\times over 127 requests with the one-off pass
One plan per rider, exactnot measurable: the exact plan did not finish within 3 hours and 190 GiB of memoryunfinished →\to 4.05 s per table—
Controlled response laws: 18 laws ×\times 8 replicates, 33 budgets each (Section 5 )
Randomized trial over all offersrelative regret (planned and scored within 20% of budget) 0.113 →\to 0.121 ( ++ 0.007 [ −- 0.012, ++ 0.027]) with a 4-week pilot; a trial with the pilot’s rider-weeks: 0.421 with 1 week (framework 0.205) and 0.225 with 49,216 →\to 2,048 nominal randomized rider-weeks per group (18 →\to 4 weeks)4.5 ×\times fewer
Public retail (M5) and three payment rules (Section 6 )
Table 1: Accuracy given up and time gained against the most accurate way to answer the same request without the framework, on the same inputs. Speed-up: median of paired ratios (mixed-integer program: ratio of geometric-mean times); brackets: interquartile range or 95% interval over laws.
StageFull frameworkCheaper alternativeAccuracy change
Predictionper rider-day distributionpoint forecastplan-cost error ×\times 3.08 [2.48, 3.81]
Reductionactivity layersequal-width bucketsregret 0.37% →\to 0.78%
Integrationjoint law across daysindependent daysplan-cost error ×\times 1.51 [1.33, 1.74]
  simplified normal×\times 2.46 [2.02, 3.01]; 5–6-day award -10.31 pp
  plug-in mean×\times 1.85 [1.58, 2.19]
Allocationbudget-grid programequal budget splitvalue −- 15.7% [13.8, 17.4]
Table 2: Rider records, 45 weekly origins ×\times 3 seeds, five activity layers. Each row replaces one stage of the full framework by a cheaper alternative. Ratios are of origin means with 95% intervals over origins. Regret at budget fraction 0.5. ∗ 45 origins ×\times 3 budget fractions.
 Within 20% of budgetWithin 10% (pre-specified)
Method1 wk2 wk4 wk1 wk2 wk4 wk
Long RCT, 18 weeks (business alternative)0.1130.130
Same-calendar RCT0.4210.3130.2250.5420.3830.264
Framework (tilt, exact sum)0.2050.1590.1210.2520.1860.136
sampled integration ( S=2,048S=2{,}048 )0.2450.1970.1630.2860.2230.189
without history (pilot frequencies)0.6030.5600.5270.6580.6110.569
Table 3: Relative regret (lower is better), 18 response laws ×\times 8 replicates ×\times 33 budgets, pilot of 1/2/4 weeks; planning, optimum and scoring within 20% of the budget or 10% (pre-specified). Indented rows change one stage (others: Table 9 ). Validity (%): issued plans with true expected spend in the 10% band, all 24 laws.
 A: integrated studyB: new generator
Pilot1 wk2 wk4 wk1 wk2 wk4 wk
Same-calendar RCT0.3910.2960.2160.4770.3260.240
Framework0.2200.1720.1480.1880.1500.116
Dose interpolation (same information)0.2270.1770.1540.2100.1710.131
Independent days (least costly replacement)0.3110.2920.2960.1980.1580.124
Budgets that bind (%, all 48 laws)72.375.0
Table 4: Two registered studies with changing requests and binding budgets (registered primary condition, 36 response laws each; relative regret within 20% of the budget). A: fresh laws from the generator of Table 3 ; B: a new rider-level generator with a rule-dependent response. Speed-up: 60 requests, one-off preparation charged, against re-running the pipeline per request.
Appendix figures & tables18 assets

Supplementary material from the paper’s appendix.

Appendix

StepFrameworkRe-draw per request
Model fitting (once per origin) a221.3 once221.3 once
Per-day draws, 7 days108.7 once108.7
Cross-day rank template3.5 oncereused
Cross-day coupling1.6 once1.8
Window reduction0.540.54
Integration, 540 plans2.522.52
Appendix
Table 5: Where the time of rider requests goes (seconds; medians over origins, one CPU core). Top: one week-long request, by step. Bottom: KK requests per origin in the fixed order (whole week, seven single days, remaining subsets); the framework total includes its once-per-origin sampling pass; speed-ups are medians of per-origin ratios. a Gradient-boosted node quantiles (29.8 s) plus share and concentration models (189.1 s), needed equally by both approaches. b Speed-up when model fitting is charged once to both sides.
StageShortcutAccuracy cost of the shortcutTime the shortcut saves
Predictionpoint forecastplan-cost error ×\times 3.08 [2.48, 3.81]not timed separately
Integrationindependent daysplan-cost error ×\times 1.51 [1.33, 1.74]1.4% of sampling; integration ×\times 1.001
Integrationsimplified normalplan-cost error ×\times 2.46 [2.02, 3.01]; 5–6-day plans ×\times 3.29integration 85.8 ×\times (3.86 s per table)
Allocationequal budget splitvalue −- 15.7% [13.8, 17.4]0.51 →\to 0.01 ms
Appendix
Table 6: Cheaper shortcuts the framework declines, on rider records (five layers, main plan grid). Accuracy: plan-cost error of the shortcut over the framework’s, or value lost (95% intervals over origins). Time: the framework’s stage time over the shortcut’s.
 Origins /RidersPlan-cost error Within
Conditionrunsper runHeld-outReferenceChangemargin
Unseen cities (four folds)45 / 1803110.3430.340++ 0.003 [ −- 0.018, ++ 0.020]yes
Appendix
Table 7: Cities left out of training, rider records (five-band system, budget fraction 1). Plan-cost error is the median absolute relative error of predicted plan cost; change = held-out minus reference on the same riders and outcomes, 95% interval over origins. Pre-set margin: upper bound ≤\leq 0.045.
ReplacementEndpointFit onceUnseen cities
Point forecastplan-cost error++ 0.416 [ ++ 0.359, ++ 0.472]++ 0.380 [ ++ 0.292, ++ 0.467]
Plug-in integrationplan-cost error++ 0.317 [ ++ 0.235, ++ 0.382]++ 0.123 [ ++ 0.097, ++ 0.148]
Two-moment normalplan-cost error++ 0.459 [ ++ 0.382, ++ 0.523]++ 0.205 [ ++ 0.167, ++ 0.240]
Independent daysplan-cost error++ 0.202 [ ++ 0.142, ++ 0.250]++ 0.058 [ ++ 0.035, ++ 0.080]
Equal-width bucketsregret++ 0.0017 [ ++ 0.0006, ++ 0.0028]++ 0.0046 [ ++ 0.0024, ++ 0.0070]
Equal budget splitregret++ 0.048 [ ++ 0.021, ++ 0.080]++ 0.122 [ ++ 0.099, ++ 0.145]
Appendix
Table 8: Stage replacements on held-out riders: replacement minus full framework, budget fraction 1, 95% interval over origins. Prediction and integration are scored by plan-cost error, reduction and allocation by selection regret. Unseen cities: the condition of Table 7 ; fit once: models fitted at the origin of 2026-04-21 and used for the next 16 weekly origins.
Replacement1 wk2 wk4 wk
Without moment matching (history only)0.6560.6560.656
Without joint integration (plug-in mean)0.5830.5990.616
Without budget allocation (equal split, exact tables)0.8610.8600.855
Appendix
Table 9: Further stage replacements of the framework (penalized relative regret, pre-specified 10% scoring, 18 response laws; the first two arms were not rescored at 20%, and the equal split on the exact tables is also given at 20% in the description of the methods above). Their differences from the framework are in Section 5 .
A Pilot length1 week2 weeks4 weeks
Synthetic term εsynv\varepsilon^{v}_{\mathrm{syn}} / εsync\varepsilon^{c}_{\mathrm{syn}}5.92 / 5.535.92 / 5.535.92 / 5.53
Moment term εmmv\varepsilon^{v}_{\mathrm{mm}} / εmmc\varepsilon^{c}_{\mathrm{mm}}8.82 / 7.676.29 / 5.314.48 / 3.69
Integration term εintv\varepsilon^{v}_{\mathrm{int}} / εintc\varepsilon^{c}_{\mathrm{int}} ( S=2,048S=2{,}048 )3.43 / 3.003.44 / 3.003.51 / 3.00
Sum of stage terms18.16 / 16.2015.65 / 13.8313.90 / 12.22
Direct error Eˉv\bar{E}_{v} / Eˉc\bar{E}_{c} (truth vs. final tables)11.79 / 10.8810.06 / 9.358.89 / 8.24
Requests with a comparator in B2Eˉc\mathcal{B}_{2\bar{E}_{c}}26/6336140/6336499/6336
Appendix
Table 10: Empirical check of Proposition 1 . A : controlled response laws with known truth (24 laws ×\times 8 replicates, 33 budgets each), sampled variant ( S=2,048S=2{,}048 ), which carries the synthetic, moment-matching and integration terms; under exact summation the integration term is zero and the other terms are unchanged; allocation enumerates exactly, so εdec=0\varepsilon_{\mathrm{dec}}=0 , and the last two rows of A are an allocation diagnostic comparing the budget-grid program with enumeration; value errors are uplift in completions and cost errors are currency units, both summed over five layers (the median optimal uplift of a request is 14.1). B : rider records, 45 weekly origins ×\times 3 seeds, five activity layers; each row replaces one stage of the framework and is compared with the framework on the realized records; intervals resample origins.
 Within 20% of budgetWithin 10%
Method1 wk2 wk4 wk1 wk2 wk4 wk
Same-calendar RCT0.3910.2960.2160.5270.3880.269
Framework0.2200.1720.1480.2580.1980.164
without moment matching0.5920.5920.5920.6260.6260.626
without history0.4710.4330.4040.5610.5090.470
plug-in integration0.5920.6040.6310.6690.6720.690
Appendix
Table 11: Integrated study: relative regret (lower is better) on 36 response laws ×\times 8 replicates ×\times 180 budgeted requests, with each request’s window, population, rule and budget drawn afresh. Indented rows change one stage of the framework; every difference from the framework has a 95% interval above zero. Validity: share of the framework’s issued plans whose true expected spend lies in the band, over all 48 laws.
   1-week pilot2-week pilot
 LevelLawsFrameworkminus same-calendarFrameworkminus same-calendar
Rulefixed award360.225−- 0.175 [ −- 0.208, −- 0.142]0.176−- 0.127 [ −- 0.166, −- 0.089]
 per completion360.229−- 0.172 [ −- 0.206, −- 0.138]0.180−- 0.125 [ −- 0.165, −- 0.087]
 attendance360.205−- 0.165 [ −- 0.192, −- 0.139]0.162−- 0.120 [ −- 0.151, −- 0.089]
Window days1–2360.221−- 0.193 [ −- 0.226, −- 0.159]0.173−- 0.146 [ −- 0.184, −- 0.110]
 3–5360.219−- 0.171 [ −- 0.200, −- 0.141]0.171−- 0.123 [ −- 0.159, −- 0.088]
Appendix
Table 12: Integrated study by request type (within 20% of budget, descriptive): framework regret and framework minus same-calendar RCT with 95% intervals over the laws in which the subgroup is defined.
PilotPreparationRequest (reuse)Request (re-run)K=8K=8K=32K=32K=60K=60
1 week7.270.227.575.8 ×\times11.9 ×\times14.1 ×\times [13.8, 14.5]
2 weeks7.400.227.575.8 ×\times11.9 ×\times14.2 ×\times [13.8, 14.5]
4 weeks6.250.216.615.9 ×\times12.0 ×\times14.3 ×\times [14.0, 14.7]
Appendix
Table 13: Integrated study, preparation charged (milliseconds on one core; medians over replicates for the one-off preparation and over requests otherwise). Speed-up: time to answer the first KK requests of a replicate by re-running the whole pipeline for each, over the preparation plus the reusing requests; geometric mean over the 36 response laws (replicates averaged on the log scale) with 95% intervals. Answers are identical on all 51,840 timed requests of these laws and on all 69,120 of all 48 laws.
 Within 20% of budgetWithin 10%
Method1 wk2 wk4 wk1 wk2 wk4 wk
Same-calendar RCT0.4770.3260.2400.7650.6010.415
Framework0.1880.1500.1160.2700.1970.140
without moment matching0.4330.4330.4330.7810.7810.781
without history0.4150.3540.3090.6100.5170.448
plug-in integration0.4470.4790.5100.5760.5770.591
Appendix
Table 14: New generator: relative regret (lower is better) on 36 response laws ×\times 8 replicates ×\times 180 budgeted requests. Indented rows change one stage of the framework; every difference from the framework, and that of dose interpolation, has a 95% interval above zero. Validity: share of the framework’s issued plans whose true expected spend lies in the band, over all 48 laws.
   1-week pilot4-week pilot
 LevelLawsFrameworkminus same-calendarFrameworkminus same-calendar
Rulefixed award360.179−- 0.318 [ −- 0.343, −- 0.293]0.120−- 0.127 [ −- 0.143, −- 0.112]
 per completion360.196−- 0.322 [ −- 0.343, −- 0.300]0.111−- 0.127 [ −- 0.144, −- 0.111]
 attendance360.202−- 0.183 [ −- 0.213, −- 0.155]0.132−- 0.114 [ −- 0.137, −- 0.093]
Window days1–2360.186−- 0.340 [ −- 0.365, −- 0.315]0.119−- 0.154 [ −- 0.175, −- 0.133]
 3–5360.188−- 0.285 [ −- 0.308, −- 0.263]0.116−- 0.121 [ −- 0.138, −- 0.106]
Appendix
Table 15: New generator by request type (within 20% of budget, descriptive): framework regret and framework minus same-calendar RCT with 95% intervals over laws.
ConditionMethod1 wk2 wk4 wk
IndependentFramework0.1960.1540.126
 Dose interpolation (same information)0.2190.1710.138
 Normal law, exact truncated moments0.2500.2080.179
 History-anchored tilt of the trial0.4050.2910.211
 Same-calendar RCT0.4800.3400.241
CalendarFramework , concurrent anchor0.2230.2150.207
Appendix
Table 16: Study C: relative regret within 20% of the budget, 288 response laws ×\times 8 replicates ×\times 60 requests ×\times 3 budgets. Every arm uses the same stored weeks, data and exact enumeration. Calendar: repeated riders with one shock per group and calendar week.
Method1 wk2 wk4 wk
Framework (pilot at three offers)0.1980.1540.123
Joint fit of the trial, framework’s dose curve0.2530.2060.183
History-anchored tilt of the trial, per offer0.4050.2890.211
Same-calendar RCT0.4790.3370.241
Placement share0.265 [0.252, 0.279]0.382 [0.361, 0.405]0.683 [0.623, 0.745]
Appendix
Table 17: Study E: relative regret within 20% of the budget, 288 response laws ×\times 8 replicates ×\times 60 requests ×\times 3 budgets; share of the difference to the tilted trial due to placing the observations at three offers, with 95% law-bootstrap intervals.
MethodBelowAboveExcursionValid-plan loss
Framework0.0340.0410.0350.136
Joint fit of the trial0.0650.0440.0460.165
Dose interpolation (same information)0.0440.0450.0410.149
History-anchored tilt of the trial, per offer0.1790.0100.0700.266
Same-calendar RCT0.2720.0100.0820.288
Appendix
Table 18: Risk components at one week within 20% of the budget (study E, 288 response laws; law means). Every arm issues a plan for more than 99.99% of the requests. Below/above: share of requests whose plan has true expected spend below/above the band. Excursion: mean distance outside the band, in units of the budget, over the plans that leave it. Valid-plan loss: value lost by the plans inside the band relative to the band optimum.
Method1 wk2 wk4 wkSelected, 1 wk
Request-aware reuse0.3800.3290.2970.395
Direct reuse of the full-week calibration0.5760.5840.5960.742
Appendix
Table 19: Study D: relative regret within 20% of the budget on the 216 window-dependent laws (8 replicates ×\times 60 requests ×\times 3 budgets). Selected: requests that target a subpopulation chosen on the previous week.
Layers per groupAll lawsBoth channelsIntensityThreshold
1 (group-level plan)0.825 [0.820, 0.831]0.9380.9440.594
20.900 [0.897, 0.904]0.9660.9710.764
30.926 [0.923, 0.928]0.9740.9780.825
40.938 [0.936, 0.940]0.9770.9820.856
50.946 [0.944, 0.948]0.9790.9840.875
Appendix
Table 20: Study F: share of the individual-level optimum retained by LL history-quantile layers per group, within 20% of the budget; 3,072 laws, 60 requests ×\times 3 budgets each; 95% law-bootstrap intervals. By family: saturating response on both channels, intensity only, threshold response near the goal.
StageModuleInput →\to outputDelivery instance
Predict1. Response ranking (Rank)pre-decision context and past offers →\to ranking by incremental activitydoubly robust learner with a 5% no-offer group ( Dudík et al., 2011 )
 2. Roster (Recall)ranking and eligibility →\to population fixed before attendance, incl. zero-activity ridersat least one delivery in the preceding 182 days
 3. Baseline forecast (Forecast)histories and context →\to no-offer outcome law for the requested windowrecurrent network, 60-day history, five-day zero-inflated Beta shares
Reduce4. Grouping (Activity layers)baseline capacity →\to groups and non-dominated plan menusordered buckets merged to balance dispersion and spending stability
Integrate5. Response correction (Correct)no-offer law and pilot or context-matched outcomes →\to offer-conditioned joint trajectoriesexponential tilt, Eq. ( 5 ); residual matching on incentive, time and weather
 6. Settlement (Score)trajectories, payment rule, attainable rewards →\to cost, award probability, score and reward witnessEqs. ( 6 )–( 7 )
Appendix
Table 21: Four stages and seven modules. The last column gives the instance used on the delivery platform.
Held-out conditionChange in plan-cost errorChange in regretMet
Unseen period (fit once, 16 origins)++ 0.051 [ ++ 0.002, ++ 0.098]++ 0.000 [ ++ 0.000, ++ 0.000]no
Unseen cities (four folds, 45 origins)++ 0.003 [ −- 0.018, ++ 0.020]++ 0.000 [ ++ 0.000, ++ 0.000]yes
Pilot rule1-week pilot2-week pilot4-week pilot
Near-lossless vs. 18-week trial (upper ≤0.05\leq 0.05 )++ 0.156 [ ++ 0.114, ++ 0.201] (no)++ 0.093 [ ++ 0.061, ++ 0.126] (no)++ 0.059 [ ++ 0.039, ++ 0.077] (no)
Beats same-calendar trial (upper <0<0 )−- 0.256 [ −- 0.290, −- 0.222] (yes)−- 0.160 [ −- 0.186, −- 0.134] (yes)−- 0.076 [ −- 0.109, −- 0.043] (yes)
Without history is worse (lower >0>0 )++ 0.371 [ ++ 0.266, ++ 0.488] (yes)++ 0.388 [ ++ 0.261, ++ 0.525] (yes)++ 0.380 [ ++ 0.258, ++ 0.514] (yes)
Appendix
Table 22: Pre-specified decision rules and their outcomes (rules fixed before the runs, and except for the integrated study also the analysis scripts). Rider held-out validation (Appendix A.1 ): paired change, held-out minus reference, of plan-cost error and selection regret at budget fraction 1, 95% intervals over origins; the rule requires upper bounds ≤0.045\leq 0.045 and ≤0.01\leq 0.01 . Synthetic trajectories and a short pilot (Section 5 ): the pre-specified primary arm is the framework with sampled integration ( S=2,048S=2{,}048 ); differences of penalized relative regret at 10% scoring on the 18 response laws, 95% intervals over laws. Integrated study (Appendix C ): the registered confirmatory family on 36 response laws; H1 is the framework minus the same-calendar trial in relative regret (band within 20% of the budget), H2 the geometric-mean speed-up of answering all 60 requests of an origin by reuse, one-off preparation charged, over re-running the whole pipeline per request; 95% intervals over laws. Its analysis script was completed after the run; H2 is computed on the registered 36 response laws. New generator (Appendix D ): the same family on its 36 response laws, with the rules and the analysis script fixed before the run. Stronger controls, behaviour change, joint fit and population reduction (Appendix E , studies C–F): registered families of seven, five, five and three one-sided tests, Holm step-down at level 0.025 within each family; estimates with 95% law-bootstrap intervals, relative regret within 20% of the budget (study F: retained share of the individual-level optimum). Study E: full re-run after the correction described in Appendix E .

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