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Warned alike, AI agents avoid the less-crowded road while people take it
Takahiro Ezaki, Naoto Imura, Katsuhiro Nishinari
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 94%
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Summary
This study investigates how shared forecasts influence resource allocation in a two-road congestion game involving AI agents and humans. Populations of GPT agents, when given a warning that others might follow a routing tip, exhibited persistent concentration on one road, increasing average travel time from 64 to 95 minutes. This self-defeating prophecy persisted for 100 rounds. In contrast, all-human groups remained near balance. In mixed groups, humans increasingly took the road avoided by agents, leading to unequal burdens where agents suffered higher costs despite lower collective costs. The findings suggest that shared information can sustain collective inefficiency among similar agents and highlight the need for population-level evaluations of AI systems.
Entities (9)
Relation Signals (7)
Takahiro Ezaki → authored → Warned alike, AI agents avoid the less-crowded road while people take it
confidence 99% · Warned alike, AI agents avoid the less-crowded road while people take it Takahiro Ezaki 1,∗
F3 broadcast → causes → increased average travel time
confidence 95% · Adding one sentence warning... Average travel time rose from 64 to 95 min
Human Participants → exhibitsbehavior → near balance
confidence 95% · Twelve all-human groups (240 participants) stayed near balance under numerical reports or the tip and warning.
GPT agents → exhibitsbehavior → persistent concentration on one road
confidence 95% · Adding one sentence warning... made populations of 50 GPT agents crowd one road while avoiding the nearly empty alternative.
Warned alike, AI agents avoid the less-crowded road while people take it → publishedby → University of Tokyo
confidence 95% · Takahiro Ezaki 1,∗ , Naoto Imura 1 , Katsuhiro Nishinari 1,2 1 Research Center for Advanced Science and Technology, The University of Tokyo
Human Participants → interactswith → GPT agents
confidence 90% · In 24 mixed groups... people increasingly took the road the agents avoided.
GPT agents → suffers → higher travel time
confidence 90% · with 15 agents and 5 humans, agent seats averaged 80 min, compared with 44 min for human seats.
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Abstract
Abstract:AI agents built on a few shared models increasingly act for many people. A shared forecast about others can align their choices and change how scarce capacity is allocated. We tested this feedback in a two-road congestion game. Adding one sentence warning that others might follow a routing tip made populations of 50 GPT agents crowd one road while avoiding the nearly empty alternative. Average travel time rose from 64 to 95 min, although any crowded-road agent could have saved 69 min by switching alone. The warning discouraged the very move it predicted. The pattern persisted for 100 rounds. Two other model families shifted the same way without locking onto one road. Twelve all-human groups (240 participants) stayed near balance under numerical reports or the tip and warning. In 24 mixed groups with a further 240 participants, imbalance grew with the share of agents in the registered analysis, while people increasingly took the road the agents avoided. Collective costs stayed below the allagent reference, but with 15 agents and 5 humans, agent seats averaged 80 min, compared with 44 min for human seats. Shared forecasts can thus sustain collective inefficiency among similar agents. A better group average can also hide an unequal burden. Evaluations of AI agents that share resources should test populations, treat messages as interventions and report who bears the costs.
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- Source: https://arxiv.org/abs/2609.30883v1
- Canonical: https://arxiv.org/abs/2609.30883v1
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Warned alike, AI agents avoid the less-crowded road while people take it Takahiro Ezaki 1,∗ , Naoto Imura 1 , Katsuhiro Nishinari 1,2 1 Research Center for Advanced Science and Technology, The University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8904, Japan 2 Department of Aeronautics and Astronautics, School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan ∗ Corresponding author. Email: tkezaki@g.ecc.u-tokyo.ac.jp Abstract AI agents built on a few shared models increasingly act for many people. A shared forecast about others can align their choices and change how scarce capacity is allocated. We tested this feedback in a two-road congestion game. Adding one sentence warning that others might follow a routing tip made populations of 50 GPT agents crowd one road while avoiding the nearly empty alternative. Average travel time rose from 64 to 95 min, although any crowded-road agent could have saved 69 min by switching alone. The warning discouraged the very move it predicted. The pattern persisted for 100 rounds. Two other model families shifted the same way without locking onto one road. Twelve all-human groups (240 participants) stayed near balance under numerical reports or the tip and warning. In 24 mixed groups with a further 240 participants, imbalance grew with the share of agents in the registered analysis, while people increasingly took the road the agents avoided. Collective costs stayed below the all-agent reference, but with 15 agents and 5 humans, agent seats averaged 80 min, compared with 44 min for human seats. Shared forecasts can thus sustain collective inefficiency among similar agents. A better group average can also hide an unequal burden. Evaluations of AI agents that share resources should test populations, treat messages as interventions and report who bears the costs. Keywords:language-model agents|congestion games|self-defeating prophecy|human–AI interaction| algorithmic monoculture Introduction As AI systems act on behalf of more people, their collective behavior [1–3] becomes a problem of resource allocation, not just individual accuracy. Many agents built from the same model may interpret a shared message alike, linking otherwise separate decisions [ 4,5]. This matters wherever users compete for limited capacity, from roads and computing services to electricity demand [6–8]. A forecast about others’ choices is especially consequential. Public signals carry disproportionate weight when people coordinate [9]. Once broadcast, a forecast also changes the situation it describes and can defeat its own prediction [10,11]. Can such a forecast sustain costly alignment even when agents repeatedly experience its consequences, and what happens to people who share the same resource? Congestion games make this feedback observable: the value of a choice depends on how many others make it [ 12,13]. In a symmetric two-road game, equal use minimizes travel time and no commuter can gain by switching alone [14]. Human experiments often approach this benchmark through heterogeneous choices [15] and learning [16,17]. Shared information can instead synchronize responses [18] and produce overreaction, oscillation or persistent disequilibrium [ 19–21]. Experiments and theory show why better information need not improve individual choices [22] or system performance [18,23,24]. This simple environment therefore tests a general tension between anticipating others and maintaining diversity of action. Language-model populations can develop conventions and collective biases [25], yet struggle when suc- cess requires different agents to choose different actions [26] or to restrain their use of a shared resource [ 27]. Their economic choices can resemble human choices while showing less heterogeneity and weaker 1 arXiv:2609.30883v1 [physics.soc-ph] 25 Sep 2026 coordination in repeated interaction [28–30]. Algorithmic pricing likewise shows that individually generated decisions can produce collective outcomes sensitive to learning and prompts [31,32]. Studies of model commuters have examined equilibrium, learning, memory and differences from human play [33–37], with related dynamics in the El Farol bar problem, where people choose whether to attend a bar that is enjoyable only when not too crowded [12,38]. What remains unclear is whether a shared forecast of others’ reactions can itself sustain costly avoidance, despite repeated feedback and profitable opportunities to deviate. The consequences also depend on who shares the resource. Machines can improve human coordination [ 39] and cooperate with people at levels rivaling human cooperation [40]. However, people readily exploit machines they expect to cooperate [41]. Human–AI combinations also often perform worse than the better of humans or AI alone [42]. If people adjust to aligned agents, collective costs may fall while the remaining burden is concentrated on one participant type. Measuring group averages alone would miss this redistribu- tion. Here we manipulate a daily broadcast while holding the game, model architecture and stated objective fixed. Adding a warning made the primary model population crowd one road despite profitable opportunities to switch. Broadcast changes and selective exposure test dependence on the information environment; other model families, a newer GPT model and reasoning settings test model dependence. Experiments with 480 participants in 36 groups show a different human response: all-human groups remain near balance, while humans sharing roads with agents increasingly use the road those agents avoid. Lower collective costs coexist with unequal costs for human and agent seats. Results A congestion game with four levels of shared information We consider a basic model of resource sharing among decision makers who receive the same public infor- mation. IfnofNcommuters use a road, its travel time is 20+80n/Nminutes: 60 min under equal use, approaching 100 min when almost everyone chooses it. The computational core used 50 agents, each im- plemented as a separate call to the primary model snapshot (gpt-5.4-mini-2026-03-17; Materials and Methods). In round 1, routes were assigned randomly; agents began choosing in round 2. Each prompt contained the game rules, a mild individual time-sensitivity trait, the agent’s own road and travel time on the previous five days, and the current broadcast. Every condition instructed agents to consider how others might react to the same information. F0 supplied no report; F1 reported both roads’ travel times from the previous day; F2 named the previously less-crowded road; and F3 added a warning to F2: “However, many drivers are expected to see this same information and switch to Route B, so Route B may become congested” (with the route label updated by the environment). The human experiments used the F1 and F3 broadcasts verbatim. For a single run or room, ˆp t is the realized share on physical Route A in roundt, and ˆ f t is the realized share choosing the previously less-crowded road. Hats distinguish observed fractions from choice probabili- ties. We write ̄ f=E t [ ˆ f t ]for the within-run time average over eligible rounds, and⟨ ̄ f⟩for the equal-weight mean of these averages across runs or rooms. The primary summaries use rounds 6–60: route imbalance I, the mean absolute deviation of a road’s population share from one half; switching fractionM, the mean fraction changing roads between rounds; and realized mean travel time. We report medians of run-levelI, Mand travel time unless stated otherwise;⟨ ̄ f⟩always denotes a mean across runs. Imbalance ranges from 0 under equal use to 1 2 when everyone uses one road. Mean travel time isE t [T t ]=60+160E t [(ˆp t − 1 2 ) 2 ], whereE t averages rounds within one run, not across runs (Materials and Methods). This cost penalizes both persistent concentration and repeated movement between crowded roads. One warning produces persistent concentration The prespecified computational core compared four conditions across ten paired seeds (40 runs of 50 agents and 60 rounds; 118,000 model decisions). Without a broadcast, agents still received their own travel-time history. In the representative F0 run (Fig. 1C), the random 21:29 split became 29:21 in round 2. All 29 agents on the slower road then switched, while 19 of the other 21 stayed, producing a 2:48 split. Subsequent 2 A Two identical roads; travel time on a road = 20 + 80 × (share of commuters on it) minutes Route A Route B N = 50 (or 20) commuters 60 (or 40) rounds Balanced use (½ : ½) gives everyone 60 min. If almost all crowd one road they take ≈95 min. Any one of them could save ≈69 min by switching alone. HomeWork 0½all share on a road 20 60 100 min B The four daily broadcasts (humans received F1 and F3) F0 No report (there is no traffic broadcast today) F1 Numbers “Yesterday’s travel times: Route A = 95 min, Route B = 25 min.” F2 Tip “Traffic advisory: Route B is currently less crowded.” F3 Tip + warning “Traffic advisory: Route B is currently less crowded. However, many drivers are expected to see this same information and switch to Route B, so Route B may become congested.” In every condition each agent also sees its own route and travel time on the last five days, and is told to consider how the other drivers might react to the same information. 0 1 Route A share ̂ p t , seed 3 C F0 none 13060 Round 0.0 0.5 1.0 Share choosing yesterday’s less-crowded road, ̂ f t 96 min D F1 numbers 13060 Round 71 min E F2 tip 13060 Round 64 min F F3 tip+warning 13060 Round 95 min G F3 continued to round 100 13060100 Round rounds 61–100: I = 0.47, 96 min no run escaped 0.00.10.20.30.40.5 Route imbalance, I (0 balanced, ½ all on one road) 0.0 0.2 0.4 0.6 0.8 1.0 Switching fraction per round, M H frozen (descriptive) lockstep oscillation filled: language-model agents hollow: adaptive-learning baseline F0 none F1 numbers F2 tip F3 tip+warning F0 none F1 numbers F2 tip F3 tip+warning 60 70 80 90 100 Mean travel time (min) I 96716495 balanced use 60 independent coin flips 60.8 F2 tip F3 tip + warning 60 70 80 90 100 J one added sentence Figure 1. A shared warning produces persistent concentration in the primary model population.(A) The game: two identical roads,Ncommuters choosing simultaneously every round, travel time increasing linearly with a road’s share of commuters (inset). (B) The four daily broadcasts (verbatim). (CtoF) For each condition of the prespecified core (50 GPT agents, 60 rounds, ten seeds), the upper strip shows the share of agents on physical Route A in one representative run (seed 3): lockstep oscillation under no report, damped oscillation under numbers, near-balance under the tip, and persistent concentration on one road under the tip plus warning. The main panel shows the share choosing the road that was less crowded in the previous round, ˆ f t , for all ten runs (thin lines; thick line, median; number, median of run-mean travel time). (G) The ten F3 runs continued to round 100 (shaded). (H) Route imbalanceIagainst switching fractionMfor all 40 runs (filled) and for the multinomial-logit learning baseline on the same seeds (hollow); the shaded region marks the descriptive frozen criterion. (I) Mean travel time per run (bars, medians; dashed, balanced use; dotted, expectation for independent coin flips). (J) Paired F2→F3 change in mean travel time by seed. 3 near-unanimous switching moved the crowd between roads rather than balancing it. Thus, feedback through experienced congestion can align choices even without a public report. Across runs, almost every agent changed roads each morning (M=0.99); populations alternated between highly unequal splits, giving mean travel time of 96 min (Fig. 1C, H, I). Numerical reports reduced imbalance and switching (I=0.23,M= 0.66, 71 min; Fig. 1D). The bare tip gave the lowest cost among these four conditions (I=0.13, 64 min; Fig. 1E). The references are 60 min under equal use and 60.8 min under independent fair choices (Fig. 1I): random round-to-round imbalances put more commuters on the crowded road, raising the expected cost above 60 min. Adding the warning produced persistent concentration: about 96% of agents used the same road, withI=0.47,M=0.07 and 95 min (Fig. 1F). Most agents kept choosing the previously crowded road, consistent with anticipating that others would move to the advertised alternative. Their stated reasons frequently invoked others switching or crowding (Fig. 2H), although these reports do not independently establish internal reasoning. The paired F3–F2 contrast was+0.32 inI(minimum over seeds+0.30) and −0.63 inM(maximum−0.61), with all ten pairs in the prespecified directions (claim-level sign tests with Holm adjustment,p=0.0039); mean travel time rose by 30 min (range 27–32 min; Fig. 1J). For a learning-based comparison, we also specified agents that choose by multinomial logit: they prob- abilistically favor roads with lower travel-time estimates, updated from experience and available numerical reports (Materials and Methods). Adding numerical reports (F0→F1) had opposite effects in the two pop- ulations. It reduced imbalance and switching in GPT populations (I: 0.48 to 0.23;M: 0.99 to 0.66), but increased both in the learning baseline (I: 0.06 to 0.44;M: 0.50 to 0.88), producing oscillation (Fig. 1H; SI Appendix, Table S1). The prespecified interaction tests this difference between the two F0-to-F1 changes; its direction held in all ten paired seeds for both outcomes (SI Appendix, section S4). The baseline treats the bare tip (F2) and tip plus warning (F3) identically because it has no representation of the warning sentence. Persistent concentration left large unilateral gains available. At a 47:3 split, an agent on the crowded road takes 95.2 min and could reduce its travel time to 26.4 min by switching alone, holding everyone else’s choice fixed. Agents rarely took this opportunity (M=0.07); the concentrated state is therefore not a Nash equilibrium. None of the ten F3 populations escaped when continued to round 100: rounds 61–100 had I=0.47 and 96 min (Fig. 1G). We call a run “frozen” when it combines high imbalance, low switching and a persistent majority road under the prespecified descriptive classifier (Materials and Methods). All ten F3 runs and none of the 30 other core runs met this criterion. Changing the broadcast changes the collective dynamics We changed the broadcast at round 31 in fresh runs to test whether the established pattern could be disrupted (Fig. 2A–C). Replacing the tip plus warning with numerical reports (F3→F1) immediately increased switch- ing in all five seeds. In round 31, 96–100% of agents chose the previously less-crowded road, while 86–98% changed their own road. Concentration consequently persisted on the opposite road:I=0.46–0.50 and mean travel time was 94–100 min. Costs subsequently declined. In the prespecified post-intervention win- dow (rounds 36–50), mean travel time was 67–74 min, compared with a pre-intervention mean of 92 min; stationary F1 runs gave 70–74 min over the same window. Replacing numbers with the tip plus warning (F1→F3) induced avoidance immediately: no agent chose the previously less-crowded road in round 31, and rounds 36–50 gave 91–96 min, compared with 72 min before the change. Switching 38 of 50 agents from F1 to F3 while retaining F1 for the other 12 raised travel time from 71 to 79 min. Thus, the current broadcast could override an established collective state: concentration was neither irreversible nor determined solely by the initial allocation. These switches do not establish weak initial-condition dependence in general. These in- terventions changed both the warning and the information format; they do not isolate removal of the warning sentence alone. Dose–response, and the benefit of warning a minority Selective exposure to the warning produced a non-monotonic response (Fig. 2D, E). In 20-agent populations, medianIat 0, 5, 10, 15 and 20 warned agents was 0.23, 0.14, 0.20, 0.35 and 0.47; corresponding travel times were 71, 65, 69, 80 and 95 min. The 50-agent grid showed a similar shape, with its lowest observed cost at 12 warned agents. Both exposure grids ran for 60 rounds. Why can warning a minority help? With numerical 4 13160 Round 0.0 0.5 1.0 Share choosing yesterday’s less-crowded road, ̂ f t A F3: 92 min F1: 72 min round 31: 96–100% choose the previously less-crowded road F3 → F1: tip + warning replaced by numbers 13160 Round B F1: 72 minF3: 93 min F1 → F3: numbers replaced by tip + warning 13160 Round C 0/50: 71 min38/50: 79 min F1 → tip + warning to 38 of 50 agents 0.00.20.40.60.81.0 Fraction of agents warned (the rest get numbers) 0.0 0.1 0.2 0.3 0.4 0.5 Route imbalance, I D warning a minority improves balance 20 agents, median 50 agents, median linear mixing of ⟨ ̄ f⟩ pure populations 0.00.20.40.60.81.0 Fraction of agents warned 60 65 70 75 80 85 90 95 100 Mean travel time (min) E balanced use dashed: the same mixing rule, no fitted parameter F Adding yesterday’s numbers moves both populations toward the tip. The cost moves in opposite directions F2 tip ⟨ ̄ f⟩ = 0.54 64 min 10 seeds F2 tip + numbers ⟨ ̄ f⟩ = 0.82 80 min 5 seeds F3 tip + warning ⟨ ̄ f⟩ = 0.04 95 min 10 seeds F3 + numbers ⟨ ̄ f⟩ = 0.13 86 min 5 seeds 60708090100 Run-mean travel time (min) Dot, median; bar, range (windows in caption) F3 tip + warning, core (10 seeds) F3 re-run 4 weeks later (3) F3, full history, not 5 days (5) F3, past tips added to history (5) F3, stronger warning wording (3) F3, invented road names (1) F3, paraphrase v1 (3) F3, paraphrase v2 (3) F3 + yesterday’s numbers (5) F3, Claude Haiku 4.5 (5) F3, Gemini 3.5 Flash (5) F2 tip, GPT core (10; reference) F2 tip, Claude Haiku 4.5 (5) F2 tip, Gemini 3.5 Flash (5) ⟨ ̄ f⟩ = 0.04 ⟨ ̄ f⟩ = 0.04 ⟨ ̄ f⟩ = 0.13 ⟨ ̄ f⟩ = 0.13 ⟨ ̄ f⟩ = 0.13 ⟨ ̄ f⟩ = 0.54 ⟨ ̄ f⟩ = 0.26 ⟨ ̄ f⟩ = 0.24 balanced use F1 numbers F2 tip F3 tip+warning 0.0 0.5 1.0 Share of stated reasons H 76% 25% 61% 7% 89% 55% filled: other drivers switching / crowding hollow: own slow recent experience What the agents say (seed 3, 2,750 decisions each) G Figure 2. The collective state follows the message currently in force. ( A to C ) ˆ f t when the broadcast is changed at round 31 (dashed line): the tip plus warning replaced by numerical reports (F3→F1; A, five seeds), numerical reports replaced by the tip plus warning (F1→F3; B, five seeds), and the warning given to 38 of 50 agents while the rest keep receiving numbers (C, three seeds); numbers give mean travel time in rounds 16–30 and 36–50. In A, 96–100% choose the previously less-crowded road in the first post-switch round; 86–98% actually change roads, and congestion remains high before costs subsequently decline. (D,E) Imbalance and mean travel time against the fraction of agents receiving the warning while the rest receive numbers (points, runs; lines, medians; diamonds, pure populations); dashed curves, linear mixing of the two pure populations’ tip-following shares with no fitted parameter, which approximates the shape but under-predicts cost near the minimum. (F) Appending yesterday’s travel times to the tip or to the tip plus warning: ⟨ ̄ f⟩and the median run-mean travel time per cell. (G) Run-mean travel time (dot, median across runs; bar, range) for message, history and model variants; windows are rounds 6–40 for the 20-seat history variants and 6–60 for the remaining runs.⟨ ̄ f⟩is shown where round-level data are available. (H) Share of agents’ stated reasons that invoke other drivers switching to or crowding a road (solid) or the agent’s own slow recent commute (light), seed 3, 2,750 decisions per condition (coding rules in Materials and Methods). 5 reports (F1), the previously less-crowded road attracts about 70% of agents, only moderately above the balanced share of 50%. With the warning (F3), only about 4% choose it: the avoidance response is much stronger than the original tendency to follow. Replacing roughly 30% of the numerical-report recipients with warning recipients therefore brings the mean share near 50%, if each group retains its original response (SI Appendix, section S5). A linear mixture of the two pure populations’ average responses approximates the U- shape and a minimum near 30% exposure, but under-predicts imbalance near the minimum (5 of 20 warned: predicted 0.03, observed 0.14; 12 of 50: 0.04 versus 0.13). A homogeneous independent-choice benchmark at the same mean also under-predicts imbalance (0.09 and 0.06, respectively; SI Appendix, Table S7). The mixture is therefore a descriptive approximation; it does not capture the full variability or the changes in behavior caused by mixing the two groups. Numerical feedback and message variants define the effect’s limits We next asked whether explicit numerical evidence changed the response to the warning. Yesterday’s travel times were appended to the tip and to the tip plus warning (five seeds each). Three concurrent repeats of the original warning condition (F3) showed stable outcomes relative to the core (SI Appendix, Fig. S3; Materials and Methods). Numbers increased tip-following under both messages, with different effects on cost (Fig. 2F). Added to the bare tip, they raised travel time from 64 to 80 min, and all five runs were classified as oscillating. Added to the warning, they reduced travel time from 95 to 86 min; two of five runs still met the frozen criterion. The numeric warning condition retained strong avoidance: the average tip-following share was 0.13 despite explicit feedback showing the other road’s lower previous cost. Most additional variants retained the frozen state, including full personal history, past advisories added to the history, stronger wording and invented road names (Fig. 2G). Of the two alternative phrasings of the warning, the first retained the frozen state; the second produced a partial state (83 min,M=0.25). Claude Haiku and Gemini also failed to meet the complete frozen criterion, as did the primary model with reasoning enabled (detailed below). These differences identify boundaries of the effect rather than uniform robustness. In a descriptive analysis of 2,750 stated reasons from core F3 seed 3, 89% referred to other drivers switching or crowding, 55% referred to the agent’s own slow experience, and 50% contained both codes (Fig. 2H, Fig. 3F; SI Appendix, section S4). These overlapping categories describe generated explanations, not the model’s internal computation. Average response and temporal variation account for cost differences Let ˆ f t be the fraction choosing the road that was less crowded in roundt−1. Under the bare tip (F2) and tip plus warning (F3), this is the road explicitly named in the broadcast; under numerical reports (F1), it is the road with the lower reported travel time. To measure the response to a previous imbalance, we use rounds whose preceding split was unequal, denotedR. A preceding tie leaves no uniquely less-crowded road, so that observation is excluded from ˆ f t summaries; a balanced outcome in the current round is retained. No such preceding ties occurred in the core warning runs; they were infrequent in the computational runs but more common in the nearly balanced human rooms (SI Appendix, sections S4 and S5). Since ˆp t equals either ˆ f t or 1− ˆ f t , applying the identityE[( ˆ f t − 1 2 ) 2 ]=(E[ ˆ f t ]− 1 2 ) 2 +Var( ˆ f t )to the round cost gives E R [T t ]=60+160 [ (E R [ ˆ f t ]− 1 2 ) 2 +Var R ( ˆ f t ) ] .(1) HereE R averages equally over the retained rounds within one run or room; ̄ f=E R [ ˆ f t ]. The two terms separate a sustained bias in the average response from variation in that response across rounds. Figure 3A plots travel time and ̄ fon the same tie-free rounds; the zero-variance curve is a lower reference, and each run’s vertical distance from it is exactly the variance contribution. Equation 1follows from the cost function and applies to any decision rule. All-round outcomes remain the primary summaries; their difference from the tie-free summaries is reported in SI Appendix, section S5. Under F0, the primary model almost always chose the previously less-crowded road (⟨ ̄ f⟩=0.96). Al- though physical road occupancy oscillated, ˆ f t was nearly constant: its excess cost was almost entirely the average-response term (34.1 of the 34.2 excess minutes; Fig. 3C). Numerical reports gave⟨ ̄ f⟩=0.69, with approximately 6 min from the average-response term and 5 min from temporal variation. The bare tip gave 6 0.00.20.40.60.81.0 Mean share choosing yesterday’s less-crowded road, ̄ f ← avoid the tip | follow the tip → 60 70 80 90 100 Mean travel time on tie-free rounds (min) A Zero-variance reference: 60 + 160 ( ̄ f − ½)² Claude Haiku: tip + warning Gemini: tip + warning GPT: tip + warning + numbers GPT: tip + warning Claude Haiku: tip Gemini: tip GPT: numbers (50 and 20 agents) GPT: tip + numbers GPT: no report GPT: tip mixed rooms: 5, 10 or 15 GPT agents among 20 players humans, 12 rooms of 20 tiptip + warning 0.0 0.2 0.4 0.6 0.8 ⟨ ̄ f ⟩ , mean over runs B balanced GPT GPT + numbers Claude Haiku 4.5 Gemini 3.5 Flash all end at 0.04–0.13 The warning lowers the lean in all three families GPT none GPT num. GPT tip GPT tip+w Haiku tip Haiku tip+w Gem. tip Gem. tip+w hum. F1 hum. F3 0 10 20 30 40 Excess travel time (min) C 34 11 5 34 25 28 18 23 2 2 solid: lean, 160( ̄ f−½)² light: volatility, 160·Var( ̂ f t ) Where the extra minutes come from 0.00.51.0 Individual’s share of tie-free rounds 6–40 on the tip road 0.0 0.2 0.4 Proportion of individuals D humans SD 0.25 (binomial ref. 0.09) split-half r = 0.73 GPT SD 0.09 (binomial ref. 0.08) split-half r = 0.14 F1 numbers, groups of 20 120 humans 100 GPT agents 0.00.51.0 humans SD 0.27 (binomial ref. 0.09) split-half r = 0.81 GPT SD 0.03 (binomial ref. 0.03) split-half r = 0.23 F3 tip + warning 120 humans 100 GPT agents 0.98 GPT F1 GPT F3 humans F1 humans F3 0.0 0.5 1.0 P(switch next day) E 0.77 0.46 0.03 1.00 0.45 0.39 0.47 0.43 solid: after a slow day (>60 min) light: after a fast day (<60 min) Reaction to own experience F “A has been consistently very slow for me, but the broadcast warns B will attract many switchers, so A is the safer bet.” “Route B may attract many switchers from the broadcast, so A is the safer choice despite recent congestion.” GPT agents under F3, verbatim. 89% of stated reasons invoke other drivers switching or crowding; 55% refer to the agent’s own slow experience (seed 3, 2,750 decisions). The response persists despite own experience Figure 3. Average response and temporal variation contribute to cost, while individual response patterns differ. (A) Mean travel time on tie-free analysis rounds against ̄ ffor the core, matched and cross-provider/numeric all-agent runs (color, message; marker, model family), for the twelve all-human rooms (stars) and for the 24 mixed rooms of Fig. 5 (hollow diamonds; room-level share over all seats). The dashed curve is the zero-variance reference from Eq.1. Both axes use the same rounds, so each point lies above it by exactly 160Var R ( ˆ f t ). (B)⟨ ̄ f⟩under the tip and under the tip plus warning for each model family (five to ten runs per point): the warning lowered average tip-following in all three families, from different baselines. (C) Decomposition of excess travel time into the average-response component, 160( ̄ f− 1 2 ) 2 , and temporal variation, 160Var( ˆ f t ); components are calculated within each run or room and then averaged with equal weight. For Claude and Gemini, their opposing changes reduce the net cost increase. (D) Distribution of individuals’ tendency to follow the tip (share of tie-free rounds 6–40), as a proportion of individuals, for 120 humans (20×six rooms) and 100 GPT agents (20×five runs) per condition in 20-seat groups; annotations give the SD, the SD expected for identical choosers at the same mean, and the split-half correlation of individual tendencies. (E) Mean individual probability of switching after a slow (>60 min) or fast (<60 min) day, for transitions from rounds 6–39 to 7–40. Individuals with no qualifying previous outcome are omitted from that conditional mean. (F) Verbatim reasons given by GPT agents under F3. 7 ⟨ ̄ f⟩=0.54, with 0.4 min and 4 min from the two terms. The warning shifted⟨ ̄ f⟩to 0.04, again with little temporal variation. Adding numbers increased⟨ ̄ f⟩under both messages (0.54 to 0.82 and 0.04 to 0.13). These decompositions are calculated for each run and then averaged; they distinguish variability in ˆ f t from switching between physical roads. In an extension planned after the core results, Claude Haiku 4.5 and Gemini 3.5 Flash were each tested under F2 and F3 at five paired seeds (Fig. 3B, C; SI Appendix, Fig. S2). The warning reduced switching in all ten pairs (mean∆M=−0.15 for Claude and−0.16 for Gemini), increased imbalance in three Claude pairs and all five Gemini pairs, and raised mean travel time by approximately 3 and 5 min, respectively. None met the frozen criterion. In both families the warning reduced⟨ ̄ f⟩(Claude, 0.26 to 0.13; Gemini, 0.24 to 0.13). The average-response cost increased by 12.5 and 10.7 min, while the temporal-variation cost fell by 9.8 and 5.4 min. All pairs showed this opposing movement of the two cost components. Thus, the direction of the warning response extended across the three tested families, whereas the full frozen state and the size of the cost increase depended on the model. Reasoning changes the collective outcome but does not consistently remove avoidance A second extension varied the reasoning setting and the model generation at the same five seeds (SI Appendix, section S9, Fig. S10 and Tables S13 and S14). Without reasoning, raising the primary model’s temperature from 0.7 to 1.0 left the core contrast essentially unchanged. All five F3 runs were frozen, while no F2 run was; the warning added 27.6 min to mean travel time. With low or medium reasoning effort, the warning still increased imbalance, reduced switching and raised travel time in all ten pairs. No run met the frozen criterion, however. Under the warning, reasoning agents chose the previously less-crowded road more often (⟨ ̄ f⟩=0.26 and 0.21, against 0.06 without reasoning). Their mean travel time was 70.0 and 74.7 min, against 91.5 min (means over five runs). GPT-6 Luna without reasoning also moved toward avoidance under the warning in all five pairs, without freezing. At its default medium reasoning, Luna froze under both messages (ten of ten runs;⟨ ̄ f⟩=0.05 under the tip and 0.07 under the warning; 92.4 and 89.5 min). In this setting the bare tip was enough to produce the costly avoidance. Adding the warning made no consistent difference. The warning thus moved responses toward avoidance wherever avoidance was not already complete. Whether the population froze depended on the model and on its reasoning setting. Human groups remain near balance under both tested broadcasts Two hundred forty participants recruited through the online experiment platform Prolific ( https://w. prolific.com) played the 20-seat, 40-round game in twelve rooms, six under numerical reports (F1) and six under the tip and warning (F3). They received the same broadcasts and five-day personal history as the agents, with timed decisions. A fixed payment was supplemented by a bonus that increased as their own mean game travel time over 40 rounds decreased (Materials and Methods). All twelve rooms remained near balance:I=0.09 under each message and mean travel time of 62.0 and 61.8 min, respectively (Fig. 4A, B, E, F). These values are close to the independent fair-choice references of 0.088 and 62.0 min for 20 seats. The registered descriptive blocked F3–F1 estimate was−0.005 inI(two-sidedp=0.28; constant-shift 95% interval−0.016 to+0.006) and+0.03 inM(p=0.31). The intervals and the small number of rooms do not establish equivalence. GPT populations matched for group size and duration incurred 73 and 96 min under the two messages (means across runs) (Fig. 4C, D). This comparison is contextual: the human and model studies were non-concurrent and differed in implementation and incentives. Most participants reported using the broadcast: 97 of 118 respondents under numerical reports (F1) and 96 of 119 under the warning (F3) said it influenced them moderately or strongly. Asked about their usual response, 35 and 38 reported taking the faster-looking road, whereas 62 and 47 reported taking the other road in anticipation of others switching (Fig. 4G). These retrospective reports are consistent with varied response tendencies, but do not independently identify the process that maintained balance. Individual response patterns differ between human and model populations Individual response patterns differed between the populations (Fig. 3D). We measured each person’s or agent’s share of tie-free rounds 6–40 spent on the previously less-crowded road, using 120 humans (20 8 12040 Round 0.0 0.5 1.0 Share choosing yesterday’s less-crowded road, ̂ f t A 62 min Humans, F1 numbers (6 rooms) 12040 Round B 62 min Humans, F3 tip + warning (6) 12040 Round C 73 min GPT, F1 numbers (10 runs) 12040 Round D 96 min GPT, F3 tip + warning (10) humans F1 humans F3 GPT F1 GPT F3 0.0 0.1 0.2 0.3 0.4 0.5 Route imbalance, I E 0.0910.0850.2480.469 independent coin flips 0.088 humans F1 humans F3 GPT F1 GPT F3 60 70 80 90 100 Mean travel time (min) F 62.061.872.695.5 balanced use 60 coin flips 62.0 050%100% F1 all human (118) F3 all human (119) F3 with agents (Fig. 5; 238) G 356217 38472410 1463941 Humans: “How did you usually respond to the report?” took the faster-looking road took the other road, expecting others to switch it depended ignored the report Figure 4. Humans in the same game.(AtoD) ˆ f t in six human rooms per condition (Prolific, 20 participants, 40 rounds) and in ten GPT runs of the same size; thin lines, rooms or runs; thick lines, round-wise medians; inset travel times, means across rooms or runs. Gaps mark rounds in which the previous counts were equal. (E,F) Imbalance and mean travel time per room or run (points; bars, means); dotted, expectation for independent coin flips; dashed, balanced use. (G) Participants’ self-reported usual response to the report in the all-human rooms and in the mixed rooms of Fig. 5. participants×six rooms) and 100 GPT agents (20 agents×five runs) per condition. GPT tendencies were concentrated under numerical reports (F1; mean 0.71, SD 0.09) and near zero under the warning (F3): 98 of 100 agents followed the tip on fewer than 10% of eligible rounds (mean 0.03, SD 0.03). Their dispersion was close to a homogeneous independent-choice reference at the observed mean (SD 0.08 and 0.03, respectively). Correlations between individuals’ first- and second-half tendencies were modest (r=0.14 and 0.23). Human tendencies were more dispersed (SD 0.25 and 0.27, compared with 0.09 under the reference) and more reproducible across session halves (r=0.73 and 0.81). Both consistent followers and avoiders were present, while mean tip-following was 0.51 under each message. Seven of 120 participants under numerical reports (F1) and 11 of 120 under the warning (F3) followed the tip on fewer than 10% of eligible rounds. These between-condition distributions do not establish how the warning changed any particular individual’s strategy. The reference and split-half analyses are descriptive: observed tendencies combine persistent individual differences, learning and shared room history. Responses to recent outcomes also differed (Fig. 3E). For GPT agents under numerical reports, the mean individual probability of switching after a slow day (>60 min) was 0.77, compared with 0.46 after a fast day (<60 min). Under the warning these probabilities were 0.03 and 1.00: even the occasional agent reaching the less-crowded road typically returned to the crowded road on the following round. Human switching probabilities were 0.45 and 0.39 under numerical reports (F1), and 0.47 and 0.43 under the warning (F3). These summaries average individual conditional probabilities over individuals with at least one qualifying outcome; they describe responses to experienced payoffs without directly measuring beliefs. Human choices shift in mixed groups, with unequal costs by seat type We next examined groups containingK=5, 10 or 15 GPT agents among 20 seats, all under F3. Two references aid interpretation. If the humans and agents retained their all-human and all-agent mean tip- following shares (0.51 and 0.03), linear mixing would give approximate imbalances of 0.11, 0.23 and 0.35 (Fig. 5A, dashed line). If allKagents chose one road, the smallest achievable imbalance would instead be 9 05101520 GPT agents among 20 players, K 0.0 0.2 0.4 0.6 Route imbalance, I A 0.070.090.26 all human all GPT linear mixing bound, all agents on one road bound, agents’ actual choices room to compensate observed rooms 05101520 GPT agents among 20 players, K 0.0 0.5 1.0 Share of the seat type on the tip road, ⟨ ̄ f ⟩ B 0.51 0.65 0.83 0.91 GPT seats human seats Humans choose the road agents avoid 05101520 GPT agents among 20 players, K 40 50 60 70 80 90 100 Mean travel time of the seat type (min) C 61 61 57 67 44 80 equal sharing 60 Travel costs differ by seat type human seats GPT seats room mean 210203040 Round 0.0 0.5 1.0 Share choosing the tip road (mean over rooms; tie rounds omitted) D all-human rooms 0.51 early window (6–20)late window (26–40) Within-session adaptation of the human seats humans, K = 5 humans, K = 10 humans, K = 15 GPT agents (all K) K = 5 K = 10 K = 15 0.4 0.6 0.8 1.0 Human seats on the tip road (early → late window, per room) E 5 of 8 rooms up 0.65 → 0.65 6 of 8 rooms up 0.81 → 0.86 6 of 8 rooms up 0.89 → 0.93 all-human 0.51 Registered early and late windows Figure 5. Human choices change within mixed rooms, with unequal costs by seat type (24 rooms).(A) Route imbalance in rooms of 20 withKGPT agents, all under F3 (purple, one point per completed room; bars, means), against the prediction if humans kept their all-human lean (dashed; linear mixing), the lower bound if all agents sit on one road (solid; accounting bound), and the bound given the agents’ actual choices (crosses; mean perK); non-concurrent all- human and all-agent reference groups are shown atK=0 andK=20. (B) Tip-following share of the human seats and of the agent seats in the same rooms (tie-free rounds 6–40). (C) Mean travel time of the human seats, of the agent seats and of the room; dashed, the 60 min of equal sharing. (D) Share of human seats (purple, byK) and of agent seats (orange) choosing the tipped road, round by round, averaged over rooms. (E) Human tip-following per room in the specified early (rounds 6–20) and late (26–40) windows; black segments join composition means. Panels summarize all eight rooms per composition; registered block-level inference and dependence sensitivities are reported in the text. 10 max(0,K/20− 1 2 ), irrespective of human choices (solid line). The latter is a conditional lower bound, equal to 0.25 atK=15; we also calculate it from the agents’ actual round-by-round choices (crosses; SI Appendix, section S5). The study yielded 24 valid rooms in eight blocks of three compositions (240 humans and 240 agent seats). Participants were told that some players were AI-controlled, but not their number or identities; agent prompts did not disclose human participation. The design, inclusion rules and registered analysis are summarized in Materials and Methods, with the registration and execution record in SI Appendix, section S8. Mean room imbalance wasI=0.07 atK=5 (eight rooms, range 0.06–0.09),I=0.09 atK=10 (eight rooms, 0.06–0.11) andI=0.26 atK=15 (eight rooms, 0.24–0.29). These means were below the fixed- response mixture values of 0.11, 0.23 and 0.35. AtK=15, all agents selected the same road in 80% of analysis rounds. The lower bound conditional on their actual choices averaged 0.238, compared with 0.259 observed: human choices brought the mean imbalance within 0.022 of the best allocation permitted by those agent choices. Five of these eight rooms met the frozen classifier; all eight retained the same majority road throughout the analysis window, but three fell below its imbalance threshold. Imbalance increased with agent share in every block, by a mean of 0.0951 per five additional agents; switching decreased in every block, by 0.2080 on average. In the registered one-sided sign-flip test over the eight block slopes,p=1/256=0.0039 for imbalance and for switching, the smallest value attainable with eight blocks. The inference assumes independent blocks with errors symmetric about zero. Sensitivity analyses that flip blocks sharing recruitment windows jointly have only 16 and 4 sign configurations; the ob- served configuration was the most extreme in both (p=0.0625 and 0.25, the minimum attainable values; SI Appendix, section S8). Leaving out any one block and applying the specified nuisance sensitivities retained the directions of both slopes (SI Appendix, Tables S10 and S11). Agents continued to avoid the tip road in mixed rooms, with mean tip-following between 0 and 0.02 by composition (Fig. 5B). The advisory repeatedly named the same road at high agent shares: 94% of consecutive rounds atK=10 and 100% atK=15, compared with 54% atK=5. Human tip-following averaged 0.65, 0.83 and 0.91 atK=5, 10 and 15. Within sessions, it rose from 0.64 and 0.62 in rounds 2–5 to 0.87 and 0.95 in rounds 31–40 atK=10 and 15 (Fig. 5D). Comparing the prespecified early and late windows (rounds 6–20 and 26–40), human tip-following rose in six of eightK=10 rooms and six of eight K=15 rooms (two were unchanged); theK=5 mean changed little (0.65 to 0.65; Fig. 5E). Across completed mixed rooms, 86 of 240 participants followed the tip on at least 90% of eligible rounds, including 30 of 40 atK=15; the corresponding count in all-human F3 rooms was 9 of 120. Mean individual switching probabilities were 0.73 after a slow day and 0.22 after a fast day. In the questionnaire, 146 of 238 respondents reported taking the faster-looking road and 39 the other road (Fig. 4G). Together, the within-session changes and outcome-contingent switching are consistent with adaptation to a predictably less-crowded road. They do not distinguish learning from experienced payoffs from explicit understanding of the agents’ policy. At the end of the task, 230 participants estimated the percentage of AI agents in their room. Their estimates averaged 63%, 67% and 69% when the actual percentages were 25%, 50% and 75%, respectively (SI Appendix, Fig. S9 and Table S12). Behavioural adaptation therefore coexisted with substantial error in reported composition, especially when agents were a minority. The population-level agreement with simple mean-response calculations is an accounting check. When K≥10 and all agents avoid the tip, even unanimous human tip-following cannot place more than half the room on that road. Imbalance then depends linearly on the human mean response, regardless of correlations between people. Consequently, matching this reconstruction cannot establish independent human choice (SI Appendix, section S5). Collective and individual costs differed substantially (Fig. 5C). Room-mean travel time was 61.3, 61.9 and 71.0 min atK=5, 10 and 15, compared with about 96 min in the all-agent reference of the same size. The human seats averaged 61.3, 57.2 and 43.5 min and the agent seats 61.2, 66.5 and 80.2 min. Thus, atK=15, human seats averaged 16.5 min below the 60-min equal-sharing benchmark, while agent seats averaged about 20 min above it. Of the 240 participants, 129 averaged less than 60 min; no agent did. Lower collective costs in mixed rooms therefore coexisted with a persistent cost disadvantage for agent seats at K=10 andK=15. AtK=5, both seat types averaged about 61 min. 11 Discussion A shared warning sustained a costly collective response in the primary language-model population. Agents repeatedly avoided the previously less-crowded road despite experiencing long travel times on the other road. The warning’s addition was sufficient to change the core outcome, and explicit numerical feedback weakened, but did not eliminate, avoidance. Changing the broadcast from the tip plus warning to numerical reports disrupted the fixed-road pattern immediately; costs fell in subsequent rounds. Two other model families also shifted toward avoidance, although none of their runs met the complete frozen criterion. The primary model behaved similarly when reasoning was enabled: the warning still moved agents toward avoidance, but no population froze. A newer model at its default reasoning setting froze even under the tip alone. The common behavioral direction and the different collective outcomes make the response to shared information, rather than a universal frozen state, the main result. We interpret the persistent avoidance as a consequence of model homogeneity interacting with shared information. A common model supplies similar decision rules, so a forecast about other drivers can elicit the same anticipatory response across many recipients, even when their personal histories differ. The warning then discourages the very migration it predicts, while repeated broadcasts sustain the response. The concen- tration of individual avoidance tendencies and the changes under selective exposure are consistent with this account. Homogeneity need not produce concentration under every message: populations of the same model stayed near balance under the tip alone (64 min). The proposed mechanism is the alignment induced by a shared warning across a population with similar decision rules. In GPT-6 Luna at its default reasoning set- ting, the tip alone was enough to produce this alignment. Its stated reasons referred to other drivers switching or crowding in 88% of decisions at seed 3, against 61% for the primary model under the tip. The cost decomposition helps explain the differences between model families. Bias in the average tip- following share and variation in that share across rounds both increase travel time. For Claude and Gemini, the warning increased the first contribution while reducing the second, limiting the net cost increase. Warning only a minority also reduced costs relative to uniform numerical reports or uniform warnings. These results suggest that evaluation should consider the distribution of responses to shared information: the same message can have different effects depending on the population’s starting behavior and on who else receives it. Human groups showed a different pattern. Individual tendencies were dispersed and reproducible across session halves, while the group means remained close to balance under numerical reports and warning- bearing advisories. These dispersed tendencies offer one plausible route to balanced use; the present design does not isolate their contribution. Across the eight registered blocks, imbalance rose and mobility fell with agent share; the registered inference assumes independent symmetric block errors. In mixed rooms, we also observed a change within sessions. Human choices increasingly favored the road avoided by the agents, accompanied by a tendency to retain fast choices and leave slow ones. This is consistent with learning from payoffs without requiring explicit knowledge of the agents’ policy. AtK=15, the observed allocation approached the lower bound permitted by the agents’ choices, but human and agent seats still averaged 44 and 80 min, respectively. The room average concealed this difference. The experiments extend work on algorithmic monoculture and routing behavior [ 4,26,36,37] by manip- ulating a shared anticipatory message in an environment with endogenous costs and examining adaptation in mixed human–agent groups. The collective response arises without direct communication, although choices remain coupled through congestion. This differs from conventions developed through explicit agent interac- tions [ 25]. These findings suggest three priorities for evaluating agents that share resources. First, test populations: plausible individual responses need not produce efficient collective outcomes. Second, manipulate message content and distribution, which changed performance here without changing the underlying model. Third, report costs by participant type alongside group averages. Mixed groups had lower collective costs than the all-agent reference, yet agent seats bore higher costs than human seats at high agent shares. If clients bear the costs incurred by their agents, better aggregate performance could therefore conceal a disadvantage for those who delegate. This experiment measures costs assigned to seats in a game; testing that deployment hypothesis requires comparisons under alternative messages and exposure patterns. Several limits define the scope of the findings. The game has two symmetric roads and immediate feed- back. All prompts asked agents to consider how other drivers might react, which may favor anticipatory 12 responses. The complete frozen state appeared in the primary model without reasoning and in GPT-6 Luna at its default reasoning setting. It did not appear in the two other families or in the primary model with rea- soning enabled. Populations combining different models under the same message would test whether model diversity reduces the observed alignment. The core used no reasoning; the later five-seed extension varied re- quest settings without inferring internal reasoning from generated explanations or token counts. The switch intervention changed numerical content as well as the warning. Continuing the same established state under F3, F2 and F1 would isolate those contributions more directly, while message controls could distinguish the forecast’s content from trust in its source. Human and all-agent benchmarks were non-concurrent and differ- ently incentivized. The mixed-group sample is complete, but composition was associated with recruitment order and some rooms were completed under amendments made after initial outcomes were known. Shared recruitment windows and nights limit the independence of the block comparisons (SI Appendix, section S8). Independent replication across days with random allocation of ready participants would strengthen causal interpretation of composition effects. Within these limits, the study shows how shared predictions can sustain inefficient behavior and how adaptation by other decision makers can change who bears its costs. Assessing shared resources therefore requires both a population view of efficiency and a participant view of outcomes. Materials and Methods Congestion game Two identical roads serveNcommuters, with travel timet(n)=20+80n/Nfor a road carryingncommuters. Weighting each road’s travel time by the fraction using it gives the round mean: T t =ˆp t (20+80 ˆp t )+(1−ˆp t )20+80(1−ˆp t )=60+160(ˆp t − 1 2 ) 2 . Here ˆp t is physical Route A’s share. Round 1 assigns routes randomly; decisions begin in round 2. The core, switching, exposure, cross-provider/numeric and reasoning studies use 60-round runs and, unless a pre/post window is specified, summaries over rounds 6–60. Matched 20-seat runs, human and mixed rooms, and the 20-seat history variants use 40 rounds with summaries over rounds 6–40. The persistence extension is summarized separately over rounds 61–100. Route imbalance isI=E t [|ˆp t − 1 2 |]. Switching fractionMaverages the proportion changing roads, in- cluding the transition into the first analysis round. Travel time is averaged over all rounds in the specified window. For ˆ f t , the reference road is the one with fewer commuters in roundt−1. When those counts are equal, ˆ f t is undefined and the round is excluded from summaries derived from ˆ f t , but retained inI,Mand travel time. WriteRfor the remaining rounds. Both sides of Eq. 1and Fig. 3A use this same set. Tie fre- quencies and the small differences between all-round and tie-free costs are reported in SI Appendix, sections S4, S5 and S9. The frozen classifier, fixed before the core, requiresI≥0.25,M≤0.20 and at least ten consecutive rounds with the same majority road. It is a descriptive classification of a run, distinct from equilibrium or from any individual’s tendency to switch. The SI usesp t andf t as shorthand for the same within-run realized fractions denoted here by ˆp t and ˆ f t . Language-model agents Each agent seat receives a separate stateless model call each round. Client-side request logs for the core and matched primary-model runs recordgpt-5.4-mini-2026-03-17, temperature 0.7, a maximum of 400 completion tokens,reasoning_effort = noneand JSON-object output. The cross-provider extension usedclaude-haiku-4-5-20251001with temperature 0.7 andmax_tokens = 400; its backend supplied no explicit thinking setting. Gemini calls usedgemini-3.5-flash, temperature 0.7,max_output_tokens = 400, JSON output and a zero thinking budget, which the backend recorded as accepted in all extension calls. The reasoning and current-model extension usedgpt-5.4-mini-2026-03-17without reasoning at temperature 1.0 and withreasoning_effortlow or medium, andgpt-6-lunawithreasoning_effort 13 none or medium, its default. The reasoning-enabled requests and all Luna requests omitted temperature; no numerical value of an internal sampling temperature was independently verified. The completion cap was 400 tokens without reasoning and 4,000 tokens with reasoning. These are recorded request configurations, not independent measurements of internal computation. Further implementation details and the scope of the request audit are reported in SI Appendix, section S1. The prompt (SI Appendix, section S1) gives the rules and cost schedule in words, an individual time- sensitivity trait drawn fromU(0.8,1.2)and rendered as one of three phrases, the agent’s own road and travel time for the last five days, and the broadcast. It asks the agent to consider expected travel time and other drivers’ responses, and to return a route and a one-sentence reason as JSON. A run’s seed controls initial routes, traits, label mapping and baseline random draws. No provider sampling seed was supplied, so repeating a seed reproduces the specified setup rather than the exact model output. Physical roads are mapped to displayed labels by seed parity. Broadcasts are generated from previous counts; F1 reports integer travel times, F2 names the previously less-crowded road, and F3 appends the warning. Numeric variants append F1 to F2 or F3. Agents do not receive others’ choices or explanations and do not communicate directly. If the previous counts tied, F1 showed 60 min for both roads; F2/F3 still named physical road 0 under the fixed tie-break, for humans and agents alike (SI Appendix, section S2). Reason coding uses two overlapping regular-expression categories: references to others switching or crowding, and references to one’s own slow experience. Percentages use all eligible reasons as the denom- inator; the intersection is reported separately (SI Appendix, section S4). The individual GPT analyses use seeds 3–7 of the matched F1 and F3 runs, providing 100 seats per condition. Conditional switching averages individual probabilities for transitions from rounds 6–39 to 7–40, excluding individuals without a qualifying previous outcome. This differs fromM, whose window includes the round-5-to-6 transition. Multinomial-logit learning baseline The comparison agents maintain estimatesv ir of travel time on each roadr, initialized at 60 min. After a choice, the experienced road’s estimate is updated toward its observed cost with learning rate 0.3; under nu- merical reports (F1), both estimates are then updated toward the reported times with rate 0.5. Choices follow P i (r)=exp(− β i v ∗ ir )/ ∑ s exp(− β i v ∗ is ), where β i =0.10u i ,u i ∼U(0.8,1.2), and the bare and tip plus warnings (F2/F3) both subtract 4 min from the indicated road’s effective estimatev ∗ ir . Other estimates are unchanged for choice. The fixed parameters and seeded starting conditions were shared across the core comparisons. This baseline represents adaptive probabilistic choice; it cannot interpret the additional warning sentence (SI Appendix, section S3). Computational experiments •Core (prespecified).Four conditions×ten seeds (3–12), 50 agents, 60 rounds; 118,000 decisions. Two claims were prespecified in an execution-time code freeze: L1, that(F0−F1) LLM −(F0−F1) MNL >0 for bothIandM, comparing the language-model population with a multinomial-logit learning baseline on the same seeds; L2, that F3 raisesIand lowersMrelative to F2. Each claim was tested by exact two-sided sign tests on the ten paired seeds, combined within a claim by the maximumpand Holm-adjusted across the two claims. •Persistence.The ten F3 runs were continued from their round-60 state to round 100 under F3; no run escaped the concentrated state. •Switching.Five seeds each of F3→F1 and F1→F3 with the broadcast changed from round 31, and three seeds in which the warning was given to 38 of 50 agents from round 31; prespecified comparison win- dow rounds 36–50 against rounds 16–30 and against stationary runs. Round-by-round values are in SI Appendix, section S6 and Table S9. •Exposure.Twenty-agent populations with 0, 5, 10, 15 or 20 warned agents (three seeds each) and 50-agent populations with 6–44 warned agents (five seeds each), the remaining agents receiving F1. 14 •Robustness.We varied history length, history of past tips, warning strength, road labels and message wording, and tested Claude Haiku and Gemini. Conditions and seed counts are in SI Appendix, Table S4. Except for a pre-core road-name screening run, variants were specified after the core; this was not a factorial design. •Cross-provider and numeric extension.Planned after the core and run about four weeks later: paired F2/F3 runs for Claude Haiku and Gemini, primary-model runs with appended numbers, and F3 repeats as a drift control (33 runs, 97,350 decisions). No prespecified drift flag was triggered (SI Appendix, Table S3 and Fig. S3). Sampling variation and changes over time cannot be separated; this extension was descriptive. •Reasoning and current-model extension.Planned with earlier results known and run on 23 September 2026: paired F2/F3 runs at seeds 3–7 for the five request settings above (50 runs, 147,500 decisions), referenced to core seeds 3–7. A prospective plan fixed the design and descriptive analysis; no significance criterion was added (SI Appendix, section S9). Human benchmark Both human studies used nonprobability, opt-in samples recruited through Prolific and were conducted on- line in a browser; no population representativeness is claimed. Repeat participation was prohibited; identifier checks confirmed 480 distinct participants across the two analyzed samples. Age and gender summaries are in SI Appendix, section S7. Two hundred forty adults were recruited through the online experiment plat- form Prolific ( https://w.prolific.com; English-fluent, desktop, approval rate≥95%) into twelve 20- person rooms in three blocks, two rooms under F1 and two under F3 per block by a committed within-block randomization, and played 40 rounds with timed choices and a five-round personal history (SI Appendix, sec- tion S7). Timeouts carried the previous road forward; three consecutive timeouts caused dropout with contin- ued carry-forward, and the main summaries retain this handling. Ordinary dropout did not invalidate a room. A registered per-protocol sensitivity excludes rooms with at least two permanent dropouts first flagged by round 30; one F3 room met this criterion and is excluded only from that descriptive sensitivity (SI Appendix, section S7). Participants received a base payment of GBP 4.50 plus a bonus of GBP 2(100−E t [t i (t)])/80 (at most GBP 2), whereE t [t i (t)]is their mean travel time over all 40 rounds, and answered a closed-item questionnaire on the report’s influence, their usual response to it and their understanding of the rules. The Ethics Review Committee of the Research Center for Advanced Science and Technology, The University of Tokyo, approved the study (approval E26ALS0545). All participants gave informed consent. Individual tendencies in Fig. 3D are each participant’s share of tie-free rounds 6–40 on the tip road; split-half reliability correlates the tendency over rounds 6–22 with that over rounds 23–40 across individuals. The descriptive message contrast averages within-block F3–F1 differences over three blocks; exact inference is specified in SI Appendix, section S7. The benchmark preceded the mixed study and supplies only a contextualK=0 reference. Mixed-group experiment Design and analysis.Eight blocks each contained one 20-seat room with 5, 10 and 15 GPT agents under the tip and warning (F3): 24 rooms, 240 humans and 240 agent seats. Participants knew that some players were AI-controlled, but not their number or identities. Agents used the 20-driver core prompt and were not told about human participation. Timing, history, timeout and payment rules matched the human benchmark. The same ethics approval (E26ALS0545) covered this experiment. All participants gave informed consent. The primary statistic was the mean of the eight within-block imbalance slopes,b b =(I b,15 −I b,5 )/2; switching fractionMwas supporting and seat-type decompositions were descriptive. The registered sign test, assump- tions and sensitivities to dependence between recruitment windows are specified in SI Appendix, section S8. Composition was associated with arrival order because larger human quotas were filled first; this limits causal interpretation. Execution and inclusion.Collection on 14–17 September 2026 yielded all 24 valid rooms, each complet- ing 40 rounds. Technical failures excluded two started attempts, whose records were preserved separately; authorized replacement attempts supplied new complete rooms. The 240 included participants produced 15 93 timeouts among 9,360 decisions in rounds 2–40 and six permanent dropouts. When a participant timed out, the system reused their previous road; after three consecutive timeouts, it continued to use that road for the remaining rounds. These system-assigned choices and their travel times were included in the primary analysis. No mixed room had two permanent dropouts by round 30. Outcomes use rounds 6–40, and early and late windows are rounds 6–20 and 26–40. Registration amendments, knowledge of preceding outcomes, recruitment and inclusion records are documented in SI Appendix, section S8 and Figs. S7 and S8. Composition estimates.An optional end-of-task question asked participants to estimate the automated share of their 20-person group. This registered secondary measure was summarized descriptively; missing answers were not imputed and beliefs did not affect inclusion or primary analyses. Response recovery, weighting and missingness are detailed in SI Appendix, section S8. Registration and study periods Human-study plans and amendments were registered on OSF (https://osf.io/whsyu). The all-human benchmark was collected on 1–6 September 2026 and the mixed-group data on 14–17 September 2026. The computational core hypotheses and classifier were fixed in an execution-time code freeze before the August runs; these computational plans were not public preregistrations. Subsequent computational extensions were planned with earlier results known. Some mixed-group completion and replacement attempts were autho- rized after initial outcomes were known; SI Appendix, section S8 provides the full registration and execution history. Individual-response, split-half, reason-coding and additional accounting analyses are exploratory; the registered primary definitions ofIandMwere retained. Statistics Runs and rooms are the units for treatment comparisons. Agent-rounds and individual participants are not treated as independent treatment replicates. Confirmatory analyses comprise the two prespecified computational-core claims and the registered mixed-group composition test, the latter conditional on its stated block-error model and amended collection history. Other comparisons are descriptive. Unless specified otherwise, computational condition summaries are medians over runs, human-benchmark outcome summaries are means over rooms, and mixed-group summaries are means over completed rooms within composition. Cost decompositions are calculated within runs or rooms and then averaged with equal weight. Individual histograms and split-half correlations describe participants or agent seats and are not treatment-effect tests. Independent fair choices give expected imbalance 0.088 and 0.056 and mean travel time 60+40/N= 62.0 and 60.8 min for 20 and 50 seats, respectively. The individual-dispersion reference assumes constant independent choices with a common probability equal to the observed mean tip-following share. These are sampling benchmarks, not fitted learning models or tests of independence (SI Appendix, sections S4 and S5). Use of AI tools OpenAI Codex (GPT-6, September 2026) and Anthropic Claude (September 2026) assisted with language editing and manuscript revision, and with writing, debugging and checking analysis and visualization code. The authors are responsible for the scientific content, interpretation, references and final verification of all AI- assisted material. This assistance is distinct from the language-model agents studied experimentally, whose configurations are described above. Acknowledgments This work was partially supported by JSPS KAKENHI Grant Number JP25H01000. 16 Competing interests The authors declare no competing interest. References [1]Iyad Rahwan, Manuel Cebrian, Nick Obradovich, Josh Bongard, Jean-François Bonnefon, Cynthia Breazeal, Jacob W. Crandall, Nicholas A. Christakis, Iain D. Couzin, Matthew O. Jackson, Nicholas R. Jennings, Ece Kamar, Isabel M. Kloumann, Hugo Larochelle, David Lazer, Richard McElreath, Alan Mislove, David C. 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Verbatim agent prompt and request parameters Every language-model decision was a single stateless call with the following user message (placeholders in braces;LA/LBare the road labels in alphabetical order after a seed-dependent mapping to physical roads): You are commuter #agent_id, one of n_agents drivers who all commute every weekday morning from the same suburb to the same business district. Exactly two roads connect them: Route LA and Route LB. The two roads are physically identical. Travel time on each road depends only on how many of the n_agents commuters pick it that morning: a nearly empty road takes about 20 minutes, a road carrying half of the commuters takes about 60 minutes, and if almost everyone crowds onto one road it takes about 100 minutes. Every driver decides independently before departure and cannot see actual traffic until already committed to a road, so your decision must rely on your own recent experience and on the traffic broadcast, when one is provided. About you: sensitivity_phrase You have been making this commute for a while and you know that all the other drivers face exactly the same choice and similar information every morning. Your own experience over the most recent days (most recent last): history_block Today’s traffic broadcast: broadcast_block It is day day. Decide which route to take this morning. Think about what travel time you expect on each route, including how the other n_others drivers might react to the same information you have. Respond with ONLY a JSON object in exactly this format, with no other text: "route": "LA", "reason": "one short sentence explaining your choice" The "route" value must be "LA" or "LB". The history block lists up to five lines of the form “Dayd: you chose RouteXand it tookmminutes.” (in the report-history sensitivity, each line also states which road that day’s report indicated). The sensitivity phrase is one of three sentences chosen by the agent’s time-sensitivity traits∼U(0.8,1.2): “You are fairly relaxed about your commute time; a slow day annoys you only mildly.” (s<0.93); “You care about your commute time about as much as the average driver.” (0.93≤s≤1.07); “You strongly dislike wasting time in traffic and react quickly when your commute gets slow.” (s>1.07). S1 Request parameters.The client-side request logs for the primary model record: model = gpt-5.4-mini-2026-03-17 temperature = 0.7 max_completion_tokens = 400 reasoning_effort = none response_format = type: json_object These settings are present in all 118,000 core decisions and all 7,800 decisions in the ten 20-seat traces used for individual-level analyses (7,000 decisions fall within rounds 6–40, before excluding tie rounds). They are recorded request settings, not an independent provider-side measurement of their implementation; the primary-model results concern a configuration that did not request extended reasoning. The audit also includes five seed-90 P0 screening runs (14,750 recorded decisions), conducted under an earlier frozen plan before the P1 execution freeze. These runs do not enter the analyses reported here, except the invented-label F3 screen, which Table S4 and Fig. 2G show descriptively; one run (2,950 decisions) requested temperature 0 rather than 0.7. The September cross-provider extension contains 29,500 decisions per provider. All Claude requests recordmodel = claude-haiku-4-5-20251001,temperature = 0.7andmax_tokens = 400; the backend does not send a thinking argument. Gemini logs recordmodel = gemini-3.5-flash, temperature = 0.7,max_output_tokens = 400,response_mime_type = application/jsonand thegoogle-genaiSDK. The backend records successful calls withthinking_budget = 0for every decision. These records establish the submitted configurations and successful requests, not the providers’ internal computation. The complete audit is indata/api_settings_audit.json. Responses were parsed as JSON with a fallback regular expression; first-attempt parse rates were 1.000 in all but three runs (0.998–0.999). Seeds.A run’s seed fixes the round-1 assignment, the traits, the label-to-road mapping and the random draws of the learning baseline. The model requests do not set a provider sampling seed. Paired runs therefore share an initial environment, but not a fixed sequence of model samples. The drift control repeats F3 at seeds 3–5 after four weeks (Table S3 and Fig. S3); differences between these runs combine sampling variability and any change over time. Prespecified drift flags were|∆I|>0.05,|∆M|>0.10 or a change of classified state relative to the original run at the same seed. None was triggered: observed deviations were|∆I| ≤0.012 and|∆M| ≤0.023. S2. Verbatim broadcasts LetXdenote the road that was less crowded in the previous round andYthe other road. F0“(there is no traffic broadcast today)” F1“Yesterday’s travel times: Route A = t A min, Route B = t B min.” (integer minutes, labels in alphabetical order) F2“Traffic advisory: RouteXis currently less crowded.” F3“Traffic advisory: RouteXis currently less crowded. However, many drivers are expected to see this same information and switch to RouteX, so RouteXmay become congested.” F3, stronger wording (F4)“Traffic advisory: RouteXis currently less crowded. IMPORTANT: many drivers are expected to see this exact same advisory and switch to RouteX, so RouteXmay well end up MORE congested than the other route today. Blindly following this advisory could backfire.” F3, paraphrase 1“Traffic advisory: RouteXhas lighter traffic right now. Keep in mind that many other drivers receive this same advisory and may move to RouteX, which could leave it congested.” F3, paraphrase 2“Traffic advisory: at the moment fewer cars are using RouteX. Because a large number of drivers will read this very message and could shift over to RouteX, it may not stay clear.” S2 F1, paraphrases“Yesterday’s measured travel times weret A minutes on Route A andt B minutes on Route B.”; “Traffic report: driving Route A tookt A minutes yesterday; Route B tookt B minutes.” F2 + numbers, F3 + numbersThe F2 or F3 message followed, on a new line, by the F1 sentence. When the previous round’s counts were equal, F1 reported equal times and F2/F3 still named a road, chosen by a fixed tie-break of the environment (physical road 0 in every archived trace with a tie and in all 119 previous-round ties in rounds 2–40 of the 24 included mixed rooms). These rounds are excluded from f t (Text S4). The human studies used the F1 and F3 texts verbatim, with the same label convention; the participant exports retain the advisory text and the indicated road for every round. S3. Multinomial-logit learning baseline The comparator population used in the prespecified L1 contrast consists ofNagents that hold an exponen- tially updated estimate of each road’s travel time (updated from their own experience and, under F1, from the reported times), map the F2/F3 tip to a prespecified utility bonus for the indicated road, and choose by multinomial logit over the two estimates. Its parameters were fixed before the core was run and are archived with the code. Because the baseline has no representation of the warning sentence, its F2 and F3 conditions are identical by construction (Table S1). Under F1 the baseline oscillates (I=0.44,M=0.88); the prespec- ified interaction contrast compares its F0-to-F1 change with that of the language-model population on the same seeds. This comparison characterizes one prespecified adaptive policy. It does not distinguish among possible explanations for the language model’s response to F3, such as interpreting the prediction, trusting the source or avoiding announced crowding, because the comparator does not respond to that sentence. S4. Outcomes, ties, classifier, prespecified tests and reason coding Core and exposure-grid outcomes use rounds 6–60; the matched 20-seat runs and human studies use rounds 6–40. Windows for switched and extended runs are stated with those experiments (Text S6 and the main text).I=E t [|p t − 1 2 |], wherep t is the share of seats on physical Route A.Mis the mean over rounds of the share of seats whose road differs from the previous round; a window beginning at round 6 includes the transition from round 5 to 6. Mean travel time averages the round’s mean realized travel time over every round of the stated window. Tip-following share. f t is the share choosing the road with the smaller count in roundt−1. It is undefined when the previous counts were equal. Throughout this supplement, a “tie round” means a round with equal counts in the preceding round. These rounds are excluded fromE[f t ], Var(f t )and individual tip-following tendencies in every data set, and retained inI,Mand travel time. No such rounds occurred in the F0 and F3 core runs, the 100-round extension, the 20-seat F3 runs or theK=15 mixed rooms. They occurred in 30 of the 103 runs with round-level data (at most six per run; 13 in total under F1 and 25 under F2 in the core), in 1–9 of the 35 analyzed rounds of each all-human room (Table S5), and in 3–11 rounds of theK=5 and K=10 mixed rooms, where near-balanced use makes a 10:10 count common. Classifier and tests.The descriptive frozen classifier requiresI≥0.25,M≤0.20 and at least ten consecu- tive rounds with the same majority road. After checking this criterion, the archived classifier assigns “oscilla- tion” whenI≥0.25,M≥0.60 and the lag-one correlation of the physical route share is at most−0.30; “near equilibrium” whenI≤0.15; and “partial” otherwise. These labels summarize trajectories and are not equilib- rium tests. The two prespecified claims of the core were: L1, the interaction((F0−F1) LLM −(F0−F1) MNL ) is positive for bothIandM; L2,F3−F2 is positive forIand negative forM. Each component was tested by an exact two-sided sign test on the ten paired seeds (all ten differences in the required direction give p=2 −9 =0.00195); a claim’spis the maximum of its two componentpvalues, and the two claims were Holm-adjusted, givingp=0.0039 for both. The claims, the tests, the classifier thresholds, the model and temperature, the analysis window, the prompt and message texts and the confirmatory seeds (3–12) were fixed in a release manifest with SHA-256 hashes of 29 source files, at commit31300fb(20 August 2026, 10:08 JST). The first core request followed at 10:10 JST. Earlier that day, a screening stage of five runs at seed 90 used the preceding code revision with the same thresholds. After it, the written descriptions of L1 and L2 were corrected to match their executable definitions; the executable contrasts, tests and thresholds S3 were unchanged. No confirmatory seed had been run. The registered mixed-group analysis is reported in Text S8; other comparisons are descriptive. Individual tendencies.An individual’s tendency is its share of tie-free rounds 6–40 on the tip road. The number of defined rounds is 26–34 in the all-human rooms, 32–35 in the 20-seat GPT runs and 24–35 in the mixed rooms. The reference SD, √ q ∗ (1−q ∗ )/n ∗ , approximates the dispersion of identical independent choosers at the observed mean tendencyq ∗ , using the mean number of defined roundsn ∗ . It is a sampling benchmark, not a fitted model of individual differences. Split-half correlation is the Pearson correlation, across individuals, between tendencies over rounds 6–22 and 23–40. Values are: humans under F1, SD 0.25 (reference 0.09),r=0.73; humans under F3, SD 0.27 (0.09),r=0.81; GPT under F1, SD 0.09 (0.08),r= 0.14; GPT under F3, SD 0.03 (0.03),r=0.23. In mixed rooms, human-seat SDs are 0.24, 0.16 and 0.13 and correlations are 0.66, 0.58 and 0.75 atK=5, 10 and 15. These pooled descriptive correlations retain shared room histories and are not estimates of context-independent traits. The GPT values come from five 20-seat runs per condition, seeds 3–7 (results/matched/m20_F1,F3_s3..7_trace.jsonl; 100 agents per condition); the all-human values come from twelve benchmark rooms (120 participants per condition). Conditional switching.For Fig. 3E, each seat’s switching frequency is calculated separately after a slow (>60 min) or fast (<60 min) round, using transitions from rounds 6–39 to rounds 7–40. Seats without an eligible transition for a category are omitted from that category’s mean. Reported probabilities are means of these seat-level frequencies, rather than pooled frequencies over all eligible transitions. Travel times of exactly 60 min are excluded from this conditioning. This diagnostic uses one fewer transition than the primary mobility outcomeM, which also includes round 5 to 6. Reason coding.The visible one-sentence reasons from seed 3 of the 50-agent core (rounds 6–60; 2,750 decisions per condition) were lower-cased and classified by fixed regular expressions. Code A requires a reference to other drivers followed by a switching or crowding term; code B requires both a slowness term and a self- or experience-reference. The exact expressions are inbuild/traces.pyin the reproducibility archive (Text S10). Shares for A, B and both were 0.76, 0.25 and 0.21 under F1; 0.61, 0.07 and 0.04 under F2; and 0.89, 0.55 and 0.50 under F3. These coarse codes describe text, not internal reasoning; code B can also capture reported road slowness under F1. S5. The cost identity, tie rounds, the mixing rule and the bounds Identity.WithnofNcommuters on a road, that road takes 20+80n/Nminutes, so the round’s mean travel time is 1 N [ n(20+80n/N)+(N−n)(20+80(N−n)/N) ] =20+80[p 2 +(1−p) 2 ]=60+160(p− 1 2 ) 2 with p=n/N. Because the share on the previously less-crowded road isf t and the share on the other road is 1−f t , (p t − 1 2 ) 2 =(f t − 1 2 ) 2 , and averaging over the rounds on whichf t is defined gives Eq. 1 of the main text with the population variance off t . The relation is an identity of the cost function; it holds for any population and any composition of a room, and it is exact on the tie-free rounds (largest numerical residual 5×10 −14 min). Equation 1 and both axes of Fig. 3A use the same tie-free rounds. The primary travel-time summaries elsewhere use all rounds of the analysis window, so they can differ from the tie-free mean: this difference is zero for runs without ties, below 0.5 min in 95 of the 103 runs, below 1 min in 100, and at most 1.65 min (20-agent F1, seed 5, two tie rounds); it is below 0.33 min in every all-human and mixed room. The lean and volatility components in Fig. 3C are 160(E[f t ]− 1 2 ) 2 and 160Var(f t )over tie-free rounds. Linear mixing.For the exposure grid, the mixing rule takes the pure-population leansE F1 [f t ]=0.695 andE F3 [f t ]=0.038 from the core, formsE mix [f t ]=(1−x)E F1 [f t ]+xE F3 [f t ]for a warned fractionx, and predictsI≈ |E mix [f t ]− 1 2 |andE[T t ]≈60+160(E mix [f t ]− 1 2 ) 2 . This approximation assumes that the pure- population leans persist under mixing and neglects temporal variation. Its minimum lies atx=(E F1 [f t ]− 1 2 )/(E F1 [f t ]−E F3 [f t ])=0.30. Table S7 also gives a homogeneous independent-choice benchmark at the same mean:I ind =E[|X/N− 1 2 |]andT ind =60+160[(E mix [f t ]− 1 2 ) 2 +E mix [f t ](1−E mix [f t ])/N]forX∼ Binomial(N,E mix [f t ]). This benchmark gives every seat the same choice probability; it does not model the distinct policies of the two exposure groups. Both benchmarks approximate the U-shape and the location of the minimum but under-predict imbalance near it (5 of 20 warned: 0.03 and 0.09 against 0.14 observed; 12 of 50: 0.04 and 0.06 against 0.13). These discrepancies identify limits of the constant-mean approximations, without isolating temporal adaptation or dependence as their cause. The same mixing rule with the all-human lean E h [ f t ]= 0 . 51 and the 20-seat agent lean E a [ f t ]= 0 . 03 gives the dashed line of Fig. 5A. The slight turn S4 of the Fig. 5A dashed line nearK=0 follows from the all-human mean being just above one half: adding a small agent fraction first brings the mixture to one half, before it falls below it. The plotted continuous reference includes this exact crossing; it does not imply that fractional agents were tested. Bounds and reconstruction in mixed rooms.If allKagents sit on one road in a round, the count on that road is at leastK, so|p t − 1 2 | ≥max(0,K/20− 1 2 )whatever the humans do; this conditional bound is the solid line of Fig. 5A. The bound given the agents’ actual choices is obtained as follows. Ifa t agents sit on Route A, the humans can move the Route-A count to any integer in[a t ,a t +20−K]. The smallest attainable |p t − 1 2 |is zero if this interval contains 10, and otherwise the distance of its nearer end from 10, divided by 20. Its mean over rounds 6–40 is 0 atK=5 andK=10 and 0.238 atK=15 (room range 0.226–0.246), against observed 0.241–0.293. All agents occupied one road in 63–91% of rounds atK=15 (mean 80%), in 80–97% atK=10, and in every round atK=5. On a defined-tip round, if all agents avoid that road andK≥10, its count is(20−K)f h,t ≤10. Thus |p t − 1 2 |= 1 2 −(20−K)f h,t /20, regardless of correlations among human choices. Averaging over the same defined-tip rounds givesE R [|p t − 1 2 |]= 1 2 −(20−K)E h [f t ]/20. Substituting the observed human tenden- cies gives 0.084 atK=10 and 0.272 atK=15, compared with the all-round observed means 0.085 and 0.259. These comparisons also reflect the agents’ occasional tip choices and, atK=10, the different treat- ment of tie rounds. AtK=5 the tip-road count can exceed 10; an independent-choice calculation with X∼Binomial(15,E h [f t ])gives 0.073, versus 0.069 observed over all rounds. These calculations describe consistency with mean behavior; they do not test independence among humans. Near-threshold runs.Three Claude F3 runs hadI>0.40 butM=0.205–0.209, just above the frozen criterion ofM≤0.20 (Table S3). The continuous outcomes therefore remain the primary description of their concentration. S6. Switch experiments round by round Table S9 lists, for each seed, the mean travel time in round 30 (last round under the original message), the tip-following share, imbalance, switching fraction and mean travel time in round 31 (first round under the new message), the travel time in each of rounds 32–35, and the mean over the prespecified post-intervention window (rounds 36–50). Under F3→F1, the first response was near-unanimous movement onto the road that had been almost empty (f 31 =0.96–1.00,I 31 =0.46–0.50,M 31 =0.86–0.98, 94–100 min). The persistent majority on the original road ended immediately, but congestion remained high in that round. From round 32, oscillations decreased in amplitude; outcomes in rounds 36–50 were similar to stationary F1 runs over the same window (67–74 min versus 70–74 min in the ten core F1 runs;I0.19–0.25 versus a median of 0.23). Under F1→F3, no agent chose the tipped road in round 31 in any seed (f 31 =0, 100 min), and rounds 36–50 averaged 91–96 min. F3→F1 replaces the entire warning-bearing advisory with numerical reports. It therefore estimates the response to that message replacement, rather than to deletion of the warning sentence alone. S7. Human study Participant characteristics and repeat participation.The analyzed samples contained 240 people in each study. Repeat participation was prohibited, including participation in earlier project experiments. Local research-identifier checks confirmed 240 distinct participants within each analyzed sample and no overlap between them (480 distinct people). The table reports optional age-group and gender responses as counts and percentages of all 240 participants per study; “prefer not to say” and missing responses are separate. De- mographic responses did not determine analysis inclusion. Only aggregate demographic counts are released (data/participant_demographics.json). S5 CharacteristicAll-humanMixed n=240n=240 Age, years 18–2457 (23.8%) 44 (18.3%) 25–3490 (37.5%) 99 (41.2%) 35–4456 (23.3%) 51 (21.2%) 45–5417 (7.1%) 28 (11.7%) 55+13 (5.4%)13 (5.4%) Prefer not to say0 (0.0%)2 (0.8%) Missing7 (2.9%)3 (1.2%) Gender Female84 (35.0%) 111 (46.2%) Male147 (61.3%) 122 (50.8%) Other0 (0.0%)1 (0.4%) Prefer not to say0 (0.0%)1 (0.4%) Missing9 (3.8%)5 (2.1%) Recruitment and procedure.Participants were recruited through Prolific (fluent in English, desktop de- vice, approval rate at least 95%) into 20-person rooms with an anonymous 180-s preparation lobby and a 30-s presence confirmation. Each choice round consisted of a 9-s advisory period, a decision window of 15 s in rounds 2–5 or 10 s in rounds 6–40, and 3 s of own-outcome feedback. The five most recent own outcomes remained visible. The twelve analyzed rooms formed three blocks of four, with two F1 and two F3 rooms per block assigned by a committed within-block randomization. Each room began with 20 participants and contributed 800 seat-round records, including any carried choices. Participant flow and registration.The plan was registered on 25 August 2026. A timing pilot (40 participants) and technical retests (four participants) were excluded. The twelve benchmark rooms completed on 1–6 September after retries of rooms that had not formed; no room was stopped or replaced after round 1 began, and every formed room is analyzed. Amendments revised lobby, compensation and recruitment procedures. The amendment authorizing the last B1 retry disclosed that the three completed B1 rooms had already been inspected descriptively. Detailed amendments are in the OSF registration ( https://osf.io/ whsyu; currently under embargo and not publicly accessible). Waiting compensation was settled for all unseated entrants, although the operational records do not support a complete count of entrants to failed attempts. Timeouts and payment.A timeout carried the previous road forward. After three consecutive timeouts, the last road was carried forward permanently, including after reconnection. The primary room outcomes retain these choices. The registered no-timeout sensitivity excludes records flagged as a current timeout or dropout carry-forward, recalculates road shares over the remaining seats, and averages over the same outcome window. It givesI=0.089 andM=0.474 for F3 rooms, versus 0.085 and 0.456 in the primary summaries. Pay consisted of a base payment of GBP 4.50 plus a bonus of GBP 2(100−E t [t i (t)])/80, bounded between GBP 0 and GBP 2, using the participant’s mean travel time over all 40 rounds. The registered per-protocol sensitivity excludes rooms with at least two permanent dropouts first flagged by round 30; ordinary dropout never invalidates the primary record. One F3 room met this criterion. Retaining five F3 rooms gives meanI=0.0843,M=0.4569 and travel time 61.75 min; all six F1 rooms are retained (I=0.0910,M=0.4298, 61.98 min). As registered, these are descriptive retained-room summaries, without a replacement significance test. The included all-human rooms were collected on 1–6 September 2026. Questionnaire and consent.The final questionnaire asked how much the traffic report influenced choices (not at all / slightly / moderately / strongly), how the participant usually responded (took the faster-looking road / took the other road expecting others to switch / ignored it / it depended), and whether the rules were understood (yes / mostly / no). Counts are in Table S6. Informed consent was obtained under approval E26ALS0545 from the Ethics Review Committee of the Research Center for Advanced Science and Technol- ogy, The University of Tokyo. The benchmark disclosure stated that all players were human; the mixed-study disclosure stated that the group included AI-controlled players (Text S8). Descriptive message contrast.The registered estimator averages the difference between the two F3 and two F1 rooms within each block, then averages these differences over the three blocks. Two-sided S6 randomization tests enumerate all ( 4 2 ) 3 =216 within-block assignments; intervals invert a constant-shift test. The estimates areI,−0.005 (p=0.28; 95% interval−0.016 to+0.006; block differences+0.001,−0.014, −0.004), andM,+0.026 (p=0.31;−0.037 to+0.088). Applying the same descriptive procedure to mean travel time gives−0.22 min (p=0.10;−0.53 to+0.07). These comparisons carry no confirmatory alpha; non-rejection does not establish equivalence. Instructions and questionnaire.Participants were told that their bonus depended only on their own mean game travel time, not their rank or the group’s performance, and could not reduce the base payment. They acknowledged the congestion rule, bonus rule and timeout handling before entering the waiting room. Unlike the agent prompt, the human instructions did not ask them to anticipate others’ reactions. Verbatim instructions and questionnaire items are retained inmanuscript/paper/tex/human_instructions.tex andmanuscript/paper/tex/questionnaire.texin the reproducibility archive (Text S10). S8. Mixed-group experiment: completion, analysis and limitations Design and completion in brief.Before mixed-group recruitment began, the 14 September 2026 OSF amend- ment fixed a target of 24 rooms: eight comparison blocks, each containing three 20-seat rooms, one with 5, one with 10 and one with 15 AI agents (K=5,10,15), all under F3. A block is therefore a set of three rooms to be compared, not necessarily a single collection time. This amendment replaced an earlier design that also included new all-human rooms; the existing all-human F3 data remained a contextual reference only. The definitions ofIandMand the primary analysis were fixed before mixed-group data collection. The initial collection on 14–15 September completed 18 of the 24 planned rooms. TwoK=5 rooms stopped because of a server fault, and fourK=15 rooms never started because they did not form. Following repair, amendments made before the respective further attempts authorized replacing the two technically invalid rooms and completing the four unstarted rooms. Four rooms were completed on 16 September and the last two on 17 September. These attempts filled the original 24-room design; they did not add new analysis blocks or enlarge the target sample. All 18 initially completed rooms were retained. The two aborted attempts were excluded on technical grounds, kept separately and never pooled with replacement data; their 30 participants received the specified partial payment. The final sample comprised 24 complete 40-round rooms, 240 distinct humans and 240 agent seats. The authors had seen earlier behavioral results when making the completion amendments. These amend- ments were prospective for the remaining attempts, not for the full dataset, and explicitly permitted excep- tions to the original prohibition on repeating a started room. The primary outcome definitions and eight-block analysis remained unchanged. The detailed protocol revisions and amendment history are provided in the OSF registration ( https://osf.io/whsyu; currently under embargo and not publicly accessible); room- level completion records are summarized in Figs. S7 and S8 and Table S8. Procedures and allocation.Each agent seat received a separate call togpt-5.4-mini-2026-03-17 using the 20-driver prompt (Text S1), without being told that other seats could be human. Humans were told that AI players were present, but not their number or identities. Prior project participants were excluded, and all seats received the same broadcast. Rooms requiring more humans were filled first (K=5, then 10, then 15); membership within a selected room followed a committed randomized ordering of ready partici- pants. A prespecified rule allowed feasible remaining rooms to start when recruitment was insufficient. Thus composition could be associated with arrival time and waiting, limiting causal interpretation. A valid room completed all 40 rounds. The analysis retained roads automatically carried forward after timeouts or dropout; technical failures were handled separately as described above. What the primary analysis compares.In each blockb, we compare imbalance in the room with 15 agents, I b,15 , with that in the room with 5 agents,I b,5 . The registered block slope is b b =(I b,15 −I b,5 )/2. The difference spans ten additional agents, so dividing by two expresses it per five additional agents. For example, an imbalance difference of 0.20 gives a slope of 0.10. The primary statistic is the mean of these eight slopes. TheK=10 rooms remain part of the required design and show the intermediate response; they do not enter this endpoint contrast, which is the equally spaced three-level linear contrast. It does not assume S7 that the observed intermediate response lies on a straight line. How the test works.The one-sided test asks whether the average slope is positive. Keeping the eight observed magnitudes fixed, it reverses or retains each slope’s sign and calculates the mean for all 2 8 =256 combinations. Thepvalue is the proportion of these means at least as large as the observed mean, including the observed combination. The registered criterion is a positive mean andp<0.05. This calculation assumes b b =β+ε b : block errors are independent and symmetric about zero, although their variances may differ. Under the null of no common positive shift (β=0), each sign is equally plausible. These are assumptions about variation across blocks; the recruitment procedure does not guarantee them. The test is therefore model-based, rather than an exact test justified solely by randomized composition assignment. SwitchingM is a supporting outcome tested for a decrease; it cannot substitute for failure of the primaryIcriterion. Why shared recruitment times matter.An analysis block and a recruitment window are different units. A window is a period in which participants were recruited together. Originally, each of four windows was scheduled to supply two blocks: (1,2), (3,4), (5,6) and (7,8). Blocks recruited together may share participant- pool or timing effects, so their errors need not be independent. The registered sensitivity analysis treats each pair as one unit when reversing signs: both slopes in a pair reverse together, while each of the eight blocks still receives weight 1/8. There are then only 2 4 =16 combinations. Even the most extreme result has p=1/16=0.0625, so this check cannot reach the 5% threshold. The later recruitment periods were needed to finish the missing rooms described above, not to create additional comparison blocks. On 16 September, one period supplied pending rooms from blocks 4, 5 and 7, and another from blocks 6 and 8. These periods linked blocks that belonged to different original pairs. An additional sensitivity analysis therefore grouped together all blocks connected by any shared recruitment period, directly or through another block. This gives two groups,1,2and3,4,5,6,7,8. Completion of blocks 4 and 5 on 17 September does not change those groups. Reversing all slopes within each group together leaves only 2 2 =4 combinations and a minimump=0.25, again with equal weight per original block. Both grouped checks allow dependence within a group but assume independent groups and joint sign symmetry of each group’s errors: reversing the entire error vector must be equally plausible under the null. They do not remove every possible dependence. In particular, the four initial windows occurred during one evening and the following midnight period, so effects shared across the whole night could link even different groups. These checks show how inference changes when fewer independent units are assumed; neither establishes independence nor supplies an alternative criterion for confirmatory success. Results and their interpretation.All eight imbalance slopes were positive, averaging+0.0951 per five additional agents. Only one of the 256 sign combinations gave a mean at least this large, sop=1/256= 0.00390625. The primary criterion is met under the independent-block model. All eight supporting switching slopes were negative (mean−0.2080,p=1/256). When signs were reversed by the four original recruitment pairs, both outcomes gavep=0.0625; grouping all shared-window connections gavep=0.25. The direction of the observed differences does not change, but neither dependence check provides significance at 5%. The evidence from the primary test therefore remains conditional on its independence assumption. Omitting any one block retained both directions (Table S10). The registered interval reports the smallest and largest block slopes: [ 0 . 0764 , 0 . 1150 ] for I and [−0.2300,−0.1886]forM. To interpret this range as an uncertainty interval for a shared central effect, the eight slopes must be independent and have a common median. Under those assumptions, the chance that all eight fall on the same side of that center is at most 2/256; the range therefore covers it with probability at least 254/256 (99.2%). This is not an interval for an arbitrary average of different block effects, nor is it valid without the stated assumptions. Registered secondary analyses include leave-one-block-out mean slopes, adjacent contrasts and curvature, the no-timeout sensitivity, and the perceived share of automated players compared with K/20. The recovered composition estimates are reported below. Carryover and waiting-time sen- sitivities subtract hypothetical nuisance effects from room outcomes:Y adj =Y− γcarryover for γ∈ 0,±0.005,±0.01,±0.02,±0.05and, separately,Y adj =Y−γwait forγ∈ −0.005,0,0.005 per minute. Here carryover refers to a later operational attempt relative to the original recorded attempt, including earlier zero-room attempts, and wait is the room’s mean human entry-to-start time. These grids describe sensitivity to hypothetical bias, not identified causal adjustments. Historical pure-human F3 rooms S8 remain outside the mixed-study blocks. Perceived automated share: recovery and missingness.An optional final question asked what percent- age of the 20 players, including the respondent, were automated. The 0–100% slider recorded no answer unless moved. The original exports omitted this field; on 22 September, a read-only, hash-checked recovery retrieved the original submitted responses without changing sample membership. These were original end- of-task answers, not a follow-up survey. Of 240 humans, 238 submitted the questionnaire and 230 supplied an estimate: 116/120, 76/80 and 38/40 atK=5, 10 and 15. Unsubmitted drafts were excluded and missing answers were not imputed. Recovery provenance is retained indata/h2_belief_summary.json. Descriptive belief results.With equal weight per respondent, estimated AI percentages averaged 63.1%, 67.2% and 68.9% for actual percentages of 25%, 50% and 75% (fig. S9; table S12). Medians were 65%, 70% and 70%; mean signed errors were+38.1,+17.2 and−6.1 percentage points, and mean absolute errors were 39.4, 20.4 and 17.1 points. Equal weighting of the eight room means per composition gave 63.2%, 67.4% and 69.3%, respectively. As an additional descriptive missing-data bound, assigning every missing estimate any value from 0 to 100% places the all-participant means in 61.0–64.3%, 63.9–68.9% and 65.5– 70.5%; these bounds are not imputations or confidence intervals. Reported composition therefore varied much less than actual composition, and overestimation was largest atK=5. This single post-task item does not measure beliefs during play or understanding of the agents’ route-choice policy. No belief-based hypothesis test, primary covariate adjustment or exclusion was performed. Checks across collection dates.Four of the eight primaryK=15 versusK=5 comparisons combine rooms collected on different nights. LaterK=15 rooms hadI=0.241–0.260, overlapping the initial range of 0.244–0.293; the two replacementK=5 rooms hadI=0.061 and 0.069, within the initial range of 0.059– 0.091. EveryK=15 room had higher imbalance than everyK≤10 room. These descriptive checks show consistent ordering, but do not remove confounding by collection date or participant recruitment. Room-by- room dates and outcomes are in Fig. S7 and Table S8. Exploratory uncertainty check (not registered).An equally weighted room-level OLS model with com- position as a factor gave the contrast(I 15 −I 5 )/2=0.0951. Its conventional standard error was 0.0036; CR1 cluster-robust standard errors were 0.0045 for seven recruitment windows and 0.0042 for eight original blocks. With so few clusters, these standard errors are descriptive checks of uncertainty under alternative co- variance assumptions, not evidence that errors are independent or a separate confirmatory test. The registered block-level analysis remains primary. Additional registered summaries.The 24 completed rooms are reported in Fig. 5, Fig. S7 and Table S8. Their blockwiseK=15 minusK=5 differences inIare +0.199, +0.230, +0.153, +0.190, +0.171, +0.180, +0.200 and +0.199. Mean adjacent contrasts (K=10 minus 5, thenK=15 minus 10) are+0.0161 and +0.1741 forIand−0.2991 and−0.1170 forM; mean curvature is+0.1580 and+0.1821, respectively. Table S11 reports every registered carryover and waiting-time sensitivity. Carryover is derived from the earliest actually opened window for the original block, including zero-room predecessor records for blocks 1, 2 and 5; a scheduled but unopened window is not an attempt. Across the specified grids the meanI slope remains positive and the meanMslope negative. The no-timeout sensitivity, excluding both current timeouts and dropout carry-forward as defined in Text S7, gives I = 0 . 071, 0.086 and 0.266 at K = 5, 10 and 15, versus 0.069, 0.085 and 0.259 with carry-forward retained. CorrespondingMmeans are 0.480, 0.173 and 0.056, versus 0.471, 0.172 and 0.055. For this mobility sensitivity, each retained current choice is compared with that seat’s saved previous-round choice; a missing current choice is removed rather than bridging across rounds. No-timeout slopes are+0.0974 forIand−0.2121 forM. No included mixed room had two permanent dropouts first flagged by round 30, an additional descriptive attrition check. The current H2 registration does not specify that threshold as a room-exclusion rule. The audit of dropout thresholds, benchmark collection dates and available questionnaire fields is recorded indata/revision_metadata_ audit.json. S9. Reasoning and current-model extension Design.This extension was planned after the preceding results were known and was run on 23 September 2026. Five request settings were each tested under F2 and F3 at seeds 3–7, with 50 agents and 60 rounds (50 runs, 147,500 decisions). Three settings usedgpt-5.4-mini-2026-03-17: no reasoning at temperature S9 1.0, low reasoning effort and medium reasoning effort. Two usedgpt-6-luna: no reasoning and medium reasoning effort, the model’s default. The core runs at the same seeds (no reasoning, temperature 0.7) are the reference. Prompts, broadcasts, histories, first-round draws, the parser and the recovery journal came from the unmodified core code. Before the first call of the formal extension, a plan file fixed the design, the SHA-256 hashes of the code files and a descriptive analysis plan. A five-round F3 pilot at low and medium effort, run earlier that day to estimate token use, was not included in the analyses; its records are archived inresults/reasoning_pilot_20260923/. The plan named two summaries: the number of frozen F3 runs in each setting, shown with the core, and the paired F3−F2 differences inI,Mand travel time, with sign counts. No significance criterion was added. The plan named medians; we report means, as for the cross-provider extension. The largest difference between the two summaries is 0.005 forI, 0.009 forMand 0.7 min for travel time. Ties occurred only under the tip, in 18 of the 50 runs (at most five per run). All-round and tie-free mean travel times differed by less than 0.5 min in 49 runs and by at most 0.78 min. Requests.Reasoning-enabled requests and all Luna requests omitted temperature; their internal sampling temperature is unknown. Completion caps were 400 tokens without reasoning and 4,000 with reasoning, in- cluding reasoning tokens. All requests specified JSON-object output. Logs retain the accepted parameters and returned model names; Luna is an undated alias, so future calls may use a different version. Each run saved 2,950 distinct decisions. Fourteen decisions were retried after token-cap failures; connection/server errors were also retried. Full attempt logs and token counts are archived with the experiment (see Repro- ducibility below). The provider reports reasoning-token counts but not reasoning content. Temperature.Without reasoning at temperature 1.0,gpt-5.4-minireproduced the core pattern at every seed. All five F3 runs were frozen. All five F2 runs were near equilibrium (table S14). Relative to the core runs of the same seeds, F3 travel time was 2.5 min lower (1.6 to 3.3 min) and switching 0.036 higher. These differences combine the change of temperature with any change since the core runs. Reasoning.With low or medium reasoning, no run met the frozen criterion. The warning still raised imbalance, lowered switching, loweredE[f t ]and raised travel time in each of the ten pairs (table S13). Compared with the no-reasoning runs at temperature 1.0, reasoning mainly weakened avoidance under the warning.E[f t ]was 0.26 at low and 0.21 at medium effort, against 0.06. Switching was 0.42 and 0.36, against 0.12. Travel time was 70.0 and 74.7 min, against 91.5 min. Under the bare tip, low reasoning increased tip-following (E[f t ]=0.69, against 0.50). Road shares then alternated from round to round (fig. S10E). Avoidance under the warning was shared across agents but was not complete for any of them. At low and medium effort, all 250 agents per setting took the tip road in more than 10% of their defined rounds 6–60. The spread of individual tip-following shares (SD 0.044 and 0.041) was smaller than expected for independent choosers with the same mean (0.059 and 0.054). In the frozen runs at temperature 1.0, 219 of 250 agents took the tip road in at most 10% of these rounds. GPT-6 Luna.Without reasoning, Luna moved toward avoidance under the warning in all five pairs but did not freeze (E[f t ], 0.35 to 0.19; travel time, 67.8 to 78.4 min). At its default medium reasoning, Luna froze in all ten runs, under the tip as well as under the warning (E[f t ]=0.05 and 0.07; travel time 92.4 and 89.5 min). In this setting the warning made no consistent difference: it raisedE[f t ]in three pairs, left it unchanged in one and lowered travel time in three. Cost components and stated reasons.The warning raised the average-response cost and lowered the temporal-variation cost in all five pairs at temperature 1.0, at low effort and at medium effort, and in four Luna pairs without reasoning (table S13). With the codes of text S4 applied to seed 3 (rounds 6–60), reasons under the warning referred to other drivers switching or crowding in 86% of decisions at temperature 1.0, 85% at low effort, 81% at medium effort, 80% for Luna without reasoning and 83% for Luna with reasoning, against 89% in the core. Under the bare tip, the share was 88% for Luna with reasoning, against 61% in the core. These codes describe the visible one-sentence reasons only. Reproducibility.tools/reasoning_replication_20260923.pyran the experiment. Its plan file and client-side attempt log are stored with the runs inresults/reasoning_replication_20260923/. build/reasoning_extension.pyrecomputes every number above and tables S13 and S14. It first checks that it reproduces the core run table and the core reason codes.build/figS10.pydraws fig. S10. S10 S10. Reproducibility of the analysis A private Zenodo draft (reserved DOI:10.5281/zenodo.22915642; not publicly accessible) provides reproducibility_review.zipand six computational archives,computational_records_part01_ of06.zipthroughcomputational_records_part06_of06.zip. Extract all seven into the same direc- tory. The privateGitHub repositorycontains the reproducibility materials; The complete Zenodo package is designed to support reproduction without GitHub access. These materials are not publicly accessible at the time of this preprint. Public release is planned upon journal publication; the reserved Zenodo DOI is not yet an active public record. Reuse licenses will be specified at release. Repository-rootREPRODUCING.mdgives dependencies and commands;MANIFEST.jsonand COMPUTATIONAL_MANIFEST.jsongive file hashes. De-identified round observations and closed- choice responses are inhuman_data/; scripts and summary data are inmanuscript/paper/build/ andmanuscript/paper/data/. Age and gender are released only as aggregate counts. These materials support figure reconstruction and numerical checks without restricted participant exports or new provider requests. Original-source reconstruction requires restricted exports; pinned source hashes and inclusion rules are recorded inmanuscript/paper/data/h2_source_manifest.json. Registered analysis specifications and an amendment summary are inregistration/; the full amendment history is in OSF (Text S8). Repository-rootVALIDATION.jsonrecords the verification of room-level outcomes, automation esti- mates, manuscript quantities, figures and reasoning-extension analyses. Paths elsewhere in the supporting text are relative tomanuscript/paper/unless stated otherwise. S11 Supplementary Figures Figure S1. Physical route shares in the prespecified core.Share of agents on physical Route A in every run of the four conditions (50 GPT agents, 60 rounds, ten seeds per condition). Road labels are mapped to physical roads with a seed-dependent parity; frozen runs can therefore concentrate on either physical road. S12 Figure S2. Cross-provider extension, all paired seeds.Route imbalance, switching fraction, mean travel time and tip-following share for Claude Haiku 4.5 (AtoD) and Gemini 3.5 Flash (EtoH) under the tip and under the tip plus warning, five matched seeds each. Figure S3. Drift control and numeric variants by seed.September 2026 runs compared with August 2026 runs at the same environment seed. Filled points repeat the original F3 condition; hollow points add numerical reports to F3 or F2. The diagonal marks equal outcomes, not identical model samples. S13 Figure S4. Exposure grid and exposure switch.(A,B) Every run of the 20- and 50-agent exposure grids. (C) Physical Route A share in the three runs in which the warning was given to 38 of 50 agents from round 31. Figure S5. Human rooms, physical route share.Share of participants on physical Route A in the six F1 rooms (A) and six F3 rooms (B). Figure S6. Individual tip-following tendencies of 50-agent GPT populations.Each agent’s share of tie-free rounds 6–60 on the road that was less crowded in the previous round, seed 3, under numbers (A), the tip (B) and the tip plus warning (C). S14 Figure S7. Every room of the mixed-group experiment.Columns,K=5, 10 and 15 GPT agents among 20 seats; first eight rows, blocks 1–8; bottom row, the two original aborted attempts. R and F mark valid rooms from the 16 and 17 September collections, respectively. Purple, share of human seats on the tip road; orange, share of GPT seats on it (both undefined on tie rounds); dashed, room imbalance scaled to 1 when everyone is on one road. Titles giveIand the mean travel time of the room and of each seat type over rounds 6–40. Aborted rooms show the rounds played before the server fault. S15 Figure S8. Design and execution record of the completed mixed-group experiment.Initial and current status of the 24 registered slots by block andK, showing the six valid additional rooms and retaining the two original aborted attempts in the history. All eight blocks are complete (Text S8). Figure S9. Post-task estimates of the automated share.Actual percentages of 25%, 50% and 75% correspond toK=5, 10 and 15 of 20 players. (A) Formal numeric answers from 116, 76 and 38 respondents; horizontal bars are respondent means. (B) The eight room means per composition; bars weight rooms equally. Horizontal jitter only separates points. Dashed lines show accurate estimates. Ten missing answers, including one unsubmitted numeric draft, are omitted; all 240 humans remain in the behavioral analysis. These are descriptive distributions, not independent treatment replicates. S16 Figure S10. Reasoning and current-model extension.(AtoD) Route imbalance, switching fraction, mean travel time and tip-following share (rounds 6–60) for each request setting under the tip (F2, green) and under the tip plus warning (F3, orange), five paired seeds each. Gray lines join runs of the same seed; black bars mark means over the five runs. The first setting is the core reference at the same seeds. The numbers at the top of A count runs meeting the frozen criterion, whose thresholds forIandMare dashed (it also requires ten consecutive rounds with the same majority road). Reasoning-enabled requests omitted temperature; its internal numerical value was not independently verified. (E) Share of agents on physical Route A in every run. S17 Supplementary Tables Table S1. Condition-level summary.Medians across runs or rooms of run-mean outcomes;E[f t ]is the mean across runs of the run-mean tip-following share (tie-free rounds); “frozen” counts runs meeting the descriptive classifier. MNL, multinomial-logit learning baseline. Population and messageNrounds runsIMtime (min)E[f t ]frozen GPT, F0 no report50 6–60 10 0.476 0.99396.4 0.96 0/10 GPT, F1 numbers50 6–60 10 0.230 0.66071.5 0.69 0/10 GPT, F2 tip50 6–60 10 0.134 0.71364.4 0.54 0/10 GPT, F3 tip + warning50 6–60 10 0.465 0.07094.8 0.04 10/10 MNL baseline, F050 6–60 10 0.063 0.49661.0– 0/10 MNL baseline, F150 6–60 10 0.437 0.88190.7– 0/10 MNL baseline, F250 6–60 10 0.130 0.51063.5– 0/10 MNL baseline, F350 6–60 10 0.130 0.51063.5– 0/10 GPT, F2 tip + numbers50 6–605 0.329 0.76179.6 0.82 0/5 GPT, F3 tip + warning + numbers50 6–605 0.396 0.20485.9 0.13 2/5 GPT, F3 repeat (Sep 2026)50 6–603 0.460 0.07994.1 0.04 3/3 Claude Haiku 4.5, F2 tip50 6–605 0.360 0.38285.7 0.26 0/5 Claude Haiku 4.5, F3 tip + warning 50 6–605 0.403 0.20988.8 0.13 0/5 Gemini 3.5 Flash, F2 tip50 6–605 0.303 0.39278.6 0.24 0/5 Gemini 3.5 Flash, F3 tip + warning 50 6–605 0.373 0.24383.3 0.13 0/5 GPT, F1 numbers (20 seats)20 6–40 10 0.242 0.70272.7 0.72 0/10 GPT, F3 tip + warning (20 seats)20 6–40 10 0.469 0.06295.4 0.03 10/10 Humans, F1 numbers (20 seats)20 6–406 0.091 0.43961.9 0.51 0/6 Humans, F3 tip + warning (20 seats) 20 6–406 0.084 0.44261.7 0.51 0/6 S18 Table S2. All 40 runs of the prespecified core(50 GPT agents, rounds 6–60; the core was fixed by an execution-time code freeze, not publicly registered). Condition seedIMtime (min)E[f t ]Var(f t ) F03 0.478 0.99396.7 0.978 0.0006 F04 0.433 0.99390.1 0.933 0.0005 F05 0.474 0.99596.0 0.974 0.0003 F06 0.492 0.99598.9 0.992 0.0009 F07 0.487 0.99098.3 0.987 0.0018 F08 0.414 0.98887.6 0.914 0.0011 F09 0.418 0.99687.9 0.918 0.0001 F010 0.441 0.98791.3 0.941 0.0014 F011 0.489 0.99398.4 0.989 0.0009 F012 0.479 0.99996.8 0.979 0.0000 F13 0.228 0.64071.8 0.690 0.0324 F14 0.224 0.66770.5 0.687 0.0301 F15 0.235 0.65571.6 0.678 0.0385 F16 0.239 0.66071.4 0.685 0.0369 F17 0.232 0.67871.1 0.708 0.0262 F18 0.218 0.66870.4 0.685 0.0307 F19 0.243 0.67173.3 0.733 0.0280 F110 0.218 0.64471.0 0.689 0.0333 F111 0.229 0.65272.1 0.703 0.0299 F112 0.238 0.65971.9 0.692 0.0376 F23 0.151 0.71865.3 0.575 0.0273 F24 0.133 0.70964.0 0.488 0.0242 F25 0.161 0.71766.6 0.597 0.0333 F26 0.152 0.72165.6 0.570 0.0296 F27 0.136 0.70764.7 0.546 0.0281 F28 0.120 0.68763.5 0.504 0.0215 F29 0.129 0.72564.0 0.477 0.0251 F210 0.147 0.70265.3 0.543 0.0286 F211 0.121 0.69463.5 0.513 0.0212 F212 0.133 0.71864.1 0.540 0.0243 F33 0.463 0.07594.3 0.037 0.0007 F34 0.449 0.10192.5 0.051 0.0012 F35 0.472 0.05695.7 0.028 0.0005 F36 0.449 0.10192.5 0.051 0.0018 F37 0.468 0.06395.2 0.032 0.0007 F38 0.468 0.06495.2 0.032 0.0010 F39 0.453 0.09293.3 0.047 0.0025 F310 0.468 0.06595.2 0.032 0.0011 F311 0.461 0.07694.3 0.039 0.0012 F312 0.468 0.06395.2 0.032 0.0008 S19 Table S3. All 33 runs of the September 2026 extension(50 agents, rounds 6–60). State labels follow the descriptive classifier in Text S4; “partial” denotes runs meeting none of its frozen, oscillation or near-equilibrium criteria. ModelmessageseedIMtime (min)E[f t ]state GPT-5.4-minitip + numbers3 0.334 0.75680.1 0.833 oscillation GPT-5.4-minitip + numbers4 0.287 0.71576.1 0.787 oscillation GPT-5.4-minitip + numbers5 0.337 0.76180.7 0.837 oscillation GPT-5.4-minitip + numbers6 0.329 0.77679.5 0.829 oscillation GPT-5.4-minitip + numbers7 0.329 0.76879.6 0.829 oscillation GPT-5.4-minitip + warning + numbers3 0.402 0.19286.4 0.098 frozen GPT-5.4-minitip + warning + numbers4 0.336 0.31080.1 0.165 partial GPT-5.4-minitip + warning + numbers5 0.404 0.18987.1 0.096 frozen GPT-5.4-minitip + warning + numbers6 0.319 0.33277.1 0.181 partial GPT-5.4-minitip + warning + numbers7 0.396 0.20485.9 0.104 partial GPT-5.4-minitip + warning (repeat)3 0.464 0.07294.7 0.036 frozen GPT-5.4-minitip + warning (repeat)4 0.448 0.10492.4 0.052 frozen GPT-5.4-minitip + warning (repeat)5 0.460 0.07994.1 0.040 frozen Claude Haiku 4.5 tip3 0.349 0.35885.7 0.207 partial Claude Haiku 4.5 tip4 0.401 0.38289.6 0.310 partial Claude Haiku 4.5 tip5 0.360 0.35186.5 0.209 partial Claude Haiku 4.5 tip6 0.369 0.40685.3 0.292 partial Claude Haiku 4.5 tip7 0.289 0.44579.6 0.280 partial Claude Haiku 4.5 tip + warning3 0.403 0.20988.8 0.112 partial Claude Haiku 4.5 tip + warning4 0.372 0.26784.9 0.133 partial Claude Haiku 4.5 tip + warning5 0.412 0.20890.1 0.124 partial Claude Haiku 4.5 tip + warning6 0.357 0.29484.5 0.144 partial Claude Haiku 4.5 tip + warning7 0.428 0.20591.9 0.133 partial Gemini 3.5 Flash tip3 0.307 0.37879.0 0.218 partial Gemini 3.5 Flash tip4 0.252 0.49274.1 0.289 partial Gemini 3.5 Flash tip5 0.292 0.39277.4 0.221 partial Gemini 3.5 Flash tip6 0.303 0.39978.6 0.234 partial Gemini 3.5 Flash tip7 0.326 0.37181.2 0.231 partial Gemini 3.5 Flash tip + warning3 0.373 0.24283.3 0.127 partial Gemini 3.5 Flash tip + warning4 0.370 0.24383.3 0.130 partial Gemini 3.5 Flash tip + warning5 0.373 0.24882.9 0.127 partial Gemini 3.5 Flash tip + warning6 0.366 0.27583.6 0.148 partial Gemini 3.5 Flash tip + warning7 0.373 0.24183.3 0.127 partial S20 Table S4. Robustness variants(means across runs; from the audited run table). Rounds 6–40 for 20-seat variants and 6–60 otherwise. VariantrunsIMtime (min) frozen F3, full history (rounds 6–40, 20 seats)5 0.479 0.04196.9 5/5 F1 + past tips in history (20 seats)5 0.247 0.49272.6 0/5 F3 + past tips in history (20 seats)5 0.497 0.00599.6 5/5 F3, stronger wording (50 agents)3 0.499 0.00399.8 3/3 Claude/Gemini, F0 (2 seeds each)4 0.249 0.83573.3 0/4 Claude/Gemini, F1 (2 seeds each)4 0.361 0.91783.1 0/4 Claude/Gemini, F3 (2 seeds each)4 0.381 0.25585.7 0/4 GPT, F1 paraphrase 13 0.275 0.73474.7 0/3 GPT, F3 paraphrase 13 0.468 0.06495.3 3/3 GPT, F1 paraphrase 23 0.331 0.81379.1 0/3 GPT, F3 paraphrase 23 0.373 0.25183.0 0/3 F3, invented road names Maple/Cedar (screen, seed 90)1 0.443 0.11191.6 1/1 S21 Table S5. Human rooms(rounds 6–40; carried choices retained).E[f t ]is the share of participants on the tip road over tie-free rounds; the last column counts the analyzed rounds in which the previous counts were equal. Roomblock messageIMtime (min)E[f t ]tie rounds pure-b1-21F1 0.089 0.42061.9 0.5274 pure-b1-31F1 0.086 0.46761.8 0.5085 pure-b2-roomwise-r1-2 2F1 0.094 0.35362.3 0.5054 pure-b2-roomwise-r1-4 2F1 0.093 0.45762.0 0.5035 pure-b3-roomwise-r3-1 3F1 0.081 0.39761.7 0.5244 pure-b3-roomwise-r3-2 3F1 0.103 0.48462.2 0.5172 pure-b1-11F3 0.084 0.49761.7 0.5096 pure-b1-r4x-11F3 0.091 0.45161.8 0.5121 pure-b2-roomwise-r1-1 2F3 0.083 0.41461.6 0.4855 pure-b2-roomwise-r1-3 2F3 0.077 0.42961.6 0.4937 pure-b3-roomwise-r3-3 3F3 0.081 0.43361.8 0.5159 pure-b3-roomwise-r3-4 3F3 0.096 0.51162.1 0.5274 Table S6. Questionnaire counts.All-human rooms: 120 participants per message (two F1 and one F3 participant did not answer). Mixed rooms: the 240 participants of the 24 completed rooms (two did not answer). Item / responseF1, all humanF3, all human F3, with agents Report influence: not at all / slightly / moderately / strongly 5 / 16 / 50 / 47 3 / 20 / 42 / 549 / 41 / 90 / 98 Usual response: faster-looking road / other road expecting others to switch / ignored / it depended 35 / 62 / 4 / 17 38 / 47 / 10 / 24 146 / 39 / 12 / 41 Rules understood: yes / mostly / no104 / 12 / 2108 / 11 / 0202 / 33 / 3 S22 Table S7. Exposure grids, the linear-mixing rule and the independent-choice benchmark.Medians across runs of I,Mand mean travel time;E mix [f t ]mixes the pure-population leans of the core with no fitted parameter;|E mix [f t ]− 1 2 | andT lin are the linear-mixing predictions andI ind ,T ind the independent-choice benchmark at the same mean (Text S5). Nwarned runsIMtimeE mix [f t ]|E mix [f t ]− 1 2 |T lin I ind T ind 2003 0.229 0.697 71.10.6950.195 66.1 0.198 67.8 2053 0.140 0.667 65.00.5310.031 60.2 0.091 62.1 20103 0.200 0.292 68.60.3670.133 62.8 0.145 64.7 20153 0.345 0.148 79.90.2020.298 74.2 0.298 75.5 20203 0.468 0.064 95.40.0380.462 94.1 0.462 94.4 5065 0.176 0.688 67.00.6160.116 62.2 0.119 62.9 50125 0.127 0.651 63.50.5370.037 60.2 0.064 61.0 50195 0.140 0.464 64.60.4450.055 60.5 0.072 61.3 50255 0.202 0.291 67.30.3670.133 62.8 0.135 63.6 50315 0.260 0.213 71.70.2880.212 67.2 0.212 67.9 50385 0.336 0.141 78.40.1960.304 74.8 0.304 75.3 50445 0.411 0.096 87.30.1170.383 83.5 0.383 83.8 S23 Table S8. Mixed rooms(rounds 6–40; carried choices retained). Subscripts h and a denote human and GPT seats; E[f t ]is the share of that seat type on the tip road over defined-tip rounds. “Frozen” applies the descriptive classifier to the room. Tie-round ranges are given in Text S4. All 24 slots are complete; the two invalid original attempts are retained separately in the execution record (Figs. S7 and S8). BlockKIM M h M a E[T t ]E[T h,t ]E[T a,t ]E[f h,t ]E[f a,t ]frozen 1 5 0.063 0.45 0.48 0.37 61.160.7 62.3 0.61 0.000 no 2 5 0.063 0.50 0.50 0.51 61.261.3 60.9 0.66 0.000 no 3 5 0.091 0.48 0.48 0.46 61.962.3 60.7 0.66 0.000 no 4 5 0.059 0.45 0.46 0.40 60.960.9 61.0 0.67 0.000 no 5 5 0.070 0.47 0.48 0.43 61.361.2 61.7 0.64 0.000 no 6 5 0.079 0.49 0.49 0.49 61.361.3 61.5 0.64 0.000 no 7 5 original attempt aborted after round 6; excluded from outcome summaries 7 R 5 0.069 0.44 0.43 0.49 61.161.2 60.9 0.64 0.000 no 8 5 original attempt aborted after round 4; excluded from outcome summaries 8 R 5 0.061 0.49 0.52 0.43 61.361.6 60.3 0.68 0.000 no 1 10 0.066 0.14 0.22 0.06 61.558.0 65.1 0.87 0.000 no 2 10 0.084 0.20 0.30 0.09 61.757.0 66.5 0.85 0.010 no 3 10 0.093 0.20 0.32 0.08 62.457.6 67.1 0.82 0.014 no 4 10 0.063 0.17 0.26 0.07 61.157.6 64.6 0.88 0.012 no 5 10 0.099 0.10 0.19 0.02 61.956.1 67.7 0.80 0.009 no 6 10 0.107 0.22 0.31 0.13 62.657.1 68.1 0.78 0.019 no 7 10 0.077 0.16 0.24 0.07 61.356.7 65.9 0.85 0.009 no 8 10 0.094 0.20 0.31 0.09 62.257.4 67.0 0.81 0.023 no 1 15 0.261 0.03 0.10 0.01 71.142.1 80.7 0.94 0.006 yes 2 15 0.293 0.10 0.22 0.06 74.049.7 82.1 0.73 0.032 yes 3 15 0.244 0.05 0.07 0.04 69.742.0 79.0 0.97 0.019 no 4 F 15 0.249 0.04 0.09 0.03 70.041.9 79.4 0.96 0.015 no 5 F 15 0.241 0.05 0.07 0.05 69.642.5 78.6 0.97 0.023 no 6 R 15 0.259 0.06 0.15 0.03 70.943.4 80.1 0.91 0.017 yes 7 15 0.269 0.07 0.21 0.02 72.044.4 81.2 0.89 0.011 yes 8 R 15 0.260 0.03 0.10 0.01 71.042.0 80.6 0.94 0.006 yes Unmarked rows: initial collection, 14–15 September 2026. R : 16 September; F : 17 September. The original aborted attempts are retained as separate rows. S24 Table S9. Switch experiments round by round.T 30 , mean travel time in the last round under the original message; f 31 ,I 31 ,M 31 andT 31 , the first round under the new message;T 32 toT 35 , the following rounds;E[T t ] 36–50 , the prespecified post-intervention window. ArmseedT 30 f 31 I 31 M 31 T 31 T 32 T 33 T 34 T 35 E 36:50 [T t ] F3→F1 391.0 0.96 0.46 0.94 93.9 60.3 70.8 80.7 60.672.3 F3→F1 480.7 1.00 0.50 0.86 100.0 63.1 61.6 69.2 64.171.3 F3→F1 593.9 0.96 0.46 0.92 93.9 60.1 70.8 83.1 60.673.6 F3→F1 688.2 0.98 0.48 0.90 96.9 70.8 60.3 76.4 61.073.1 F3→F1 796.9 0.96 0.46 0.98 93.9 64.1 67.7 83.1 63.167.2 F1→F3 391.0 0.00 0.50 0.06 100.0 100.0 93.9 93.9 96.995.4 F1→F3 460.1 0.00 0.50 0.48 100.0 100.0 96.9 96.9 93.990.8 F1→F3 570.8 0.00 0.50 0.24 100.0 100.0 100.0 96.9 100.093.3 F1→F3 660.3 0.00 0.50 0.46 100.0 96.9 88.2 100.0 100.096.0 F1→F3 772.5 0.00 0.50 0.22 100.0 100.0 100.0 96.9 100.092.5 S25 Table S10. Registered block slopes and leave-one-block-out estimates.Slopes are per five additional agents. Leave- one-block-out columns are mean slopes across the other seven blocks, without additional tests. The full-sample means are 0.0951 forIand−0.2080 forM(Text S8). Blockb b (I)b b (M)LOO meanIslope LOO meanMslope 1 +0.0993 -0.2086+0.0945-0.2080 2 +0.1150 -0.2007+0.0922-0.2091 3 +0.0764 -0.2150+0.0978-0.2070 4 +0.0950 -0.2007+0.0951-0.2091 5 +0.0857 -0.2079+0.0964-0.2081 6 +0.0900 -0.2129+0.0958-0.2073 7 +0.1000 -0.1886+0.0944-0.2108 8 +0.0993 -0.2300+0.0945-0.2049 Table S11. Registered nuisance sensitivity grids.Each row reports the equally weighted mean block slope after subtracting the stated hypothetical nuisance effect from room outcomes. Carryover is binary and wait is measured in minutes. These are descriptive sensitivity calculations, not estimated causal corrections (Text S8). Nuisance variableγMeanIslope MeanMslope Carryover-0.050+0.0982-0.2049 Carryover-0.020+0.0963-0.2068 Carryover-0.010+0.0957-0.2074 Carryover-0.005+0.0954-0.2077 Carryover+0.000+0.0951-0.2080 Carryover+0.005+0.0948-0.2083 Carryover+0.010+0.0945-0.2087 Carryover+0.020+0.0938-0.2093 Carryover+0.050+0.0920-0.2112 Wait (min)-0.005+0.0957-0.2074 Wait (min)+0.000+0.0951-0.2080 Wait (min)+0.005+0.0944-0.2087 Table S12. Recovered formal estimates of the automated share.Each composition comprises eight rooms. Re- sponses gives numeric respondents / all human participants. Means and SDs weight respondents equally; IQR gives the 25th and 75th percentiles. Bias is estimated minus actual percentage; MAE is mean absolute error, both in percent- age points (p). Missingness comprises eight unanswered items among submitted questionnaires and two unsubmitted questionnaires. No missing or draft-only value is imputed. KActual (%) Responses Mean (%) SDMedian [IQR] Bias (p) MAE (p) 525116/12063.1 24.9 65.0 [50.0, 80.0]+38.139.4 105076/8067.2 19.0 70.0 [50.0, 80.0]+17.220.4 157538/4068.9 21.2 70.0 [60.0, 83.0]-6.117.1 S26 Table S13. Reasoning and current-model extension, summary(50 agents, rounds 6–60, seeds 3–7; means over five runs). The first block is the core reference at the same seeds. “Response” and “variation” are the two terms of Eq. 1, 160(E[f t ]− 1 2 ) 2 and 160Var(f t )in minutes, calculated within each run. Rows “warning−tip” give mean paired differences. In parentheses is the number of pairs, of five, whose difference has the sign of the core warning effect (higherIand travel time, lowerMandE[f t ]). MessagefrozenIMtime (min)E[f t ]response variation GPT-5.4-mini, no reasoning, temperature 0.7 (core runs at the same seeds) tip0/50.1460.71465.20.5550.74.6 tip + warning5/50.4600.07994.00.04033.90.2 warning−tip+0.314 (5)−0.635 (5) +28.8 (5)−0.515 (5)+33.2−4.4 GPT-5.4-mini, no reasoning, temperature 1.0 tip0/50.1310.68763.90.4950.13.7 tip + warning5/50.4420.11591.50.05831.30.2 warning−tip+0.311 (5)−0.572 (5) +27.6 (5)−0.437 (5)+31.2−3.4 GPT-5.4-mini, low reasoning effort (provider default temperature) tip0/50.1890.65566.90.6905.81.2 tip + warning0/50.2410.41670.00.2599.30.8 warning−tip+0.051 (5)−0.240 (5)+3.2 (5)−0.430 (5)+3.5−0.5 GPT-5.4-mini, medium reasoning effort (provider default temperature) tip0/50.0970.58262.30.5821.11.3 tip + warning0/50.2950.36074.70.20513.90.9 warning−tip+0.198 (5)−0.222 (5) +12.4 (5)−0.377 (5)+12.8−0.5 GPT-6 Luna, no reasoning (no temperature sent) tip0/50.1840.62667.80.3553.44.1 tip + warning0/50.3230.35778.40.18515.92.5 warning−tip+0.139 (5)−0.269 (5) +10.6 (5)−0.170 (5)+12.5−1.6 GPT-6 Luna, medium reasoning effort, the model default (no temperature sent) tip5/50.4480.10392.40.05232.20.2 tip + warning5/50.4270.14589.50.07329.20.3 warning−tip−0.022 (1) +0.042 (0)−2.9 (2) +0.022 (1)−3.0+0.1 S27 Table S14. All 50 runs of the reasoning and current-model extension(50 agents, rounds 6–60). “Ties” counts analysis rounds whose previous counts were equal; these are excluded fromE[f t ]. State labels follow the descriptive classifier in Text S4. MessageseedIMtime (min)E[f t ]ties state GPT-5.4-mini, no reasoning, temperature 1.0 tip3 0.139 0.69164.3 0.5193 near equilibrium tip4 0.126 0.68663.6 0.4672 near equilibrium tip5 0.133 0.68164.0 0.5191 near equilibrium tip6 0.127 0.69264.0 0.4945 near equilibrium tip7 0.131 0.68863.9 0.4772 near equilibrium tip + warning3 0.451 0.09892.7 0.0490 frozen tip + warning4 0.431 0.13789.9 0.0690 frozen tip + warning5 0.454 0.09293.1 0.0460 frozen tip + warning6 0.430 0.13989.9 0.0700 frozen tip + warning7 0.445 0.10991.9 0.0550 frozen GPT-5.4-mini, low reasoning effort (provider default temperature) tip3 0.185 0.65166.7 0.6862 partial tip4 0.200 0.65967.4 0.7001 partial tip5 0.186 0.65166.8 0.6902 partial tip6 0.185 0.65366.6 0.6871 partial tip7 0.190 0.66266.9 0.6860 partial tip + warning3 0.240 0.41970.1 0.2600 partial tip + warning4 0.251 0.40470.6 0.2490 partial tip + warning5 0.227 0.42068.9 0.2730 partial tip + warning6 0.255 0.40771.1 0.2450 partial tip + warning7 0.232 0.42969.5 0.2680 partial GPT-5.4-mini, medium reasoning effort (provider default temperature) tip3 0.108 0.58863.4 0.6045 near equilibrium tip4 0.107 0.58662.6 0.5882 near equilibrium tip5 0.085 0.57961.7 0.5673 near equilibrium tip6 0.094 0.57262.1 0.5784 near equilibrium tip7 0.091 0.58361.9 0.5733 near equilibrium tip + warning3 0.291 0.37174.4 0.2090 partial tip + warning4 0.299 0.34675.2 0.2010 partial tip + warning5 0.287 0.37573.9 0.2130 partial tip + warning6 0.299 0.34975.1 0.2010 partial tip + warning7 0.297 0.35775.2 0.2030 partial GPT-6 Luna, no reasoning (no temperature sent) tip3 0.177 0.62967.2 0.3543 partial tip4 0.170 0.65066.4 0.3570 partial tip5 0.192 0.62168.4 0.3532 partial tip6 0.199 0.61069.2 0.3552 partial tip7 0.181 0.62067.7 0.3553 partial tip + warning3 0.328 0.36179.8 0.1980 partial tip + warning4 0.313 0.37277.7 0.1870 partial tip + warning5 0.339 0.33080.2 0.1740 partial tip + warning6 0.316 0.36377.5 0.1840 partial tip + warning7 0.318 0.35976.9 0.1820 partial GPT-6 Luna, medium reasoning effort, the model default (no temperature sent) tip3 0.456 0.08893.4 0.0440 frozen tip4 0.436 0.12890.5 0.0640 frozen tip5 0.463 0.07494.4 0.0370 frozen tip6 0.435 0.12990.5 0.0650 frozen tip7 0.453 0.09493.0 0.0470 frozen tip + warning3 0.420 0.15788.6 0.0800 frozen tip + warning4 0.436 0.12990.6 0.0640 frozen tip + warning5 0.419 0.16188.5 0.0810 frozen tip + warning6 0.435 0.12990.5 0.0650 frozen tip + warning7 0.424 0.14889.2 0.0760 frozen S28