Persistent Partners Raise Prices Among Learning Agents

Paul-Peter Arslan; Yubin Kim; Xiao Xiao

cs.LGSep 28, 2026

Persistent Partners Raise Prices Among Learning Agents

Authors: Paul-Peter Arslan, Yubin Kim, Xiao Xiao

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

Abstract

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.

Figures & tables

contrast ( fixed minus shuffled )	status	estimate	95% CI	$p$	verdict
H1, level $C_{\Delta}$	registered	$+0.274$	$[+0.195,+0.350]$	$0.00034$	substantial
H6, enforcement $C_{E}^{\mathrm{BR}_{1}}$	registered	$+1.48$	$[-1.50,+4.75]$	$0.145$	inconclusive
H6, enforcement $C_{E}^{\mathrm{BR}_{K}}$	registered	$+1.36$	$[-1.25,+4.50]$	$0.156$	inconclusive
swap, fixed agents vs a non-partner, $\Delta$	stage 2, Holm	$-0.177$	$[-0.275,-0.080]$	$0.0003$	rejected
mask rival history, $E_{\mathrm{net}}$	stage 2, Holm	$-0.70$	$[-10.75,+10.75]$	$0.43$	not rejected
level, rival hidden	post hoc	$+0.295$	$[+0.240,+0.350]$	–	–

Table 1: Results at a glance. Registered tests first (

p

one-sided), then post hoc readings that spend no

\alpha

. Stage 2 rows instead compare probed with standard final policies. Intervals are two-sided 95% sets. Levels are run-level means over the eight checkpoints unless marked final.

$\Delta$	level at the rest point, on profits, $0$ Nash, $1$ joint monopoly, $0.378$ random play
$E$ , $E_{\mathrm{myopic}}^{r}$ , $E_{\mathrm{net}}^{r}$	raw enforcement, its static benchmark ( $r\in\{\mathrm{BR}_{1},\mathrm{BR}_{K}\}$ ), their difference
$C_{\Delta}$ , $C_{E}$	fixed minus shuffled contrasts on run-level $\Delta$ and $E_{\mathrm{net}}$ , averaged over the other factors
$\alpha_{\Delta}$ , $\alpha_{E}$ , $\alpha_{M}$	test levels of H1, H6 and stage 2
SESOI Δ , SESOI E	smallest effect sizes of interest, $0.10$ on the level and $4$ bins $\cdot$ rounds on enforcement
$M$ , $\eta_{Q}$ , $\delta$	population size ( $4$ ), learning rate and discount of the value module

Table 2: Notation. SESOI stands for the smallest effect size of interest, fixed at registration.

Appendix figures & tables6 assets

Supplementary material from the paper’s appendix.

Appendix

cell	$\Delta$	$E$	$E_{\mathrm{net}}^{\mathrm{BR}_{1}}$	$E_{\mathrm{net}}^{\mathrm{BR}_{K}}$	$P_{1}$	$P_{2}$	pattern	spread	gain	clamp	$L$	drift
fixed / visible / none	$0.673$	$6.99$	$3.61$	$3.75$	$0.23$	$1.00$	$0.29$	$0.36$	$0.015$	$0.01$	$5.3$	$5.6$
shuffled / visible / none	$0.467$	$4.85$	$1.40$	$1.22$	$0.17$	$0.80$	$0.41$	$0.58$	$0.015$	$0.04$	$7.2$	$5.5$
fixed / visible / free_text	$0.697$	$3.47$	$1.69$	$1.43$	$0.26$	$1.00$	$0.33$	$0.25$	$0.014$	$0.00$	$4.0$	$4.6$
shuffled / visible / free_text	$0.399$	$5.16$	$-2.04$	$-1.47$	$0.01$	$0.20$	$0.28$	$0.63$	$0.016$	$0.05$	$11.1$	$6.6$
fixed / hidden / none	$0.534$	$0$	$0$	$0$	$0$	$0$	$0$	$0.45$	$0.018$	$0.06$	$6.9$	$4.0$
shuffled / hidden / none	$0.257$	$0$	$0$	$0$	$0$	$0$	$0$	$0.73$	$0.015$	$0.11$	$7.8$	$4.3$

Appendix

Table 3: Per-cell means over the five blocks (registered descriptive table).

P_{1}

is the share of collusive points per run,

P_{2}

the share of runs with at least one collusive point, and pattern the share of points with a punishment pattern on at least half the starts, followed by the mandatory diagnostics, one-shot gain of the deviation (profit units), clamp frequency, deviator’s persistence

L

(rounds) and sham drift (bins).

variant	$\Delta$ fixed	$\Delta$ shuffled	contrast	swap	signature
trained agent (reference)	$0.673$	$0.467$	$+0.206$	$-0.293$	$+0.012$
head removed (ablation)	$0.681$	$0.477$	$+0.204$	$-0.161$	$+0.010$
table weight lowered to $5.5$	$0.395$	$0.198$	$+0.198$	$-0.020$	$+0.028$
weight $5.5$ , blocks 4 and 5, added	$0.437$	$0.256$	$+0.181$	$+0.155$	$+0.117$
table weight lowered to $3$	$0.351$	$0.192$	$+0.159$	–	–
language model alone, policy gradient	$0.178$	$0.113$	$+0.064$	–	–

Appendix

Table 4: Which part carries what, in the visible-rival cell without a channel, with the confirmatory seeds. The first five rows are means over all eight checkpoints of 8 000 rounds, on five blocks for the first two, on the three planned blocks for the table weight of

5.5

, on its blocks 4 and 5, added after reading blocks 1 to 3, in their own row, and on five planned blocks for the weight of

3

. The language model trained alone is read on its four checkpoints of a 4 000-round run (the hybrid gives

+0.258

on the same blocks and checkpoints). Prompted models are read on the realised level of their last 50 of 200 rounds (five planned blocks) or their last 100 of 500 rounds (planned blocks only), with the rematched arm re-paired every 20 rounds or every round among four agents. Rows below the first five are comparable in sign and size only. The last two columns give the swap-opponent drop and the coupling signature of fixed runs. Exploratory, no

\alpha

id	prediction	observed	direction / interval
A1	$M=2$ , $C_{\Delta}\approx 0$	$-0.032$ , $[-0.190,+0.130]$	held / inconclusive at 5, precise null at 25
A2	$C_{\Delta}(8)\geq C_{\Delta}(4)>C_{\Delta}(2)$	$+0.396$ , $+0.363$ , $-0.032$	held / only $C_{\Delta}(4)>C_{\Delta}(2)$
A3	synchronous, $C_{\Delta}\approx 0$ , levels near Nash	$-0.042$ , levels $0.29$ and $0.33$	level clause failed / small negative at 25
A4	floor lowers $C_{\Delta}$	$+0.264$ against $+0.363$	held / intervals overlap
A5x	$10^{6}$ rounds, two outcomes recorded	hidden gap persists	not a test
C1	signature $>0$ under fixed , $\approx 0$ under shuffled	Table 7	held / –

Appendix

Table 5: The fifteen predictions A1 to D4, scored twice (C4b and the head read-out are reported with their paragraphs). D1 and D4 are scored on the three planned blocks of the lowered table weight, and its two added blocks leave both verdicts unchanged. Direction asks whether the point estimate lies where the prediction said. Interval applies the standard of the confirmatory tests. A prediction of no effect holds only if its 95% interval lies inside

\pm\mathrm{SESOI}_{\Delta}=\pm 0.10

, and an ordering only if the two intervals do not overlap. A dash means that no interval exists (single pairs) or that the item is descriptive.

variation	cell	$\Delta$	$\Delta_{\mathrm{final}}$	$E_{\mathrm{net}}^{\mathrm{BR}_{1}}$	pattern	$P_{1}$
$M=4$ (reference)	fixed / visible	$0.661$	$0.773$	$1.28$	$0.34$	$0.17$
	shuffled / visible	$0.365$	$0.308$	$6.43$	$0.35$	$0.07$
	fixed / hidden	$0.571$	$0.667$	$0$	$0$	$0$
	shuffled / hidden	$0.141$	$0.036$	$0$	$0$	$0$
$M=2$	fixed / visible	$0.688$	$0.764$	$1.59$	$0.45$	$0.25$
	shuffled / visible	$0.721$	$0.848$	$3.95$	$0.36$	$0.30$

Appendix

Table 6: Mechanism runs on the tabular learner, per-cell means over five blocks, giving the level averaged over checkpoints, the level at the final checkpoint

\Delta_{\mathrm{final}}

(round 8 000, or round

10^{6}

in the last four rows),

E_{\mathrm{net}}^{\mathrm{BR}_{1}}

, pattern share and

P_{1}

cell	1k	2k	3k	4k	5k	6k	7k	8k	$\rho_{\text{part}}$	$\rho_{\text{non}}$	residual
fixed / visible / none	$.003$	$.004$	$.013$	$.009$	$.011$	$.012$	$.013$	$.013$	$.905$	$.892$	$+0.16$
shuffled / visible / none	$-.001$	$.001$	$.002$	$.001$	$.000$	$.000$	$.001$	$.001$	$.910$	$.909$	$\leq 0.12$
fixed / visible / free_text	$.000$	$.006$	$.011$	$.015$	$.012$	$.012$	$.013$	$.013$	$.922$	$.909$	$+0.24$
shuffled / visible / free_text	$.001$	$.005$	$.006$	$.006$	$.008$	$.009$	$.009$	$.008$	$.909$	$.901$	$\leq 0.12$
fixed / hidden / none	$.004$	$.012$	$.040$	$.079$	$.121$	$.113$	$.145$	$.170$	$.920$	$.749$	$+1.04$
shuffled / hidden / none	$-.002$	$.002$	$.002$	$-.001$	$-.002$	$-.002$	$.000$	$-.001$	$.988$	$.989$	$\leq 0.12$

Appendix

Table 7: C1, the coupling signature

\rho(\text{partners})-\rho(\text{non-partners})

in the hybrid agent’s tables, blocks 1 to 4, by round, and the two correlations at round 8 000. The residual variant (correlating each table’s residual around the run’s mean table) was defined after seeing these numbers and is post hoc.

$\mu$	$0.05$	$0.10$	$0.20$	$0.25$	$0.30$	$0.40$	$0.45$	$0.49$	$0.50$
slope	$0.835$	$0.674$	$0.436$	$0.357$	$0.299$	$0.222$	$0.196$	$0.179$	$0.175$
$d^{\ast}$	$0.013$	$0.035$	$0.110$	$0.166$	$0.234$	$0.409$	$0.530$	$0.681$	$0.774$
$R$	$0.011$	$0.024$	$0.050$	$0.062$	$0.074$	$0.098$	$0.114$	$0.136$	$0.153$

Appendix

Table 8: The validity curve, continuous (no grid),

m_{k}=0.10

. The slope is

\partial\mathrm{BR}/\partial p_{\mathrm{rival}}

at the monopoly price, and

d^{\ast}

the required deviation depth as a fraction of the Nash-to-monopoly gap. The calibration used in this paper is

\mu=0.25

, where the discrete grid gives

R=0.050

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