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A data-driven heuristic-ensemble method for behavioural inference in collective-risk dilemma experiments

Marc Sadurní, Martín F. Díaz, Josep Perelló, Miquel Montero

arXiv:2610.13595v1physics.soc-phcs.CYphysics.data-an

TL;DR

Collective-risk dilemma experiments generate detailed records, yet aggregate outcomes can mask the behavioural pathways behind them. This paper estimates state-conditioned heuristics from those records, combines their proposals by plurality vote, and validates the resulting simulations against empirical macro-indicators. In the xAire example, the common fund is reproduced almost exactly, while the authors note limits on transfer and validation beyond the matched-size comparison.

  • Problem

    Aggregate outcomes such as total contributions, target achievement, or payoff distributions provide only a partial picture because similar outcomes may arise from distinct behavioural pathways.

    Source: Background · p. 3

  • Method

    The method estimates contribution probabilities conditioned on observable game-state features, simulates agents using single heuristics or their plurality-vote ensemble, and validates synthetic macro-indicators against empirical ones.

    Source: Method details · p. 4, Ensemble decision rule (plurality… · p. 10

  • Results

    The common fund is reproduced almost exactly, but Hpeer and Hbudget are no closer than chance on average for the two inequality indicators; Gini difference and Lorenz-area metrics rank the five models in the same order.

    Source: Method validation · pp. 13–14

  • Takeaways & Limitations

    The framework provides an empirical, reproducible way to characterise behavioural heuristics and generate synthetic data for tested or not-yet-implemented experimental configurations.

    Source: Background · p. 4

    Conditional distributions must be re-estimated for new datasets, and the worked-example validation does not establish transfer to other populations or game structures; large-scale simulations lack validation beyond the matched-size comparison.

    Source: Limitations · pp. 14, 17

Abstract

from arXiv · show

The collective-risk dilemma has become a paradigm for studying cooperation in a wide variety of social experiments, initiatives that generate increasingly rich datasets describing individual actions over successive decision-making processes. Beyond the purely descriptive analysis of such empirical datasets, a methodological framework is required to interpret and reproduce the observed dynamics, a scheme that may be skewed by exogenous biases. Here, we introduce a data-driven, non-parametric method for simulating individual decisions together with a validation protocol for assessing the fidelity level of the outcomes they generate. Our methodological contribution is based on an empirical heuristic ensemble: agents are represented by a finite set of behavioural heuristics, probabilities conditioned on different observable features of the game state, endowed with a hard-voting arbitration that determines their per-round final decisions. This reproducible generative framework allows for behavioural inference in different collective-risk dilemma setups: Behavioural heuristics are seeded directly from empirical data assessing conditional probabilities, without a priori assumptions; a plurality vote with uniform random tie-breaking resolves between competing heuristics each round, adding no free parameters; validation follows a pattern-oriented approach, comparing synthetic and empirical macro-indicators against single-heuristic and uniform-random baselines; synthetic data generated allow exploration of scenarios or experimental setups that have not been empirically realised; as a worked example, we instantiate our methodological approach on the xAire citizen science collective-risk dilemma experiment run in Barcelona.

Background

Collective-risk dilemma experiments provide rich records of repeated contributions, but aggregate outcomes can conceal the different behavioural pathways that produce them. The paper presents an empirical, state-conditioned approach to characterising those pathways and generating synthetic data, illustrated with the xAire experiment.

Background

  • Individuals may react simultaneously to their own history, others’ actions, collective progress, and resources, so similar aggregate outcomes can arise from distinct behavioural pathways.The xAire dataset records decisions and contextual information across rounds, including different resource distributions and inequality treatments.Source: Background · p. 3
  • The method estimates conditional action probabilities directly from observed state–action records rather than assigning predefined behavioural classes or specifying strategy distributions a priori.It requires no external calibration data or pre-specified behavioural models.Source: Background · p. 3
  • It represents behaviour as state-conditioned decision rules linking actions to individual history, social information, collective progress, and resource availability.The approach can generate synthetic data for tested scenarios or experimental configurations not yet implemented.Source: Background · p. 4
  • The xAire CRD dataset is the worked example; the perspective is also applicable to behavioural experiments with decisions and observable contextual variables.The authors describe the approach as a transparent and reproducible way to characterise mechanisms, compare heuristics, and build observation-grounded generative models.Source: Background · p. 4

Method details

The method builds agents from empirically estimated, state-conditioned heuristics and combines their per-round proposals through plurality voting. Its workflow generates synthetic populations and compares their macro-indicators with empirical ones; the four-heuristic example is tailored to xAire’s observed feedback channels.

Method details

  • The four-step workflow identifies heuristics, estimates their conditional probabilities from experimental data, simulates single-heuristic or ensemble agents, and validates synthetic macro-indicators against empirical ones.The method’s contribution is the arbitration that combines competing heuristic proposals into one action.Source: Method details · p. 4
  • Each heuristic reads one feature of the previous-round state, produces a contribution probability, and samples a synthetic dataset whose macro-indicators are compared with experiment data.Figure 1 depicts this per-heuristic pipeline and the ensemble’s connection to the plurality vote.Source: Figure 1 · p. 5
  • The four heuristics in the worked example map to xAire’s feedback channels and are not a limitation; the framework accepts threshold-public-goods or collective-risk datasets with participant-round logs and game information.Behavioural parameters are estimated from those logs without external data sources or guidelines.Source: Method details · p. 5
  • The ensemble executes the plurality of the heuristics’ proposed contributions, breaking ties by drawing uniformly at random.Figure 2 depicts the per-round decision rule.Source: Figure 2 · p. 6

Game setting and notation

The worked example uses six players over ten rounds, with contributions drawn from remaining endowments and a target set relative to the initial resources. Three endowment treatments vary the distribution while holding the group’s initial total fixed.

Game setting and notation

  • In the xAire experiment used here as a worked example, each game involves N = 6 players interacting over R = 10 rounds; these values are specific to the illustrative dataset, not requirements of the general methodology.Source: Game setting and notation · p. 6
  • Players choose contributions c_{k,i} from {0, 2, 4} monetary units, and the collective target is half the total initial endowment.The passage introduces the contribution choices and target for the worked example.Source: Game setting and notation · p. 6
  • The game tracks the cumulative common fund and each player’s evolving individual endowment.Source: Game setting and notation · p. 6
  • Equal, Unequal-L, and Unequal-H treatments each hold initial total endowment at 240 MU but distribute it differently among players.Equal assigns 40 MU to each player; Unequal-L assigns 24 MU to two and 48 MU to four; Unequal-H assigns 30 MU to four and 60 MU to two.Source: Game setting and notation · p. 7

State-conditioned contribution heuristics

Each heuristic maps a different observable feature of the previous-round game state to a probability distribution over contribution actions. In the worked example, the four features correspond to the feedback channels presented to participants.

State-conditioned contribution heuristics

Estimating the conditional distributions from data

The method estimates contribution probabilities from observed actions, conditional on binned game-state features. It initializes first-round behavior from empirical frequencies, including games that both achieved and missed the target.

Estimating the conditional distributions from data

  • Each heuristic maps a binned state feature to probabilities of contributing 0, 2, or 4 MU, stratified by treatment and initial endowment.The features are own previous contribution, peers’ mean previous contribution, cumulative fund relative to the 120 MU target, and fraction of endowment spent.Source: Estimating the conditional… · pp. 7–8
  • Figure 3 illustrates the four conditional-probability panels at a representative 60 MU endowment.Each cell gives action probabilities for a conditioning bin; the colour scale is set separately for each panel.Source: Figure 3 · p. 8
  • Synthetic players’ first contributions are sampled from observed frequencies, including both successful and unsuccessful games.The authors state this is intended to capture the full spectrum of decision tendencies and avoid selection bias.Source: Estimating the conditional… · p. 8, Table 1 · p. 9

Simulation pipeline

The workflow uses a dataset-matched simulation run for Table 2 scores and a larger run to characterize the method and estimate ensemble trajectories.

Simulation pipeline

  • The matched run contains 77 games across three treatments and supplies the scores reported in Table 2.Its game and treatment counts match the original dataset.Source: Simulation pipeline · p. 9
  • A 30,000-game extensive run, with 10,000 games per treatment, characterizes the method and supplies expected ensemble trajectories for Figure 4.The larger run is used because 77 games are too noisy for those trajectories, with only six Equal-treatment games.Source: Simulation pipeline · p. 9
  • The empirical curves in Figure 4 use all 77 games, while Table 2 scores use only the matched run.Source: Simulation pipeline · p. 9

Ensemble decision rule (plurality arbitration)

In the ensemble, each player evaluates the heuristics each round, and a plurality vote selects the executed contribution. Ties between most-frequent actions are resolved uniformly at random.

Ensemble decision rule (plurality arbitration)

  • Each round, every player evaluates all heuristics, whose components represent own history, peers’ actions, collective progress, and own resources.These four feedback channels match those displayed by the experimental interface after each round.Source: Ensemble decision rule (plurality… · pp. 9, 11
  • The simulation pipeline initializes first-round actions empirically, then uses plurality arbitration in ensemble mode and updates contribution history and the common fund.Source: Ensemble decision rule (plurality… · p. 10
  • The executed contribution is the most frequent action among the heuristics’ hypothetical contributions, with tied modes resolved by a uniform random draw.With four heuristics and three actions, the decision vector spans 81 combinations.Source: Ensemble decision rule (plurality… · pp. 9–10
  • Plurality voting adds no parameters or pre-set strategy proportions; the heuristics remain fixed conditional distributions rather than updating within a game.The paper says it did not test jointly fitting one model on all four state variables or using weighted or softmax votes.Source: Ensemble decision rule (plurality… · pp. 10–11

Method validation

Validation compares synthetic and empirical patterns across multiple scales against random and single-heuristic baselines. The ensemble is closer to the data than the mean random null on all eight indicators, though some apparent fit is built in and score uncertainty remains.

Method validation

  • The pattern-oriented validation matched game counts and treatment configurations, compared multiple indicator scales, and restricted comparisons to goal-reaching games.Source: Method validation · p. 11
  • Trajectory fit uses nMSE and DTW; payoff distributions use Jensen–Shannon and Wasserstein-1, while inequality uses Gini and Lorenz-area differences.An nMSE of 1 equals the empirical trajectory’s own variability; payoff metrics average across endowment levels, inequality metrics across treatments, and temporal scores use concatenated vectors.Source: Method validation · pp. 11–13
  • Across all eight indicators, the ensemble beat the mean random null by 1.6–2.2 null standard deviations on payoff and inequality indicators and 4.2–7.5 on dynamic indicators.Source: Method validation · p. 13
  • Common-fund nMSE was 0.010 but partly built in; average-contribution nMSE was 0.283, and the ensemble smoothed over the late-round contribution decline.Empirical contribution probabilities make total contributions over ten rounds near the data before arbitration.Source: Method validation · p. 13
  • Hpeer and Hbudget were no closer than chance on the two inequality indicators, while Gini-difference and Lorenz-area metrics ranked the five models in the same order.The Lorenz-area metric compares the whole curve, unlike the coarser scalar Gini difference.Source: Method validation · pp. 13–14
  • Strong majorities occurred in about 37% of decisions; at matched-run sizes, scores fluctuated 13–34% around converged values, so small model differences merit caution.Both common-fund scores settle after a few hundred simulated games per treatment.Source: Method validation · p. 14

Limitations

The worked-example validation is limited by its dataset and heuristic choices, and the ensemble’s round-by-round fit misses the late fall in contributions. Large-scale simulations are generative and are not validated beyond the matched-size comparison.

Limitations

  • The conditional distributions are estimated from one dataset, so validation does not establish transfer to other populations or game structures.New datasets require re-estimation; the heuristic set may also need to change.Source: Limitations · p. 14
  • Because each heuristic uses only the previous round, the ensemble cannot generate the coordinated end-game contribution drop seen in the data.It tracks the fund’s build-up but misses the late fall.Source: Limitations · pp. 14, 17
  • The estimated probabilities depend on a discretization binwidth, and plurality voting is one arbitration choice among possible alternatives.Weighted, softmax, and priority-ordered arbitration are also possible.Source: Limitations · p. 14
  • The large-scale simulations are not validated against empirical data beyond the matched-size comparison.Source: Limitations · p. 17

Ethics statements

The xAire data were collected under ethics approval, with informed consent and anonymised data use.

Ethics statements

  • The experiment was approved by the Universitat de Barcelona Ethics Committee under IRB00003099.Source: Ethics statements · p. 17
  • Participants signed informed consent forms, with parental or legal guardian consent where appropriate.Source: Ethics statements · p. 17
  • The data were used in anonymised form, and no privacy issues were observed to conflict with public release of the processed data.Source: Ethics statements · p. 17

Data and code availability

The project repository provides code to regenerate key tables and figures from the deposited data, while the worked-example dataset is available separately.

Data and code availability

  • The Project-xAire-EHE GitHub repository regenerates Tables 1 and 2 and data-driven Figures 3–5, including synthetic heuristic and ensemble runs.It also checks outcomes against stored reference values.Source: Data and code availability · p. 18
  • Without the 30,000-game extensive run, the pipeline takes about two minutes; including it takes about two hours on one core.Source: Data and code availability · p. 18
  • The empirical data are not duplicated in the code repository; the worked-example dataset is available in the companion data article and CORA repository.Source: Data and code availability · p. 18
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