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How language model agents reshape price competition

Martin Bichler, Philip Hierhager, Kassian Köck, Fabian Pieroth

arXiv:2610.21711v1cs.GT

TL;DR

Repeated Bertrand competition offers a sharp benchmark, but little is known about the long-run regimes selected by LLM pricing agents. The paper examines how deployment interfaces affect competition and finds that they can produce distinct self-play and mixed-market price regimes.

  • Problem

    Although standard learners often approach the competitive Bertrand outcome, little is known about the long-run regimes selected when LLM pricing agents compete.

    Source: §1 · p. 2, §1.2 · p. 4

  • Method

    The paper compares LLM pricing deployments and standard learners, examining how interfaces and opponent behavior shape pricing dynamics.

    Source: §2.4 · p. 13, §3 · p. 21

  • Results

    P-R-1 can sustain high prices, whereas D-S-25 and D-S-50 move toward the finite-grid Nash region in homogeneous self-play; mixed deployments can produce anchor-and-harvest dynamics.

    Source: §3 · p. 21

  • Takeaways & Limitations

    Deployment choices alter the adaptive system being studied, so pricing outcomes warrant analysis as dynamic regimes rather than only stationary equilibria.

    Source: §3 · p. 21

    The reported dynamic regimes are not formally certified as equilibria; such certification would require counterfactual deviation analysis.

    Source: §4.8 · p. 31

Abstract

from arXiv · show

Repeated Bertrand competition is a stringent benchmark for algorithmic pricing: in homogeneous-goods markets, standard learning procedures typically select the stage-game Nash outcome with prices near marginal cost. We show that LLM pricing agents can escape this benchmark, depending on deployment decisions. LLMs outperform standard online learners through faster adaptation, but this advantage largely disappears against full-information learners. An extracted surrogate algorithm illustrates how LLMs in specific deployments profit from regularly undercutting slower learners. Agents that post one price every period from raw market histories that include rival prices can sustain supracompetitive regimes in self-play. Agents that periodically emit probability distributions from summarized histories of their own profits instead move toward the competitive region in self-play. Markets mixing these interfaces can generate dynamic regimes in which LLMs establish a high-price anchor and regularly undercut it to capture demand. These regimes can keep both competitors' prices above marginal cost. Deployment choices shape these outcomes, shifting the focus from stationary equilibria to dynamic pricing regimes.

1 Introduction

The paper asks which long-run regimes LLM pricing agents select, because repeated-game theory identifies outcomes that can be sustained but does not predict which adaptive processes reach them. It uses the standard-learning benchmark in Bertrand competition to frame how LLM deployment choices shape pricing dynamics.

1.1 Standard learners widely converge to Nash in all-or-nothing Bertrand competition

  • In homogeneous-goods Bertrand competition, the stage-game Nash outcome is marginal-cost pricing, and standard learners often converge toward it; repeated-game theory characterizes sustainable outcomes but not which one is selected.Source: §1 · p. 2, §1.1 · p. 3

1.2 LLMs adapt quickly against standard learners

  • LLM deployments can earn higher finite-horizon profits by exploiting bandit learners’ slow early adjustment; D-S-25 undercuts above-floor rivals, a pattern DBBS explains through ordered payoff-feedback search with inertia.Source: §1.2 · pp. 4–5

1.3 Competing LLMs aim to escape the Bertrand paradox

  • P-R-1 agents using raw histories and point prices can sustain high-price regimes, while summarized-history, distributional D-S-25 and D-S-50 agents move toward the finite-grid competitive region in homogeneous self-play.Source: §1.3 · p. 5
  • In mixed deployments, P-R-1 can anchor prices high while D-S-1 learns a profitable lower threshold, producing recurrent anchor-and-harvest dynamics whose form depends on interface and opponent.Source: §1.3 · p. 5
  • Because the deployment design space cannot be tested exhaustively, competition authorities should examine deployment decisions as well as prompts and foundation models when auditing pricing agents.Source: §1.3 · pp. 5–6

2 Results

The results characterize how deployment interfaces shape LLM pricing against standard learners and in self-play, including sustained high-price play, movement toward competition, and heterogeneous anchor-and-harvest dynamics. Decision traces and a D-S-25 surrogate distinguish opponent-facing anchoring from payoff-feedback search; these observed price regimes are descriptive, not established equilibria.

2.1 Standard learners converge toward the competitive benchmark

  • Most standard learners end near the finite-grid competitive region, but UCB can sustain high prices; PPO moves toward competition in five of its seven markets.Markets with two full-information learners reach the competitive region within the first 10^4 periods, faster than markets with bandit or reinforcement learners alone; the competitive benchmark is therefore strong but not uniform across update rules.Source: §2.1 · pp. 7–8, Figure 1 · p. 8

2.2 LLM pricing agents earn private profit advantages against standard learners

  • D-S-25 earns 71–72% of cumulative industry profit against Exp3, BM-RM, PPO, and UCB over 2,000 periods, primarily by undercutting slower learners.This is a finite-horizon, path-conditioned advantage, not evidence of asymptotic dominance; P-R-1’s advantage against full-information learners disappears. In an illustrative P-R-1–Exp3 run, it earns 63% of industry profit, while against a win-up–lose-down heuristic it matches prices to the ceiling by period 21.Source: §2.2 · pp. 9–11, Figure 2 · p. 10

2.3 Decision traces reveal opponent-facing anchoring and payoff-feedback search

  • P-R-1 traces more often describe opponent-facing anchoring than D-S-25 traces, while D-S-25 more often describes searching payoff regions and probing prices.At least one opponent-facing motif appears in 94% of P-R-1 traces versus 34% of D-S-25 traces; payoff-region motifs appear in 98% of D-S-25 traces versus 21% of P-R-1 traces. These traces are behavioral evidence, not direct access to hidden computation.Source: §2.3 · p. 11

2.4 DBBS explains payoff-feedback search under the D-S-25 interface

  • DBBS reproduces D-S-25’s held-out price updates and final price levels, explaining its movement toward Nash in self-play and undercutting of slower learners.The surrogate combines recency-weighted payoff search with inertia; without inertia, it undercuts in 81% of late periods versus the agent’s 59% and prices about 0.04 too low. It does not explain P-R-1’s response-sensitive anchoring.Source: §2.4 · p. 13, Figure 3 · p. 14

2.5 Deployment interfaces shift LLM competition between high-price coordination and undercutting

  • P-R-1 sustains high prices in homogeneous Bertrand self-play, while D-S-25 and D-S-50 move toward the finite-grid competitive region; information and history interfaces matter more than output format alone.Changing point-price output to grid or distributional output alone leaves price elevation high, whereas hiding rival prices and summarizing histories produce larger movements away from the high-price control.Source: §2.5 · p. 15
  • In self-play, P-R-1 sustains supracompetitive prices while D-S-25 and D-S-50 move toward competition; ablations identify rival-price visibility and history representation as key regime determinants.Changing the action representation alone leaves prices at the monopoly level, whereas hiding rival prices or summarizing histories lowers the price-elevation index; the full D-S-50 bundle reaches 0.13.Source: §2.5 · p. 16, Figure 4 · p. 16

2.6 Heterogeneous LLM deployments generate anchor-and-harvest dynamics

  • With rival prices visible, P-R-1 anchors at the 0.36 ceiling while D-S-1 repeatedly undercuts, yielding a 0.356 market-clearing price, 98% of monopoly joint profit, and 66% of industry profit for D-S-1.In this run, D-S-1 matched the ceiling in two thirds of the final 100 periods and priced one grid step below it in the rest, capturing the whole market in those periods.Source: §2.6 · p. 17, Figure 6 · p. 19
  • Hiding rival prices prevents precise undercuts but does not eliminate supracompetitive pricing: D-S-1 learns from payoff feedback, and recurrent cycles average 0.26 rather than 0.36 with visible prices.D-S-1 earns 71% of industry profit in the hidden-rival run, compared with 66% when rival prices are visible.Source: §2.6 · pp. 17–18, Figure 6 · p. 19
  • Against P-S-1, P-R-1 and P-S-1 both end in anchor-and-harvest regimes, but who harvests varies while market-clearing prices remain above competition at 0.28 and 0.25.With rival prices visible, P-S-1 earns 86% of industry profit; when P-S-1 cannot see the rival price, P-R-1 earns 75%.Source: §2.6 · p. 19
  • Adding firms does not itself lower prices: three-firm P-R-1/D-S-1 markets end near the finite-grid Nash region, while P-R-1-only markets remain elevated.In mixed three-firm markets, P-R-1 earns 89% of industry profit with rival prices visible and all of it with rival prices hidden; three- and four-agent P-R-1-only runs end at 0.26 and the 0.36 ceiling.Source: §2.6 · p. 19
  • Point-price-only outcomes are path-dependent: three P-R-1 agents repeatedly reset and undercut high prices, while four eventually hold the ceiling and split profit equally.These are single runs, not estimates of market-size effects.Source: §2.6 · pp. 19–20
  • Elevated-price regimes recur across models, but the harvester is not fixed: GPT-5.6 Terra with medium reasoning ends at 0.34 with P-R-1 earning 96%, while without reasoning agents split profits at the ceiling.Two medium-reasoning P-R-1 agents instead alternate between restoring and undercutting a high price, ending at 0.29 with nearly equal profit shares.Source: §2.6 · p. 20
  • These interactions are descriptive regimes, not equilibrium claims; the experiments use representative single-run paths, and equilibrium interpretation would require counterfactual deviation analysis.High-price regimes persist when preserving an anchor outweighs immediate harvesting; repeated aggressive harvesting can instead drive prices toward competition.Source: §2.6 · pp. 17, 20

3 Discussion

LLM pricing outcomes depend on the deployed interface and opponent ecology, shifting attention from equilibrium learning to selection among price regimes. The discussion connects those regimes to seller incentives, welfare, and the need for system-level evaluation.

3 Discussion

  • P-R-1 can earn higher cumulative profit against Exp3 and exploit simple undercut-last heuristics, while D-S-25 can profit by undercutting standard learners and capturing demand.Thus, adoption incentives can arise from either high prices or faster adaptation along structured opponent price paths.Source: §3 · p. 21
  • Raw rival-price histories and point-price actions can support high-price P-R-1 regimes, while summarized histories, distributional output, and blockwise updating move D-S-25 and D-S-50 toward the competitive region in homogeneous self-play.Mixed P-R-1–D-S-1 markets can generate recurrent anchoring, threshold learning, and anchor-and-harvest cycles.Source: §3 · p. 21
  • LLM deployments act as induced learning rules: prompts, memory, history representation, observation policy, action representation, and query schedule help determine observed long-run regimes.Changing these components changes the adaptive system, rather than merely its implementation.Source: §3 · pp. 21–22
  • The relevant game-theoretic object may be a selected price regime or recurrent attractor, not a stationary equilibrium; recurrent behavior is not automatically a dynamic equilibrium.The paper does not claim that all documented recurrent regimes satisfy the counterfactual deviation condition for an ε-dynamic equilibrium.Source: §3 · p. 22
  • Cumulative-reward optimization alone does not explain high prices: PPO does not generally sustain them, while P-R-1 can do so without an explicit dynamic program.The adaptive process is shaped by the prompt, memory, observation stream, action interface, and update cadence; DBBS explains D-S-25 but is not a model of P-R-1.Source: §3 · p. 22
  • Welfare depends on the regime: consumers retain 95% of surplus under D-S-25 and D-S-50 self-play, 72–84% when D-S-25 undercuts standard learners, and 15% under P-R-1 self-play.Average prices indicate consumer harm in the homogeneous market but cannot distinguish stable plateaus, anchor-and-harvest, or undercutting; transaction-level price paths are needed to detect those patterns.Source: §3 · p. 23
  • Evaluation and governance should target the deployed pricing system across market structures, opponent ecologies, and interface designs, rather than checking prompts alone.The experiments use stylized Bertrand markets; real systems add inventory, search, consumer heterogeneity, multi-product portfolios, platforms, and asynchronous updates.Source: §3 · pp. 23–24
  • Summary histories, distributional output, and blockwise updating can make simple high-price conventions harder to sustain, but their effects must be evaluated jointly with market structure and opponent ecology.Interface constraints can alter the likelihood and form of supracompetitive regimes, but do not guarantee competitive outcomes.Source: §3 · pp. 23–24

4 Methods

The methods compare standard learners and LLM pricing interfaces across Bertrand markets, information settings, and opponent mixes. They evaluate held-out outcomes and describe recurrent price patterns without treating persistence alone as equilibrium evidence.

4.1 Market environments

  • Homogeneous and differentiated Bertrand markets provide the test environments, with finite-grid Nash benchmarks marked for homogeneous price paths.The homogeneous setup includes continuous- and grid-price agents; differentiated-market results are normalized against Nash and joint-profit prices.Source: §4.1 · pp. 24–25

4.2 Feedback structures

  • Visible rival prices allow reconstruction of counterfactual payoffs in the two-firm homogeneous market, while hidden-price treatments provide bandit feedback with some inferable structure.Visible-price experiments also include full-information learners; hidden-price treatments leave agents with their own price, quantity, and profit.Source: §4.2 · p. 26

LLM deployments.

  • LLM deployments vary history representation, output type, and intervention interval, alongside persistent state and prompt configuration.P-R-1 emits a price each period from raw history; D-S deployments emit a price distribution from per-price summaries, sampled during the following intervention block.Source: LLM deployments · p. 27

4.4 Configuration screening

  • The configuration screen selects learning and aggressive configurations using regret, NashConv, volatility, profit, and price-elevation criteria.The frozen D-S-25 and D-S-50 configurations and learner baselines are then used in held-out evaluations.Source: §4.4 · p. 28

4.5 Experiment families

4.6 Held-out evaluation

  • Held-out evaluations compute price, quantity, profit, and profit-share outcomes, plus realized external regret and welfare measures, using seed-level estimates and bootstrap intervals.The primary homogeneous estimates use the final 10% of the trajectory; regret is a finite-horizon, path-conditioned diagnostic, not policy regret against a strategically responding opponent.Source: §4.6 · p. 29, §4.7 · p. 30

4.8 Price regimes

  • Price-regime labels describe recurrent price paths, including competitive undercutting, high-price lock-in, and anchor-and-harvest patterns.In anchor-and-harvest runs, P-R-1 stays near the upper price range, D-S-1 earns positive profits below it, and P-R-1 intermittently lowers its price to regain demand; persistence alone does not establish equilibrium.Source: §4.8 · p. 31

4.9 Reasoning traces and DBBS

  • Trace analysis treats plans and insights as observable behavioral traces, distinguishing opponent-facing language in P-R-1 from payoff-region language in D-S-25 and D-S-50.DBBS is constructed only as a surrogate for D-S-25 updates; it maintains recency-weighted payoff and failure beliefs, and is fitted and evaluated with teacher-forced and free-running tests.Source: §4.9 · pp. 31–32

Supplementary Note 1: Experimental setup

The supplementary setup specifies the market environments, deployment interfaces, learner baselines, and evaluation metrics. It distinguishes how agents receive histories and choose actions, and bounds the interpretation of realized external regret.

Homogeneous winner-takes-all Bertrand market.

1.2 Deployment parametrization

  • A deployment specifies history, action space, output, intervention interval, persistent state, and prompt configuration.D-S-25, for example, emits a distribution from summarized history every 25 periods; deployment codes are shorthand, not complete specifications.Source: §1.2 · pp. 36–37

1.3 Distribution–summary deployments D-S-25 and D-S-50

  • D-S-25 and D-S-50 are screened using one-factor-at-a-time variations from a common baseline, including model, temperature, intervention interval, and history representation.Their prompts provide market history and the agent’s own cost, but not the rival’s cost, price, or profit.Source: §1.3 · p. 38

History representations.

  • Summary histories aggregate observations by price tick, whereas raw histories list period-level prices, quantities, and profits.Summary rows include profit statistics and trends; the stated window length scales with the intervention interval.Source: History representations · pp. 38–39

Objective framings.

  • Prompt 0 uses neutral framing, while Prompts 1 and 2 use a profit-oriented base; Prompt 2 additionally permits aggressive exploration and gives a directional undercutting hint.Prompt 1 adds profit-protective exploration to the shared profit-oriented framing.Source: Objective framings · p. 39

1.4 Deployments in the Fish harness: P-R-1, P-S-1 and D-S-1

  • P-R-1 uses a point-price response from raw history and a persistent scratchpad, while P-S-1 summarizes history and D-S-1 additionally emits a price distribution.Against non-LLM learners, P-R-1 is restricted to the shared 21-point grid so exact ties remain possible.Source: §1.4 · pp. 39–40, Action representation · p. 40

1.5 Standard learners

  • The four D-S-25 and D-S-50 learner baselines use the same grid and local information scope, but differ in their learning paradigms and retained history.Exp3, BM-RM, and UCB update from recent realized payoffs; PPO conditions on a fixed-length encoding of recent own-market observations.Source: §1.5 · p. 41

1.6 Evaluation metrics

  • Realized external regret compares observed-path profit with the best fixed grid price, and is a finite-horizon diagnostic rather than policy regret against a responsive opponent.The metric does not account for how the opponent would have reacted to counterfactual actions.Source: §1.6 · p. 42

Supplementary Note 2: Configuration screening

The configuration screen compares candidate agents in homogeneous-goods self-play, then freezes selected configurations for later experiments. It specifies candidate grids for the standard learners and records scope and screening-window qualifications.

Supplementary Note 2: Configuration screening

  • The screen evaluates 67 candidate cells on a 21-price grid over 2,000 periods and five seeds, with rival prices hidden except for reactive heuristics.The 14 distributional-LLM and learner cells use local information; the three strategic-heuristic cells use full information, a market scope shared by both firms.Source: Supplementary Note 2: Configuration… · p. 43
  • Screening statistics average across firms, runs, and four time windows, so they include initial exploration and are not comparable to held-out final-10% estimates.Selected configuration aliases are held fixed afterward, and screening trajectories are not reused in held-out evaluation.Source: Supplementary Note 2: Configuration… · p. 43

2.1 Exp3 candidates

  • Exp3 candidates vary the exploration rate γ, which controls both uniform exploration and the multiplicative-weight update step size η = γ/K.The grid includes a horizon-dependent reference bracketed by 0.125 and 0.15, plus an anytime candidate whose exploration declines with elapsed time.Source: §2.1 · pp. 43–44

2.2 BM-RM candidates

  • BM-RM samples from the stationary distribution of an explored replacement matrix and updates row learners using only the realized price’s profit.All eleven candidates couple η = γ/21; the grid also covers lower exploration rates because the theoretical anchor prescribes near-uniform exploration at this horizon.Source: §2.2 · pp. 44–45

2.3 PPO candidates

  • PPO screening varies rollout length, learning rate, and entropy regularization through a staged local search around a manually tuned baseline.The six candidates include a doubled rollout, lower learning rates, and entropy-regularized variants; other settings remain at defaults.Source: §2.3 · p. 45

2.4 UCB candidates

  • UCB screening varies the optimism coefficient α across 22 candidates, while forcing each of the 21 prices to be played once before using its index.The grid brackets the UCB1 reference and extends toward near-greedy behavior.Source: §2.4 · p. 45

2.5 Selection criteria

The section defines two within-family rules for freezing configurations: one balances learning measures, while the other favors profit and elevated prices subject to learning-quality filters. The resulting choices and benchmark outcomes vary by agent family and evaluation window.

2.5 Selection criteria

  • The learning rule ranks candidates equally across external regret, swap regret, NashConv, and cross-firm price dispersion; the aggressive rule filters poor learning candidates before ranking profit and price elevation.Ranks are computed within each agent family, and the two rules select from different eligible sets in general.Source: §2.5 · pp. 46–47

2.6 Screening results and frozen configurations

  • The LLM family’s frozen learning and aggressive configurations differ, and each scores poorly under the other rule’s objective.The learning rule selects neutral framing at interval 25; the aggressive rule selects Prompt 1 at interval 50. BM-RM’s two rules select the same candidate.Source: §2.6 · p. 47, Table 2 · p. 48

Sensitivity to the metric window.

  • The LLM learning-rule choice shifts from interval 25 under aggregate screening to interval 50 in single-window alternatives, while BM-RM and PPO choices remain stable.The UCB learning choice also shifts across windows; the authors note that replicated baseline candidates contribute to the LLM rank movement.Source: Sensitivity to the metric window · pp. 48–49

Supplementary Note 3: Standard learners in homogeneous competition

  • The standard-learner round robin uses seven learners, rival-price visibility, ten seeds, and a 10^7-period horizon to examine long-run outcomes.Bandit and reinforcement learners alone take a median 6.4 × 10^5 periods to settle, while markets with two full-information learners settle within 10^4 periods.Source: Supplementary Note 3: Standard… · pp. 51–52
  • Most standard-learner markets—25 of 28—end in the finite-grid competitive region after 10^7 periods, with three exceptions involving UCB or PPO.The same 25 markets have NashConv below 5% of one-firm monopoly profit; exceptions are UCB self-play, UCB against PPO, and PPO against WFP.Source: Supplementary Note 3: Standard… · p. 51

Supplementary Note 4: LLMs against standard learners

  • Held-out D-S-25 replications pair the agent with Exp3, BM-RM, PPO, and UCB for 2,000 periods, while its aggressive-UCB pairings serve as stress tests.The P-R-1 runs against standard learners and simple pricing rules are single runs.Source: Supplementary Note 4: LLMs against… · p. 55

4.1 Held-out price dynamics

Against each tested standard learner, D-S-25’s median price falls below the learner’s within the first few hundred periods and remains lower; aggressive-UCB stress tests show the same ordering.

4.1 Held-out price dynamics

  • D-S-25’s median price falls below each standard learner’s within the first few hundred periods and stays lower.This price ordering is the undercutting summarized in the main-text figure.Source: §4.1 · p. 55
  • The undercutting pattern appears against every tested learner, rather than only one opponent.The passage names the median price paths across the learner matchups.Source: §4.1 · p. 55
  • The same price ordering appears in the aggressive-UCB stress cells.Source: §4.1 · p. 55

4.2 Evaluation-window robustness

The evaluation reports robustness across window lengths and illustrates how P-R-1 behaves against learners and simple pricing rules. Its advantage is concentrated against slow, exploring learners and predictable rules, rather than full-information learners.

4.2 Evaluation-window robustness

  • D-S-25’s advantages are reported as robust to evaluation-window length, which is assessed across final windows from 5% to the full horizon.The analysis tracks profit and price gaps, lower-price share, and external-regret gap across four learner matchups.Source: §4.2 · p. 55, §4.3 · p. 56

4.3 P-R-1 against standard learners and pricing rules

  • Against bandit learners and exploitable rules, P-R-1 earns the larger industry-profit share, but it gains no profitable advantage against full-information learners.Coin betting and full-information regret matching illustrate that a large profit share can coexist with low joint profit.Source: §4.3 · pp. 56–57
  • Against Exp3, P-R-1 revises its price more often early than later, while Exp3 continues exploring the grid.P-R-1 changes price in 30% of the first 150 periods and 4% afterward; against undercut-last it periodically resets to the ceiling.Source: §4.3 · pp. 56–57
  • P-R-1 matches win-up–lose-down to the price ceiling, while randomized win-up–lose-down produces less stable supracompetitive coordination.Against undercut-last, P-R-1 exploits the predictable rule; these examples are single-run illustrations.Source: §4.3 · p. 57

Supplementary Note 5: Reasoning traces and motif extraction

  • The observable plans-and-insights text is treated as a behavioral trace for identifying candidate mechanisms, not as direct access to hidden computation.The surrogate algorithm is validated separately against emitted distributions and realized price paths.Source: Supplementary Note 5: Reasoning traces… · p. 59

5.1 Motif-extraction procedure

The authors extract recurring decision motifs from LLM pricing traces, code their frequency with a fixed lexicon, and compare the patterns across deployed interfaces and protocol ablations. The trace patterns differ sharply by interface, and the authors report that the lexicon measures wording rather than intent.

5.1 Motif-extraction procedure

  • The extraction prompt asks for recurring rules explaining probability updates and examines feedback evaluation, recency, price ordering, focal-region movement, exploration, and neighbor asymmetry.The procedure first requests recurring patterns supported by examples from different runs and treatments, before a qualitative rule.Source: §5.1 · p. 59

Keyword lexicon.

  • A fixed keyword lexicon codes ten motifs in two families: opponent-facing reasoning and reasoning about the firm’s own payoff regions.The lexicon uses case-insensitive patterns and exclusions intended to remove systematic false matches.Source: Keyword lexicon · p. 60

From matches to shares.

  • Motif shares are averaged across firm-runs, which receive equal weight regardless of call count; uncertainty intervals bootstrap games because firms within a game interact.A trace can match multiple motifs, so the “any” family rows are not sums of the individual motif rows.Source: From matches to shares · pp. 60–61

Precision audit.

  • The precision audit found 58 of 66 sampled matches expressed the coded motif, but the lexicon measures wording rather than intent and recall was not estimated.Most false positives worked against the reported P-R-1–D-S-25 contrast.Source: Precision audit · p. 62
  • Opponent-facing motifs appear in 94% of P-R-1 traces versus 34% of D-S-25 traces, while payoff-region motifs appear in 21% versus 98%, respectively.D-S-50 is close to D-S-25; D-S-1 lies between the two.Source: Precision audit · p. 62
  • Protocol ablations show 96–99% opponent-facing traces with raw histories and visible rival prices, versus 6% and 3% with summarized histories and hidden prices for point and distribution outputs.Hiding rival prices alone lowers the share to 45%; summarizing history alone lowers it to 82%.Source: Precision audit · p. 62

Supplementary Note 6: DBBS construction and validation

  • DBBS is fitted and validated only in the homogeneous winner-takes-all market, because zero-sales failures would not occur in the differentiated logit market.Without that failure signal, the failure-penalty parameter would be inert.Source: Supplementary Note 6: DBBS… · p. 64

6.1 DBBS update

DBBS treats prices as an ordered grid and updates recency-weighted payoff and failure scores from observed play. It turns those scores into a base policy, then adds local upward exploration and a pre-observation cold-start distribution.

6.1 DBBS update

  • At each intervention epoch, DBBS identifies the modal played tick and updates recency-weighted payoff and failure scores for each played tick using its normalized profit.Unplayed ticks retain their scores.Source: §6.1 · p. 64
  • The failure signal marks zero-sales outcomes above marginal cost; a price at marginal cost is never counted as a failure.In the winner-takes-all market, a price above marginal cost earns zero profit exactly when it does not sell.Source: §6.1 · p. 64
  • DBBS converts tick scores into a base policy with a softmax, then mixes it with a local search distribution over ticks above the focal tick.The local search distribution has weight ρ and is uniform over the Gup ticks above the focal tick m.Source: §6.1 · p. 64
  • Before its first observation, DBBS emits a cold-start distribution estimated per treatment from the training seeds of the fold.Source: §6.1 · p. 64

6.2 Inertia

The surrogate adds persistence to a payoff-feedback search policy, with its inertia weight chosen using training-seed predictions. The paper evaluates both next-distribution prediction and free-running reproduction, fitting one shared parameter vector across treatments.

6.2 Inertia

  • The surrogate mixes its search policy with a recency-weighted histogram of the agent’s recent prices; the inertia weight is selected by training-seed one-step RPS, not free-running outcomes.The authors add inertia because search alone undercuts standard learners more than D-S-25 does in free running.Source: §6.2 · pp. 64–65

6.3 Teacher forcing and free-running reproduction

  • Teacher forcing predicts the LLM’s full next emitted distribution from the same realized history, while free-running reproduction replaces the LLM and acts endogenously.Recorded trajectories span 2,000 periods, with 79 prediction targets after the cold start.Source: §6.3 · p. 65

6.4 Fit set

  • One parameter vector per fold is jointly fitted across seven treatments, requiring opponent-specific free-game price levels to emerge from a shared mechanism.The fit uses leave-one-seed-out cross-fitting across the ten environment seeds.Source: §6.4 · p. 65

6.5 Prediction metrics and parameter selection

  • The fitted failure penalty is close to zero, so DBBS’s downward probability shift comes mainly from its one-sided locality penalty combined with wide upward probing.The parameter search widened its bounds after selected exploration parameters reached their initial limits.Source: §6.5 · p. 66

6.6 Heuristic baselines

  • The evaluation compares DBBS with own-information baselines, full-information heuristics, and a same-treatment training-set mean reference.The full-information group includes undercut-last and weighted fictitious play; the LLM itself did not observe the rival prices or demand model.Source: §6.6 · p. 66

6.7 Changes, persistence and free-running controls

DBBS with inertia tracks D-S-25’s free-running prices more closely than the comparison mechanisms, while one-step prediction favors payoff-feedback search on changing blocks. The theoretical results also clarify that DBBS does not guarantee adversarial no-regret behavior, despite idealized convergence results.

6.7 Changes, persistence and free-running controls

  • On changing blocks, DBBS without inertia is the most accurate model in all seven treatments against persistence, win-up–lose-down, and weighted fictitious play.The LLM’s mean price moves by at least one grid step in 22% of blocks; the cross-fitted inertia weight lowers held-out RPS versus persistence alone in 6 of 7 treatments.Source: §6.7 · p. 67
  • In 70 paired games, DBBS has mean absolute final-window price error 0.020, versus 0.030 without inertia and 0.074 for persistence.The reported error is the mean absolute difference from the D-S-25 agent’s mean price over the final 25% of periods.Source: §6.7 · p. 68
  • With inertia, DBBS’s price bias shrinks to 0.013, and its treatment-level ranking of LLM final prices has Spearman correlation 0.89.DBBS is closer to the LLM than every tested mechanism, including full-information references.Source: §6.7 · p. 68

6.8 Theoretical statements and proofs

  • The propositions show that DBBS is not an adversarial no-regret algorithm, while an idealized full-information version tracks the lowest best response and enters and remains in the finite-grid Nash bracket in self-play.Source: §6.8 · p. 68

Case 1: fixed τ > 0.

  • With fixed τ > 0, an opponent always playing tick 20 can cause DBBS constant expected per-period regret, so cumulative regret grows linearly in T.This is one of the stated cases establishing failure of no external adversarial regret.Source: §6.8 · p. 69, Case 1: fixed τ > 0 · p. 69

Case 2: decreasing τn.

With decreasing temperature, DBBS can retain a suboptimal action long enough to incur linear expected regret, while idealized updates track best responses or enter the finite-grid Nash bracket under their stated conditions.

Case 2: decreasing τn.

  • A positive-probability bait-and-switch path makes DBBS incur linear expected regret: it can keep choosing tick 18 after the opponent switches to 20, although tick 19 then earns more.The fixed-length bait phase's advantage is eventually outweighed by tick 18's per-period shortfall of 1/20; the argument also applies with inertia weight w < 1.Source: Case 2: decreasing τn · pp. 69–70
  • With correct payoff beliefs and no failure or locality penalties, an idealized DBBS update selects the opponent policy's minimum best response and keeps that search centre.This holds for fixed opponent policy σ2 and τ > 0.Source: Case 2: decreasing τn · p. 70
  • With perfect beliefs and positive failure penalties, idealized self-play enters and remains in the finite-grid Nash bracket when τ is sufficiently small.For pure opponent actions k ≥ 3, the unique score maximizer is k − 1, moving high-price anchors downward.Source: Case 2: decreasing τn · pp. 70–71

Supplementary Note 7: Deployment ablations

Deployment components jointly determine whether the Fish-style pricing control remains near the joint-profit benchmark or moves toward competition. The clearest movement comes from changing how history is summarized and whether rival prices are visible, not from changing output representation alone.

7.1 Fish-style positive control.

  • The Fish-style positive control's market-average price elevation is close to the joint-profit benchmark across independent runs.It emits a continuous single price, observes rival prices, receives raw long-memory histories, and uses profit-oriented prompt language.Source: §7.1 · p. 72

Outcome measure and the robust-mover criterion.

  • The ablation screen holds key conditions fixed where appropriate, scores the final 100 periods, and calls an arm a robust mover only when every seed falls below the control's [0.92, 1.01] band.The index is normalized from 0 at one-shot Nash to 1 at joint profit; grid and continuous arms use different benchmark anchors, so small cross-arm level differences should not be over-read.Source: Outcome measure and the robust-mover… · pp. 72–73, §7.2 · p. 72

Single-factor ablations.

  • Among single-component changes, only summarized per-price history robustly moves prices down: A6 falls from 0.97 to 0.53, while hiding rival prices alone is not a robust mover.Changing only the action representation instead raises the index: A3 is 1.09 and A4 is 1.07.Source: Single-factor ablations · p. 73

Combined ablations.

  • Only summarized history combined with hidden rival prices robustly moves every seed below the control band, regardless of output representation.The continuous-output pair scores 0.31, the grid-point version 0.34, the grid-distribution version 0.24, and the interval-50 distribution version 0.14.Source: Combined ablations · p. 74
  • Combined arms without summarized history do not robustly move toward competition; shortening history with grid-point output scores 1.13, while interval-50 grid arms score 1.00 and 1.04.The interval-50 scores use only two decisions in the fixed 100-period scoring window, so the authors caution against reading them closely.Source: Combined ablations · p. 74

The movers do not reach competition.

  • Even the strongest mover, A13 at 0.14, remains above Nash at 0; the deployment bundle lowers price elevation but does not deliver competitive pricing.Robust movers range from 0.14 to 0.34, whereas non-moving arms are at 0.88 or above.Source: The movers do not reach competition · p. 74

The movement is genuine competition.

  • Low-price arms also forgo profit and have lower NashConv, while high-elevation arms concentrate on fewer prices and robust movers keep revising their actions.Robust movers' NashConv ranges from 0.81 to 2.78 versus 13.75 to 17.82 for the control and non-movers; their repeated-action shares are 55%, 32%, and 2% for A7, A10, and A13.Source: The movement is genuine competition · pp. 74–75
  • The evidence points to a deployment bundle or information interface: changing action representation alone moves outcomes slightly upward, whereas how agents see history and rivals moves them down.Low-price arms' lower profit and NashConv distinguish this movement from merely sliding along a flat profit region or coordination failure.Source: The movement is genuine competition · pp. 74–75

Supplementary Note 8: Heterogeneous LLM markets

P-R-1 matchups and heterogeneous LLM deployments produce different price outcomes, including near-Nash play, high-price lock-in, and recurrent anchor-and-harvest paths. The matchup estimates are small-sample orderings, and the regime labels describe individual runs rather than frequencies.

8.1 P-R-1 matchups in the logit market

  • With only two to six seeds per matchup and uncontrolled LLM sampling, the table supports opponent-class ordering, not precise point estimates or differences of a few hundredths.The separation between high-elevation and near-Nash cells is large and consistent across seeds.Source: §8.1 · p. 77

8.2 Heterogeneous-deployment trajectories

  • In heterogeneous deployments, a P-R-1 agent can maintain or restore a high-price anchor while a D-S-1 agent prices below it and captures demand, producing recurrent anchor-and-harvest paths rather than a characterized stationary equilibrium.The P-R-1 agent may briefly lower its price to harvest demand at an elevated margin.Source: §8.2 · p. 77

Opponent set.

  • P-R-1 price elevation ranges from 1.11 against P-R-1 and 1.04 against an undercut heuristic to 0.04 against full-information regret matching.The matchup set also includes distributional LLMs, bandit and full-information learners, and a static monopoly control.Source: Opponent set · p. 77, §8.2 · p. 78

8.2 Heterogeneous-deployment trajectories

  • The visible-rival P-R-1–D-S-1 run locks in near the ceiling as D-S-1 harvests one-step undercuts; with rival prices hidden, the run instead cycles through anchor-and-harvest.Both are single runs.Source: §8.2 · p. 78

8.3 Regime labels

  • The twelve heterogeneous-run labels distinguish competitive undercutting, high-price lock-in, and anchor-and-harvest, but each label describes a representative 300-period path, not regime frequency.The GPT-5.6 Terra medium-reasoning runs use provider-default sampling rather than the temperature setting used without reasoning.Source: §8.3 · pp. 78, 80

Supplementary Note 9: Welfare

The welfare accounting distinguishes homogeneous markets, where prices divide a fixed surplus, from the logit market, where higher prices can destroy surplus. The supplementary materials also document the prompt interfaces used for the pricing agents and the authors' generative-AI use.

Homogeneous market.

  • In the homogeneous market, total welfare is fixed at 260 per period with no deadweight loss, so price changes redistribute surplus between consumers and producers.Consumer share equals one minus the price-elevation index measured from marginal cost.Source: Homogeneous market · p. 81

Logit market.

  • In the logit market, changing the information interface cuts welfare loss to 2–6%, and the full D-S-50 bundle cuts it to 0.8%, raising consumer surplus from 33.7 to 66.1 per period against Nash's 71.5.Grid-point and grid-distribution arms lose 15%, more than joint monopoly's 13.7%, because their prices overshoot the joint-profit price.Source: Logit market · pp. 81, 83

10.1 Distributional wrapper prompt

  • The distributional wrapper receives prior pricing distributions and market observations, then returns a probability distribution over available actions and updated plans and insights for later decisions.The market history may include own prices, quantities, profits, and competitor prices when available.Source: §10.1 · p. 84

10.2 Configuration-screen framings

  • The configuration-screen framings vary the stated objective and exploration guidance, including profit-oriented instructions and an addendum permitting risky or aggressive exploration.The prompt directs the agent to use prior price, quantity, and profit data and prior plans when choosing the next interval's pricing strategy.Source: §10.2 · p. 85

10.3 P-R-1 prompt

  • The P-R-1 point-action prompt gives the model product information, persistent plans and insights, and market history, then requests a single chosen price.Depending on the condition, the action set is specified by a maximum price or an explicit grid, and the prompt states how long the choice remains in force.Source: §10.3 · pp. 85–87

10.4 Generative-AI use

  • The authors used GPT-5.6 Sol for qualitative motif extraction, editing, and proof-development assistance, while checking generated material and computing market outcomes and numerical results from experimental code.The language model did not generate the reported market outcomes or numerical results.Source: §10.4 · p. 87
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