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Persistent Partners Raise Prices Among Learning Agents

Paul-Peter Arslan, Yubin Kim, Xiao Xiao

arXiv:2609.35402v1cs.LGcs.MA

TL;DR

The paper asks whether a platform’s choice of repeated counterparties changes prices learned by pricing agents and whether higher prices depend on learned punishment. In a pre-registered randomized experiment, it finds that keeping the same partner raises prices, including when rival prices are hidden.

  • Problem

    Prior Q-learning pricing studies keep counterparties fixed, leaving unclear whether the platform’s choice of who faces whom changes learned prices or enforcement.

    Source: §1 · p. 2

  • Method

    A pre-registered randomized Bertrand-duopoly experiment varies partner persistence, rival-price visibility, and messaging for agents whose prices are set by tabular Q-learning.

    Source: §1 · p. 2

  • Results

    +0.27 of the competitive-to-monopoly profit gap is the registered average-training increase from keeping the same partner, with all twenty paired runs positive.

    Source: §1 · p. 2

  • Takeaways & Limitations

    Where rival prices are hidden, persistent partners still raise prices despite the policy’s inability to retaliate, so punishment-only tests can miss the rise.

    Source: §1 · p. 2, ETHICS STATEMENT · p. 10

    The contrasts are conditional on one environment, one trained 0.5B model, and prompted models under one prompt; hiding the rival also changes state size.

    Source: §8 · p. 9

Abstract

from arXiv · show

When pricing agents meet repeatedly on a platform, the platform decides who faces whom. We ask whether that choice moves the prices the agents learn, and whether a rise comes with learned punishment. In a pre-registered randomised experiment in the Bertrand duopoly of Calvano et al., each agent's price is set by a tabular Q-learning module, not by the small language model attached to it, and we randomise whether each agent keeps its partner, sees its rival's prices and can send messages. Keeping the same partner raises the level of profits, averaged over training, by 0.27 of the gap between competitive and monopoly profit (95% CI 0.20 to 0.35, all twenty paired runs positive), our registered primary result, and the resting price by 0.17 of the Nash-to-monopoly range (post hoc). A plain tabular learner reproduces the effect in all 25 further blocks, and there one permanent partner raises the level more than about three do (+0.23 against +0.05, exploratory). Where rival prices are hidden, the price-setting module cannot see a cut, so cannot punish it, yet the resting price rises as much and the rise lasts to the end of training, while with visible rivals it shrinks with longer training (post hoc). Where the rival is visible, a static best responder accounts for a third to a half of what a forced-deviation probe reads as punishment, on the starts where the rival can see the cut, and net of it the registered test of learned punishment is inconclusive. A test that looks only for punishment would thus miss the rise where the rival is hidden, while a check for profitable deviations flags most of those prices (post hoc). In an exploratory extension, untrained Qwen2.5 7B and 14B models under one prompt show the effect when the rival's price is left out of the prompt and inconsistently when it is shown, the 7B result replicating on fresh blocks, while two other model families show none.

1 INTRODUCTION

The paper tests whether platform-controlled partner persistence changes the prices learning agents reach, and whether those prices rely on punishment. In a pre-registered randomized experiment, keeping a partner raises learned prices, including where the rival cannot observe a deviation.

Prices without retaliation

  • With rival prices hidden, persistent partners still raise resting prices, although the price-setting policy cannot see or punish a cut.The rise lasts to the final training checkpoint; a punishment-only test would miss it, while a profitable-deviation check flags most such rest points.Source: §1 · p. 2

Research design

  • The study randomizes who faces whom, rival-price visibility, and messaging to test a platform-controlled environmental factor that prior Q-learning pricing studies held fixed.Its endpoints are profit level and the rival’s response to a forced deviation net of its static best response.Source: §1 · p. 2

Main result

  • +0.27 of the competitive-to-monopoly profit gap is the registered average-training increase from keeping the same partner; all twenty paired runs are positive.A plain tabular learner reproduces the effect in 25 further blocks; one permanent partner raises the level by +0.23 versus +0.05 for about three partners (exploratory).Source: §1 · pp. 2–3

Enforcement

  • Where rivals are visible, a static best responder explains a third to a half of raw probe readings on starts where the rival sees the cut, leaving the registered learned-punishment test inconclusive.The registered test was planned at about one-half power.Source: §1 · pp. 2–3

Language-model extension

  • In an exploratory extension, prompted Qwen2.5 7B and 14B show the effect when rival prices are omitted, while two other model families do not.The 7B result replicated on fresh blocks; the hybrid experiment’s prices were set by its tabular Q-learning module, not its language model.Source: §1 · pp. 2–3

Mechanism

  • Committed-ahead predictions are consistent with spontaneous coupling of learners’ value estimates, but do not establish that mechanism.The proposed account is that learners meeting repeatedly can receive correlated rewards and have their value estimates drift upward together.Source: §1 · p. 3

2 RELATED WORK

Prior work studies supra-competitive pricing, partner persistence, and language-model agents, but the paper identifies randomized assignment of counterparties and a net-of-static-response enforcement endpoint as its contribution.

Q-learning and the origin of the environment

  • No known algorithmic-pricing study randomizes who faces whom, and this paper adds that contrast with an endpoint subtracting the static response.Calvano et al. (2020) established supra-competitive prices for tabular Q-learners in the calibrated duopoly; later work varies other pricing factors.Source: §2 · p. 3

Partner persistence in multi-agent learning

  • Partner persistence is studied as a possible influence on price levels, but the tests do not show that partner adaptation causes the increase.Prior multi-agent work treats failure with fresh partners as co-adaptation and evaluates it with cross-play.Source: §2 · p. 3

Supra-competitive prices without threats

  • Supra-competitive prices can arise without threats in several settings, and this study tests whether persistence-associated prices require observable rival actions.The paper reports an effect where the agent’s state has no signal of rival actions, while distinguishing those prices from reward-punishment.Source: §2 · p. 4

LLM agents

  • The language-model literature varies algorithms, prompts, and data access; the paper registers a rival prediction that in-context inference could reduce the persistence effect.Prior human random-rematching experiments observe match behavior, unlike the hidden-rival setting discussed here.Source: §2 · p. 4

3 EXPERIMENTAL SETUP

The experiment uses a calibrated Bertrand duopoly and a hybrid agent, then randomizes partner persistence, observability, and communication across seeded blocks.

3.1 ENVIRONMENT

  • The environment is a differentiated-product Bertrand duopoly with logit demand, 33 price bins, and 8,000 training rounds.The static Nash price is bin 6, the joint-monopoly price is bin 26, and each run has sixteen two-seat markets.Source: §3.1 · p. 4

3.2 THE AGENT, A LANGUAGE BACKBONE COUPLED TO A VALUE MODULE

  • Each hybrid agent couples a language backbone trained by policy gradient with a tabular value module learned by the Calvano rule.The paper reports that the backbone trained alone did not reach a collusive point, whereas adding the value module did.Source: §3.2 · p. 4

3.3 THREE FACTORS, DEFINED AS AVAILABILITY

  • The three randomized factors are partner persistence, rival-price observability, and a free-text channel; hidden rival prices make enforcement quantities zero by design.Under shuffled pairing, markets rematch at episode ends; under fixed pairing, agents keep their partners throughout the run.Source: §3.3 · p. 5

3.4 BLOCKS, SLOT SEEDS AND THE RANDOM ASSIGNMENT

  • Eight experimental cells are run across five seed blocks, with slot-seed assignments permuted in advance and the analysis inverting that assignment.A sender check found the trained and untrained senders wrote the same sentence regardless of market state; shuffled pairing also changes update frequency and partner exposure.Source: §3.4 · p. 5, §3.3 · p. 5

4 ENDPOINTS AND INFERENCE

The section defines a frozen-policy deviation probe, a net enforcement measure corrected for static response, and registered decision rules for level, enforcement, and descriptive collusion.

4 ENDPOINTS AND INFERENCE

  • At each checkpoint, frozen policies branch from a rest point into sham and forced-deviation paths, measuring greedy-policy profits and responses.The deviation forces one firm four price bins below its rival for 25 rounds; the level uses rest-point profits relative to Nash and monopoly profits.Source: §4 · p. 5
  • Net enforcement subtracts a static best-responder’s response along the deviator’s realised path from the rival’s raw response.The benchmark never enters the market; the correction addresses cuts a static best responder would make without learned punishment.Source: §4 · pp. 5–6
  • A point is called collusive only if uncorrected punishment appears on at least half the starts, level exceeds 0.6, and firm-level spread is below 0.3.This descriptive verdict uses the uncorrected punishment pattern and has no separate error budget.Source: §4 · p. 6
  • The registered primary test asks whether the fixed-versus-shuffled profit-level contrast is positive; enforcement must reject against both static benchmarks.The exact sign-flip inference enumerates 220 patterns; studentising is intended to support weak-null validity asymptotically, only approximately with five blocks.Source: §4 · p. 6

5 CONFIRMATORY RESULTS

The registered level test finds a substantial fixed-partner effect, while the enforcement test is inconclusive. Hidden-rival prices also rise, and the opponent-swap test cannot separate partner adaptation from the cost of facing a stranger.

5 CONFIRMATORY RESULTS

  • H6 enforcement is inconclusive: neither component rejects, and both intervals extend across the smallest effect of interest.Power at that threshold was about one half as planned; enforcement is zero by construction when the rival is hidden.Source: §5 · p. 6
  • H1’s level interval lies wholly above SESOI∆=0.10 (T=9.56), with all five block contrasts and twenty run pairs positive.At the final checkpoint, the visible-rival contrast is not distinguishable from zero; much of the registered effect there reflects how fast prices rise (post hoc).Source: §5 · p. 6, Figure 2 · p. 7
  • A static best responder accounts for 48% of the raw response in fixed/visible/none, reducing E=6.99 bins·rounds to net 3.61.Across starts where the rival can see the cut, the static response accounts for 40%, 50%, or 32%; in shuffled/visible/free_text the response changes from 5.16 to −2.04.Source: §5 · p. 7
  • With the rival hidden, persistent pairs have higher level and resting price than rematched pairs, although most rest points permit a profitable unanswered cut.The level is 0.53 against 0.23, and the resting price is 1.66 against 1.58; all ten block-by-channel differences are positive.Source: §5 · p. 7
  • The swap-opponent test rejects, but it cannot distinguish partner-specific adaptation from the cost of facing a stranger.On the tabular learner, rematched agents lose 0.18 and persistent agents 0.28 against true strangers, a difference of −0.10 with interval [−0.26, +0.07].Source: §5 · pp. 7–8

6 MECHANISM, PARTLY CONSISTENT WITH SPONTANEOUS COUPLING

The tabular learner reproduces the persistence effect, but evidence for learned enforcement is inconclusive. Several tests fit spontaneous coupling, while a rival account and imperfect correlation signatures leave the mechanism only partly resolved.

6 MECHANISM, PARTLY CONSISTENT WITH SPONTANEOUS COUPLING

  • The plain tabular learner reproduces a larger persistence effect: C∆=+0.363 [ +0.215, +0.520 ] and +0.306 [ +0.270, +0.340 ] across 25 further blocks.Its enforcement contrast is inconclusive under the two-benchmark rule; with a visible rival, a deeper cut gives intervals above zero on both benchmarks, while the hidden-rival dissociation remains clean.Source: §6 · p. 8
  • With one possible partner, rematching has a precise null effect; one permanent partner raises level more than about three do in an exploratory comparison.The contrast is −0.032 [−0.190, +0.130] and −0.025 [−0.065, +0.015] on 25 further blocks; the exploratory contrasts are +0.229 versus +0.048.Source: §6 · p. 8
  • Partner table correlations grow under fixed matching and stay near zero under shuffled matching, but the average-partner best-response account also fits switch-off tests.In tabular blocks, never-met pairs also correlate positively, so the hidden-rival signature is only partly specific to partners and the placebo prediction fails.Source: §6 · p. 8

7 WHICH AGENTS RESPOND TO PERSISTENCE

The hybrid’s value module carries the level effect in the tested ablation, while prompted Qwen2.5 models show a persistence effect when rival prices are omitted. Other model families show no effect under the tested prompt.

7 WHICH AGENTS RESPOND TO PERSISTENCE

  • Zeroing the language-model head leaves the hybrid’s contrast unchanged in the tested visible-rival cell, while lowering table weight lets the head overturn the table more often.At weight 5.5 the head overturns the table in 49% of settled seats, versus 0 of 408 in the ablation; at weight 3 it does so in 79%.Source: §7 · p. 9
  • With rival price omitted from the prompt, Qwen2.5 7B and 14B show positive persistence contrasts, and the 7B result replicates on fresh blocks.The 7B persistent arm is only slightly above random play, and mixing or stale rival history accounts for about half its contrast; with rival price shown, 7B results are mixed.Source: §7 · p. 9
  • Mistral-7B sits at a monopoly ceiling, while OLMo-2-7B shows a precise null; neither model’s response to its environment was tested under this prompt.The backbone trained alone by policy gradient prices near Nash in the three-block comparison.Source: §7 · p. 9

8 LIMITATIONS AND CONCLUSION

The paper concludes that partner persistence can raise prices even where agents cannot retaliate, while its policy implications and empirical scope remain bounded. It also documents deviations from the registered plan and details the agent and training setup.

8 LIMITATIONS AND CONCLUSION

  • The findings are conditional on one market environment, M = 4, 8 000 training rounds, one trained 0.5B model, and prompted models under one prompt.The hidden-rival contrast also mixes observability with state size; persistence is confounded with per-partner experience and update frequency, and the swap probe cannot distinguish adaptation from stranger costs.Source: §8 · pp. 9–10
  • A hidden rival can coincide with higher prices in policies that cannot retaliate, so punishment-only tests may miss the rise; profitable-deviation checks flag most such rest points.The authors limit this conclusion to the studied trained agents, market, and horizon, and do not claim that rematching is an effective or costless remedy.Source: ETHICS STATEMENT · p. 10
  • The policy discussion does not establish legal agreement, deployed-market effects, or rematching’s welfare benefits or costs; those questions remain outside the evidence or scope.The authors also note that visible-rival persistence effects shrink with longer training and that rematching costs are not modelled.Source: ETHICS STATEMENT · p. 10

A DEVIATIONS FROM THE REGISTERED PLAN, AND AMENDMENTS

  • The appendix records registered-plan deviations, including unimplemented probes and deferred sweeps, and reports a sensitivity analysis specified after blocks 1 to 4 were read.The sensitivity analysis was exploratory and did not affect the registered decision.Source: A DEVIATIONS FROM THE REGISTERED PLAN,… · pp. 14–15

B AGENT AND TRAINING DETAILS

  • The trained hybrid uses a Qwen2.5-0.5B-Instruct backbone, a price head, and a 21-state tabular Q-learning module, with 8 000 rounds across sixteen markets per run.The prompt supplies own and rival price histories or an unobserved placeholder, along with a message or a not-transmitted placeholder.Source: B AGENT AND TRAINING DETAILS · p. 16

C THE NEGATIVE RESULT ON POLICY GRADIENT ALONE

The paper investigates why the hybrid agent’s policy-gradient-only control does not reach collusive points. Its evidence separates representation limits, learning-rule performance, and the ability to maintain an already installed equilibrium.

C THE NEGATIVE RESULT ON POLICY GRADIENT ALONE

  • In a five-seed toy, shared-representation variants cut price with probability 1.00 after a revealed high-price state, versus 0.01 for the explicit-cell control.The shared variants had 50–56% regret, compared with 1.2 ± 0.1% for the explicit-cell control; the authors infer that the head’s representation does not linearly encode the relevant conjunction.Source: C THE NEGATIVE RESULT ON POLICY… · p. 16
  • 0 of 265 final probe points reached a collusive point under on-policy policy gradient in the tabular toy, while the Calvano update rule did so on the same table.The authors limit the claim to this update rule, budget, and tested configurations, not every possible gradient-training policy.Source: C THE NEGATIVE RESULT ON POLICY… · p. 16
  • The tabular Q-learner passes the learnability gate under the identical probe, supporting a reading of the null as about the learning rule rather than the environment.The claim rests on the comparison described in the toy experiments.Source: C THE NEGATIVE RESULT ON POLICY… · p. 16
  • The gradient-only agent sustains an installed equilibrium under a sharp policy in 80 of 80 points, but under a soft policy in 0 of 5 seeds.The language model alone destroys an installed equilibrium within 250 rounds in 0 of 8 points, while the hybrid sustains it under exploration in 28 of 30 points.Source: C THE NEGATIVE RESULT ON POLICY… · p. 16

D PER-CELL RESULTS AND EXPLORATORY ANALYSES OF BLOCKS 1 TO 4

These per-cell and exploratory analyses distinguish price levels from enforcement, examine training and welfare patterns, and qualify what the probes can establish. The hidden-rival cells show the clearest separation: persistence raises levels despite structurally zero enforcement.

D PER-CELL RESULTS AND EXPLORATORY ANALYSES OF BLOCKS 1 TO 4

  • With a hidden rival, enforcement quantities are identically zero, yet persistence raises the level to 0.53 against 0.23; the swap-opponent drop is near zero there.Where the rival is visible, the swap-opponent drop averages −0.33, compared with −0.02 where hidden.Source: D PER-CELL RESULTS AND EXPLORATORY… · p. 17
  • At round 8 000, fixed agents receive 927 to 933 policy-gradient updates and shuffled agents 152 to 164, despite equal transition counts; the tabular learner also reproduces the effect.A visible-rival, no-channel ablation with the head and backbone frozen yields +0.204 against +0.206, indicating the two-agent test isolates the clock.Source: D PER-CELL RESULTS AND EXPLORATORY… · p. 17
  • On the last three checkpoints, joint profit is 0.62 versus 0.58 and consumer surplus 0.49 versus 0.52 in fixed/visible/none; with a hidden rival, the corresponding values are 0.58 versus 0.49 and 0.54 versus 0.63.These are blocks 1 to 4; the reported surplus gaps are smaller than the joint-profit gaps because of price-index movement and greater price asymmetry in rematched pairs.Source: D PER-CELL RESULTS AND EXPLORATORY… · p. 17
  • On four blocks, the O × C interaction is −0.04, the shuffled-hidden minus fixed-visible contrast is −0.10, and the communication main effect is −0.03; all intervals cover the search grid.On five blocks, the registered analysis gives −0.01 and −0.07 for the first two contrasts; muting messages leaves Enet unchanged on all 20 channel runs.Source: D PER-CELL RESULTS AND EXPLORATORY… · p. 18
  • After 106 rounds, the exploratory visible-rival contrast is +0.08 at the endpoint and +0.12 over the converged regime, while the hidden-rival contrast is +0.59 and +0.51, respectively.The stand-alone learner agrees in direction; enforcement is structurally zero with a hidden rival, and nine of ten arms end with a complete punishment pattern.Source: D PER-CELL RESULTS AND EXPLORATORY… · p. 18
  • The forced-deviation probe leaves the deviator’s own price bucket unchanged on 43% of visible-rival starts, making the probe blind by construction on those starts.The resulting Enet = −Emyopic is a downward bias, larger in the fixed arm; the matched ablation reproduces the level but not the enforcement contrast.Source: D PER-CELL RESULTS AND EXPLORATORY… · p. 18
  • Dropping starts with sham-branch drift of at least 8 bins gives C∆ = +0.264, [+0.150, +0.380], against the registered +0.274; the level replicates within ±0.035 across machines.The cross-hardware enforcement endpoint does not agree in sign, limiting the precision of H6 at this sample size.Source: D PER-CELL RESULTS AND EXPLORATORY… · p. 18
  • In the channel-use check, messages differ only 0.02 across senders and carry 0.000 mutual information about the sender’s next price bucket; the receiver’s price shift does not grow with sender size.The check covers one block, one checkpoint, and greedy decoding; the channel is unused because messages do not depend on state.Source: D PER-CELL RESULTS AND EXPLORATORY… · pp. 18–19

E AGENTS IN WHICH THE LANGUAGE MODEL SETS THE PRICE

The exploratory extension tests prompted language models that set prices, first under shown-rival prompts and then with the rival hidden. Effects vary across prompt conditions and model families, with several hidden-rival Qwen results but nulls in two other families.

E AGENTS IN WHICH THE LANGUAGE MODEL SETS THE PRICE

  • The prompted-agent experiments were exploratory, and their prompt supplied own history plus rival history or an unobserved placeholder, with nothing else changed between those variants.The prompt specified long-run profit and the allowed price choices; in the first round, history and round number were replaced by a no-history statement.Source: E AGENTS IN WHICH THE LANGUAGE MODEL… · p. 19
  • With the value module removed, the visible-rival language-model-only contrast is +0.064 on three blocks, and both arms remain below 0.50 at 0.178 and 0.113.The authors describe this result as descriptive given three blocks and a bimodal learner.Source: E AGENTS IN WHICH THE LANGUAGE MODEL… · p. 19
  • In the initial 200-round prompted Qwen2.5-7B design, the five planned blocks give a +0.075 contrast, while ten separately reported added blocks give −0.067.The fixed arm kept pairs for 200 rounds and the shuffled arm re-paired every 20 rounds; both arms priced above Nash in the planned blocks.Source: E AGENTS IN WHICH THE LANGUAGE MODEL… · p. 19
  • In the 500-round sharper Qwen2.5-7B design, persistent agents rest at 0.475 and rematched agents at 0.257, a contrast of +0.217, positive in 8 of 10 blocks.The separate ten-block replication gives a +0.006 contrast, [−0.135, +0.155], and is too imprecise for a precise null.Source: E AGENTS IN WHICH THE LANGUAGE MODEL… · p. 20
  • With the rival shown, Qwen2.5-14B rests at 0.829 with a persistent partner and 0.738 when re-paired, a +0.091 contrast; its interval-rule prediction is inconclusive.An anchoring control finds no run above bin 10 against a rival fixed at the Nash bin, supporting response to that rival in this control.Source: E AGENTS IN WHICH THE LANGUAGE MODEL… · pp. 20–21
  • With the rival hidden, Qwen2.5-7B rests at 0.460 versus 0.076, a +0.384 contrast, and Qwen2.5-14B at 0.734 versus 0.188, a +0.546 contrast.The 7B forced-deviation probe’s two enforcement benchmarks disagree in sign, so net enforcement is neither established nor excluded.Source: E AGENTS IN WHICH THE LANGUAGE MODEL… · pp. 20–21
  • On fresh hidden-rival Qwen2.5-7B seeds, persistent agents average 0.481 and rematched agents 0.039, a +0.442 contrast, positive in all eight blocks.A history control raises the rematched arm to 0.249; the persistent arm still leads by +0.232, while partner-only history remains older and sparser.Source: E AGENTS IN WHICH THE LANGUAGE MODEL… · p. 21
  • With the rival hidden, Mistral-7B-Instruct averages 0.968 and 0.979, a −0.011 contrast; OLMo-2 averages 0.835 and 0.843, a −0.008 contrast.Both are reported as precise nulls, though the Mistral arms are at the joint-monopoly level and the OLMo arms are high but below the prespecified ceiling threshold.Source: E AGENTS IN WHICH THE LANGUAGE MODEL… · p. 21

F TABULAR REPLICATIONS ON 25 FURTHER BLOCKS

The further-block tests reproduce the partner-persistence effect and probe whether it depends on partner identity, enforcement, or learned coupling. The evidence supports a persistent-partner effect but leaves the enforcement mechanism inconclusive.

Reference and switch-off tests

  • +0.306, [+0.270, +0.340]: the M = 4 reference effect is positive in all 25 further blocks.With two agents the contrast is −0.025, [−0.065, +0.015], a precise null; the synchronous learner gives −0.036, [−0.065, −0.010], a small negative effect.Source: F TABULAR REPLICATIONS ON 25 FURTHER… · p. 22

Enforcement

  • +3.75, [+0.20, +7.30] against BR1 and +3.15, [−0.45, +6.70] against BRK: registered-cut enforcement contrasts fall below the smallest effect of interest.The deeper cut changes the estimand, and the five-block Section 6 estimates are −2.58, [−7.25, +1.75] against BR1 and −5.16 for the visible-rival contrast alone.Source: F TABULAR REPLICATIONS ON 25 FURTHER… · p. 22

Swap probe

  • 0.337, [0.240, 0.435]: swapping a fixed agent’s final partner lowers ∆ with a visible rival, but by only 0.001 with a hidden rival.Against a true stranger, the fixed-minus-shuffled difference is −0.096, [−0.260, +0.070], which includes zero; neither arm loses with a hidden rival.Source: F TABULAR REPLICATIONS ON 25 FURTHER… · p. 22

Placebo of the coupling signature

  • Partners that never met correlate at +0.0014, [−0.0025, +0.0055], with a visible rival and +0.0169, [+0.0045, +0.0300], with a hidden rival.Both placebo correlations fall outside the predicted ±0.005 range, narrowly for visible rivals and clearly for hidden rivals.Source: F TABULAR REPLICATIONS ON 25 FURTHER… · p. 22

G MECHANISM TESTS, PREDICTIONS AS COMMITTED AND FULL TABLES

The mechanism tests yield evidence consistent with spontaneous coupling, but do not establish it. They also show that rematching frequency, rival visibility, and which component sets the policy matter to the observed patterns.

G MECHANISM TESTS, PREDICTIONS AS COMMITTED AND FULL TABLES

  • Figure 4’s rematching sweep shows the level dropping to the shuffled value at p = 0.25 and not moving thereafter; almost all of the coupling-signature rise occurs with a hidden rival.Panel (c) measures partner-minus-non-partner correlation in the hybrid agent’s Q tables on the rows used by each condition.Source: Figure 4 · p. 22
  • Table 4 compares policy components and prompted models in the visible-rival, no-channel cell, but later rows are comparable only in sign and size.Its final two columns report swap-opponent drop and the coupling signature of fixed runs.Source: Table 4 · p. 23
  • At w = 0, the head-only registered four-cell contrast is −0.140, negative in four of five blocks, versus +0.274 for the full policy.In the visible-rival cell without a channel, the head-only rest-level contrast is −0.231, while the full policy gives +0.206.Source: G MECHANISM TESTS, PREDICTIONS AS… · p. 23
  • Seven predictions held by direction, one partly held, four did not, and three could not be scored; the mechanism evidence is consistent with spontaneous coupling but does not establish it.By interval criteria, none of the no-effect predictions is a precise null at five blocks, and only the M = 4 above M = 2 ordering is established.Source: G MECHANISM TESTS, PREDICTIONS AS… · p. 23, Table 5 · p. 24
  • +0.464, with confidence-set hull [+0.135, +0.775]: at 106 rounds the two-pair contrast is positive, and the shuffled/visible cell eventually shows a probe-read punishment pattern.That rematched learner still does not reach the level of the persistent learner.Source: G MECHANISM TESTS, PREDICTIONS AS… · p. 24, Table 6 · p. 25
  • C1’s coupling signature prediction is held, C2 is partly held, C3 fails, and C5 is held; the M = 8 signature gap is smaller, not larger, than at M = 4.At M = 4 the signature gaps are +0.018 with a visible rival and +0.057 with a hidden one; at M = 8 they are +0.010 and +0.026, respectively.Source: G MECHANISM TESTS, PREDICTIONS AS… · p. 24
  • With a fixed initial pairing, C4b gives a p = 0 minus p = 1 contrast of +0.229, [+0.125, +0.325], rather than the gradient predicted across rematching probabilities.The earlier implementation used multiple permanent partners at p = 0; the corrected version produces a step between p = 0 and p = 0.25.Source: G MECHANISM TESTS, PREDICTIONS AS… · pp. 24–25

H.1 A REFERENCE THAT SEES WHAT THE AGENT SEES

This section tests how the static reference used to measure punishment should match what the agent can observe, and examines when the resulting benchmark is valid. It finds that reference mismatches can bias the net score, while the calibrated deviation rule itself loses meaning beyond a differentiation threshold.

H.1 A REFERENCE THAT SEES WHAT THE AGENT SEES

  • The reference can bias the net punishment score when its information differs from the agent’s: on 43% of starts the exact-price reference moves while the bucketed agent cannot, mechanically lowering Enet.Using a bucket-aware reference on those starts changes the five block contrasts to −0.63, −0.67, +1.23, +3.29, and +5.36.Source: §H.1 · p. 26
  • Under a matched causal linear-filter reference, Enet = (G − 1) Emyopic; unit-gain filters therefore give Enet = 0 regardless of delay or smoothing.At finite horizons the result is asymptotic; the conservative truncation-bias sign is established only for non-negative kernels.Source: §H.1 · p. 26
  • The reference must match the rival’s response structure: BR1 handles post-best-response inertia, while BRK handles best response to a window of past prices.On the calibration, mismatched references leave opposite-signed residuals, so H6 rejects only if both references reject; no tested sweep case made both components positive.Source: §H.1 · pp. 26–27
  • At µ = 0.25, the discrete validity-curve value is R = 0.050; R rises with differentiation, but the deviation rule has no solution beyond µ* = 0.50046.Beyond that threshold, no undercut of the monopoly price yields a one-shot gain of at least 10%, so the forced-deviation test loses its meaning.Source: §H.1 · p. 27, Table 8 · p. 27
  • The registered reference-comparison experiments held D1, D2, and F1–F2 but failed D3–D4 and G2′.The authority test covered blocks 1–3; two later blocks were reported separately, and the five-block mean contrast was +0.191 at weight 5.5 and +0.206 at weight 16.5.Source: §H.1 · pp. 27–28

I IMPLEMENTATION NOTES ON THE INFERENCE

These implementation notes qualify how the registered inference procedure behaves at the paper’s small block counts. They describe limits on exactness, confidence-set resolution, and the power of the joint test.

I IMPLEMENTATION NOTES ON THE INFERENCE

  • At five blocks, studentisation is intended to improve validity for the weak null as blocks grow, but its justification here is approximate and not proved for the sign-flip test used.The unstudentised test is exact only under the sharp null that no unit’s outcome depends on persistence.Source: I IMPLEMENTATION NOTES ON THE INFERENCE · p. 28
  • The inverted confidence set may be non-contiguous at these block counts, so the paper reports its hull, rejected grid points inside it, and whether it reaches the grid edge.The only reported interval with rejected grid points inside its hull is the Appendix G 106-round two-pair contrast.Source: I IMPLEMENTATION NOTES ON THE INFERENCE · p. 28
  • For one-sample sign-flip tests, studentisation cannot change p-values because the statistic ordering matches the raw means; at block level, it can change the ordering.The paper computes the invariant p-values from means as the numerically stable route.Source: I IMPLEMENTATION NOTES ON THE INFERENCE · p. 28
  • At the registered smallest effect of interest, H6 power was 0.53 in the registration simulation and 0.48 and 0.47 for its two benchmarks after re-centring observed differences.Source: I IMPLEMENTATION NOTES ON THE INFERENCE · p. 29
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