How language model agents reshape price competition

Martin Bichler; Philip Hierhager; Kassian Köck; Fabian Pieroth

cs.GTOct 15, 2026

How language model agents reshape price competition

Authors: Martin Bichler, Philip Hierhager, Kassian Köck, Fabian Pieroth

Organizations: School of Computation, Information and Technology, Technical University of Munich, Germany.

Abstract

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.

Figures & tables

Figure 1: Standard learners mostly select competitive outcomes in homogeneous Bertrand competition. Round robin of seven standard learners (all unordered pairs including self-play; ten seeds per pair; final window = last 250 periods), each with one frozen configuration. All firms observe the rival’s price, which FTRL, FI-RM and WFP require; Exp3, BM-RM and UCB learn from their own profit only. a , Market-clearing price index, 0 = marginal cost, 1 = price ceiling; with winner-takes-all demand it equals joint normalized profit. b , Period after which each market’s price stays in the competitive region (block means of $10^{4}$ periods at most half a grid step above $0.126$ ); rows are markets sorted by the median over seeds (large markers), small dots are seeds, colour gives how many of the two learners use full-information feedback; $10^{4}$ means competitive from the first block. Right column, number of seeds still outside the region after $10^{7}$ periods. FTRL, follow-the-regularized-leader with a Euclidean regularizer; BM-RM, Blum–Mansour bandit regret matching; FI-RM, full-information regret matching; WFP, weighted fictitious play (best response to a discounted belief over the rival’s price, discount 0.9; grouped with the full-information learners because it uses the rival’s price). Horizon $10^{7}$ periods; dashed lines in b , the horizons of the P-R-1 (at most 300 periods) and D-S-25 (2,000 periods) runs against standard learners.

Figure 2: LLM pricing agents can earn private profit advantages against standard learners and rapidly adapt to opponent behavior. a , D-S-25 earns more than Exp3, BM-RM, PPO and UCB over the full 2,000-period horizon. Small points denote independent seeds, large points denote matchup means, and the dashed line marks equal cumulative industry profit. Each matchup contains ten independent seeds. b , In the final 10 percent of periods, D-S-25 agents post lower prices than their opponents more often than they post higher prices. Percentages may not sum to 100 because of rounding. c , Illustrative P-R-1 –Exp3 trajectory. Exp3 explores broadly over the price range, while the P-R-1 agent moves through more persistent price regions and earns 63 percent of cumulative industry profit. d , Illustrative P-R-1 interaction with a win-up–lose-down heuristic. From period 3 onward, the P-R-1 agent matches the heuristic’s price. Because each tie generates positive demand, the heuristic raises its price by one grid step in the following period and the P-R-1 agent follows. Both agents reach the price ceiling in period 21 and remain there, splitting cumulative industry profit approximately equally. The shaded areas in panels c and d denote the finite-grid competitive region. Panels c and d show single illustrative runs (

n=1

); panels a and b use ten independent seeds per matchup.

Figure 3: DBBS reproduces how D-S-25 updates and where it prices. DBBS uses one parameter vector fitted jointly on the seven D-S-25 treatments (ten environment seeds each, leave-one-seed-out cross-fitting); every alternative is selected under the same folds and criterion, so all results are held-out. DBBS, search policy mixed with the agent’s own recent price distribution (inertia weight cross-fitted on training seeds); DBBS without inertia, the search policy alone. Own-information mechanisms use only what the agent observed (its own prices, sales and profits). † Undercut-last and weighted fictitious play need full information—the rival’s prices and the demand model and costs—which the agent never observed. ‡ Train-mean emits the agent’s own average distribution for that treatment and encodes the outcome rather than a mechanism. a , One-step prediction of the next emitted distribution: held-out Wasserstein-1 distance on all blocks (ticks) and on the 22% of blocks in which the agent’s mean price moves by at least one grid step (markers). Persistence is most accurate overall because emitted distributions are highly persistent; on changing blocks the DBBS search policy is the most accurate model. b , Free-running reproduction: each model replaces the agent against the same opponent on the same held-out environment seed (7 treatments

\times

10 seeds); absolute difference between the model’s and the agent’s final-window mean price. c , Mean price per 50-period block, averaged over the ten held-out seeds, for four treatments. Without inertia DBBS undercuts the learners too far; with inertia it follows the agent. Grey band, finite-grid competitive region.

Figure 4: The information interface, not the action representation, separates high-price P-R-1 play from competitive D-S-25 and D-S-50 play. Values are the price-elevation indices of Methods,

\Delta

in the logit market (0 = continuous one-shot Nash price, 1 = joint-profit price) and

\Delta^{H}

in the homogeneous market (0 = marginal cost, 1 = the price ceiling, which is the joint-monopoly price), from the mean posted price over the final third of each run; small points are independent runs, large points their means. Indices above 1 occur in the logit market, where firms can price above the joint-profit price. a , Self-play in the same homogeneous winner-takes-all market: P-R-1 (continuous price every period, raw history, rival price visible;

n=5

runs of 300 periods) and the distributional deployments D-S-25 (learning) and D-S-50 (aggressive) (

n=10

runs of 2,000 periods each). b , c , Starting from the P-R-1 control of the protocol ablations in the differentiated logit market, which differs from the Fish-style positive control only in using one merged scratchpad file (Supplementary Note 1) (orange; grey band, range across its runs), the action representation ( b ) or the information interface ( c ) is changed. Grid-point or distributional output leaves prices at the monopoly level. Hiding the rival’s price or summarizing the history roughly halves the index, their combination lowers it further, and the full D-S-50 bundle (distribution, summary, hidden rival price, 50-period interval; blue) reaches 0.13. A short raw history does not lower prices. Each row

n=5

runs.

Figure 5: Heterogeneous and multi-agent LLM deployments exhibit distinct price regimes. Homogeneous winner-takes-all Bertrand market; each row is one representative 300-period run (

n=1

). Deployments P-R-1 , P-S-1 and D-S-1 as in Table 1 ; “rival visible/hidden” refers to the information setting, whether the LLMs observe competitors’ prices. Upper block: Gemini 3 Flash, varying the deployment; lower block: GPT-5.6 Terra, varying reasoning. a , Market-clearing price: grey bar, 10th–90th percentile over the run; open marker, full-run mean; filled marker, mean over the final 100 periods; grey band, grid-based competitive benchmark (three lowest grid prices with two firms, two lowest with three or four); dotted line, price ceiling. b , Share of industry profit per firm in the final 100 periods ( P-R-1 and P-S-1 agents orange, D-S-1 agents blue). Regime labels describe each run’s price path qualitatively (Methods; Supplementary Note 8). Single-run rows are representative paths, not treatment-effect estimates.

Figure 6: Rival-price visibility shifts the balance between anchoring and harvesting. One P-R-1 agent against one D-S-1 agent in the homogeneous Bertrand market; single run per panel (

n=1

), 300 periods. Top: P-R-1 posted price (orange) and the D-S-1 agent’s emitted policy (blue line, mean; shading, 10th–90th percentile); grey band, grid-based competitive benchmark; dotted line, price ceiling. Bottom: per-period profit as a share of one-firm monopoly profit, P-R-1 agent upwards and D-S-1 agent downwards. a , With rival prices visible both firms settle near the ceiling after about 45 periods; the D-S-1 agent earns 66% of profit in the final 100 periods. b , With the rival price hidden from D-S-1 the P-R-1 agent repeatedly re-anchors high while the D-S-1 agent follows below it, producing recurring cycles in which the D-S-1 agent earns 71% of profit in the final 100 periods.

Interface code	Output	History	Interval	Prompt	Used in
P-R-1	point	raw, 100	1	1	Figs. 2c,d, 4, 5, 6
P-S-1	point	summary, 100	1	1	Fig. 5
D-S-1	distribution	summary, 100	1	1	Figs. 5, 6
D-S-25	distribution	summary, 75	25	0	Figs. 2a,b, 3, 4a
D-S-50	distribution	summary, 150	50	1	Fig. 4a,c

Table 1: LLM deployments. Interval, periods between LLM decisions; window, periods summarized or listed in the history. All deployments use Gemini 3 Flash at temperature 0.7 unless stated otherwise; rival-price observability is set by the information setting and varies within deployments.

Interface code	$\mathcal{H}$	$\mathcal{A}$	$\mathcal{O}$	$\Delta$	$\mathcal{M}$	$\mathcal{P}$
P-R-1	raw, 100	continuous; 21-point grid against standard learners	point	1	scratchpad, one file (two in the logit positive control)	Prompt 1, Fish template; prices $\times 20$ in the homogeneous market
P-S-1	summary, 100	continuous	point	1	scratchpad, one file	Prompt 1, Fish template, $\times 20$
D-S-1	summary, 100	21-point grid	distribution	1	scratchpad, one file	Prompt 1, Fish template, $\times 20$
D-S-25	summary, 75 ( $3\Delta$ )	21-point grid	distribution	25	plans_and_insights , previous distribution	Prompt 0, structured, $\times 20$
D-S-50	summary, 150 ( $3\Delta$ )	21-point grid	distribution	50	plans_and_insights , previous distribution	Prompt 1, structured, $\times 20$

Supplementary Table 1 : Components of the LLM deployments. Windows in periods. Fish template: the free-text reply template of [ 23 ] with observations, scratchpad and price fields; structured: a JSON response validated against a schema. Rival-price observability is part of the information setting and is reported per experiment.

Class	$S^{\mathrm{learn}}$	$S^{\mathrm{aggr}}$
LLM	1.25	1.00
Exp3	1.25	2.50
BM-RM	1.00	1.00
PPO	2.25	1.00
UCB	6.25	1.00

Supplementary Table 2 : Selection scores of the frozen configurations, each evaluated under the rule that selected it. Lower is better. The two columns refer to different configurations within a family, not to one configuration scored twice; the only exception is BM-RM, for which both rules select the same candidate. Scores are averages of one-indexed within-family ranks, so the attainable minimum is

1.00

Class	Configuration
D-S-25	Prompt 0 (neutral), Gemini 3 Flash Preview, temperature $0.7$ , intervention interval $25$ , summary history $3\Delta$
Exp3	Constant $\gamma=0.30$
BM-RM	Constant $\gamma=0.50$ , $\eta=\gamma/21$
PPO	$R=50$ , $\eta=0.005$ , $\lambda_{H}=0$
UCB	$\alpha=1.7$

Supplementary Table 3 : Frozen learning-quality configurations. The history window

3\Delta

is relative to the intervention interval, so it resolves to

75

periods at

\Delta=25

here and to

150

periods for the aggressive configuration at

\Delta=50

Class	Configuration
D-S-50	Prompt 1, Gemini 3 Flash Preview, temperature $0.7$ , intervention interval $50$ , summary history $3\Delta$
Exp3	Constant $\gamma=0.125$
BM-RM	Constant $\gamma=0.50$ , $\eta=\gamma/21$
PPO	$R=50$ , $\eta=0.010$ , $\lambda_{H}=0$
UCB	$\alpha=1.0$

Supplementary Table 4 : Frozen aggressive configurations.

Class	Rule	Aggregate (frozen)	Final $20\%$	Full horizon
LLM	learning	interval $\Delta=25$	baseline ( $\Delta=50$ )	baseline ( $\Delta=50$ )
	aggressive	Prompt 1	Prompt 2	Prompt 1
Exp3	learning	$\gamma=0.30$	$\gamma=0.20$	$\gamma=0.30$
	aggressive	$\gamma=0.125$	$\gamma=0.10$	$\gamma=0.125$
BM-RM	learning	$\gamma=0.50$	$\gamma=0.50$	$\gamma=0.50$
	aggressive	$\gamma=0.50$	$\gamma=0.50$	$\gamma=0.50$

Supplementary Table 5 : Configuration selected by each rule when the identical selection rules are applied to a single metric window rather than to the aggregate. We report the final

20\%

and the full horizon, the two windows on which a selection could reasonably be based; the two early windows measure behavior before learning has settled and are not offered as alternatives. Every entry is one selected candidate. The LLM candidates are one-factor deviations from a common baseline ( gemini-3-flash-preview , temperature

0.7

, interval

\Delta=50

, summary history, Prompt 0), so an LLM entry names the dimension in which the selected candidate differs from that baseline, and the two rules may select along different dimensions. All PPO entries use

R=50

and

\lambda_{H}=0

; all BM-RM entries use

\eta=\gamma/21

Supplementary Fig. 1 : LLM configuration screen. a , Self-play price elevation and profit per period of the 14 screened distributional cells (five seeds, 2,000 periods each). b , Learning-quality and aggressive-outcome rank scores (lower is better). The star and diamond mark the configurations frozen by the learning and aggressive rules.

Supplementary Fig. 2 : Standard-learner round robin: full matrices. Rows are the focal learner, columns the opponent. a , Focal normalized profit. b , Profit advantage over the opponent. c , d , External and internal regret per period. e , Consumer surplus. f , Focal mean posted price. Final-window means over ten seeds; regret is per period over the full horizon.

Supplementary Fig. 3 : Standard-learner round robin: empirical NashConv of the final-window product of empirical price distributions (profit units; one-firm monopoly profit is 260). Colour uses $\log(1+\mathrm{NashConv})$ ; printed values are untransformed. Means over ten seeds.

Supplementary Fig. 4 : Standard-learner round robin: market-clearing price over time for every pair (log time axis). Thin lines are seeds, thick lines their median; grey band, finite-grid competitive region.

Supplementary Fig. 5 : Held-out D-S-25 price paths against standard learners. Median over ten seeds of 25-period rolling mean prices; bands span the 10th–90th percentiles across seeds. Grey band, finite-grid competitive region; light shading, final 10% of periods used in main-text Fig. 2b.

Metric	Final 5%	10%	20%	50%	Full
Profit gap	24.64	24.92	25.69	25.99	27.70
Price gap	-0.04	-0.04	-0.04	-0.04	-0.04
ER gap (cumul.)	-2198.30	-4007.74	-7739.55	-16364.73	-33867.11

Supplementary Table 6 : Robustness to evaluation-window length. Entries average cell-level means across the four learner matchups. The ER gap is cumulative over the window, not divided by its length.

Supplementary Fig. 6 : Robustness to the evaluation window. a , Profit gap; b , price gap; c , share of periods in which the D-S-25 agent posts the lower price; d , external-regret gap per period; for final windows of 5–100% of the 2,000 periods. Means over ten seeds with 95% seed-bootstrap intervals.

Opponent	Periods	P-R-1 share	Profit	Behaviour
Bandit feedback
Exp3	300	62.7%	0.20/0.12	P-R-1 agent adapts to Exp3’s exploration (main-text Fig. 2c)
Exp3 ( P-R-25 )	2,000	62.1%	0.17/0.10	advantage persists over a longer horizon with blockwise calls
PPO	230 ∗	66.1%	0.25/0.13	P-R-1 agent prices just below PPO’s exploratory prices
Full information
Coin betting	300	28.9%	0.16/0.38	learner undercuts the P-R-1 agent’s anchor for 200 periods; prices then fall to about 0.18

Supplementary Table 7 : P-R-1 against standard learners and simple pricing rules in the homogeneous market. Cumulative P-R-1 share of industry profit and mean per-period profit of each firm as a share of the one-firm monopoly profit ( P-R-1 /opponent) over the run; single runs (

n=1

). Learners use the configurations frozen in the configuration screen (Supplementary Note Supplementary Note 2: Configuration screening ) and are far from converged at these horizons (Supplementary Note Supplementary Note 3: Standard learners in homogeneous competition ).

Supplementary Fig. 7 : P-R-1 price paths against standard learners and simple pricing rules. Homogeneous market; one run per panel, the runs of Supplementary Table 7 . Orange, P-R-1 posted price; grey, opponent posted price; grey band, finite-grid competitive region; dotted line, price ceiling. Rows group the opponents by feedback: bandit or RL learners, full-information learners and simple pricing rules. Each panel states the cumulative P-R-1 share of industry profit. The Exp3 and win-up–lose-down panels repeat the runs of main-text Fig. 2c,d for comparison; the second panel is the blockwise P-R-25 variant.

Motif	Trace-level pattern	DBBS component
Broad initialization	Before receiving feedback, the LLM distributes probability broadly around the middle of the price grid.	Cold-start distribution.
Recency weighting	Recent profitable outcomes receive more emphasis than older averages or historical peaks.	Recency-weighted payoff score $v_{i}$ .
Failure sensitivity	Zero-profit or failed outcomes reduce probability on the corresponding price region.	Failure score $\phi_{i}$ and penalty $\beta_{\mathrm{fail}}\phi_{i}$ .
Ordered-price reasoning	Failures at relatively high prices are interpreted as evidence that the profitable region lies lower.	One-sided locality penalty $\beta_{\mathrm{loc}}(i-m)_{+}$ .
Local asymmetric search	Probability concentrates around a focal price; higher neighbours serve as margin probes and lower neighbours as safer fallbacks.	Search centre $m$ with local upward probing $\{m+1,\ldots,m+G_{\mathrm{up}}\}$ .

Supplementary Table 8 : Recurring behavioural motifs extracted from LLM decision traces and their representation in DBBS.

Motif	Keywords and patterns
Opponent-facing
Rival response	competitor(s)/rival(s)/opponent(s)/other firm(s) $\ldots$ ( $\leq 80$ characters) follow/match/respond/react/retaliat/mirror/copy/punish/reciprocat* ; the verbs follow/match/respond/react/retaliat/mirror/copy* $\ldots$ ( $\leq 40$ characters) competitor/rival/opponent/other firm ; tit-for-tat , retaliat* , forgiv* , punish* .
Price war	price war(s) , race to the bottom , downward spiral , mutually destructive .
Signal or reset	signal(s/ed/ing) , unless followed by (a/the) need/shift/change/instability/potential/that/weak/declin/drop/deteriorat/saturat ; reset* ; price leader* ; lead a/the (price) increase/rise/move .
Preserving a high price	cooperat* , collu* , tacit* , truce ; avoid(ing) (one word) undercut* ; do not/don’t/never undercut/price below ; maintain(ing) (the) high/ceiling/parity/price level ; preserv* (the) high/margin/price/ceiling .
Undercutting or harvesting	undercut* , unless preceded by avoid/avoiding/not ; harvest* ; capture (the) (entire/full/whole/all) market/demand/volume .

Supplementary Table 9 : Keyword lexicon for the motif frequencies of Supplementary Table 10 . Matching is case-insensitive. An asterisk marks a word stem that matches any continuation; slashes separate alternatives; parentheses mark optional words; “

\ldots

” requires both terms in the same sentence within the stated number of characters or words. A trace exhibits a motif if any listed pattern matches.

Folds	$\lambda$	$\beta_{\mathrm{fail}}$	$\beta_{\mathrm{loc}}$	$\tau$	$\rho$	$G_{\mathrm{up}}$	$w$
all 10	0.822	0.018	0.096	0.049	0.176	4	0.55–0.60

Supplementary Table 11 : Selected DBBS parameters by fold.

w

, inertia weight on the persistence component.

Model	All	Changing	Stable
DBBS	4.9	10.7	3.4
DBBS without inertia	6.0	10.2	4.8
Persistence	4.9	12.1	3.1
Win-up–lose-down	7.1	11.5	6.0
Randomized win-up–lose-down	8.1	11.5	7.3
Weighted fictitious play †	15.6	18.1	15.6

Supplementary Table 12 : Held-out one-step accuracy on D-S-25 on all blocks, on the 22% of blocks in which the LLM’s mean price moves by at least one grid step, and on stable blocks (W 1 in % of the price range; means over the seven treatments).

Model	Error	DBBS closer
DBBS	0.020	–
DBBS without inertia	0.030	59/70 ( $p=5\times 10^{-7}$ )
Win-up–lose-down	0.063	56/70 ( $p=4\times 10^{-8}$ )
Randomized win-up–lose-down	0.064	55/70 ( $p=3\times 10^{-8}$ )
Persistence	0.074	58/70 ( $p=3\times 10^{-10}$ )
Weighted fictitious play †	0.042	55/70 ( $p=1\times 10^{-6}$ )

Supplementary Table 13 : Free-running reproduction of final-window prices (last 25% of 2,000 periods). Error, mean absolute difference between the model’s and the D-S-25 agent’s mean price over 70 games; DBBS closer, games in which DBBS with inertia is closer to the LLM than the model (paired Wilcoxon test).

Arm	Output	Market	$I$	History	$H$	Rival price
A1	Continuous point	logit	1	Raw	100	Visible
A2	Continuous point	homogeneous	1	Raw	100	Visible
A3	Grid point	logit	1	Raw	100	Visible
A4	Grid distribution	logit	1	Raw	100	Visible
A5	Continuous point	logit	1	Raw	100	Hidden
A6	Continuous point	logit	1	Summary	100	Visible

Supplementary Table 14 : Direct protocol ablations. All treatments use the same model, temperature, price bounds, persistent text memory, and number of LLM calls unless otherwise stated.

I

is the intervention interval and

H

the history depth in periods. Arms A1–A6 change exactly one component relative to the A1 control, A7–A10 change two or more at interval

1

, and A11–A13 are the interval-

50

variants.

Supplementary Fig. 8 : Ablation arms that lower prices compete genuinely. Seed means over the last 100 periods against the price-elevation index; grey band, range of the control’s seeds; orange, control; blue, robust movers (every seed below the control band); grey, arms without robust movement. a , Profit-elevation index with the parity line: low-price arms genuinely forgo profit, and arms above 1 earn no more than the control. b , Share of the window at the modal price: high-elevation arms lock in on few prices (A11 on a single action), robust movers use 12–48 distinct prices. c , NashConv in profit units falls with elevation, from 13.7–17.8 for the control and non-movers to 0.8–2.8 for robust movers; A2 is omitted because in the winner-takes-all market NashConv is set by the resolution of the response grid.

Supplementary Fig. 9 : Per-call actions of every ablation arm. Mean price index of each LLM decision (the mean price over its intervention block) across seeds and firms, band

\pm 1

s.d.; 300 calls per arm. Non-moving arms freeze on a single action (A3, A8, A11, A12 repeat their action in all of the last 50 calls); robust movers keep revising (A7, A10, A13).

Cell	Seeds	P-R-1 $\Delta$	Opponent $\Delta$	Interpretation
P-R-1 vs P-R-1	3	1.11	0.91	high-price regime under like LLM interaction
P-R-1 vs undercut heuristic	2	1.04	0.97	high P-R-1 prices are a best response to the fixed rule
P-R-1 vs D-R-1	6	0.91	0.93	cross-deployment LLM interaction remains high
P-R-1 vs bandit regret matching	2	0.57	0.53	partial persistence under slow bandit adaptation
P-R-1 vs static monopoly	3	0.44	1.03	P-R-1 exploits a fixed high-price anchor
P-R-1 vs Exp3	3	0.36	0.46	intermediate; seeds range from 0.19 to 0.64

Supplementary Table 15 : P-R-1 matchups in the differentiated-product market. Values are focal P-R-1 price elevations and opponent elevations, averaged over all seeds of each cell; the same runs as Supplementary Fig. 10 . Given that sample size and the uncontrolled LLM sampling discussed above, the rows should be read as an ordering of opponent classes rather than as precise point estimates.

Supplementary Fig. 10 : P-R-1 matchups in the logit market. Focal P-R-1 price-elevation index (0 = continuous Nash price, 1 = joint-profit price) over 300 periods; grey points are seeds, large points their means. Orange, LLM or fixed-rule opponents; black, learning algorithms.

Supplementary Fig. 11 : All heterogeneous-deployment runs. Posted prices of the point-price agents P-R-1 and P-S-1 (orange) and the emitted policy mean of D-S-1 agents (blue) in each run of main-text Fig. 5; shades distinguish firms of the same type. Grey band, finite-grid competitive region (three lowest grid prices with two firms, two lowest with three or four); dotted line, price ceiling. Rows 1–3, Gemini 3 Flash; row 4, GPT-5.6 Terra.

Regime	Runs	Price	Consumer (%)	Producer (%)	Leading firm (%)
Competitive benchmark (p = c)		0.100	100	0
Joint-monopoly benchmark (p = v)		0.360	0	100
Standard learners, markets without UCB	21 markets, 10 seeds	0.124	91	9
Standard learners, markets with UCB	7 markets, 10 seeds	0.158	78	22
D-S-25 self-play	10 seeds	0.113	95	5
D-S-50 self-play	10 seeds	0.113	95	5

Supplementary Table 16 : Welfare in the homogeneous market. Price, mean market-clearing (lowest) price; consumer and producer surplus as percentages of the maximal per-period surplus

M(v-c)=260

, which is also total welfare in every row; leading firm, share of industry profit of the firm earning most (the D-S-25 agent in the rows against learners). In the heterogeneous runs, P-R-1 always observes its rivals’ prices and “rival hidden” applies to the other firms. Single-run rows are representative paths, not treatment estimates.

Arm	Price	CS	PS	TW	DWL (%)
Continuous Nash benchmark	1.47	71.5	44.6	116.1	0.0
Joint-monopoly benchmark	1.93	32.7	67.5	100.2	13.7
Control (continuous point, raw, visible)	1.91	33.7	67.4	101.1	12.9
Grid point	1.95	31.2	67.3	98.5	15.1
Grid distribution	1.94	31.8	66.7	98.5	15.2
Rival hidden	1.73	49.6	59.4	109.0	6.1

Supplementary Table 17 : Welfare per period in the logit market of the protocol ablations (five seeds per arm, final third). Deadweight loss is the shortfall of total welfare relative to the continuous Nash benchmark.

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Sep 28, 2026

Persistent Partners Raise Prices Among Learning Agents

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.

Paul-Peter Arslan, Yubin Kim, Xiao Xiao

Institute For Future Technologies Devinci Higher Education · Massachusetts Institute of Technology

Algorithmic Collusion Multi-Agent Systems Dynamic Pricing Multi-Agent Reinforcement Learning

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Abstract

Algorithmic Collusion

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Persistent Partners Raise Prices Among Learning Agents