Theory of Scene: Breaking the Symmetry Trap in Multi-Agent LLM Coordination

Liangqi Yuan; Wenzhi Fang; Shiqiang Wang; Christopher G. Brinton

cs.LGSep 26, 2026

Theory of Scene: Breaking the Symmetry Trap in Multi-Agent LLM Coordination

Authors: Liangqi Yuan, Wenzhi Fang, Shiqiang Wang, Christopher G. Brinton

Organizations: Purdue University · University of Exeter

Abstract

Multi-agent systems built on large language models (LLMs) are largely homogeneous, as their agents behave alike even across distinct LLMs. We show that when such agents act concurrently without communication, they collide on targets they must split and diverge on targets they must take together, a double failure we term the symmetry trap. Theory of Mind (ToM), widely used for coordination without communication, cannot escape this trap, since homogeneous agents form the same prediction of one another and respond to it in the same way. We propose Theory of Scene (ToS), a training-free reasoning schema in which each agent reads its public role, the only difference between the agents, and the task context they all observe. Homogeneous agents thereby derive one division of labor, each taking the part its role fixes, which turns homogeneity from the cause of the trap into the cure. ToS reads the role together with the scene through role gating, which determines whether ownership overlaps or is already divided, and the task context through task coupling, which infers whether the team must converge on each target, divide it, or take its stages in turn. We evaluate on DivvyBench, a controlled environment we introduce, whose target types make an episode Competitive, Cooperative, or Mixed across Tabletop, Airspace, and Household scenarios, and on two established agentic benchmarks, GovSim and Overcooked. ToS outperforms all six baselines on every benchmark, and each baseline falls far behind it in at least one setting. Against ToM given the same inputs, ToS raises the DivvyBench success rate from 71.1% to 99.6%, the GovSim total gain from 207 to 400, and the Overcooked level-normalized throughput from 1.41 to 1.67.

Figures & tables

Figure 1: Symmetry trap: ToM vs. ToS. Left: DivvyBench covers three scenarios, Tabletop picking, Airspace survey, and Household assistance. Top: Under ToM, homogeneous agents reason identically at every step, so they collide, correct, and collide again. Bottom: Under ToS, the agents read the division of labor from the shared scene and the public role, diverging onto the Competitive targets and converging on the Cooperative targets.

Figure 2: Each baseline succeeds in one setting and fails in the other. DivvyBench Tabletop with 2 agents, 8 balls, and a 10-step budget. Left: Competitive success rate against the step budget. Right: Cooperative success rate at the full budget.

Figure 3

	Competitive		Cooperative		Mixed		Avg.
Method	Success (%) $\uparrow$	Stuck (%) $\downarrow$	Success (%) $\uparrow$	Stuck (%) $\downarrow$	Success (%) $\uparrow$	Stuck (%) $\downarrow$	Success (%) $\uparrow$	Stuck (%) $\downarrow$
ReAct	40.7 $\pm$ 4.0	48.7 $\pm$ 1.4	96.7 $\pm$ 1.5	2.5 $\pm$ 0.7	56.7 $\pm$ 4.1	26.9 $\pm$ 1.1	64.7 $\pm$ 2.3	26.0 $\pm$ 1.1
Focal	47.3 $\pm$ 4.1	45.4 $\pm$ 1.4	42.0 $\pm$ 4.0	36.6 $\pm$ 2.4	50.0 $\pm$ 4.1	29.0 $\pm$ 1.1	46.4 $\pm$ 2.4	37.0 $\pm$ 1.1
CFD	57.3 $\pm$ 4.1	31.8 $\pm$ 1.8	66.7 $\pm$ 3.9	20.5 $\pm$ 2.1	36.0 $\pm$ 3.9	33.9 $\pm$ 1.7	53.3 $\pm$ 2.4	28.7 $\pm$ 1.1
Classic ToM	72.7 $\pm$ 3.7	35.6 $\pm$ 1.5	86.7 $\pm$ 2.8	8.2 $\pm$ 1.6	54.0 $\pm$ 4.1	27.1 $\pm$ 1.6	71.1 $\pm$ 2.1	23.6 $\pm$ 1.1
ProAgent	68.7 $\pm$ 3.8	37.2 $\pm$ 1.7	72.0 $\pm$ 3.7	20.1 $\pm$ 2.7	34.7 $\pm$ 3.9	35.0 $\pm$ 1.5	58.4 $\pm$ 2.3	30.8 $\pm$ 1.2

Table 1: ToS is the only method that succeeds on all three benchmarks, and it outperforms every baseline on almost every metric. Best per column in bold, ToS shaded, and

\uparrow

and

\downarrow

mark the improving direction. Appendix A.10 reports a two-sided 95% confidence interval and a significance test for every gap on the headline statistic of each benchmark.

Figure 3: ToS breaks symmetry on the first step, recovers after a failure, and sustains the resource pool. Left: On DivvyBench, the progress rate at the first step and at the step after the first failure, in each setting. The progress rate is defined as the fraction of episodes in which the team takes at least one target on that step. Right: On GovSim, the resource remaining over the horizon.

Ablation Study
	DivvyBench - Success (%) $\uparrow$			GovSim		Overcooked
Method	Competitive	Cooperative	Mixed	Survival (%) $\uparrow$	Total Gain $\uparrow$	$\times$ ReAct $\uparrow$
ToS (Ours)	99.3 $\pm$ 0.7	100.0 $\pm$ 0.0	99.3 $\pm$ 0.7	85.2 $\pm$ 2.3	400 $\pm$ 14	1.67 $\pm$ 0.08
w/o role	97.3 $\pm$ 1.3	100.0 $\pm$ 0.0	99.3 $\pm$ 0.7	58.1 $\pm$ 3.0	242 $\pm$ 10	1.45 $\pm$ 0.09
w/o task	83.3 $\pm$ 3.1	76.0 $\pm$ 3.5	18.0 $\pm$ 3.1	67.9 $\pm$ 3.0	344 $\pm$ 17	1.61 $\pm$ 0.07
Robustness Study

Table 2: Ablation Study and Robustness Study. Removing role gating or task coupling lowers performance, while reordering the two readings or withholding the other agents’ actions does not. Red shaded cells are significantly worse than ToS (one-sided two-sample

z

-test,

p<0.05

	DivvyBench - Success (%) $\uparrow$
Method	Competitive	Cooperative	Mixed
Classic ToM	72.7 $\pm$ 3.7	86.7 $\pm$ 2.8	54.0 $\pm$ 4.1
+ role	64.7 $\pm$ 3.9	80.0 $\pm$ 3.3	47.3 $\pm$ 4.1
+ task	91.3 $\pm$ 2.3	68.7 $\pm$ 3.8	82.7 $\pm$ 3.1
+ role + task	88.7 $\pm$ 2.6	68.7 $\pm$ 3.8	78.7 $\pm$ 3.4
ToS (Ours)	99.3 $\pm$ 0.7	100.0 $\pm$ 0.0	99.3 $\pm$ 0.7

Table 3: Classic ToM with ToS’s role and task readings.

Appendix figures & tables25 assets

Supplementary material from the paper’s appendix.

Appendix

		Competitive		Cooperative		Mixed
Temp	Public Role	Success (%) $\uparrow$	Stuck (%) $\downarrow$	Success (%) $\uparrow$	Stuck (%) $\downarrow$	Success (%) $\uparrow$	Stuck (%) $\downarrow$
0.0	None	0.7 $\pm$ 0.7	80.5 $\pm$ 1.8	100.0 $\pm$ 0.0	0.0 $\pm$ 0.0	8.0 $\pm$ 2.2	48.9 $\pm$ 1.2
	1 Bit (index)	92.7 $\pm$ 2.1	24.4 $\pm$ 1.9	100.0 $\pm$ 0.0	0.0 $\pm$ 0.0	99.3 $\pm$ 0.7	9.8 $\pm$ 0.9
	1 Bit (role)	100.0 $\pm$ 0.0	9.3 $\pm$ 1.0	100.0 $\pm$ 0.0	0.0 $\pm$ 0.0	100.0 $\pm$ 0.0	6.4 $\pm$ 0.8
0.7	None	12.7 $\pm$ 2.7	53.1 $\pm$ 1.4	100.0 $\pm$ 0.0	0.2 $\pm$ 0.1	28.0 $\pm$ 3.7	33.0 $\pm$ 1.1
	1 Bit (index)	99.3 $\pm$ 0.7	19.5 $\pm$ 1.3	100.0 $\pm$ 0.0	0.2 $\pm$ 0.1	98.7 $\pm$ 0.9	13.7 $\pm$ 1.0

Appendix

Table 4: One bit of distinction suffices to break the symmetry. The ToS reasoning is held fixed on DivvyBench and only the public role differs. Under None the agents receive the same wording, under 1 Bit (index) they are told apart by “A0” and “A1”, and under 1 Bit (role) by the leader and follower labels ToS uses, the last two carrying the same 1 bit. Red marks a cell significantly worse than 1 Bit (role) in the same setting at the same temperature (one-sided two-sample

z

-test,

p<0.05

		Competitive		Cooperative		Mixed
	Method	Success (%) $\uparrow$	Stuck (%) $\downarrow$	Success (%) $\uparrow$	Stuck (%) $\downarrow$	Success (%) $\uparrow$	Stuck (%) $\downarrow$
Shared	ReAct	40.7 $\pm$ 4.0	48.7 $\pm$ 1.4	96.7 $\pm$ 1.5	2.5 $\pm$ 0.7	56.7 $\pm$ 4.1	26.9 $\pm$ 1.1
	Focal	47.3 $\pm$ 4.1	45.4 $\pm$ 1.4	42.0 $\pm$ 4.0	36.6 $\pm$ 2.4	50.0 $\pm$ 4.1	29.0 $\pm$ 1.1
	CFD	57.3 $\pm$ 4.1	31.8 $\pm$ 1.8	66.7 $\pm$ 3.9	20.5 $\pm$ 2.1	36.0 $\pm$ 3.9	33.9 $\pm$ 1.7
	Classic ToM	72.7 $\pm$ 3.7	35.6 $\pm$ 1.5	86.7 $\pm$ 2.8	8.2 $\pm$ 1.6	54.0 $\pm$ 4.1	27.1 $\pm$ 1.6
	ProAgent	68.7 $\pm$ 3.8	37.2 $\pm$ 1.7	72.0 $\pm$ 3.7	20.1 $\pm$ 2.7	34.7 $\pm$ 3.9	35.0 $\pm$ 1.5

Appendix

Table 5: Only ToS succeeds in every setting under disagreeing observations. Under Shared, the agents are shown the targets in the presentation order, as in Table 1 . Under Reversed and Shuffled, A1 is shown them in reverse or in a fixed derangement of that order, with the targets and their types unchanged, and the rules state that the orders may differ. Red marks a setting whose Success is 20% or below.

		Competitive		Cooperative		Mixed		Avg.
	Method	Success (%) $\uparrow$	Stuck (%) $\downarrow$	Success (%) $\uparrow$	Stuck (%) $\downarrow$	Success (%) $\uparrow$	Stuck (%) $\downarrow$	Success (%) $\uparrow$	Stuck (%) $\downarrow$
Qwen3.5-35B	ReAct	40.7 $\pm$ 4.0	48.7 $\pm$ 1.4	96.7 $\pm$ 1.5	2.5 $\pm$ 0.7	56.7 $\pm$ 4.1	26.9 $\pm$ 1.1	64.7 $\pm$ 2.3	26.0 $\pm$ 1.1
	Focal	47.3 $\pm$ 4.1	45.4 $\pm$ 1.4	42.0 $\pm$ 4.0	36.6 $\pm$ 2.4	50.0 $\pm$ 4.1	29.0 $\pm$ 1.1	46.4 $\pm$ 2.4	37.0 $\pm$ 1.1
	CFD	57.3 $\pm$ 4.1	31.8 $\pm$ 1.8	66.7 $\pm$ 3.9	20.5 $\pm$ 2.1	36.0 $\pm$ 3.9	33.9 $\pm$ 1.7	53.3 $\pm$ 2.4	28.7 $\pm$ 1.1
	Classic ToM	72.7 $\pm$ 3.7	35.6 $\pm$ 1.5	86.7 $\pm$ 2.8	8.2 $\pm$ 1.6	54.0 $\pm$ 4.1	27.1 $\pm$ 1.6	71.1 $\pm$ 2.1	23.6 $\pm$ 1.1
	ProAgent	68.7 $\pm$ 3.8	37.2 $\pm$ 1.7	72.0 $\pm$ 3.7	20.1 $\pm$ 2.7	34.7 $\pm$ 3.9	35.0 $\pm$ 1.5	58.4 $\pm$ 2.3	30.8 $\pm$ 1.2

Appendix

Table 6: ToS’s advantage holds across model backbones. Three further backbones (Gemma-4-26B, GPT-OSS-20B, and Qwen3.5-9B) stand in for the main Qwen3.5-35B-A3B across the three DivvyBench settings, with all agents in a run sharing one backbone.

		Competitive		Cooperative		Mixed		Avg.
A1	Method	Success (%) $\uparrow$	Stuck (%) $\downarrow$	Success (%) $\uparrow$	Stuck (%) $\downarrow$	Success (%) $\uparrow$	Stuck (%) $\downarrow$	Success (%) $\uparrow$	Stuck (%) $\downarrow$
Gemma-4-26B	ReAct	53.3 $\pm$ 4.1	44.2 $\pm$ 1.8	98.7 $\pm$ 0.9	1.4 $\pm$ 0.5	52.7 $\pm$ 4.1	29.3 $\pm$ 1.0	68.2 $\pm$ 2.2	25.0 $\pm$ 1.1
	Focal	65.3 $\pm$ 3.9	37.0 $\pm$ 1.4	58.0 $\pm$ 4.0	24.7 $\pm$ 2.0	61.3 $\pm$ 4.0	27.9 $\pm$ 1.0	61.6 $\pm$ 2.3	29.8 $\pm$ 0.9
	CFD	86.7 $\pm$ 2.8	19.0 $\pm$ 1.3	77.3 $\pm$ 3.4	13.5 $\pm$ 1.7	82.7 $\pm$ 3.1	15.1 $\pm$ 1.0	82.2 $\pm$ 1.8	15.9 $\pm$ 0.8
	Classic ToM	78.7 $\pm$ 3.4	36.4 $\pm$ 1.3	97.3 $\pm$ 1.3	1.6 $\pm$ 0.6	75.3 $\pm$ 3.5	21.0 $\pm$ 1.4	83.8 $\pm$ 1.7	19.7 $\pm$ 0.9
	ProAgent	84.7 $\pm$ 3.0	37.1 $\pm$ 1.6	90.7 $\pm$ 2.4	6.2 $\pm$ 1.1	62.7 $\pm$ 4.0	26.6 $\pm$ 1.5	79.3 $\pm$ 1.9	23.3 $\pm$ 1.0

Appendix

Table 7: ToS’s advantage holds when the agents run different model backbones. The agents run the method named in each row on DivvyBench, A0 on the main Qwen3.5-35B-A3B backbone and A1 on the backbone named in each panel.

		Competitive		Cooperative		Mixed		Avg.
A1	A0 Method	Success (%) $\uparrow$	Stuck (%) $\downarrow$	Success (%) $\uparrow$	Stuck (%) $\downarrow$	Success (%) $\uparrow$	Stuck (%) $\downarrow$	Success (%) $\uparrow$	Stuck (%) $\downarrow$
ReAct	ReAct	40.7 $\pm$ 4.0	48.7 $\pm$ 1.4	96.7 $\pm$ 1.5	2.5 $\pm$ 0.7	56.7 $\pm$ 4.1	26.9 $\pm$ 1.1	64.7 $\pm$ 2.3	26.0 $\pm$ 1.1
	Focal	63.3 $\pm$ 3.9	38.5 $\pm$ 1.6	56.0 $\pm$ 4.1	26.1 $\pm$ 2.2	50.7 $\pm$ 4.1	28.3 $\pm$ 1.1	56.7 $\pm$ 2.3	30.9 $\pm$ 1.0
	CFD	60.0 $\pm$ 4.0	31.4 $\pm$ 1.4	77.3 $\pm$ 3.4	14.3 $\pm$ 1.9	66.7 $\pm$ 3.9	18.7 $\pm$ 1.2	68.0 $\pm$ 2.2	21.5 $\pm$ 0.9
	Classic ToM	74.0 $\pm$ 3.6	29.8 $\pm$ 1.7	94.7 $\pm$ 1.8	4.5 $\pm$ 1.2	73.3 $\pm$ 3.6	21.1 $\pm$ 1.3	80.7 $\pm$ 1.9	18.5 $\pm$ 1.0
	ProAgent	82.0 $\pm$ 3.1	31.6 $\pm$ 1.6	85.3 $\pm$ 2.9	9.8 $\pm$ 1.7	66.0 $\pm$ 3.9	23.8 $\pm$ 1.2	77.8 $\pm$ 2.0	21.7 $\pm$ 1.0

Appendix

Table 8: ToS achieves the highest success in every setting beside an agent that reasons differently. A0 runs the method named in each row, and A1 runs the reasoning schema named in each panel.

Figure 4: Team size leaves ToS unchanged and lowers the baselines in DivvyBench Cooperative and Mixed. DivvyBench success rate versus step budget from 2 to 5 agents, one panel per setting and team size.

Figure 5: ToS degrades more slowly than the baselines as the population grows. GovSim survived rounds and total gain over populations from 5 to 100 on a pool of fixed capacity.

Figure 6: The baselines leave over-takers at every population size and ToS leaves almost none. Per-agent demand on GovSim (Pasture, first round), the fraction of the population at each demand level for every method and population, with the dashed line at the sustainable per-agent share. An over-taker asks for half the pool or more, which is the whole team’s sustainable allowance and $N$ times the dashed line.

Figure 7: ToS’s advantage compounds over a long horizon. GovSim run for 120 rounds, $10\times$ the default horizon, reporting the survival rate (the fraction of runs whose commons is still alive at each round) and the cumulative total gain.

	Competitive		Cooperative		Mixed
Method	No History	With History	No History	With History	No History	With History
Classic ToM	34.3 $\pm$ 2.8	30.5 $\pm$ 1.1	91.7 $\pm$ 1.6	91.5 $\pm$ 0.6	45.8 $\pm$ 3.0	56.2 $\pm$ 1.1
ProAgent	29.5 $\pm$ 2.6	30.3 $\pm$ 1.1	90.2 $\pm$ 1.7	78.4 $\pm$ 0.9	46.3 $\pm$ 3.1	49.0 $\pm$ 1.0
HypMinds	43.1 $\pm$ 2.9	29.6 $\pm$ 1.1	79.7 $\pm$ 2.3	73.9 $\pm$ 0.9	48.6 $\pm$ 3.0	44.4 $\pm$ 1.0

Appendix

Table 9: ToM prediction accuracy depends on the setting and does not improve with history. The percentage of DivvyBench steps on which a baseline predicts the target the other agent then selected. No History is the first step, and With History every step after it.

Figure 8: Predicting a team does not scale. Each bar is the mean number of other agents a ToM method names in a single prediction step, and the rounded cap marks the number of other agents ( $N-1$ ). The hatched part is what the method predicted in responses whose reasoning failed to parse, so the agent fell back to a default move. Responses cut off at the generation budget are excluded from the mean.

	DivvyBench - $\delta_{ij}$ at the first step
Method	Competitive $\uparrow$	Cooperative $\downarrow$
ReAct	0.307 $\pm$ 0.043	0.007 $\pm$ 0.013
Focal	0.187 $\pm$ 0.039	0.100 $\pm$ 0.031
CFD	0.480 $\pm$ 0.035	0.040 $\pm$ 0.024
Classic ToM	0.327 $\pm$ 0.041	0.047 $\pm$ 0.020
ProAgent	0.280 $\pm$ 0.043	0.067 $\pm$ 0.027

Appendix

Table 10: Only ToS moves the distance toward the value each step requires. The total variation distance

\delta_{ij}

between the target distributions of A0 and A1 on the first step of DivvyBench. The arrows give the value Proposition 4.2 requires,

1

where the step’s targets have to be divided and

0

where one of them has to be selected together.

	DivvyBench - Success (%) $\uparrow$
Method	Competitive	Cooperative	Mixed
ReAct	+58.7 [51.3, 66.0] ✓	+3.3 [0.7, 6.7] ✗	+42.7 [34.7, 50.7] ✓
Focal	+52.0 [44.0, 59.3] ✓	+58.0 [52.7, 62.7] ✓	+49.3 [41.3, 57.3] ✓
CFD	+42.0 [36.0, 48.0] ✓	+33.3 [31.3, 35.3] ✓	+63.3 [56.7, 70.0] ✓
Classic ToM	+26.7 [19.3, 34.0] ✓	+13.3 [8.7, 18.0] ✓	+45.3 [37.3, 53.3] ✓
ProAgent	+30.7 [23.3, 38.0] ✓	+28.0 [22.0, 34.0] ✓	+64.7 [57.3, 72.0] ✓

Appendix

Table 11: ToS’s lead is significant in 35 of the 36 comparisons. Each baseline row gives ToS’s lead over that baseline on the statistic Table 1 reports, and the brackets hold the two-sided 95% interval of the number before them. The ToS row gives its own score. ✓ marks a gap significant at

p<0.05

and ✗ one that is not.

Parameter	Value
Backbone	Qwen3.5-35B-A3B
Temperature	0.7
Generation budget	8192 tokens
Max model length	16384 tokens
Thinking mode	Disabled
Serving	vLLM on 4 NVIDIA H100

Appendix

Table 12: Implementation details.

Agents	2	5	2
	DivvyBench	GovSim ( Piatti et al., 2024 )	Overcooked ( Sun et al., 2025 )
Public role	Leader, Follower	A0 to A4	Chef, Assistant
Scenarios	Tabletop, Airspace, Household	Fishery, Pasture, Pollution	Levels 1–6
Settings	Competitive, Cooperative, Mixed	-	-
Action	One target	An integer harvest	One operation (e.g., pickup , cut )
Episode budget	10 steps	12 rounds	50 steps

Appendix

Table 13: Benchmark details.

Panel of Figure 9 — Figure 9: One Tabletop episode in each setting. The agents act at the same step, and the same joint action has opposite outcomes in the two settings. On the left, two agents on one ball collect neither ( ✗ ) and leave the table unchanged, and on the right they collect it together ( ✓ ), where a split would collect nothing.

Panel of Figure 10 — Figure 10: Task prompt for DivvyBench Competitive in Tabletop.

Panel of Figure 11 — Figure 11: Task prompt for DivvyBench Cooperative in Tabletop.

Panel of Figure 12 — Figure 12: Task prompt for DivvyBench Mixed in Tabletop.

Panel of Figure 13 — Figure 13: Task prompt for DivvyBench Competitive in Airspace.

Panel of Figure 14 — Figure 14: Task prompt for DivvyBench Competitive in Household.

Panel of Figure 15 — Figure 15: Prompt for ToS.

Panel of Figure 16 — Figure 16: ToS reasoning on DivvyBench Tabletop (Mixed), where one table carries both couplings. On Step 1 the agents read the Cooperative balls as Joint and commit to the first one in the shared order, since a Cooperative ball is taken only when all agents select it together. On Step 5 the Cooperative balls are gone, the agents read the remaining Competitive balls as Exclusive , and they split them without exchanging a message.

Panel of Figure 17 — Figure 17: ToS reasoning on GovSim (Fishery). The agents read the pool as shared and each asks for its per-agent share of what the pool can regrow, so the total stays sustainable without anyone proposing it. Only A0 is expanded, and the other four reach the same two readings and the same catch.

Panel of Figure 18 — Figure 18: ToS reasoning on Overcooked. The recipe is a chain and the public role gives each agent its own stations, so ownership reads as divided and no agent settles a division. On Step 1, A1 opens the chain by fetching the pepper while A0, whose stations have nothing to work on yet, fetches the dish the order will be served on. On Step 2, A1 loads the board and A0 waits, since nothing in its own part is ready.

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