Organizations: University of Edinburgh, Edinburgh, United Kingdom · Imperial College London, London, United Kingdom · University of Southampton, Southampton, United Kingdom
Many collective-action systems are neither fully decentralized nor fully centralized: agents voluntarily organize into institutions that share organizational costs and delegate decisions to representatives. Delegation can reduce individual costs, but it also transfers control to representatives whose private objectives need not align with those of their constituents. We study this tension in a repeated game with endogenous delegation to self-interested centers. A simple reduced-form model yields a scale--accountability tradeoff: larger constituencies offer greater cost-sharing benefits to members, yet the marginal value to a center of retaining any one constituent decreases as group size grows. Consequently, the prospect of individual exit can sustain non-exploitative delegation only up to an endogenous accountable constituency size. Additionally, we identify parameter regions in which fully delegated, accountable governance, when feasible, must be polycentric. We investigate these predictions through simulations with large language model (LLM) agents. Across five matched pairs of 30-agent simulations, polycentric structures repeatedly emerge but remain fluid, exhibiting cycles of concentration, exploitation, exit, and re-formation. We further find that models of different capacity exhibit similar willingness to assume leadership but differ substantially in their exercise of authority, with Sol initiating exploitation more frequently than Luna in matched decision states.
Figures & tables
Figure 1. Overview of our delegation setting. Agents choose whether to remain independent or join a constituency (to share costs) led by a self-interested center. Each center decides whether to exploit its delegates for private gain. Delegates may detect exploitation and exit the delegation.
More reliable detection and exit strengthen accountability
δ
–
↑
Future exit matters more
L
–
↓
Longer lock-in weakens exit discipline
Table 1. Comparative statics for the fixed-cost participation and accountability thresholds.
Figure 2. Constituency partitions at the start of each epoch. Left: centralized initialization, Right: decentralized initialization Each panel partitions all 30 agents; labels show constituency sizes of at least three. Multiple viable constituencies formed, dissolved, and re-formed under both initialization conditions. Appendix 8 shows all five repeats. Six panels show complete constituency partitions over twenty epochs for three matched repeats. Centralized episodes occupy the first row and decentralized episodes the second. Colours distinguish singleton, size-two, viable, and oversized constituencies.
Outcome
Centralized
Decentralized
Polycentric epochs
55/100
62/100
Agent-rounds within theoretical bounds
24.6%
29.2%
Exploited delegate-rounds after epoch 1
52.9%
42.2%
Table 2. Polycentricity and exploitation by initialization condition, aggregated over five matched pairs of 30-agent simulations (20 governance epochs per episode). An epoch is classified as polycentric when at least two constituencies have sizes within the theoretical benchmark interval 3≤m≤7 . Exploitation rates are calculated in epochs 2–20 to avoid the distinct initialization epoch.
Figure 3. Post-initialization matched-state comparison of exploitation decisions across constituency sizes. The shaded area corresponds to the theoretical accountable interval 3≤m≤7 . Sol generally initiated exploitation more frequently as compared to Luna. Two line charts compare Luna and Sol exploitation incidence and selected delegate shares across constituency sizes.
Figure 4. Timing of new exploitation within a governance epoch. New exploitation refers to targeting at least one delegate who was not previously being exploited during that epoch. Luna and Sol behaved similarly in the first operational round, but Sol initiated new exploitation more frequently in subsequent rounds, particularly rounds two to four. A line chart compares the share of Luna and Sol decisions that initiated new exploitation in each of the five operational rounds. The rates are similar in round one, after which Sol remains substantially higher through round four.
Figure 5. Themes in model-generated explanations for matched states where Sol initiated exploitation but Luna did not. Position and marker size represent each model’s mean topic weight. Luna emphasized long-term fee losses, whereas Sol emphasized membership turnover and retained constituency size. Comparison of reason in the cases where Sol exploited and Luna did not. Luna strongly emphasizes long-term fee loss, while Sol strongly emphasizes turnover and retained membership.
Appendix figures & tables9 assets
Supplementary material from the paper’s appendix.
Appendix
Quantity
Value
Meaning
n
30
Persistent agents
κ
1
Constituency fee per round
ℓ
3/5
Delegation cost per round
d
2/5
Per-target transfer to the center
δ
99/100
Continuation probability after each round
L
5
Operational rounds per epoch
Appendix
Table 3. Simulation parameters.
ℓ\d
0.2
0.4
0.6
0.8
1.0
0.2
[2,10]
[2,7]
[2,6]
[2,5]
[2,4]
0.4
[2,10]
[2,7]
[2,6]
[2,5]
[2,4]
0.6
[3,10]
[3,7]
[3,6]
[3,5]
[3,4]
0.8
[5,10]
[5,7]
[5,6]
[5,5]
–
0.9
[10,10]
–
–
–
–
Appendix
Table 4. Viable interval [mmin,mmax] as ℓ and d vary, with κ=1 , α=1 , δ=0.99 , and L=5 . A dash denotes an empty interval; bold marks the simulated setting.
δ\L
1
5
10
20
0.900
5
2
1
1
0.950
7
3
2
1
0.980
11
5
3
2
0.990
16
7
5
3
0.995
22
10
7
5
Appendix
Table 5. mmax as δ and L vary, with κ=1 , d=0.4 , and α=1 . Bold marks the simulated setting.
α
0.00
0.25
0.50
0.75
1.00
mmax
1
4
5
6
7
Appendix
Table 6. mmax as the accountability probability α varies, with κ=1 , d=0.4 , δ=0.99 , and L=5 . Bold marks the simulated setting.
κ
mmin
mmax
0.50
–
5
0.75
5
6
1.00
3
7
1.25
2
8
1.50
2
9
Appendix
Table 7. Thresholds as κ varies, with ℓ=0.6 , d=0.4 , α=1 , δ=0.99 , and L=5 . A dash denotes that no finite mmin exists; bold marks the simulated setting.
Table 8. Reasoning-topic weights by matched action outcome. The middle columns report Sol minus Luna in percentage points; positive values indicate greater Sol emphasis. The final columns give the models’ absolute mean weights in the states where only Sol exploited.
Figure 6. Distribution of delegation surplus. Bars aggregate the five repeats and points show individual repeats. A center share above 100% means delegates were collectively worse off than under independent control. Bars show center and delegate shares of delegation surplus under centralized and decentralized initialization, with points for each of five matched repeats.
Figure 7. Intensity of new exploitation by constituency size. Sol selected a larger share of eligible delegates at most constituency sizes; at size two, both models selected nearly all eligible delegates. A line chart compares the share of eligible delegates selected by Luna and Sol across constituency sizes.
Figure 8. Complete constituency partitions for all five matched repeats. Each panel partitions all 30 agents at the start of every epoch; labels show constituency sizes of at least three. Ten panels show complete constituency partitions over twenty epochs for all five matched centralized and decentralized episode pairs. Colours distinguish singleton, size-two, viable, and oversized constituencies.
In this study, we propose a generalization of the classic model of liquid democracy that allows fractional delegation of voting weight, while simultaneously allowing for the existence of equilibrium states. Our approach empowers agents to partition and delegate their votes to multiple representatives, all while retaining a fraction of the voting power for themselves. We introduce a penalty mechanism for the length of delegation chains. We discuss the desirable properties of a reasonable generalization of the classic model, and prove that smaller penalty factors bring the model closer to satisfying these properties. In the subsequent section, we explore the presence of equilibrium states in a general delegation game utilizing the proposed voting measure. In contrast to the classical model, we demonstrate that this game exhibits pure strategy Nash equilibria, contingent upon the imposition of a penalty on the length of delegation chains.
Francisco M. Bersetche
Centro Atómico Bariloche; CONICET · ANID, FONDECYT Project 3220254
Large language model (LLM)-based multi-agent systems (MAS) are commonly organized around two coordination regimes. Centralized coordination provides explicit control but can create a bottleneck at a single coordination locus, whereas decentralized coordination offers local autonomy but must achieve effective collaboration through distributed decisions and experience. These trade-offs motivate a systematic study of the space between the two regimes. We introduce Degrees of Freedom (DoF), a continuous configuration framework that quantifies the size of the decision space delegated to downstream agents. DoF controls the granularity of delegated work and the target probability that each task unit is assigned to the branches. Its endpoints recover the two coordination regimes. Across multiple task domains, DoF reveals a hierarchy of organizational effects. Delegated granularity and within-regime routing produce structured but bounded profile reallocations, whereas task domain and retained experience are associated with stronger differences between the centralized- and decentralized-branch profiles. The resulting performance landscape is task-dependent: different domains favor different DoF configurations, while atomic-target conditions are more consistent across runs. In-domain retained experience brings the two branch profiles closer together. The same controlled sweep reveals a task-dependent coordination landscape: some domains exhibit higher-performing configurations, whereas others show flatter responses and no single configuration dominates across domains. In five of the six evaluated state, an intermediate DoF configuration achieves higher mean accuracy than both endpoints, demonstrating the value of exploring the coordination space between them. Together, these findings provide empirical guidance for studying and designing intermediate coordination regimes in LLM-based MAS.
Yiqing Peng, Hehai Lin, Zixuan Wang +7
The Hong Kong University of Science and Technology (Guangzhou) · Wake Forest University · University of Science and Technology of China +1
Across millennia, complex societies have faced the same coordination problem of how to organize collective action among cognitively bounded and informationally incomplete individuals. Different civilizations developed different political institutions to answer the same basic questions of who proposes, who reviews, who executes, and how errors are corrected. We argue that multi-agent systems built on large language models face the same challenge. Their central problem is not only individual intelligence, but collective organization. Historical institutions therefore provide a structured design space for multi-agent architectures, making key trade-offs between efficiency and error correction, centralization and distribution, and specialization and redundancy empirically testable. We translate seven historical political institutions, spanning four canonical governance patterns, into executable multi-agent architectures and evaluate them under identical conditions across three large language models and two benchmarks. We find that governance topology strongly shapes collective performance. Within a single model, the gap between the best and worst institution exceeds 57 percentage points, while the optimal architecture shifts systematically with model capability and task characteristics. These results suggest that collective intelligence will not advance through a single optimal organizational form, but through governance mechanisms that can be reselected and reconfigured as tasks and capabilities evolve. More broadly, this points to a transition from self-evolving agents to the self-evolving multi-agent system. The code is available on GitHub.