Source-linked AI summary

Large Language Model Orchestration under Heterogeneous Preferences via Explicit Persona Inference

Shuqing Shi, Ziyan Wang, Milind Tambe, Yali Du

arXiv:2610.07587v1cs.CL

TL;DR

Existing LLM orchestrators carry beliefs about hidden agent preferences as prompt text, without an explicit update rule. HARP instead keeps per-agent Bayesian posteriors outside the prompt, and HARP+ achieves the lowest non-oracle regret on every evaluated backbone.

  • Problem

    Existing LLM orchestrators carry persona beliefs as prompt text, while joint-profile inference grows exponentially with the number of agents.

    Source: §1 · p. 1
  • Method

    HARP maintains a numeric posterior per agent outside the prompt and updates it by Bayes’ rule from actions and candidate likelihoods.

    Source: §1 · p. 1
  • Results

    HARP+ attains the lowest non-oracle regret on every backbone, and the factored posterior matches joint inference at far smaller storage.

    Source: §1 · p. 2, §5 · p. 7
  • Takeaways & Limitations

    When personas are hidden, orchestration is an estimation problem, and the estimate belongs outside the prompt.

    Source: §5 · p. 7

    HARP assumes a fixed persona library, finite actions, and an offline-calibrated scorer; continuous control and persona drift remain open.

    Source: Limitations · p. 7

Abstract

from arXiv · show

LLM orchestration investigates how an orchestrator coordinates a group of autonomous agents to achieve common goals or maximize collective welfare. The agents are typically heterogeneous, each holding a private preference that it pursues but does not reveal. Inferring such hidden preferences from behavior has been a subject of long-standing research in game theory and multi-agent systems. The core challenge lies in maintaining a belief over every agent's preference and updating it from the agents' observed actions. Existing LLM orchestrators carry that belief as prompt text with no explicit update rule. This lets early errors persist and propagate rather than be corrected. We therefore propose \textbf{HARP} (Heterogeneous-preference Agent oRchestration via Preference inference), a novel framework that moves the belief out of the prompt. Specifically, HARP maintains one numeric posterior per agent over a finite set of candidate preferences and updates it in closed form by Bayes' rule. The language model supplies only actions and per-candidate likelihoods, so estimation is decoupled from its reasoning. We prove that HARP attains the same $\tilde O(\sqrt K)$ Bayesian regret as explicit joint inference when the factorization is exact. Furthermore, HARP\textsuperscript{+} augments planning with a bonus for actions that distinguish the candidates, so inference continues even when the optimal action is uninformative. Empirical results on three substrates, ranging from payoffs the preferences fully determine, through payoffs that depend on more than them, to scales where explicit joint inference is infeasible, demonstrate that HARP\textsuperscript{+} is the strongest non-oracle method across the class our theory identifies.

1 Introduction

HARP addresses the challenge of inferring hidden agent personas when prompt-carried beliefs can preserve early errors and joint persona profiles grow exponentially. It keeps numeric per-agent posteriors outside the prompt, and HARP+ adds an incentive to choose persona-distinguishing assignments; the paper reports the lowest non-oracle regret on every evaluated backbone.

1 Introduction

  • Existing prompt-based methods carry natural-language persona descriptions between turns, while joint-profile inference scales exponentially with the number of agents.Prompt-carried descriptions can preserve early errors, and the joint profile has |Θi|^n entries for n agents with |Θi| candidates each.Source: §1 · p. 1
  • HARP stores one numeric posterior per agent outside the prompt and updates it by Bayes’ rule using actions and candidate likelihoods.The language model supplies actions and likelihoods, decoupling estimation from its reasoning.Source: §1 · p. 1
  • HARP+ adds a planning bonus for assignments that distinguish candidate personas, bounding early loss by a constant.The overview describes per-agent posteriors, persona sampling, planning, scoring actions under each candidate, and Bayesian updates.Source: §1 · pp. 1–2
  • HARP+ attains the lowest non-oracle regret on every evaluated backbone, including when structural conditions are deliberately violated.Under those violations, HARP remains within a small constant factor of explicit joint inference.Source: §1 · p. 2

2 Related Work

Related work spans posterior sampling and latent-type learning, structural decompositions, and language-model approaches to partner modeling. HARP's distinction is exact online factorization of partner-persona beliefs under stated conditions.

2 Related Work

  • Prior posterior-sampling work establishes near-optimal Bayesian regret for a single agent, while multi-agent extensions often assume known types or zero-sum structure.The paper positions its structural conditions as serving the identifiability role in latent-MDP learning.Source: §2 · p. 2
  • HARP's related-work contribution is the conditions for exact per-teammate beliefs and finite-time regret at linear storage, not the separate-belief representation itself.Prior type-based ad hoc teamwork already maintains a separate belief per teammate (Albrecht and Ramamoorthy 2015, 2019; Barrett, Stone, and Kraus 2011; Albrecht and Stone 2018).Source: §2 · p. 2
  • Existing structural methods decompose values, allocations, or games, whereas HARP factors the posterior over partner personas without approximation and infers them online.The compared methods include networked distributed POMDPs, restless-bandit formulations, and Bayesian Stackelberg solvers.Source: §2 · p. 2
  • Language-model partner-modeling methods use learned beliefs, natural-language hypotheses, or reasoner selection, while HARP maintains an explicit posterior over a fixed persona library.The cited approaches include ECON (Yi et al. 2025), Hypothetical Minds (Cross et al. 2025; Li et al. 2023b), and A-ToM.Source: §2 · pp. 2–3

3 Methodology

The paper formulates multi-agent coordination as Bayesian inference over fixed agent personas and presents HARP, which maintains and updates one posterior per agent while planning centrally over joint assignments. It proves exact posterior factorization and unchanged Bayesian regret under structural conditions, then characterizes how regret changes when those conditions fail.

3.1 Preliminaries and Definitions

  • The model has an orchestrator assigning actions to agents over K episodes, with team welfare as its objective and each agent responding according to a fixed persona.Each candidate persona specifies an action distribution for an agent's context; the orchestrator knows the candidate library but not the true personas.Source: §3.1 · p. 3

3.2 The HARP Framework

  • HARP keeps one probability vector per agent, samples a persona from each, plans a joint policy, and updates the marginals by Bayes’ rule from observed actions.The offline-calibrated scorer evaluates actions under each candidate, so updates require no online model call.Source: The HARP loop · p. 3
  • HARP+ adds a bonus for assignments where candidate personas disagree about predicted rewards, encouraging informative choices when welfare-optimal assignments do not distinguish them.Without that incentive, repeated uninformative assignments can delay identification exponentially in the described worst case.Source: HARP+ · p. 4

3.3 Theoretical Results

  • Exact factorization requires transition independence, response locality, and a prior that factors across agents; violating any one can break exactness.Response locality excludes an agent's response depending on other agents' personas.Source: §3.3 · p. 4
  • Under those conditions, each per-agent Bayes update preserves exact joint-posterior factorization, with storage O(n|Θi|) rather than O(|Θi|^n).The paper characterizes the factorization as an exact reparameterization, not an approximation.Source: §3.3 · p. 4
  • Under the structural conditions, HARP matches single-agent posterior-sampling regret; outside them, the bound degrades with the violation, while removing response locality alone restores an exponential rate.With response locality and a separation margin, the degradation is a K-independent constant; HARP+ bounds early loss by a constant when the separation margin vanishes.Source: §3.3 · p. 4

4 Numerical Evaluation

The evaluation tests representation, exact-inference parity, and component contributions across controlled payoff tables, Concordia games, and ride-hailing dispatch.

4.1 HP-SPGG: Controlled Diagnostic Benchmark

  • HARP+ attains the lowest non-oracle mean regret on all four backbones; the strongest text-belief baseline incurs 1.63–43.77 times that regret, and LLM-PSRL 13.8–50.8 times.The advantage holds seed by seed, with one loss in forty pairings and sign tests significant on three of four backbones.Source: §4.1 · p. 5, Figure 2 · p. 5
  • HARP and Joint-PSRL agree within seed-level error, while factored storage is 5.3× lower at n=3 and 51× lower at n=5.The storage advantage grows with agent count without a corresponding regret separation.Source: §4.1 · p. 5, Figure 3 · p. 5
  • Removing updates, correct identity attachment, the discrimination bonus, or centralized dispatch raises regret; the bonus is inactive with exact analytic likelihoods.Removing the update costs +0.66 paired regret; removing the bonus raises DeepSeek regret from 0.360 to 0.985.Source: §4.1 · pp. 5–6, Figure 3 · p. 5

4.2 Concordia-Derived Compact Benchmarks

  • On Pub Coordination HARP+ stays within 1% of oracle_joint; on Haggling it wins or ties every non-oracle method across nine configurations.HARP+ reaches oracle_focal exactly on five Haggling configurations.Source: §4.2 · p. 6, Figure 4 · p. 6

4.3 Ride-Hailing Dispatch on MaaSSim

  • On MaaSSim, HARP stays within 2.7–4.4 oracle-regret units while the strongest text-belief baseline rises from 4.5 to 10.7 as personas diverge.The advantage grows across the conflict-strength sweep.Source: §4.3 · p. 6, Figure 5 · p. 7
  • At fleet sizes n=2–4, marginal posteriors match within 2.5×10−14 total variation; the learned posterior adds +16.5 utility over the prior on a fixed replay.At n=4, the joint update takes 497 µs per event versus 7 µs flat for the factored posterior; the joint table is infeasible at the full fleet.Source: §4.3 · pp. 6–7

5 Conclusion

HARP keeps persona beliefs as explicit per-agent posteriors, matching joint inference under stated structural conditions with lower storage cost. HARP+ is the strongest non-oracle method across backbones, and its gains depend on updating and acting on correctly attached beliefs.

5 Conclusion

  • Under three structural conditions, per-agent posteriors reconstruct the joint posterior exactly at linear rather than exponential cost and unchanged regret.The framework carries the persona posterior outside the language model.Source: §5 · p. 7
  • HARP+ attains the lowest non-oracle regret on every backbone, and its gain requires updating the belief, attaching it to the right agent, and centralized dispatch.The factored posterior matches joint inference at far smaller storage.Source: §5 · p. 7
  • Figure 7 reports relative oracle-regret reduction and per-event update time, with additional panels on utility by belief source and posterior gain by event type.The joint table is not run beyond n=4.Source: Figure 7 · p. 7, §5 · p. 7

Limitations

The stated scope assumes a fixed persona library, finite actions, and an offline-calibrated scorer; continuous control and persona drift remain open.

Limitations

  • HARP assumes a fixed persona library, finite actions, and an offline-calibrated scorer, while continuous control and persona drift remain open.Personas outside the library are measured at the open-text boundary.Source: Limitations · p. 7
  • Because profiling risk accompanies preference inference, deployments should disclose it.Source: Limitations · p. 7

A.0 Reading Guide

  • A.0 maps the main-text research questions to evidence bundles and locations; it certifies no standalone result.Source: §A.0 · p. 10, Table 1 · p. 10

A.1 RQ1: Numeric versus In-Prompt Representation

  • A.1 identifies the environment-matched control for measuring the representational advantage and the open-text boundary where that advantage stops.Source: §A.1 · p. 10

Environment-Matched Control (C1)

The environment-matched control tests HARP+ on a constructed finite-model benchmark and on open-ended SOTOPIA-Hard. Results support a transfer boundary: the SOTOPIA-Hard setting does not verify the conditions for exact guarantees, and corrected recurrent updates yield no focal-score gain.

Environment-Matched Control (C1)

  • In the HP-SPGG control, HARP+ is compared under common environments, profiles, seeds, and a pinned outcome tensor, but provider sampling seeds are unavailable for pathwise matching.HP-SPGG is constructed to satisfy TI, RL, and PF with an exactly calibrated persona-conditioned scorer.Source: Environment-Matched Control (C1) · p. 10
  • The open-text SOTOPIA-Hard setting does not verify TI, RL, or PF, so the exact finite-model guarantees do not transfer to it.The corrected recurrent rerun removes the evidence-path implementation liability but produces no focal-score gain.Source: Open-Text Transfer Boundary (C5) · p. 11
  • Historical SOTOPIA-Hard results are descriptive rather than evidence for recurrent belief updating, because the original evidence path supplied zero updates and reduced posterior sampling to prior sampling.HARP+ ranked second to fourth by backbone; family margins over the strongest baseline were +0.06, +0.10, and +0.02.Source: Open-Text Transfer Boundary (C5) · p. 11, Table 3 · p. 11
  • On SOTOPIA-Hard, HARP+ ranks second to fourth across backbones and remains within 0.10 of the best alternative, while all five non-oracle methods fit within a 0.12 envelope.The one-hot oracle_belief also lies within that envelope; best-of-five oracle_policy reaches 3.540.Source: Open-Text Transfer Boundary (C5) · p. 11, Table 4 · p. 12
  • The corrected recurrent rerun updates its posterior, but proxy-persona concentration varies by family and its focal means trail retained GPT-nano best-baseline means by 0.095, 0.157, and 0.457.The proxy is profile-derived, not a native SOTOPIA truth label.Source: Open-Text Transfer Boundary (C5) · p. 11, Figure 9 · p. 13
  • Corrected component controls show no attributable focal-score gain from the numeric update: none of the paired HARP differences has a positive lower confidence bound.The rerun pairs 30 cases across four replicate IDs, but provider and judge randomness are not pathwise controlled.Source: Open-Text Transfer Boundary (C5) · p. 12, Figure 10 · p. 13

A.2 RQ2: Factored versus Joint Computation

Under the stated structural conditions, factored persona posteriors match explicit joint inference while reducing storage and update work. The section also probes the limits of that result and shows that a discrimination bonus can help when identification is slow.

A.2 RQ2: Factored versus Joint Computation

  • Storage savings grow from 5.3× to 51× over n=3–5 without a corresponding regret separation in the representation-only comparison.The comparison holds likelihood, planner, and seeds fixed while changing posterior representation.Source: Exactness and Storage (C1) · p. 13, Table 6 · p. 14
  • Across 17 joint-feasible cells, HARP and Joint-PSRL produce identical action sequences and regret trajectories, with zero action mismatches over 170 seed-cells.At n=6 with 16 personas, the joint-to-factored storage ratio reaches 174,763×.Source: Exactness and Storage (C1) · p. 13, Analytic Scaling to the Joint… · p. 14
  • Joint-PSRL update work grows from 7.5 µs at n=2 to 68.8 ms at n=6, while HARP stays within 1.3–4.8 µs across the reported sweeps.Source: Analytic Scaling to the Joint… · p. 14
  • At n=7 and n=8, HARP+ remains within 0.06 of the pinned oracle after Joint-PSRL exceeds the update-time and memory limits, respectively.Joint-PSRL’s first update takes 1.02 s at n=7; at n=8 its table requires 34.4 GB.Source: Analytic Scaling to the Joint… · p. 14, Table 8 · p. 15
  • HARP+ reduces regret under a weakly discriminating library: at η=0, it cuts mean regret 6–10× versus HARP at n=5 and n=7.The minimum pairwise Hellinger gap is 1.2 × 10^-3, making identification slow even with exact likelihoods.Source: Analytic Scaling to the Joint… · p. 14
  • When response locality is violated, factored and exact-joint posteriors diverge, but the tested geometry shows no detectable regret disadvantage for HARP.At α=4, marginal posterior TV is 0.0071 ± 0.0025 analytically and 0.0171 ± 0.0047 on the live grid; paired regret gaps include zero.Source: Structural Class (C4) · p. 15, Figure 13 · p. 16
  • Under a coupled shared-type data-generating process, HARP remains within 1.52× of correctly supported Joint-PSRL-Aware on every reported cell and has lower regret on 9 of 15.This is robustness under prior-factorization violation, not exact decoupling.Source: Structural Class (C4) · p. 15

A.3 RQ3: Component Attribution

  • On HP-SPGG, the bonus shortens burn-in on slowly concentrating backbones without degrading others; the Concordia-derived diagnostic also shows HARP+ flattening after identification while PSRL-NoType keeps accumulating regret.HARP+ reaches 0.352 ± 0.035 cumulative regret versus 1.361 ± 0.172 for PSRL-NoType in the diagnostic.Source: §A.3 · p. 15, Update Value and Rate (C2) · p. 16

Discrimination Bonus (C3)

This section examines the settings where the discrimination bonus helps and the assumptions behind HARP’s theoretical guarantees, including exact factorization of per-agent posteriors and trajectory-equivalence to joint inference. It also delineates limits of those claims: factorization cannot generally be dropped, the guarantees assume centralized assignment, and unknown transition kernels are excluded from the regret analysis.

Discrimination Bonus (C3)

  • β = 0.25 reduces Llama-Maverick regret from 0.97 to 0.31 and is the smallest value in the no-degradation interval; downstream evidence shows no one-shot Concordia effect and only descriptive SOTOPIA differences.Only HP-SPGG supports interpreting the bonus as accelerated posterior concentration; the SOTOPIA posterior remained at its prior.Source: Discrimination Bonus (C3) · p. 16, Table 12 · p. 17, Figure 18 · p. 19
  • Centralized dispatch yields a 3–4× lower cumulative-regret slope than decentralized modes, with the same ordering on the five-player welfare sum.Across dispatch modes using the same LLM, centralized modes cluster near the joint oracle; the decentralized gap compounds over five episodes.Source: Mechanism: Identity and Dispatch (C6) · pp. 17, 19, Figure 19 · p. 20
  • On fixed MaaSSim states, the learned posterior raises utility by 16.50 over a uniform prior, while attaching it to the wrong driver collapses utility below the prior.The learned posterior closes 59.3% of the prior-to-oracle utility gap; rejection events provide greater rule-accuracy gains than accepts, and the comparison trades additional pickup wait for fewer rejections.Source: Mechanism: Identity and Dispatch (C6) · pp. 19–20, Table 13 · p. 19
  • In the designed full-conflict MaaSSim stress test, the hybrid exceeds the best pure prompt baseline by +39.69, remaining positive at 8.79 against an oracle at 22.44.At λ = 1, every pure prompt baseline has negative utility; this endpoint is a designed stress test, not an estimate of normal operations.Source: Mechanism: Identity and Dispatch (C6) · p. 20, Table 14 · p. 21
  • MaaSSim findings are limited to synthetic driver rules, one road network, and one LLM backbone; passenger inference is weak and the exploration bonus did not alter assignments.The LLM tables aggregate ten seeds without retained per-seed outcomes, precluding paired significance tests.Source: Mechanism: Identity and Dispatch (C6) · p. 21

A.4 Oracle Constructions (Shared Infrastructure)

  • Oracle references differ by benchmark: HP-SPGG and compact Concordia menus permit exact enumeration, while SOTOPIA-Hard uses non-strict oracle references because its open-ended actions and judge prevent exact enumeration.For Haggling, oracle_focal is the focal-payoff ceiling; oracle_joint instead maximizes joint welfare. SOTOPIA-Hard’s oracle-policy gap reflects both action-search and judge-variance headroom.Source: §A.4 · p. 21

B.1 Overview and Claim-to-Theorem Map

  • The formal appendix maps the paper’s theoretical claims to statements and proofs across posterior decoupling, regret bounds, misspecification, and HARP+ burn-in improvement.It frames these results as extensions of the Section 3.1 setting.Source: §B.1 · p. 21

B.2 Formal Setting

  • The formal setting has agents with finite persona libraries and a fixed latent persona drawn from a prior; the orchestrator observes their emitted actions, whose known response laws provide evidence about those personas.The orchestrator chooses joint assignments using public history and its own randomization; rewards, state transitions, and response rules are known in the core setting.Source: §B.2 · pp. 21–22
  • The formal results assume centralized dispatch, with one planner assigning all agents and observing their joint emitted action.An approximately optimal planner incurs at most Kϵ_plan additional regret; the stated extension covers known persona-independent transition kernels, not unknown ones.Source: §B.2 · p. 23

B.3 Posterior Decoupling

  • Under Theorem B.11’s assumptions, the joint posterior factorizes into per-agent posteriors, so HARP and Joint-PSRL have identical trajectory laws and regrets.The equivalence holds under the stated assumptions and a deterministic centralized planner; with shared random numbers, their trajectories coincide pathwise.Source: §B.3 · p. 24
  • HARP requires O(Σ_i |Θ_i|) posterior storage and likelihood evaluations per update, versus O(∏_i |Θ_i|) for an explicit joint table.For uniform persona libraries, the stated comparison is O(n|Θ_i|) versus O(|Θ_i|^n).Source: §B.3 · p. 24

B.4 No Condition Can Be Dropped

  • The factorization conditions (TI), (RL), and (ISI) each cannot be dropped generally: violating any one can produce a non-product posterior after one observation.The conditions are sufficient but not necessary for every individual instance; unknown transition kernels add a model-learning term not analyzed here.Source: §B.4 · pp. 24–25

B.5 Regret Bound

Under the stated assumptions, HARP’s factored analysis bounds Bayesian regret through per-agent information and influence sets, without a joint persona-space term. The bound also clarifies how response structure and prior information affect regret.

B.5 Regret Bound

  • The regret bound sums per-agent terms over each agent’s influence set, with each term controlled by that agent’s persona library and prior entropy.The proof decomposes the policy-value gap agent by agent and bounds each term by information revealed about that agent’s persona.Source: §B.5 · p. 25
  • The joint persona-space product does not appear in the bound, and a uniform prior gives the worst-case entropy term.The bound’s per-agent terms depend on prior entropy; informative priors can lower the terms for agents whose influence sets include that persona.Source: §B.5 · p. 25
  • When rewards depend only on an agent’s own response and emitted actions do not enter the state, that agent’s regret is governed by its own persona alone.This is the additional condition in Corollary B.16; HP-SPGG has this form.Source: §B.5 · p. 25
  • Under the theorem’s assumptions, Joint-PSRL and HARP have the same law, so their regret is the same.The factorization sharpens the analysis rather than changing the algorithm’s regret in this regime.Source: §B.5 · p. 25
  • With TID, every response separates candidates and regret stops growing; without that separation, the no-gap bound still applies.Theorem B.15 covers cases where some contexts do not separate candidates, while the TID regime yields a constant bound.Source: §B.5 · p. 27

B.6 Robustness to Scorer Misspecification

The scorer used for live posterior updates may be misspecified relative to the true response rule. The analysis bounds this error’s effect on posterior contraction and identifies when the discrimination bonus matters.

B.6 Robustness to Scorer Misspecification

  • The misspecification analysis isolates the posterior update while retaining the same planner and regret comparator.The planner still returns P(θ) for every profile, and regret is measured against P(θ⋆).Source: §B.6 · p. 27
  • With pointwise log-likelihood error at most η, posterior odds contract at rate e^−(ρ−η) per turn when η < ρ.The regret bound then has a K-independent constant with ρ−η replacing ρ.Source: §B.6 · p. 28
  • A scorer-error threshold is necessary: once η reaches the log-ratio radius L, a label-swapped scorer can drive the posterior toward the wrong persona.The proposition guarantees contraction for η < ρ, and the thresholds leave a gap.Source: §B.6 · p. 28
  • When the effective separation gap is small, HARP+’s discrimination bonus carries the burn-in and cuts mean regret by 6–10× on the analytic scaling suite.The reported gap is ρ̂ = 1.2 × 10^−3.Source: §B.6 · p. 28

B.7 Approximate Decoupling under Bounded Violations

This section quantifies how bounded violations of the factorization conditions affect posterior decoupling and regret. Prior coupling contributes an initial discrepancy, while response-law error can accumulate over turns.

B.7 Approximate Decoupling under Bounded Violations

  • Under bounded prior coupling and local response-law error, the true posterior and HARP’s factored belief remain within a trajectory-wise discrepancy that grows as Δk = Δ0 + (k−1)Hεℓ.The posterior ratio is bounded using this accumulated discrepancy.Source: §B.7 · p. 28
  • With exact response locality and TID, regret remains bounded independently of K even when the prior coupling is unbounded.The result applies to every prior for which the product prior dominates.Source: §B.7 · p. 29
  • When response-law error is bounded, regret degradation is linear in the violation size over the finite horizon; with exact response locality, the team bound sharpens under a welfare-maximizing planner.The second regime uses the prior-ratio factor κ.Source: §B.7 · p. 29
  • In the reported diagnostic, marginal total variation stayed at most 0.0171 at α = 4, with every paired regret gap covering zero.The authors note that this diagnostic is not an instance of the corollary because the Gaussian score-law log-ratio error is not small.Source: §B.7 · p. 29

B.8 Unbounded Coupling and the RL Threshold

Under response locality and TID, arbitrary prior coupling costs only a K-independent burn-in constant; without TID, coupling can instead produce a large or growing cost. A separate construction shows online persona inference can outperform type-agnostic sampling in regret rate.

B.8 Unbounded Coupling and the RL Threshold

  • Under response locality and TID, HARP’s regret is bounded independently of K for any full-support product prior, even when the true prior has arbitrary coupling.The bound holds conditionally on every realized profile.Source: §B.8 · p. 29
  • Without TID, a shared-type example attains a coupling penalty of κ = m^(n−1), while HARP+ has zero regret on that family for any β > 0.The comparison conditions on the all-jackpot profile; averaged over the shared-type prior, HARP’s team regret is κ times its factored-model regret.Source: §B.8 · pp. 29–30
  • Without TID, the cost of coupling can grow with K, and whether HARP+ or another factored method avoids that growth remains open.The passage gives HARP team regret values of 0.46, 1.72, and 3.66 at K = 10, 50, and 200 in one example.Source: §B.8 · p. 30

B.9 Rate Separation

  • HARP attains Õ(√K) regret on an instance where type-agnostic sampling without updates has Ω(K) regret.The gap is presented as the value of online persona inference relative to type-agnostic sampling.Source: §B.9 · p. 30
  • HARP’s marginals are point masses from episode two onward, giving Reg_i(K)=1; by contrast, selecting i with probability 1/2 independently at each k gives Reg_i(K)=K/2.Source: §B.9 · p. 30

B.10 HARP+ Burn-in Improvement

HARP+ uses an information bonus to avoid uninformative burn-in, while the lower-bound analysis shows that dropping response likelihood-ratio control permits exponential hardness even with a factored prior.

B.10 HARP+ Burn-in Improvement

  • The jackpot family satisfies TI, RL, and PF but violates TID because all personas respond identically at the start state.With a uniform prior and ties broken toward hold, HARP can repeat an uninformative assignment until it samples the all-jackpot profile.Source: §B.10 · pp. 30–31
  • Under TID, the bonus adds at most a K-independent constant because its contribution decays geometrically as the posterior concentrates.Sufficiently large β can over-perturb planning, but the β = 0.25 boundary is empirically inactive on every backbone.Source: §B.10 · p. 30
  • HARP+ has zero regret on the jackpot construction for any β > 0, while HARP and Joint-PSRL incur burn-in regret growing to Θ(mnH).The bonus breaks the assignment tie, advances in the first turn, and identifies every persona by the second episode.Source: §B.10 · pp. 30–31

B.11 Lower Bound Without RL

  • The lower bound reduces to a Bayesian M-armed bandit with M = mn, using a uniform needle over joint assignments.The prior is factored, and averaging the minimax bandit lower bound over that needle prior yields a Bayesian lower bound (Lattimore and Szepesvári 2020, §15).Source: §B.11 · p. 31
  • Replacing the stochastic response with a deterministic one gives only Θ(mn) burn-in and cannot certify the mnK rate.Each assignment eliminates one profile, so the stochastic construction is essential to the stronger lower-bound result.Source: §B.11 · p. 31
  • Dropping RL while retaining PF allows exponential hardness for every algorithm, whereas the cited RL results provide a √K rate under PF.The lower-bound construction couples the response to the full persona profile; the finite RL-only diagnostic is not this worst case.Source: §B.11 · p. 31

B.12 Empirical Validation

The validation reports well-specified and coupled-DGP comparisons, then tests the contraction study with a corrected preregistered design. Its measured burn-in scales with inverse information and logarithmically with population size.

B.12 Empirical Validation

  • Under the same iid DGP, well-specified Joint-PSRL-Uniform and HARP remain within sample noise; the complementary coupled-DGP table is not a realization of the needle lower bound.The passage identifies the coupled-DGP result as the finite-sample counterpart of Theorem B.32.Source: §B.12 · p. 31
  • The corrected confirmatory study passes all five gates, while the initial pooled predictor (n+1) log m fit was null with slope −0.25 and R2=0.007.The corrected design samples the channel and uses the proof's constants; its fifth gate checks transfer to the adaptive HARP planner.Source: Preregistered Contraction Study · p. 31
  • Across 15 cells, the empirical likelihood-ratio fit has slope 1.001 (95% CI [0.991, 1.009]) and R2=0.99996; per-agent burn-in follows log(m√m)/(ρactionH) across 28 cells with R2=0.906.The bootstrap slope CI is [0.264, 0.273], with zero censoring at 2,048 episodes and within-library turn-equivalent ratios at most 1.10.Source: Preregistered Contraction Study · p. 32
  • A 32× increase in population n raises all-agent burn-in by 2.5×, and the corrected log n predictor reaches R2=0.994 versus 0.849 for the retired linear-n comparator.The preregistered advantage gate passes, and the posterior-error proxy changes by at most 1.8% from episode 1,024 to 2,048.Source: Preregistered Contraction Study · p. 32, Table 17 · p. 32

C Cost, Compute, and Prompts

The appendix accounts for where language models enter the three substrates, their call and wall-clock costs, and the compute and prompt setup. HARP's posterior update is arithmetic rather than an online model call.

C Cost, Compute, and Prompts

  • The cost section aggregates four backbones across HP-SPGG, Concordia, and SOTOPIA-Hard, reporting calls, wall-clock cost, hardware/software, and prompts.The suite includes K=20 episodes per backbone-baseline-scenario cell.Source: C Cost, Compute, and Prompts · p. 32, §C.2 · p. 32

C.1 Where the Language Model Enters

  • HP-SPGG uses an offline persona-conditioned score tensor, MaaSSim uses an LLM dispatcher, and SOTOPIA-Hard uses an LLM to write utterances.SOTOPIA-Hard has no finite planner; its utterances use the numeric posterior in the prompt and a keyword-weighted intent surrogate replaces the scorer.Source: §C.1 · p. 32

C.2 LLM-Call Accounting

  • HARP evaluates the scorer nH|Θi| times per episode, but its closed-form Bayes update and analytic bonus add no LLM calls.Live LLM-coordination baselines issue n · H action calls per episode; Concordia and SOTOPIA-Hard add one evaluator or judge call per episode.Source: §C.2 · p. 32
  • The aggregate experiment budget is on the order of 5×10^5 LLM calls, and the reported token consumption is approximately 6.5×10^8 tokens.About 95% of tokens are prompts; the closed-source backbones account for roughly 90% of the bill.Source: §C.2 · pp. 32–33

C.3 Wall-Clock Cost

  • Median HARP+ episode time at concurrency 5 is 1.8 minutes on HP-SPGG, 2.4 on Concordia, and 3.5 on SOTOPIA-Hard.The full suite completed in roughly 480 wall-clock hours of LLM-API time.Source: §C.3 · p. 33

C.4 Compute Environment

  • The experiments ran on one CPU workstation without a local GPU, and finite planner searches took under 1 ms per decision.Per-agent posterior updates cost O(n|Θi|H) per episode and were negligible relative to LLM cost.Source: §C.4 · pp. 33–34

C.5 Prompt Templates

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