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Parallelism or Concession? Concurrency-Aware Procurement Negotiation for Agentic Commerce

Xiaolin Xu, Donghao Zhu

arXiv:2610.06017v1cs.GTcs.AIcs.MA

TL;DR

Agentic buyers can parallelize procurement negotiations, but thread costs and duplicate commitments make concurrency and price concessions interdependent. The paper develops CANO, an optimizer that jointly selects concurrency and a shared procurement cap. Across the full factorial evaluation, common heuristic policies recover less surplus than CANO, and the analysis is limited to a single negotiation window.

  • Problem

    Agentic buyers can launch parallel negotiations, but thread costs and simultaneous agreements create trade-offs that make the number of negotiators and their shared price cap consequential.

    Source: §1 · p. 1

  • Method

    CANO jointly optimizes negotiation concurrency and a common terminal price cap for one-unit procurement, balancing fulfillment, failure loss, thread cost, and excess commitments.

    Source: §1 · p. 2, §3.1 · p. 3, §3.2 · p. 3

  • Results

    Across the full factorial, Single-agent, All-parallel, Mean-cap, and Fixed-q recover 61.4%, 85.7%, 72.7%, and 44.0% of Cano’s recoverable surplus, respectively.

    Source: §5.2 · p. 6

  • Takeaways & Limitations

    Parallelism can reduce per-conversation concession, but its fulfillment gains are eventually overtaken by thread and commitment costs at a fixed target.

    Source: §6.3 · p. 8

    The results assume one negotiation window; when retries are possible immediately at negligible cost, sequential launch weakly dominates parallel launch.

    Source: §6.1 · p. 8

Abstract

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Agentic buyers can cheaply fork a procurement task into many parallel negotiations, but concurrency is not free: every thread consumes resources, and simultaneous agreements create cancellation and commitment risk. We study a one-unit post-order sourcing problem with a single hard-deadline negotiation window, in which a planner jointly chooses the number of seller-facing negotiators and a common procurement price cap. The model combines a product-specific acceptance curve with fulfillment loss, per-thread cost, and excess-commitment cost. We establish three structural results. First, holding the per-thread acceptance target fixed, the marginal value of another negotiator decays geometrically, yielding a conditional concurrency threshold. Second, under a convex quantile curve, parallelism substitutes for concession: more concurrent negotiators imply a weakly lower per-thread acceptance target and price cap. Third, when prices are more dispersed, Agentic buyers benefit by searching harder for bargains, but suffer when they instead try to guarantee procurement by offering higher prices. We operationalize these results in the Concurrency-Aware Negotiation Optimizer (CANO), a deterministic optimizer that jointly determines the optimal negotiation concurrency and procurement price cap for an agentic procurement system. Across different analytic market configurations and extensive Monte Carlo, finite-data, non-Gaussian, and correlated-seller stress tests, CANO consistently outperforms common heuristic policies while validating the predicted structural properties.

1 Introduction

Agentic procurement can cheaply parallelize seller negotiations, but threads consume resources and simultaneous acceptances create commitment risk. The paper frames concurrency and a shared price cap as a joint, auditable control problem for one-unit procurement.

1 Introduction

  • Parallel negotiations can raise the chance of finding an acceptable seller and reduce per-thread concession, but thread costs and near-simultaneous agreements make contacting every seller nonoptimal.Each thread consumes tokens, time, and seller attention; multiple agreements can create commitments for a single purchase.Source: §1 · p. 1
  • Prior approaches often fix thread count or reservation price, or learn dialogue policies, leaving the ex ante choice of how many negotiators to launch and what shared cap to set unresolved.The paper identifies this orchestration decision as distinct from dialogue-level bargaining.Source: §1 · p. 1
  • Concurrency and the price cap are coupled: more threads improve fulfillment but raise duplicate-acceptance risk, while a higher cap improves per-thread success but reduces transaction surplus.Optimizing either control alone can produce a high deal rate while destroying value.Source: §1 · p. 1
  • For one-unit post-order sourcing, the planner chooses concurrent negotiators and a common terminal cap within one negotiation window, accounting for failure loss, thread cost, and excess-commitment cost.All contacted sellers launch before terminal responses arrive, and excess commitments capture residual race costs even with a commitment manager.Source: §1 · p. 1
  • CANO is presented as a small, auditable control problem that jointly optimizes concurrency and concession, with analytic comparisons and simulation and stress tests supporting its evaluation.The evaluation includes a 1944-configuration factorial sweep and one million simulated orders.Source: §1 · p. 2

2 Background and System Framework

The framework positions Cano as an ex ante economic controller, distinct from dialogue policies and negotiation benchmarks. Its architecture separates market estimation, joint planning, constrained parallel negotiation, and commitment handling.

2.1 Related Work

  • Related work covers concurrent protocols and commitment models, multi-resource tentative-agreement coordination, one-to-many learned policies [4], and language-mediated bargaining models and [8].Recent evaluations include NegotiationArena, buyer-side bargaining, and self-play [5] [7] [15], while AgenticPay compares sequential and parallel modes [13].Source: §2.1 · p. 2

2.2 Positioning

  • Cano treats unit-demand successes as substitutes and chooses thread count and a shared reservation cap before seller-facing text is generated, rather than optimizing utterances or agreement rate alone.Its objective jointly values fulfillment, price, computation, and duplicate commitments.Source: §2.2 · p. 2, §2.1 · p. 2

2.3 System Architecture

  • The architecture uses historical offer-response data to estimate seller availability and acceptance, then plans concurrency, acceptance target, and cap; execution validates agent actions and retains at most one transaction.Outcome records support recalibration, audits, and review, while estimator design, language policy, and execution engineering are outside the paper's optimization focus.Source: §2.2 · p. 2, §2.3 · pp. 2–3

3 Procurement Model

The procurement model makes a one-window, one-unit sourcing decision over concurrent threads and a common terminal cap. Its expected-profit objective balances fulfillment and failure outcomes against operating and excess-commitment costs.

3.1 Market and Decisions

  • After a customer order is accepted, the planner chooses how many eligible sellers to contact and a common walk-away cap; a seller accepts when its reservation price is at or below the cap, which must remain below net order value.The cap is charged whenever at least one seller accepts, a conservative abstraction when a multi-round protocol can close below the cap.Source: §3.1 · p. 3
  • The model assumes conditionally i.i.d. seller reservation prices within a frozen product-state estimate and one negotiation window, with no replacement thread after failure; the common-cap restriction is not claimed optimal for arbitrary asymmetric mechanisms.Seller-specific caps and joint heterogeneous assignment are outside the structural claims.Source: §3.1 · p. 3

3.2 Expected Objective

  • Expected profit combines fulfilled-order value, procurement-failure loss, per-thread operating cost, and the expected cost of acceptances beyond the first.The planner maximizes over thread count and per-thread acceptance probability, then compares the best policy with immediate cancellation; duplicate-acceptance probability complements expected excess commitments.Source: §3.2 · pp. 3–4

4 Structural Results and Cano

The structural results characterize when added concurrency is worthwhile, how it changes the optimal acceptance target, and how price dispersion affects profit. CANO implements the joint optimization using a search procedure designed to handle empirical acceptance curves.

4.1 A Threshold for Concurrency

  • For a fixed acceptance target, the planner adds negotiators while each next thread has positive marginal value, which decays geometrically with the number already launched.The threshold weakly rises with fulfillment value and failure loss, and falls with per-thread and excess-commitment costs; a large seller pool alone does not make contacting every seller optimal.Source: §4.1 · p. 4

4.2 Parallelism Substitutes for Concession

  • Under a convex quantile curve, increasing concurrency lowers the optimal per-thread acceptance target and procurement cap, so parallelism substitutes for concession.The cap balances the marginal gain from procurement reliability against marginal procurement and duplicate-commitment costs; thread cost affects the outer choice of concurrency, not this fixed-concurrency condition.Source: §4.2 · p. 4

4.3 When Dispersion Helps

  • Within a mean-preserving symmetric location-scale family, the optimized target q*=.5 separates lower-tail search from reliability-through-concession; the result is local to that family.The lower-tail regime benefits from dispersion when the target is below the median, while thin markets or critical fulfillment reliability may require a target above the median; the joint-optimization conclusion applies piecewise when optimal concurrency remains constant.Source: §4.3 · p. 5

4.4 Cano Optimizer

  • CANO enumerates feasible concurrency levels, solves each fixed-concurrency problem, and compares the best objective with the cancellation option.Because empirical acceptance curves and mixtures need not meet the convexity assumption, the implementation uses a bounded global grid scan followed by golden-section refinement, at cost O(NG + Nlog(1/ε)).Source: §4.4 · p. 5

5 Computational Evaluation

The evaluation tests CANO’s structural predictions, compares joint optimization with restricted policies across a full factorial, and examines estimation and dependence stress tests. It reports that joint optimization outperforms common restricted policies, while identifying seller dependence as a key source of misspecification.

5.1 Design

  • The design compares CANO with four policies that restrict one or both decision dimensions, measuring regret, surplus recovery, near-oracle frequency, and cancellation.The analytic factorial covers 1,944 configurations; main comparisons use analytic expected values rather than sampling intervals.Source: §5.1 · p. 5

5.2 Structural and Policy Results

  • 61.4%, 85.7%, 72.7%, and 44.0% of Cano’s recoverable surplus is recovered by Single-agent, All-parallel, Mean-cap, and Fixed-q, respectively, across 1,944 configurations.All-parallel is within 1% of the joint solution in 62.8% of configurations but cancels in 10.9%.Source: §5.2 · p. 6, Table 2 · p. 7
  • As seller count rises from 1 to 50, mean optimal concurrency grows from 1.00 to 17.06 while the mean target falls from .610 to .291.Aggregate fulfillment rises from .610 to .888; in the reference market, the joint optimum is (m*, q*, b*) = (13, 0.161, 0.648), with expected profit 0.465.Source: §5.2 · pp. 6–7

5.3 Estimation and Dependence Stress Tests

  • At n=25 versus n=1,000 histories, mean oracle regret falls from 0.0686 to 0.0033, while mean acceptance-calibration error falls from .059 to .010.The experiment directly observes latent reservation thresholds, a favorable estimation setting compared with interval-censored negotiation logs.Source: §5.3 · p. 7
  • At correlation ρ=.7, independence-planned profit falls from 0.519 to 0.042, while a correlation-aware planner recovers 0.169 by reducing concurrency from 14 to 4.The planner also raises the target from .151 to .363; seller marginals remain fixed while shared shocks weaken parallel diversification.Source: §5.3 · p. 7

5.4 What the Evaluation Establishes

  • The evaluation validates the economic controller against theoretical predictions and Monte Carlo, not a complete agentic procurement system.The paper does not establish LLM instruction following, stationary seller behavior, or buyer profit as the appropriate social objective.Source: §5.4 · pp. 7–8, §5.3 · p. 7

6 Limitations, Responsible Use, and Conclusion

The paper limits its claims to a one-unit, single-cap procurement setting and identifies operational and calibration boundaries. It concludes that agentic systems should delegate language while centrally optimizing and enforcing economic authority.

6.1 Limitations

  • The model isolates one-unit, single-cap procurement with conditionally i.i.d. sellers and risk-neutral expected profit, excluding bundles, repeated interaction, endogenous seller strategies, and dynamic bargaining.Acceptance curves must be learned from offer–response data; the evaluation concerns the economic controller, not LLM persuasion quality.Source: §6.1 · p. 8
  • Results depend on a single negotiation window; if failed negotiations can be retried immediately at negligible cost, sequential launch weakly dominates parallel launch.Multi-window settings require dynamic batch-size decisions.Source: §6.1 · p. 8
  • The common cap treats all prices below it as operationally equivalent, while real procurement may optimize price alongside delivery, warranty, or other attributes.A multi-attribute decision would require multidimensional acceptance models.Source: §6.1 · p. 8
  • The factorial grid covers representative market regimes, not their empirical frequency, and its normalization does not calibrate κ or δ to currency units.Deployment therefore requires parameter estimation, periodic recalibration, and robust fallbacks under distribution shift.Source: §6.1 · p. 8

6.2 Responsible Use

  • Responsible deployment calls for rate limits, disclosed automation where required, clear separation of tentative and binding actions, externally enforced caps, auditable logs, and human oversight in high-stakes cases.Seller-level limits and cooldowns should constrain N before optimization, and κ should include seller-attention costs.Source: §6.2 · p. 8

6.3 Conclusion.

  • Parallelism can lower per-conversation concession, but only until thread and commitment costs overtake geometric fulfillment gains at a fixed target.CANO jointly re-optimizes the target and makes the trade-off an auditable policy.Source: §6.3 · p. 8
  • The authors’ design principle is to delegate language while centrally optimizing and enforcing economic authority.Parallel agents can amplify spam, bargaining-power asymmetry, deception, accidental commitments, and seller-contact externalities.Source: §6.3 · p. 8, §6.2 · p. 8

A Derivations and Complete Proofs

The appendix derives the expected-payoff setup for a fixed cap and acceptance probability, then handles its boundary cases. It also relates the cap to the acceptance target through the quantile curve.

A.1 Expected Objective

  • Conditional on the product state, the number of acceptances K is binomial with parameters m and q, and the payoff distinguishes failure from one-or-more acceptances.The stated payoff includes fulfillment value or loss, the per-thread cost κm, and excess-commitment cost δ(K−1) when K≥1.Source: §A.1 · p. 9
  • The excess-commitment identity rewrites (K−1)+ as K−1{K≥1}, and substituting b=C(q) yields the objective in (3).The supplied derivation identifies the identity and substitution, without giving further steps here.Source: §A.1 · p. 9
  • At q=0, launching threads is weakly worse than immediate cancellation; at q=1 with finite C(1), one thread dominates greater concurrency.For unbounded support, no finite cap attains q=1, so optimization uses q∈[0,1) and interprets the upper boundary by a limit.Source: §A.1 · p. 9

A.2 Proof of Proposition 4.1

The proof characterizes when adding a negotiator raises expected profit at a fixed acceptance target. It uses the sign of the marginal change to locate the preferred thread count.

A.2 Proof of Proposition 4.1

  • When G(q)>0, the proof introduces the marginal profit change from adding a thread; when G(q)≤0, that change is nonpositive.The supplied passage defines G(q)=V+L+δ−C(q), but does not include the full expression for the positive-G case.Source: §A.2 · p. 9
  • Profit rises while the marginal change is positive, is flat when it is zero, and falls once it becomes negative.Resolving a zero marginal toward the smaller thread count selects the smallest m with a nonpositive marginal, unless the maximum N is reached first.Source: §A.2 · p. 9
  • Moving every marginal in one direction moves its crossing threshold weakly in that direction, while N only truncates the resulting threshold..Source: §A.2 · p. 9

A.3 Proof of Theorem 4.2

The appendix proves comparative statics for the acceptance target and price cap, establishes conditions for the dispersion result, and describes the planner and its stress tests. It also states when the implementation’s global optimality is guaranteed.

A.3 Proof of Theorem 4.2

  • Under the stated conditions, strict concavity gives a unique optimum, and increasing concurrency weakly lowers the optimal acceptance target and corresponding cap.The proof compares roots using a term that is nondecreasing in m and strictly increasing in q; the cap comparison follows because C is increasing.Source: §A.3 · p. 9

A.4 Proof of Theorem 4.3

  • For the dispersion result, the fixed-m envelope derivative has three sign cases: negative below q=1/2, zero at 1/2, and positive above it.At an integer concurrency switch, the upper envelope remains continuous but need not be differentiable; the derivative also requires a unique interior solution.Source: §A.4 · p. 9

B.1 Planner Output and Failure Behavior

  • The planner compares positive-concurrency policies against cancellation, which it selects when all yield lower expected value; global optimality is guaranteed only under Assumption 1.On a reference uniform instance, the implementation matched an exhaustive 100,001-point grid search within 10−8 as a regression check, not a universal guarantee.Source: §B.1 · pp. 9–10

B.2 Algorithm

  • For each m, the algorithm scans an acceptance-target grid, refines the best bracket by golden-section maximization, converts the selected target to a cap, and compares against cancellation.Under Assumption 1, strict concavity permits boundary derivative checks and bisection, requiring O(log(1/ε)) evaluations per m.Source: §B.2 · p. 10

B.3 Correlated Sellers

  • In the correlated-seller stress test, positive dependence lowers the diversification benefit of concurrency, shifting the optimizer toward fewer, more accommodating threads.The test uses an exchangeable Gaussian copula and a 32-node Gauss–Hermite quadrature evaluation of the one-dimensional expectation.Source: §B.3 · p. 10

C Additional Experimental Detail

The additional experiments document the analytic-market design, validate expected profit by simulation, test distributional robustness, and examine parameter comparative statics.

C.1 Full Protocol

  • The experiments are reproducible from one Python script that generates analytic rows, figures, and result macros; no proprietary data, model API, or human-subject data are used.The script uses NumPy and Matplotlib, with main seed 20,260,803.Source: §C.1 · p. 10
  • The 1,944-market analytic factorial design and Figure 4 stress tests cover finite-data regret, shared seller shocks, and four reservation-price distributions.The reported stress-test panels include mean regret with 95% bootstrap intervals over 100 replicates and partial recovery under shared seller shocks.Source: Table 4 · p. 10, Figure 4 · p. 10

C.2 Monte Carlo Validation

  • One million independent orders produce simulated mean profit of 0.519354 (standard error 0.000293), versus analytic expected profit of 0.518956.For the stated reference configuration, absolute error is 0.000397; simulated fulfillment and duplicate probabilities are 0.89924 and 0.64661.Source: §C.2 · p. 10

D.1 Distribution Shape

  • The distribution-shape tests use four markets with mean one and standard deviation approximately 0.30, applying grid scan and local refinement when quantile convexity fails globally.The two-component market represents low-price and smaller high-price segments, with its quantile estimated from 100,000 fixed-seed draws.Source: §D.1 · p. 10, Table 5 · p. 11

D.2 Parameter Comparative Statics

  • Raising per-thread cost from .001 to .06 reduces optimal concurrency from 20 to 4 and raises the target from .117 to .325.Raising excess-commitment cost from 0 to .20 reduces concurrency from 15 to 9 and excess commitments from 1.659 to .637; raising failure loss from 0 to .60 increases concurrency from 12 to 14 and fulfillment from .869 to .929.Source: §D.2 · p. 11

D.3 Scope of the Agentic Evaluation

The evaluation targets the economic control layer rather than language-agent behavior. It does not test LLM negotiation or attribute profit improvements to language generation.

D.3 Scope of the Agentic Evaluation

  • The evaluation samples seller acceptance and commitment outcomes but does not call an LLM, measure rhetoric, or compare prompts, so it makes no claim that language generation improves profit.The stated scope is the economic control layer; a future controlled LLM evaluation is proposed with matched market seeds, seller messages, and model snapshots.Source: §D.3 · p. 11
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