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.