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A General Framework for Budgeted Threshold Incentives on Request

Zhuolin Wu, Chengrui Zhu, Wenhua Nie, Kenny Ye Liang, Junming Lin, Haiyang Li, Zhilin Li, Wenjia Geng, Zeyu Wu, Yinan Wu, Jinghua Hao, Renqing He

arXiv:2609.29724v1cs.AI

TL;DR

On-demand platforms need incentive plans for changing windows, populations, rules, and budgets, while randomized trials can take longer than the campaign they inform. The paper builds a request-driven framework that reuses no-offer history with short-pilot response correction. Its reported comparisons show faster request answering and lower regret than several trial and same-information controls.

  • Problem

    Plans must cover changing rider populations, windows, rules, and budgets, but trials are slow and experimental data are scarce relative to no-offer history.

    Source: §1 · p. 2

  • Method

    A four-stage framework composes rider-day prediction, population reduction, response-corrected trajectory integration, and budget allocation, reusing no-offer history with short-pilot moments.

    Source: §1 · p. 2, §2 · p. 4

  • Results

    11.04× faster with almost no value loss; in registered controlled studies, regret is below equal-effort trials, same-information controls, and a trial fitted with the framework’s own dose curve.

    Source: §8 · p. 10

  • Takeaways & Limitations

    The framework answers changing requests while the response law remains reusable, including behavior-changing windows and subpopulations when correction is window-aware.

    Source: §8 · p. 9

    Reusing a response correction assumes the response is unchanged; a new city, season, or rule that may change behavior calls for its own pilot.

    Source: §2 · p. 4

Abstract

from arXiv · show

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.

1 INTRODUCTION

On-demand platforms need to place threshold rewards for changing rider populations and activity windows, despite scarce, slow trials and abundant no-offer history. The paper proposes a reusable four-stage framework and evaluates its theoretical guarantees and practical alternatives.

1 INTRODUCTION

  • Operators must place incentive tiers for thousands of riders within a fixed budget, but relevant campaigns can fall during holidays or severe weather when history is thinnest.A trial with nine offers across five rider groups requires 18 weeks, longer than the campaign it should inform.Source: §1 · p. 2
  • The framework composes four stages and seven replaceable modules to turn a request into executable rewards for each rider layer.Prediction produces rider-day completion laws; reduction forms activity layers; integration scores plans; and allocation selects plans within the budget band.Source: §1 · p. 2, Figure 1 · p. 3
  • New periods, populations, rules, or budgets can reuse upstream outputs when their inputs remain unchanged.For response correction, abundant no-offer history supplies trajectory shape while a one-to-four-week pilot at three offers supplies offer-specific moments.Source: §1 · p. 2
  • The paper contributes a four-term loss decomposition and evaluates accuracy, time savings, and performance on cities excluded from training.The decomposition includes synthetic data, moment matching, integration, and decision terms; the introduction also states that each stage has omission cases whose error floors the others cannot remove.Source: §1 · pp. 2–3

2 REQUESTS AND THE FOUR-STAGE FRAMEWORK

A request specifies the population, activity window, context, rule, and budget. The framework processes it through prediction, reduction, integration, and allocation, with response correction reusing no-offer history and short-pilot moments.

2 REQUESTS AND THE FOUR-STAGE FRAMEWORK

  • A request names an eligible population, activity window, context, activity rule, and budget set; each layer receives one plan from its menu.Plans can specify ordered thresholds, rewards or rates, eligible orders, and attendance conditions.Source: §2 · p. 3
  • The stages reduce distinct difficulties: composing windows, shrinking individual search, integrating different rules from shared trajectories, and allocating across layers within the budget band.Estimation and allocation errors add at the allocation interface on a fixed menu, while the cost of restricting riders to layer menus is measured rather than bounded.Source: §2 · pp. 3–4
  • Prediction composes any requested window from rider-day laws; reduction forms activity layers and trims menus without removing a feasible optimum under the stated rules.Reduction drops dominated plans under a sole cap, or only same-cost dominated plans when the budget band has a floor.Source: §2 · p. 4
  • Integration tilts the no-offer trajectory law to match short-pilot moments, then evaluates each rule over stored trajectories, exactly when trajectory types can be enumerated.Offers between pilot points interpolate the tilt parameter piecewise linearly in the reward increment.Source: §2 · p. 4
  • A new window, population, budget, or rule recomputes only the stages affected by that request field, but reusing a response correction assumes behavior is unchanged.A new city, season, or rule that may change behavior calls for its own pilot; group-additive allocation assumes no response interference between groups.Source: §2 · p. 4

3 AWARD PROBABILITIES AND THE FOUR-TERM LOSS DECOMPOSITION

Award probabilities and award-marked moments let the framework evaluate diverse payment rules from trajectory laws. On a fixed plan menu, its loss bound separates four stage errors, while stated conditions and omission cases delimit that guarantee.

3 AWARD PROBABILITIES AND THE FOUR-TERM LOSS DECOMPOSITION

  • Nested award probabilities and award-marked moments encode fixed prizes or completion rates, while joint trajectories capture budget risk that marginal expectations miss.Expected spend is additive over riders without an independence assumption; its variance and probability of staying within a band depend on the joint law.Source: §3 · pp. 4–5
  • The four-term decomposition tracks errors from synthetic response, moment matching, integration, and allocation across successive trajectory laws and compiled tables.Under exact summation, the integrated law equals the moment-matched law; allocation tables also match integrated tables under exact enumeration.Source: §3 · p. 5
  • On a fixed menu and given stage errors, the conditional bound adds value and cost errors, with cost error controlled by a growth condition and a tightened budget band.The bound concerns the best value within the layer menus; restricting individuals to those menus is not itself bounded by the proposition.Source: §3 · p. 5
  • Omitting any stage can leave an error floor: uplift may be invisible, threshold probabilities undetermined, threshold prizes mispriced, or value lost to equal budget splits.These are instances with the other stages exact; the bound is conditional on stage errors and does not bound the value lost by restricting riders to layer menus.Source: §3 · p. 5
  • The four stage errors arise from distinct resources: unmatched response, pilot moment estimation, trajectory integration, and budget-grid decisions.The moment term shrinks as n^-1/2 in pilot size for a given base, integration as S^-1/2 and to zero under exact summation, and decision error with grid step size.Source: §3 · pp. 5–6

4 RIDER RECORDS: ACCURACY GIVEN UP AND TIME GAINED

The rider-record replay tests pricing accuracy before behavioral response, while a separate held-out-city condition checks transfer. The framework reuses stored trajectories to answer all 127 weekly windows 11.04× faster, with nearly unchanged allocation value.

Data and protocol

  • The replay scores plans against recorded completions, so it tests pricing and computation rather than behavioral response or reward-level choice.It covers 3,000 riders across 45 weekly origins; the budget binds in hindsight only in the Spring Festival week at the 0.5 budget fraction.Source: §4 · p. 6

Near-lossless and much faster

  • 11.04× faster: answering all 127 weekly windows takes 12.2 rather than 131.1 minutes, charging the one-off sampling pass.Against re-drawing trajectories and solving exactly for each request, scenarios are identical and allocation gives up almost nothing.Source: §4 · p. 6

Every stage earns its place

  • Distributional prediction is the largest single gain: point forecasts raise plan-cost error from 0.215 to 0.660 and median award-rate error from 4.0 to 66.9 percentage points.Independent-day composition raises plan-cost error by 1.51×; the simplified normal baseline raises it by 2.46×, and five- and six-day award rates fall 10.31 points.Source: §4 · p. 6, Table 2 · p. 7
  • Five history layers retain 94.6% of the individual-level optimum, while equal-width buckets raise regret from 0.37% to 0.78%.Splitting the budget equally across layers gives up 15.7% of the value found by the budget program; the grid program matches the exact optimum in 134 of 135 main-grid instances, with a largest loss of 0.69%.Source: §4 · p. 6, Table 2 · p. 7

Cities left out of training

  • Held-out cities have plan-cost error 0.343 versus 0.340 for the reference trained on them, a +0.003 difference within the pre-set 0.045 margin.The held-out forecasts use the riders’ own histories; stage replacements again lose accuracy.Source: §4 · pp. 6–7

5 SYNTHETIC TRAJECTORIES AND A SHORT PILOT VERSUS A LONG TRIAL

Controlled response laws compare short-pilot correction with randomized trials, while registered studies test changing requests and binding budgets. The framework beats same-calendar trials across the registered pilot lengths and a four-week pilot with exact summation comes within +0.007 of an 18-week trial.

Design

  • The controlled-law study tests 24 response laws, with five rider groups, three payment rules, nine offers, and 32 weekly trajectory types.The pilot uses three offers for one, two, or four weeks; exact summation is possible because the trajectory types can be enumerated.Source: §5 · pp. 7–8

Results

  • A four-week pilot with exact summation comes within +0.007 of the 18-week trial despite using 4.5 times fewer randomized rider-weeks.With a one-week pilot, regret falls by 0.215 (51.2%) versus randomizing the same nominal rider-weeks across all offers.Source: §5 · p. 8, Table 3 · p. 8
  • Dropping history, moment matching, joint integration, or budget optimization raises one-week regret; sampling 2,048 trajectories instead of exact summation also raises it.The reported regret increases are +0.406, +0.404, +0.331, and +0.610 for the first four replacements, respectively; sampled integration raises regret by +0.039.Source: §5 · p. 8

Changing requests, binding budgets and a new generator

  • With changing windows, populations, rules, and binding budgets, regret is below the same-calendar trial at every pilot length and below dose interpolation of the same data.With one week, regret differences from the same-calendar trial are −0.171 and −0.289 in the two studies; reusing prepared laws answers 60 requests faster with identical answers.Source: Table 4 · p. 8, §5 · p. 9
  • Across the registered studies, the one-week framework has lower regret than dose interpolation of the same history and pilot data, by +0.007 and +0.022 for the interpolation comparison.The studies use 60 requests per replicate and include a new rider-level generator with rule-dependent response.Source: §5 · p. 9

6 TRANSFER ACROSS DOMAINS AND RULES

The authors test reuse of the rider framework on sales rebates and alternative activity rules, reporting faster responses with identical scenarios or exact values. The budget program also resolves requests that a per-request mixed-integer program leaves unresolved.

Transfer to sales

  • On 3,000 item–store series, law reuse cuts the median request time from 84.8 s to 4.36 s, a paired 20.0× speed-up with identical scenarios and exact values on all 405 tables.Charging the 80.63 s preparation, it is 1.66×, 3.92×, and 7.11× faster for 8, 32, and 127 requests per origin.Source: §6 · p. 9

Transfer across rules

  • Joint histograms let an active-day rule reuse the same laws, with all 14,580 answers identical to direct settlement.The transfer evaluation also covers fixed prizes, per-completion rates, and attendance awards across 396 conditions.Source: §6 · p. 9

Budget allocation

  • Across 65,340 requests, the budget program resolves every request exactly, while a per-request mixed-integer program leaves 25 raw and 5 envelope requests unresolved.At 33 requests, the framework is 20.9× and 5.5× faster, respectively.Source: §6 · p. 9

7 RELATED WORK

Related work covers budgeted uplift allocation, incentive planning, and combining small experiments with observational data. The paper distinguishes its multi-day trajectory-dependent threshold activities and its pilot-based correction for planning.

Budgeted allocation

Threshold activities and allocation

  • Here, each decision concerns a multi-day threshold activity whose payment depends on the whole trajectory, and requests can change the window, population, rule, and budget.The allocation stage extends multiple-choice knapsacks and graphical Bellman algorithms to two-sided budget bands and reward traceback.Source: §7 · p. 9

Pilot and observational data

  • Where prior work combines experiments with observational data to estimate effects, this paper uses a short pilot to correct history for planning by entropy balancing on offer-conditioned trajectories coupled across days.The correction’s error is priced as a separate term in the end-to-end bound.Source: §7 · p. 9

8 CONCLUSION

The framework reuses conditional trajectory laws to answer changing requests, with a correction designed to use a short pilot for behaviour-changing offers. On rider records, reuse answered every weekly window much faster with identical scenarios and little allocation loss.

8 CONCLUSION

  • Four stages exchange conditional trajectory laws to answer all windows of a week 11.04× faster at almost no loss; controlled studies beat equal-effort trials and same-information controls.Source: §8 · pp. 9–10
  • Four stages exchange conditional trajectory laws so changing requests can reuse unchanged upstream outputs; a window-aware response correction addresses behaviour-changing windows and subpopulations.The correction reweights abundant no-offer history to match moments from a one-to-four-week pilot at three offers.Source: §8 · p. 9

A RIDER REPLAY: PROTOCOL AND TIMING

  • The rider replay used 3,000 riders across 45 weekly origins, refitting models at each origin before evaluating seven-day windows.The evaluated plans varied tier shape, active-day requirement and reward level; the budget was an upper cap based on realized reference cost.Source: A RIDER REPLAY: PROTOCOL AND TIMING · p. 12
  • The timing comparison charged the framework for its one-off sampling pass and compared it with re-drawing, re-coupling and integrating the same scenarios for each request.Model fitting was shown separately because both approaches needed it once per origin.Source: A RIDER REPLAY: PROTOCOL AND TIMING · p. 13, Table 5 · p. 13
  • The evaluation also compared cheaper shortcuts on rider records, reporting their accuracy costs and time savings against the framework.The shortcut comparison covered plan-cost error or value lost, with intervals over origins.Source: A RIDER REPLAY: PROTOCOL AND TIMING · p. 13, Table 6 · p. 14

A.1 CITIES LEFT OUT OF TRAINING

The held-out-city evaluation withheld each city fold from model fitting while leaving later stages unchanged. It prespecified margins for judging changes in plan-cost error and selection regret.

A.1 CITIES LEFT OUT OF TRAINING

  • The held-out design removed each fold’s city rows and contexts from fitting, forecasting those riders from their own histories while keeping later stages, menus and budgets unchanged.The 17 separately modelled city nodes were divided into four folds; the pooled small-city node remained in training.Source: §A.1 · p. 14
  • The pre-set immaterial-loss rule required upper 95% bounds no greater than 0.045 for increased plan-cost error and 0.01 for increased selection regret.The plan-cost-error margin was half the width of the reference’s own interval.Source: §A.1 · p. 14, Table 7 · p. 14
  • The held-out comparison reports replacement-minus-reference changes on the same riders and outcomes, alongside stage replacements scored by plan-cost error or selection regret.The stage-replacement table also identifies unseen-city and fit-once conditions.Source: Table 8 · p. 14, Table 7 · p. 14

B SYNTHETIC TRAJECTORIES AND A SHORT PILOT: DESIGN AND FURTHER

The controlled-law study tests whether short-pilot correction and the framework’s stages reduce regret, and measures the proposed decomposition against known truth. Its findings distinguish pilot-sensitive moment error from other stage errors.

B SYNTHETIC TRAJECTORIES AND A SHORT PILOT: DESIGN AND FURTHER

  • The study used 24 controlled laws with five groups, 32 weekly trajectory types and nine offers per group; methods shared exact portfolio enumeration and 33 budgets.The framework matched normalized-completion and active-day moments at three piloted offers, interpolated the tilt, then summed payment and value over the types.Source: B SYNTHETIC TRAJECTORIES AND A SHORT… · pp. 14–15
  • Excluding the development law left the reported framework-versus-same-calendar RCT gaps at −0.296, −0.198 and −0.135 for one-, two- and four-week pilots.The four-week gap to the long RCT was +0.004; one-week ablations raised regret by +0.386, +0.391, +0.328 and +0.034 for sampling.Source: B SYNTHETIC TRAJECTORIES AND A SHORT… · p. 16
  • The reported ablation regrets were 0.656 without moment matching, 0.583–0.616 without joint integration, and 0.855–0.861 without budget allocation.The table caption describes the further replacements as penalized relative regret under pre-specified 10% scoring on 18 response laws.Source: B SYNTHETIC TRAJECTORIES AND A SHORT… · p. 16, Table 9 · p. 16

B.1 EMPIRICAL CHECK OF THE DECOMPOSITION

  • Moment-matching error fell 1.40 and 1.41 times per pilot doubling, while synthetic and integration terms stayed fixed; the direct error was below the sum of stage terms.The moment-term declines were close to the predicted √2 rate; the bound check uses truth and is therefore an after-the-fact analysis.Source: §B.1 · p. 16
  • Among 4,968 requests with a feasible true-cost portfolio, the tightened band held for 0.5%, 2.8% and 10.0% at one, two and four weeks, respectively.The tightened-band bound was below the comparator value for 0.3%, 1.2% and 5.8%, respectively.Source: §B.1 · p. 16
  • Independent days misestimated active-day requirements under shared shocks, while a normal shortcut failed where attendance bound; dropping distributional prediction or joint allocation cost most.These replayed-plan comparisons do not involve behavioural response.Source: §B.1 · p. 16

C INTEGRATED STUDY: CHANGING REQUESTS AND BINDING BUDGETS

The registered study varies windows, populations, rules and binding budgets across requests, comparing the framework with same-calendar trials and other methods using the same pilot information. The framework had lower regret than the same-calendar RCT and dose interpolation, while stage replacements increased regret.

C INTEGRATED STUDY: CHANGING REQUESTS AND BINDING BUDGETS

  • Each replicate issued 60 requests with independently drawn windows, populations, activity rules and three budgets; 72.3% of budgeted requests bound across all 48 laws.The primary analysis used 36 response laws; 12 null laws checked validity and timing only.Source: C INTEGRATED STUDY: CHANGING REQUESTS… · pp. 17–18
  • The framework prepared history and pilot-corrected laws once, then answered each request by exact summation over trajectory types and exact portfolio enumeration.A same-calendar RCT used empirical offer laws; the study also timed re-running the whole pipeline, including preparation, for each request.Source: C INTEGRATED STUDY: CHANGING REQUESTS… · p. 18
  • Within 20% of budget, framework-minus-same-calendar RCT regret was −0.171, −0.124 and −0.069 for one-, two- and four-week pilots, respectively.The corresponding reductions were 43.8%, 42.0% and 31.8%; the gain held across every rule, window length, population size and budget level.Source: C INTEGRATED STUDY: CHANGING REQUESTS… · pp. 18–19, Table 11 · p. 18, Table 12 · p. 19
  • For a one-week pilot, removing moment matching, history, plug-in integration, day coupling or joint allocation raised regret by +0.373, +0.251, +0.372, +0.091 or +0.479, respectively.The study’s table describes these as one-stage replacements, with each difference from the framework having a 95% interval above zero.Source: Table 11 · p. 18, C INTEGRATED STUDY: CHANGING REQUESTS… · p. 19
  • Dose interpolation had higher regret than the framework by +0.007, +0.005 and +0.006 at one, two and four weeks, respectively.This was a same-information control using history and pilot laws interpolated in the reward increment.Source: C INTEGRATED STUDY: CHANGING REQUESTS… · pp. 18–19

D RESPONSE VALIDATION WITH A NEW GENERATOR

A registered evaluation on a new controlled response generator finds lower regret for the framework than a same-calendar RCT across tested rules, windows, populations, and budgets. Replacing key framework stages with simpler practices raises regret.

D RESPONSE VALIDATION WITH A NEW GENERATOR

  • The registered comparison spreads the same nominal rider-weeks across three rules and nine offers; actual same-calendar RCT cell counts are slightly lower than the pilot counts.The pilot uses one, two, or four weeks at 512 riders per group and week, spread across three rules and offers 2, 5, and 8.Source: D RESPONSE VALIDATION WITH A NEW… · pp. 19–20
  • The new generator uses 48 laws with six rider groups per law, each group mixing 16 latent rider types, and includes rule-dependent responses and several response families.Activity and completion processes also vary by type and day, with a weekly shock; the truth is computed exactly rather than by enumerating all 167 daily patterns per rider.Source: D RESPONSE VALIDATION WITH A NEW… · pp. 19–20
  • Framework minus same-calendar RCT regret was −0.289, −0.176, and −0.123 within 20% across one-, two-, and four-week pilots, respectively; gains held across rules, windows, populations, and budgets.These correspond to 60.6%, 53.9%, and 51.4% lower regret; Table 15 breaks results down by request type.Source: D RESPONSE VALIDATION WITH A NEW… · p. 20, Table 15 · p. 21
  • Removing moment matching, removing history, and using plug-in integration raised one-week-pilot regret by +0.245, +0.228, and +0.260, respectively.The passage reports these as stage replacements.Source: D RESPONSE VALIDATION WITH A NEW… · p. 20

E STRONGER CONTROLS, WEIGHT DIAGNOSTICS, BEHAVIOUR-CHANGING

Across stronger controls, changing-behaviour studies, and population-reduction tests, the framework outperforms comparison methods on regret or retained value, while its calibration diagnostics remain heuristic. The public retail-data passage describes a computation and settlement-transfer test, not incentive-response validation.

E STRONGER CONTROLS, WEIGHT DIAGNOSTICS, BEHAVIOUR-CHANGING

  • In Study C, framework regret was lower than dose interpolation by 0.022 and the tilted trial by 0.209 at one week; reuse returned identical answers in 184,320 comparisons per condition and was 2.73× faster.The comparison uses shared stored weeks, data, and exact enumeration; Table 16 gives the study scope.Source: E STRONGER CONTROLS, WEIGHT… · p. 22, Table 16 · p. 22
  • At one week, sharing the tilt across offers accounts for 73.5% of the framework’s advantage over the tilted trial, while placement accounts for 26.5%.At the same parameterization, framework regret is below the joint fit by 0.055 at one week; Table 17 reports the source split.Source: E STRONGER CONTROLS, WEIGHT… · p. 23, Table 17 · p. 23
  • At one week, framework plans leave the budget band less often than joint-fit plans, with below-band rates of 3.4% versus 6.5% and valid-plan losses of 13.6% versus 16.5%.Table 18 defines the risk components and reports the corresponding above-band and excursion measures.Source: E STRONGER CONTROLS, WEIGHT… · p. 23, Table 18 · p. 23
  • The weight diagnostics are heuristic: fits converge and weights are not concentrated, but they do not verify the interior-hull and eigenvalue conditions along the parameter path.The ESS deficit tracks cost-table error, while the stated sufficient conditions remain unchecked.Source: E STRONGER CONTROLS, WEIGHT… · p. 24
  • For window-dependent laws, request-aware reuse lowers one-week regret by 0.195 versus direct reuse and by 0.129 versus same-calendar RCT.For selected-subpopulation requests, its one-week reduction versus direct reuse is 0.347; reuse also matches refitted tilts in all 18,432 comparisons at each pilot length.Source: E STRONGER CONTROLS, WEIGHT… · p. 24, Table 19 · p. 24
  • Population reduction is evaluated with exact per-type tables and binding budgets, isolating the structural loss from assigning one plan to each history layer.The study compares layered plans against an individual-level optimum across 3,072 laws.Source: E STRONGER CONTROLS, WEIGHT… · p. 25
  • Five history layers retain 94.6% of the individual-level optimum, versus 82.5% for one group-level plan; each added layer raises retained value, though threshold responses near the goal retain less.The fourth and fifth layers add 0.013 and 0.008; retention is 87.5% under threshold response near the goal. Table 20 gives the study scope.Source: E STRONGER CONTROLS, WEIGHT… · p. 25, Table 20 · p. 25

F PUBLIC RETAIL DATA

  • The public retail-data test uses 3,000 riders’ histories over 45 weekly origins to assess computation and settlement transfer; sales have no incentive response.The passage describes M5 forecasting and aligned scenarios, but reports no performance outcome here.Source: F PUBLIC RETAIL DATA · p. 25

G COMPUTATION AND APPROXIMATION BOUNDARIES

The computation and allocation procedures reuse cached value information, settle activity rules from trajectory statistics, and preserve exactness under specified assumptions. The passages also delimit what these tests establish, including when response corrections may be reused and which allocation approximations can lose value.

G COMPUTATION AND APPROXIMATION BOUNDARIES

  • Reusing a valid law lowers median request time from 84.78 to 4.36 seconds, with identical scenarios and exact values on all 405 tables.With preparation charged, 127 requests per origin are 7.11× faster; on all 30,490 series, the measured speedup is 13.88×.Source: G COMPUTATION AND APPROXIMATION… · pp. 25–26
  • The dynamic program reuses its value curve across requests and is 5.52× faster than envelope MIP at 33 budgets, with the same optimal value on every MIP-resolved request.Across 396 conditions, raw and envelope MIP leave 25 and 5 requests unresolved, while all dynamic-programming variants resolve every request.Source: G COMPUTATION AND APPROXIMATION… · p. 26
  • For the active-day rule, all 14,580 answers equal direct settlement, and shared preparation cuts complete time by an average of 70.98% across origins.The comparison covers 45 origins, 2,048 scenarios, five groups, and 270 reward paths.Source: G COMPUTATION AND APPROXIMATION… · p. 26

H CONTROLLED RESPONSE INSTANCE

  • The controlled response instance defines trajectory-based uplift, payment, and spending stability, while its generality condition requires a new pilot when a rule or setting may change behaviour.A new window, population, or budget can reuse specified upstream outputs; reusing stored laws is exact for computation, but reusing a response correction assumes unchanged response.Source: H CONTROLLED RESPONSE INSTANCE · p. 26

I SETTLEMENT INTERFACE AND EXACT ALLOCATION

  • Award probabilities and award-marked moments provide rule-specific settlement quantities, but joint trajectories are needed for variance and spending-band risk.Equal expected payment and award probability need not imply equal probability of staying within a spending band.Source: I SETTLEMENT INTERFACE AND EXACT… · pp. 27–28

I.1 EXACT ALLOCATION OVER ATTAINING SCORE SEGMENTS

  • The budget-grid program loses value on 3 of 405 predicted five-layer main-grid instances, with a maximum loss of 0.92%, and never exceeds the cap.Independent complete enumeration agrees on feasibility and optimal value for all re-solved tables small enough for enumeration.Source: §I.1 · p. 28
  • For fixed laws, additive scores, group-wise menus, and exact rational arithmetic, segment recursion returns the same value curve and canonical rewards as dense allocation.The recursion handles reward-domain gaps, two-sided budgets, rule-dependent event scores, and a reward witness for every scored spend.Source: §I.1 · p. 29

J FOUR-TERM LOSS DECOMPOSITION: PROOFS AND STAGE BOUNDS

  • Population reduction adds no error to expected tables when a layer law is the mixture of its members’ laws; its effect is through the available menus.The decomposition statement concerns expectations of bounded trajectory functions and does not remove menu-related effects on the optimum.Source: J FOUR-TERM LOSS DECOMPOSITION: PROOFS… · p. 30

J.1 PROOF OF PROPOSITION 1

The proof bounds value loss in terms of estimation and stage errors, and gives instances showing that omitting a stage can leave an error floor. It also distinguishes attainable finite-sample rates from conditional guarantees and lower bounds that establish hard instances rather than rule out error cancellation on any particular instance.

J.1 PROOF OF PROPOSITION 1

  • A deterministic planner can earn zero against V⋆(0)=ω(2Ec)+2Ev on an instance with identical final tables, even when it sees the stage-error magnitudes.The construction also shows that a planner maximizing estimated value without tightening may select a portfolio whose true spend is infeasible.Source: §J.1 · p. 31
  • Without pilot correction, two offers can have identical estimated tables despite an uplift U, causing a deterministic planner to lose the entire optimal uplift.Using only the first two moments can also misprice a threshold prize by at least R/(2(1+t²)).Source: §J.1 · p. 31
  • Splitting a budget equally across L layers can earn zero when the joint optimum earns U, because layer 1 then receives less than its required budget B.The proof also notes that a comparator over a finer reward domain adds the difference between the two optima at 2Ec to η.Source: §J.1 · pp. 30–31

J.2 BOUNDS ON THE FOUR STAGE TERMS

  • Threshold-rule table errors depend on award-probability errors, while multi-day attendance requirements need joint trajectories: shared marginals can produce G(3)=1/2 versus 1/4.Thus separate completion and attendance marginals do not determine the joint eligibility probability.Source: §J.2 · p. 32
  • Response correction removes the KL term from the synthetic-error bound and makes it vanish when the response is a feature tilt, but unpiloted-offer interpolation bias persists with larger pilots.The correction bound covers piloted offers; controlling the interpolation bias requires more pilot offers or smoothness in the offer parameter.Source: §J.2 · p. 33
  • Estimated-base stability is claimed only when support overlap, weight range, and feature-covariance conditioning are controlled; losing support can make the bound grow or prevent the refit.When the corrected law is the tilt of the empirical stored-week law, its weighted sum is exact for that law.Source: §J.2 · p. 35

J.3 RATES OF THE MOMENT-MATCHING AND INTEGRATION TERMS

  • The stated rates hold for fixed-integrand table entries, not for mean-centred stability scores, which can retain constant error as pilot moments converge.When riders in a week share a shock, n counts independent weeks or blocks rather than rider-weeks.Source: §J.3 · p. 36
  • For fixed integrands, moment-matching and sampled-integration errors are O(n^-1/2) and O(S^-1/2), respectively; exact summation gives zero integration error.The moment-matching rate with a moving monetary stability band requires a margin and sets in only once n ≳ dβ^2ȳ^2/(κργ^2); a small margin delays it.Source: §J.3 · p. 37

J.4 WHAT THE LOWER BOUNDS COVER

  • The synthetic-stage constant 2 and the n^-1/2 and S^-1/2 moment-matching and sampling-integration orders are attained by constructed instances.The synthetic construction makes the regret bound tight for the actual I-projection; exact summation over a finite trajectory space removes integration error.Source: §J.4 · pp. 39–40
  • The decomposition bounds perturbation loss against the same layer menus, but not the loss from restricting individual plans or the tested ranking and roster choices.Budget-tightening loss is bounded only under a margin or growth condition; the proposition’s table-level lower bounds do not imply that stage errors cannot cancel on a given instance.Source: §J.4 · p. 40

K REGISTERED TEST FAMILIES

  • Table 22 specifies registered decision rules and test families across validation, synthetic-pilot, integrated, new-generator, and stronger-control studies.Rules were fixed before runs, with the integrated-study analysis script completed afterward; reported intervals and test procedures vary by study family.Source: Table 22 · p. 42
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