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Strategic Multi-Agent Resource Allocation via Statistical Contracts

Ben Chugg, Michael I. Jordan, Aaditya Ramdas

arXiv:2610.18336v1cs.MAcs.GTstat.ME

TL;DR

The paper asks how a principal can allocate resources among agents with unknown distributions while limiting allocations to null agents. It develops e-value-based allocation regions and mechanisms, including tractable optimization for null allocation rate. It also studies agent entry and sequential allocation, with an acknowledged computational limitation for general-loss regions.

  • Problem

    A principal must limit allocations to null agents while distributing resources among agents whose distributions and null status may be unknown.

    Source: §1 · pp. 1–2

  • Method

    The paper uses e-value-based continuous-closure allocation regions for risk control, alongside explicit policies and an online mechanism.

    Source: §1 · p. 3

  • Results

    The paper establishes that general-loss allocation regions are not efficiently computable in general, while null-allocation-rate optimization admits polynomial-time linear optimization.

    Source: §4 · p. 9, §5.1 · p. 10

  • Takeaways & Limitations

    For sequential arrivals and irrevocable allocations, the framework provides time-uniform null allocation rate control while permitting incremental risk allocation through a predictable policy.

    Source: §1 · p. 3

    The general-loss allocation regions are not efficiently computable in general.

    Source: §4 · p. 9

Abstract

from arXiv · show

We study a class of strategic hypothesis testing problems in which a principal aims to allocate resources among competing agents who possess data or data-generating mechanisms that may be relevant to the principal's goals. Each agent can pay a participation cost to opt into a statistical contract, in which case they generate data from a private distribution and receive an allocation based on a predetermined (and publicly known) mechanism. Each agent is either "null" or "non-null" and the principal's goal is to design the contract to limit the allocation to the former while maximizing transactions with the latter. Using e-values, we define a "continuous-closure," a set of allocations which prevents null agents from profiting in expectation and provides post-hoc risk control for general loss functions under arbitrary dependence. When the loss is the fraction of resources allocated to null agents, the region admits polynomial-time linear optimization. We also give explicit, interpretable closed-form approximations, and we analyze participation from a non-null agent's perspective, giving conditions under which entry into the contract is profitable. Finally, we extend the framework to support online arrival of the agents, in which case the principal has to make irrevocable on-the-fly decisions, and provide a method which gives time-uniform error-control in this setting.

1 Introduction

The paper studies contracts that allocate resources among agents with unknown distributions, seeking to favor non-nulls while limiting allocations to nulls. It introduces risk-controlled allocation mechanisms and studies their optimization, agent participation, and online use.

1 Introduction

  • The contract publicly specifies an allocation mechanism and participation costs; entrants pay the cost, generate data, and receive allocations based on the mechanism.The principal's goal is to design mechanisms that yield controllable incentive structures.Source: §1 · p. 2, Figure 1 · p. 3
  • The principal must limit allocations to null agents while distributing resources to non-nulls whose distributions and null status may be unknown.Agents may decide whether to enter based on their private information and beliefs about competitors.Source: §1 · pp. 1–2
  • The continuous-closure supports arbitrary loss functions, null deterrence, and post-hoc risk control; for null allocation rate, linear objectives are polynomial-time optimizable and simple mechanisms approximate the optimum.The paper characterizes tradeoffs analytically and experimentally.Source: §1 · pp. 2–3
  • Strong evidence alone may not make entry profitable for a non-null agent, since faster-evidence competitors can absorb nearly all the budget.Truncating evidence yields explicit lower bounds on payout, and is necessary for such guarantees in a particular formal sense.Source: §1 · p. 3
  • For sequential arrivals with irrevocable allocations, the paper establishes time-uniform control of the null allocation rate while allowing incremental risk allocation through a predictable policy.The allocation can also fund another round of experimentation.Source: §1 · p. 3

2 Related Work

The paper situates its framework among statistical contracts, contract theory, multiple testing, and resource-allocation methods. It distinguishes its statistical guarantees and participation setting from neighboring approaches.

Statistical contract theory

  • Prior statistical contract work studied principal-agent hypothesis testing with a single agent; this paper extends the setting to multiple agents.Statistical contracts address information asymmetry involving data analysis and statistical quantities.Source: §2 · p. 4

Algorithmic contract theory

  • Unlike algorithmic contract theory's hidden costly actions that shape outcomes, agents here choose whether to participate without changing their data distribution.The paper notes that connections between the subfields may merit further investigation.Source: §2 · p. 4

Multiple testing and e-values

  • The continuous-closure generalizes the e-closure principle [48], while the e-BH and online mechanisms draw inspiration from Wang and Ramdas and Xu and Ramdas [47].The paper frames multiple testing as binary resource allocation, with hypotheses acting as agents.Source: §2 · p. 5

Resource allocation

  • Bandits and auctions address allocation under uncertainty or strategic behavior, but the paper contrasts them with its statistical guarantees on null-agent behavior.Bandits do not typically model alternatives' strategic behavior, while auctions do not impose statistical guarantees on null agents.Source: §2 · p. 5

3 Multi-Agent Statistical Contracts

The paper formalizes a contract in which agents pay to enter, generate data, and receive allocations, then defines individual deterrence and portfolio risk objectives. It emphasizes post-hoc control and focuses subsequent analysis on the null allocation rate.

3 Multi-Agent Statistical Contracts

  • A contract may have a finite total-allocation budget or an “infinite” budget, meaning no natural bound on allocations rather than unlimited finances.Examples include profit licenses, market access, and exposure multipliers.Source: §3 · p. 5
  • Null deterrence requires each null agent's expected allocation not to exceed its participation cost.Because allocations are nonnegative, this is equivalent to each null agent's allocation divided by its cost being an e-value.Source: §3 · p. 6
  • Post-hoc risk control permits the principal to choose both the loss function and error bound based on data, while the null set remains fixed.It is stricter than expected-loss control at a fixed level and single loss.Source: §3 · pp. 6–7
  • The paper focuses on null allocation rate, the expected fraction of total allocation going to null agents, as a portfolio-level loss.With binary allocations, null allocation rate control is equivalent to false discovery rate control, with strategic entry of hypotheses.Source: §3 · p. 7
  • Null deterrence and null allocation rate control do not imply one another; allocations equal to costs satisfy deterrence but can yield a large null allocation proportion.Conversely, allocations of cost plus one can violate deterrence while the null allocation rate is small under the stated cost-and-count condition.Source: §3 · p. 7
  • The allocation regions in Section 4 support optimization while preserving null deterrence and post-hoc risk control, and Section 5 studies computational tractability.The paper also identifies agent utility and non-null entry incentives as a further concern.Source: §3 · p. 7

4 Continuous-Closure for General Losses

The continuous-closure uses e-collections to define allocations that satisfy null deterrence and provide post-hoc risk control for general losses. Its general regions are not efficiently computable, motivating tractable specializations for particular losses.

4 Continuous-Closure for General Losses

  • An e-collection provides subset-indexed e-values, and the continuous-closure uses them to constrain allocations for a specified loss family.Each subset's e-value has expectation at most one when all agents in that subset are null.Source: §4 · p. 8
  • Taking the constraint indexed by the true null set yields an expected-loss bound of Δ for the corresponding selection.The passage derives this bound from the continuous-closure constraint for that null set.Source: §4 · p. 8
  • Every measurable continuous-closure selection satisfies null deterrence and the stated post-hoc risk guarantee for each fixed null set.The null set remains fixed throughout the relevant suprema and expectation.Source: §4 · p. 8
  • The same e-collection must be used across losses and thresholds; separate collections may not preserve the post-hoc guarantee.The allocation itself may change with the loss and threshold.Source: §4 · p. 9
  • General continuous-closure regions are not efficiently computable, motivating tractable specializations for the null allocation rate.The constraints can be challenging because they range over every subset of agents.Source: §4 · pp. 8–9
  • Agents may choose among e-values before their data are generated, with guarantees unchanged; the choice must precede data generation.The paper notes that non-null agents may select an e-value suited to their data-generating process if they know it.Source: §4 · p. 9

5 Optimal and Approximate Allocations for NAP

For null allocation rate control, the paper develops a polynomial-time exact optimization method and simpler explicit policies, then compares their allocation quality experimentally. The explicit policies are interpretable, while optimization over the simpler region QΔ loses little relative to the full region PΔ.

5.1 Exact optimization in polynomial time

  • A separation oracle based on max-flow/min-cut gives polynomial-time linear optimization over PΔ(E), with O(m^3) arithmetic operations per separation call.The oracle finds a maximally violated subset constraint or certifies that all such constraints hold.Source: §5.1 · p. 10

5.2 Approximate-but-explicit policies for NAR control

  • The three explicit policies select allocations from QΔ(E), inheriting null deterrence and post-hoc risk control; its NAP risk bound is at most CN/C.The simpler region replaces exponentially many subset constraints with a single constraint per agent.Source: §5.2 · pp. 11, 13
  • The cumulative-evidence, all-or-nothing, and e-BH policies provide interpretable alternatives to optimizing directly over QΔ(E), which requires one linear program with O(m) constraints.The policies also support analysis of non-null agents’ expected utility when they enter the contract.Source: §5.2 · p. 11
  • The cumulative-evidence rule funds the smallest number of agents whose aggregate potential allocation reaches the global evidence threshold; it allocates zero if that threshold is unmet.All-or-nothing instead allocates to every agent once the same activation threshold is met, while e-BH withholds funds from agents whose potential allocations are too small.Source: §5.2 · pp. 11–13
  • The policies optimize different objectives: cumulative evidence minimizes funded agents, all-or-nothing maximizes the minimum allocation-to-potential ratio, and e-BH selects the largest self-consistent set.Neither all-or-nothing nor e-BH uniformly dominates the other; the choice depends on the principal’s objective.Source: §5.2 · pp. 13–14

5.3 Experimental comparison

  • At σv = 0.8, QΔ’s median gaps are 2.9% for linear and 6.2% for concave utility, versus 57.0%–65.4% and 54.3%–85.6% for the explicit policies.Across weight dispersions, QΔ’s median gap reaches at most 4.7% for linear and 8.7% for concave utility.Source: §5.3 · p. 14, Figure 3 · p. 15
  • Across tested weight dispersions, average realized null allocation shares stayed at most 3.9% under PΔ optimization, 2.9% under QΔ optimization, and 0.22% under explicit policies, all below Δ = 0.1.The optimizers allocate more to null agents than the explicit policies do, though the reported shares remain below Δ.Source: §5.3 · p. 15
  • With m increasing from 10 to 100, QΔ’s median gap rose from 3.7% to 6.8% for linear utility and from 5.1% to 7.5% for concave utility.At m = 100, explicit-policy gaps reached 67.7%–73.2% for linear and 64.5%–91.7% for concave utility; B remained fixed at 30.Source: §5.3 · p. 15, Figure 4 · p. 16

6 Non-Null Participation

Strong evidence may still leave a non-null agent with little or no payout when competitors’ evidence grows faster. Truncating evidence gives payout guarantees for MAoN and Me-bh, while payment floors trade some objective value for participation protection.

6 Non-Null Participation

  • With a finite budget, agent 1’s allocation tends to zero despite both agents being selected when the competitor’s evidence grows faster.In Example 3, the potential-allocation share c1E1,n/(c1E1,n + c2E2,n) tends to zero.Source: §6 · p. 16

6.1 Participation under truncated evidence

  • For MAoN and Me-bh, truncating e-values yields a lower bound on a non-null agent’s expected allocation when its e-value reaches L and ciL ≥ C/∆.The threshold condition is necessary for the stated guarantee; the result holds under arbitrary dependence.Source: §6.1 · p. 17
  • When non-null agents’ evidence grows exponentially and log Ln = o(n), truncation can give agent 1 limiting allocation Bc1/(c1 + c2), despite faster-growing competitor evidence.The theorem also gives eventual strictly profitable entry when B > CS under its stated conditions.Source: §6.1 · pp. 17–18
  • Proposition 6 and Theorem 2’s guarantees apply only to MAoN and Me-bh; Mcum can give an agent zero despite reaching the truncation level.This can occur when both agents reach L and c1 > c2, with Mcum funding only agent 1 once c1L ≥ C/∆.Source: §6.1 · p. 18
  • Without truncation, a non-null agent’s payout can depend on its evidence-growth rate; logarithmic transformation prevents vanishing share but retains dependence on limiting growth rates.Under the example’s rates, agent 1’s limiting payment is Bc1/(c1 + 3c2).Source: §6.1 · p. 18

6.2 Optimizing with payment floors

  • Payment floors protect a fraction λ of MAoN’s baseline payment while preserving safety, and choosing λ > CS/B when B > CS guarantees eventual profitability at a cost of at most fraction λ of the optimum-baseline gap.The principal optimizes its objective subject to Ai ≥ λbi(E) for every agent.Source: §6.2 · p. 19

7 Extensions to Online Arrival and Two-Stage Contracts

The paper extends statistical contracts to irrevocable online allocations and to contracts with a second experimentation stage. Online control is time-uniform for a fixed, prespecified ∆, while valid second-stage evidence must account for first-round selection.

7.1 Online multi-agent statistical contracts

  • Online contracts allow strategic entry over time, irrevocable allocations, and a budget that can increase at each timestep.The principal decides each allocation on the fly rather than deferring or revoking it.Source: §7.1 · pp. 19–20
  • Online validity requires e-values valid conditional on past information, and the time-uniform guarantee is for a fixed, prespecified ∆, not post-hoc choice of ∆.The online null-deterrence condition is conditional on a null agent entering.Source: §7.1 · pp. 20–21
  • The online mechanism uses predictable risk increments and conditional e-values to control the null allocation rate simultaneously at every time, including stopping times.An arriving agent receives ctEt if its e-value crosses the time-specific threshold and the remaining budget permits it; otherwise it receives zero.Source: §7.1 · p. 20

7.2 Two-stage contracts

  • Second-stage evidence can be multiplied by first-stage evidence and used with batch mechanisms while retaining risk control, provided the second round includes the original set of agents.The second-stage e-value must be conditionally valid given first-round information.Source: §7.2 · p. 21
  • Restricting the second round to agents funded in the first can invalidate guarantees because selection on first-round e-values biases them upward.The passage also notes an alternative using second-stage e-values alone, which remain conditionally valid.Source: §7.2 · p. 21

8 Conclusion

The conclusion frames statistical contracts as a way to coordinate competing agents with private information while addressing the principal’s allocation goals.

8 Conclusion

  • The paper studies resource allocation between a principal and competing agents with private information using statistical contracts.Its objectives include disincentivizing ineffective agents from entering and bounding the share paid to them.Source: §8 · p. 21
  • The authors present statistical contract theory as a lens combining statistical inference, multiple testing, game theory, and contract theory.They describe the theory as being in its infancy.Source: §8 · p. 22
  • The paper’s broader outlook treats data and data analysis as shared resources and trading opportunities in networked collectives.The authors see this research direction as addressing problems at the intersection of statistics and economics.Source: §8 · p. 22

A Omitted Proofs

The omitted proofs establish the paper’s optimization and allocation guarantees through subset-constraint arguments, maximum-flow reductions, and asymptotic bounds. They also show how online risk control follows across times and stopping times.

A Omitted Proofs

  • A maximum-flow/minimum-cut separation oracle checks the subset constraints in O(m^3) arithmetic operations per call, yielding polynomial-time linear optimization over the rational allocation region.The graph has m + 2 vertices and O(m^2) edges.Source: A Omitted Proofs · p. 26
  • The passage establishes that ˆA is a maximizer; if v=t* and t*=1 or t*=B/∥F∥1<1, then A=ˆA, and the same argument applies over Q∆(E).Source: A Omitted Proofs · p. 27
  • If each non-null agent meets the asymptotic condition, their allocations converge to Bci/CS, null allocations converge to zero, and the expected null allocation rate tends to zero.The budget binds with probability tending to one.Source: A Omitted Proofs · p. 28
  • With truncated evidence, a non-null agent who reaches L receives a lower-bounded allocation under MAoN or e-BH; under the theorem’s conditions, its limiting expected allocation is at least Bci/CS.The proof uses the fact that the agent reaching L alone triggers selection, followed by budget rescaling.Source: A Omitted Proofs · pp. 27–28
  • Adding payment floors preserves null deterrence and post-hoc risk control; the objective loss is bounded by λ times the gap between the optimum and baseline objective.The proof uses a convex mixture of the optimal allocation and the baseline.Source: A Omitted Proofs · p. 28
  • The online proof bounds the null allocation rate simultaneously over time and therefore also at stopping times.The stopping-time claim follows from the same pathwise bound used for the supremum over times.Source: A Omitted Proofs · p. 29

B.1 Alternative scaling method for finite budgets

The appendix compares evidence-adaptive reserve rescaling with hard-cap rescaling for finite budgets. The reserve rule preserves null deterrence and post-hoc NAP guarantees, while the utility comparison depends on whether raw utility is negative or positive.

B.1 Alternative scaling method for finite budgets

  • Evidence-adaptive reserve rescaling preserves null deterrence and leaves NAP unchanged, so the explicit policies retain their post-hoc NAP risk guarantees.The reserve is treated as a dummy agent; its share of the budget decreases as total raw allocation grows.Source: §B.1 · p. 30
  • For positive raw utility, hard-cap rescaling gives higher utility; for negative raw utility, reserve rescaling gives higher utility.When raw utility is zero, the two rules yield equal utility; when total raw allocation is zero, both allocate zero.Source: §B.1 · p. 30
  • For coordinate-wise nondecreasing objectives, hard-cap rescaling is weakly preferable because it pays at least as much to every agent.The reserve rule can instead be useful when false allocations are especially costly.Source: §B.1 · p. 30

B.2 Extension to approximate and asymptotic e-values

The paper extends its contract guarantees to approximate and asymptotic e-values, with bounds that depend on approximation error. Strong asymptotic validity supports limiting guarantees, while truncation can recover that stronger property from uniformly asymptotic e-values.

B.2 Extension to approximate and asymptotic e-values

  • Many guarantees degrade gracefully with approximation error, provided the approximation notion is chosen appropriately.A positive δ allows the expectation bound to fail on an event of probability at most δ.Source: §B.2 · p. 30
  • With approximate e-values and δi = 0 for null agents, null allocation is bounded by ci(1 + εi), with corresponding post-hoc risk bounds inflated by average null approximation error.For general approximation, the NAP bound is ∆(1 + ε̄N) + δN; with finite budget, expected null allocation is bounded by ci(1 + εi) + δiB.Source: §B.2 · p. 31
  • Uniformly strongly asymptotic e-values yield limiting null deterrence and post-hoc risk control, whereas uniformly asymptotic e-values alone may not provide post-hoc risk control.For uniformly asymptotic e-values, null deterrence holds asymptotically when B is finite, and expected NAP is at most ∆ in the limit for each fixed ∆.Source: §B.2 · pp. 30–31
  • An asymptotic e-value need not be strongly asymptotic: the exponential t-statistic construction has infinite expectation under Gaussian nulls, causing infinite post-hoc risk and failure of null deterrence when B = ∞.Replacing the sample standard deviation with the uncentered root mean square, or using the cited Bentkus-type construction, gives strongly asymptotic alternatives.Source: §B.2 · p. 32
  • Truncating approximate e-values makes them strongly approximate, and fixed truncation therefore permits post-hoc risk control for uniformly asymptotic sequences.More generally, truncation levels Ln work when δnLn → 0.Source: §B.2 · pp. 30, 32

B.3 Truncation is necessary for growth-independent allocations

The appendix establishes that truncation can provide non-null agents growth-independent allocation guarantees that unbounded evidence transformations cannot. It also examines selective entry: an ex-post wrapper preserves post-hoc risk control, but null deterrence in the participation-conditioned sense is generally restrictive.

B.3 Truncation is necessary for growth-independent allocations

  • Any unbounded nondecreasing transformation bounded above by the identity can be defeated by a competitor whose evidence grows sufficiently quickly, so diverging transformed evidence alone does not guarantee profitable entry.By contrast, truncation guarantees a limiting allocation based on an agent’s own power, independent of competitors’ evidence growth.Source: §B.3 · pp. 32–33

B.4 Selective entry and the ex-post wrapper

  • The ex-post wrapper treats nonparticipation as zero evidence and preserves post-hoc NAP risk control, with a sharper bound for the three explicit mechanisms.Continuous-closure also gives a nonempty region and risk control for losses that vanish at zero; these results hold for any budget.Source: §B.4 · p. 34

B.4.1 Limits of null deterrence under selective entry

  • Under selective entry, expected allocation at most cost does not ensure nonpositive expected gain conditional on entry, which generally requires null allocations never to exceed cost.For e-value allocations Ai = ciEi, that condition requires the e-value never to exceed one, allowing only trivial choices.Source: §B.4.1 · pp. 34–35
  • This impossibility argument assumes the entering agent can compute its counterfactual allocation from other agents’ behavior; less-informed agents may admit an appropriate version of null deterrence.The paper notes that agents may instead estimate that allocation when they do not know one another’s actions.Source: §B.4.1 · p. 35

C Examples of Loss Functions

  • Continuous-closure applies to losses with nonempty feasible regions; losses vanishing at zero ensure nonemptiness, and finite-budget feasibility is assessed on the final allocation.Table 1 lists examples of losses the principal may consider.Source: C Examples of Loss Functions · p. 35

D Experimental Details

The experiments use simulated agents with Gaussian null and alternative observations and compare allocation objectives and methods on active instances. The section also specifies optimization procedures and implementation checks.

D Experimental Details

  • Each simulated agent is null with probability 1/2 and supplies 30 independent observations from N(0, σ²) or N(1, σ²), with σ² = 4.For Figure 3, the setup uses m = 30, B = 30, and ∆ = 0.1, retaining 150 instances where the safe polytope admits nonzero allocation.Source: D Experimental Details · p. 35
  • The Figure 3 comparison varies weight dispersion σv from 0 to 1.2 on the same 150 instances, isolating allocation quality from whether evidence suffices to allocate.The methods include full-polytope optimization, optimization over Q∆(E), and cumulative-evidence, AoN, and e-BH allocations, evaluated using the same objective.Source: D Experimental Details · pp. 35–36
  • For Figure 4, utility comparisons use 150 active instances at each of five agent counts, m ∈ {10, 20, 40, 80, 100}, with B = 30 and ∆ = 0.1.The implementation uses SciPy optimization routines and a NetworkX minimum-cut oracle, with a stated objective-gap tolerance of 10^-6 max{1, Vlog(A)}.Source: D Experimental Details · p. 36
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