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An evolutionary origin of collective decision making in humans and machines

Guocheng Wang, Qi Su, Joshua B. Plotkin

arXiv:2610.05676v1physics.soc-phcs.LGq-bio.PE

TL;DR

Prior theory explained how individual incentives can evolve accurate collective predictions only for linear problems. This paper analyzes layered majority voting with marginal feedback that rewards pivotal individuals across higher layers. It shows that single-layer voting cannot solve nonlinear classifications, while two-layer voting with marginal feedback reaches high accuracy on generalized XOR.

  • Problem

    Whether individual incentives can evolve accurate collective predictions for nonlinear problems remains unresolved beyond prior theoretical understanding limited to linear tasks.

    Source: Discussion · p. 8

  • Method

    The paper analyzes recursively layered majority voting and introduces marginal feedback, rewarding individuals when pivotal within their group and at every higher layer.

    Source: Discussion · p. 8

  • Results

    Single-layer voting cannot solve nonlinear classification, whereas two-layer voting under marginal feedback reaches high accuracy on generalized XOR and reconstructs several classification boundaries.

    Source: Single-layer voting cannot solve… · p. 4, Layered voting with marginal feedback… · p. 4

  • Takeaways & Limitations

    The results offer a naturalistic account of hierarchical institutions, linking layered majority voting on nonlinear classifications to the importance of pivotal actors.

    Source: Discussion · p. 8

Abstract

from arXiv · show

Groups of individuals can solve collective problems more accurately than any single member, by aggregating their opinions. Recent theoretical work has identified individual-level reward schemes that allow uninformed individuals to evolve collective intelligence from the bottom up, through social learning. Yet these results are restricted to linear prediction problems and simple averaging, while the decision tasks that real groups confront are often non-linear, and the institutions that aggregate opinions are seldom single-layer averages: districts elect representatives who in turn vote on policy, referees advise editors who decide on publication. Here we develop a framework for the evolution of collective intelligence in multi-layer voting populations, where individuals observe limited information and groups recursively aggregate their opinions by majority rule. We prove that single-layer voting cannot solve non-linear classification problems under any individual reward scheme. We then identify a "marginal feedback" payoff structure, which rewards individuals only when their opinion is pivotal in their group, and at every layer above them. This reward scheme induces a layered population to evolve accurate collective solutions to complex, non-linear decision tasks through individual-level peer imitation alone. The collective behavior that emerges is equivalent to a multi-layer perceptron in machine learning. Our results provide a naturalistic account of hierarchical institutions, in which the outsize importance of swing voters is the incentive that sustains collective accuracy; and they identify the credit-assignment rule in machine learning as not just an engineered solution but a natural evolutionary outcome.

A model of collective prediction

This section models how populations learn complex binary predictions through imitation and layered voting, examining how marginal feedback and voting-group structure relate to accuracy. It also considers swing-voter signals and credit assignment across layers, while noting unresolved real-world conditions and quantitative prediction tasks.

A model of collective prediction

  • Individuals observe one factor and form a linear opinion about it; groups and then the population aggregate opinions by majority rule, while individuals imitate strategies within their group.The outcome depends on multiple randomly redrawn factors through an unknown classification rule, and payoffs are averaged over rounds to guide payoff-biased imitation.Source: A model of collective prediction · p. 2, Single-layer and two-layer voting · pp. 2–3

Single-layer voting cannot solve non-linear problems

  • Single-layer voting cannot solve any non-linear classification problem, regardless of the reward scheme, because its aggregate remains a linear function of the observed factors.With feedback payoffs, single-layer voting performs no better than chance on generalized XOR, although it can achieve near-perfect accuracy on linear problems.Source: Single-layer voting cannot solve… · p. 4

Layered voting with marginal feedback solves non-linear problems

  • Two-layer voting with marginal feedback evolves accurate predictions on non-linear problems, while either ingredient alone performs no better than chance.On generalized XOR, accuracy declines as the number of factors increases; evolved predictions also reproduce several non-linear decision boundaries. The mechanism rewards individuals whose opinions are pivotal within their group and across the higher-level vote.Source: Layered voting with marginal feedback… · p. 4, Figure 2 · p. 5

An intermediate number of groups performs best

  • For non-linear tasks, an intermediate number of groups maximizes accuracy by balancing within-group and inter-group diversity.At fixed population size, too few groups reduce layered voting to a single layer, while too many small groups limit the strategies imitation can explore. Larger populations improve chiefly by forming more groups.Source: An intermediate number of groups… · pp. 5–6, Figure 3 · p. 6

Deeper hierarchies solve more complex problems

  • For m=6, maximum achievable accuracy rises from 0.57 with two layers to 0.87 with three and 0.90 with four under global marginal feedback.Individuals must be pivotal at every layer above them; more layers can represent more problems but produce sparser rewards, slowing convergence and requiring stronger selection.Source: Figure 4 · p. 7, Deeper hierarchies solve more complex… · pp. 7–8

Discussion

  • The authors interpret layered aggregation and attention to pivotal actors as functional features of institutions, while noting that the model is deliberately simple and not a quantitative account of any particular institution.The conclusions persist with binary or thresholded opinions, multiple observed factors, unequal group sizes, varying innovation rates, and growing populations.Source: Discussion · p. 8
  • The framework links layered voting to a multi-layer perceptron, with marginal feedback providing an implicit credit-assignment signal analogous to backpropagation.A change to an individual’s opinion receives reward only when it can propagate through every relevant layer to affect the final verdict.Source: Discussion · p. 8
  • Layered institutions and deep networks may solve the same basic problem: assigning credit across layers so local imitation can improve a global judgment.In the model, swing voters provide the learning signal for accurate collective decision-making.Source: Discussion · p. 9
  • Whether the diversity tradeoff applies in real groups is unknown, and institutions may misjudge who was pivotal.The authors also note that real deliberative bodies can be partisan and that some collective problems require quantitative predictions rather than binary verdicts.Source: Discussion · p. 9
  • Even partisan crowds can converge on accurate estimates.Source: Discussion · p. 9

Supplementary Materials

The supplementary materials include Materials and Methods supplementary text and Figures S1–S12.

machines

The supplied passages identify the paper’s authors and note that the PDF includes supplementary text and Figures S1 to S12.

machines

  • The author line names Guocheng Wang, Qi Su, and Joshua B. Plotkin.Source: machines · p. 11

This PDF file includes:

Materials and Methods

The methods link pivotality to the sensitivity of a smoothed vote, use evolutionary dynamics and a Lyapunov function to analyze collective accuracy, and estimate accuracy and diversity through simulations.

Continuous approximation of the vote

Evolutionary dynamics and Lyapunov functions

Diversity

  • Diversity is measured by joint entropy of within-group opinion compositions and across-group binary predictions; simulations discretize opinions into λ = 2 bins.Source: Diversity · pp. 12–13

Simulations

  • Monte Carlo simulations draw factors afresh each round, allow stationary payoffs to accumulate before payoff-biased within-group imitation with innovation rate u, and estimate accuracy from correct predictions.Source: Simulations · p. 13

Supplementary Text · S1 Model description

The model specifies how individuals form predictions from limited factor observations, how voting aggregates them across one or two layers, and how payoff-based imitation changes strategies over time.

S1.1 Population prediction

  • The binary outcome is the sign of a function of m randomly sampled factors, with linear and non-linear tasks distinguished by whether the classification boundary is flat or curved.The factors may be correlated or uncorrelated and are sampled from a distribution with finite variance.Source: S1.1 Population prediction · p. 14
  • Voting aggregates opinions by taking the sign of their sum, and an opinion’s magnitude indicates assurance and can increase its influence on the collective prediction.Opinions may be any real number rather than only −1 or 1.Source: S1.1 Population prediction · pp. 14–15
  • Each individual observes at most one factor and expresses an opinion as its observed value times a believed correlation strength; stereotypical individuals instead give constant opinions.The strategy is represented by the chosen factor index and correlation-strength coefficient, with a virtual constant factor for stereotypical individuals.Source: S1.1 Population prediction · p. 15
  • In a two-layer population, individuals’ opinions are first aggregated within groups and group predictions are then aggregated; equal group sizes are assumed for simplicity, and the number of groups is odd.The model can also apply to unequal group sizes.Source: S1.1 Population prediction · p. 15
  • The model treats R = 1 as perfect accuracy, meaning the population prediction always equals the real outcome Y.Source: S1.1 Population prediction · p. 15

S1.2 Social learning

  • Individuals receive payoffs after predictions, and their stationary payoff is the expected payoff over re-sampled factors when prediction occurs many times before imitation.The model assumes imitation is much slower than prediction.Source: S1.2 Social learning · p. 16
  • In each update, two individuals are randomly selected; with probability 1 − u one may imitate the other according to payoff, while probability u brings a randomly generated strategy, and selection intensity controls payoff influence.The random strategy represents innovation or mutation.Source: S1.2 Social learning · p. 16

S2 Single-layer voting

The section defines payoff schemes and a continuous approximation for analyzing single-layer voting, then shows that sufficiently large, diverse populations can evolve perfect linear predictions, while single-layer voting cannot accurately solve nonlinear prediction problems.

S2.1 Payoff structures

  • Marginal feedback multiplies the feedback payoff by a pivotality indicator, rewarding or penalizing individuals only when their opinion can change the collective prediction.When opinions nearly balance, many individuals may be pivotal at once.Source: S2.1 Payoff structures · p. 17

S2.2 Continuous approximation of the voting function

  • For evolutionary analysis, sign voting is approximated by a tanh function with transition bandwidth ε; assuming opinions are of order P, the approximation uses characteristic length P.Source: S2.2 Continuous approximation of the… · p. 18

S2.3 Evolutionary dynamics

  • For linear problems, strategy imitation can reach perfect prediction in sufficiently large, diverse populations under feedback, while marginal feedback increases accuracy monotonically to perfection.Feedback had previously been shown to evolve accurate collective prediction for linear outcomes.Source: S2.1 Payoff structures · p. 17, S2.3 Evolutionary dynamics · pp. 19–20
  • Single-layer voting cannot accurately solve nonlinear prediction problems because its collective prediction is always a linear combination of factors.Source: S2.3 Evolutionary dynamics · p. 20

S3 Two-layer voting

In two-layer voting, marginal feedback supports evolving accuracy on nonlinear prediction tasks through peer imitation, while ordinary feedback fails to improve accuracy. An alternative marginal-feedback reward also increases accuracy, but more slowly.

S3.1 Payoff structures

S3.2 Evolutionary dynamics

  • Feedback fails to improve accuracy on nonlinear prediction problems under either single-layer or two-layer voting, so the analysis focuses on marginal feedback.Source: S3.2 Evolutionary dynamics · p. 21
  • Under marginal feedback, accuracy increases over evolution in nonlinear two-layer prediction, although perfect accuracy (R = 1) may not be reachable.When a higher-payoff individual is imitated, the analysis shows the Lyapunov function decreases, implying long-run accuracy gains.Source: S3.2 Evolutionary dynamics · p. 21
  • V-induced and U-induced marginal feedback both guarantee monotone accuracy increases, but U-induced feedback yields smaller payoffs and slower convergence.The alternative structure equals zero more often than Eq. S39b, so the analysis focuses on Eq. S39b.Source: S3.2 Evolutionary dynamics · p. 22

S4 Multi-layer voting model

The model extends voting to L equal-sized layers of groups that pass binary verdicts upward. It shows that global marginal feedback, which rewards individuals when pivotal at every layer above them, increases prediction accuracy over time.

S4.1 Population structure

  • The framework has L layers of equal-sized groups: individuals form layer-1 groups, whose binary verdicts pass upward until layer L casts the final vote.The number of layer-J entities is NJ = nJ nJ+1 · · · nL.Source: S4.1 Population structure · p. 23

S4.2 Payoff structures and evolutionary dynamics

  • Global marginal feedback makes the Lyapunov function V decrease consistently, guaranteeing increased prediction accuracy in the long run.An individual receives nonzero payoff only when their opinion is pivotal at every layer above them.Source: S4.2 Payoff structures and… · p. 24
  • Simulations confirm that global marginal feedback outperforms local marginal feedback, which drops one or more pivotality indicators.Global feedback contains L −1 pivotality indicators.Source: S4.2 Payoff structures and… · p. 24
  • Both V-induced and U-induced global marginal feedback increase prediction accuracy, but V-induced feedback has fewer indicators and enables more frequent payoffs and faster evolutionary convergence.The paper focuses on V-induced feedback.Source: S4.2 Payoff structures and… · p. 24

S5 Extensions

The section defines diversity separately at each level of the hierarchy and extends voting to even numbers of groups, retaining two-layer accuracy gains while modifying multi-layer marginal feedback for parity.

S5.1.1 Diversity in two-layer population

S5.1.2 Diversity in multi-layer population

  • In multi-layer populations, diversity is computed layer by layer, discretizing predictions only in the individual layer.Higher layers have binary predictions and use the group-level entropy measure.Source: S5.1.2 Diversity in multi-layer… · p. 25

S5.2 When the number of groups is even

  • With an even number of groups, two-layer marginal feedback still guarantees increases in collective accuracy.For multi-layer voting, global marginal feedback must be modified to account for whether the number of groups in the preceding layer is odd or even.Source: S5.2 When the number of groups is even · p. 26
  • For even-sized groups, pivotal states shift to ϕx = 0 and −2, so the feedback indicator must reward either group verdict.The standard condition would otherwise leave individuals without incentives when the group prediction is −1.Source: S5.2 When the number of groups is even · p. 26

Supplementary Figures

Supplementary results show that non-linear prediction requires both layered voting and marginal feedback. Collective accuracy also persists across several variations in opinion formation, group structure, population size, and innovation.

Figure S1

  • Two-layer voting evolves collective intelligence across four opinion-formation methods, with peak accuracy at an intermediate number of groups.Discarding factor magnitude lowers accuracy, while thresholded opinions can exceed the baseline, especially with few groups; results also hold in a mixed-method population.Source: Figure S1 · p. 28

Figure S2

  • Observing more factors raises collective prediction accuracy because individuals process more information about the prediction problem.The figure compares individuals observing one, two, or three factors.Source: Figure S2 · p. 29

Figure S3

  • Unequal group sizes produce qualitatively similar stationary accuracies under both update rules; inequality lowers accuracy with few groups but raises it with many.With many groups, concentrating individuals in larger groups restores within-group diversity that small groups lack.Source: Figure S3 · p. 30

Figure S4

  • Single-layer voting with marginal feedback and two-layer voting with ordinary feedback both fail on non-linear prediction, although each solves linear prediction.Layered voting and marginal feedback are each necessary, and neither alone is sufficient, for non-linear prediction.Source: Figure S4 · p. 31

Figures S5 and S9

  • Marginal feedback rewards or penalizes only individuals whose removal or duplication could flip their group verdict, and global marginal feedback adds pivotality conditions at each higher layer.In deeper hierarchies, individuals receive nonzero payoffs less often because their reward must satisfy additional pivotality indicators.Source: Figure S5 · p. 32, Figure S9 · p. 36

Figure S6

  • Either present decisiveness or potential decisiveness alone raises accuracy to the combined marginal-feedback scheme’s stationary level, while the combination converges faster.Present decisiveness means an opinion determines the group verdict; potential decisiveness means adding a second copy could correct a wrong verdict or disrupt a correct one.Source: Figure S6 · p. 33

Figures S7 and S8

  • Larger two-layer populations reach higher stationary accuracy, whereas enlarging a single-layer population does not solve non-linear prediction.The optimal number of groups increases with population size and decreases with the number of factors; optimal group size grows only slowly or remains roughly constant.Source: Figure S7 · p. 34, Figure S8 · p. 35

Figure S10

  • The effect of innovation depends on group size: higher innovation raises accuracy in many small groups but has little effect in few large groups.A very high innovation rate lowers accuracy by adding noise.Source: Figure S10 · p. 37

A B Two-layer voting Three-layer voting

In two- and three-layer voting populations, V-induced and U-induced marginal feedback reach the same stationary accuracy. V-induced feedback pays individuals more often and converges faster.

Two-layer voting; Three-layer voting

  • V-induced and U-induced marginal feedback reach the same stationary accuracy in both two- and three-layer populations.The U-induced forms add a top-layer margin indicator, which is redundant for stationary accuracy.Source: Figure S11 · p. 38
  • V-induced feedback pays individuals more often and converges faster because it uses one fewer indicator.Source: Figure S11 · p. 38

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