physics.soc-phOct 5, 2026
SaveAn evolutionary origin of collective decision making in humans and machines
Organizations: Department of Biology, University of Pennsylvania, Philadelphia, PA 19104, USA · School of Automation and Intelligent Sensing, Shanghai Jiao Tong University, Shanghai 200240, China · Meiji Institute for Advanced Study of Mathematical Sciences, Meiji University, Nakano, Tokyo 164-8525, Japan
Abstract
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.
Figures & tables
Figure 1: A model of collective prediction by layered voting. The population predicts a binary outcome by aggregating the opinions of individuals, either in a single vote or in layers. ( A ) A binary outcome is determined by factors through a fixed but unknown true classification rule . The factors are redrawn at random in each round, so the outcome varies over time. ( B ) Each individual observes a single factor and issues an opinion , encoded by their strategy : which factor to observe, , and the believed correlation between that factor and the outcome, . ( C ) Under single-layer voting the collective prediction is the sign of the total opinion, . ( D ) Under two-layer voting each group of individuals first returns a majority verdict, and then the majority of the group verdicts produces the collective prediction.
Figure 2: A two-layer population whose members are rewarded only for pivotal opinions evolves, through peer imitation alone, to solve non-linear prediction problems. Panels A to D show the collective accuracy as individuals in the population evolve their strategies in response to marginal feedback rewards; panels E to L compare a prediction sampled from an evolved two-layer population to the true outcome, with predictions shown in blue ( ) and white ( ). ( A and B ) Single-layer voting under the feedback reward scheme evolves near-perfect accuracy for a linear prediction problem ( A ) but performs no better than chance in a non-linear XOR problem ( B ). The linear problem is shown for factors and the non-linear XOR problem for . ( C and D ) Two-layer voting with the marginal feedback reward scheme evolves accurate prediction for both linear ( C ) and non-linear ( D ) problems. Accuracy on the non-linear problem declines as grows, because sets both the number of factors and the degree of the classification boundary. ( E to L ) The stationary prediction of a two-layer population ( F , H , J , and L ) closely matches the true outcome ( E , G , I , and K ) as seen in four two-factor problems: a linear boundary ( E and F ), a product boundary ( G and H ), a parabola ( I and J ), and a circle ( K and L ), reaching accuracies of – . Parameters: (so that is close to ), ( C to L ), , (uniform distribution), , , .
Figure 3: An intermediate number of groups predicts best, by balancing two kinds of diversity. For a fixed population size ( ) we vary the number of groups and measure the stationary accuracy ( A and B ) and the diversity of predictions ( C ). ( A ) For a linear prediction problem, accuracy is highest for a single group and declines only slightly as the group number increases. ( B ) For the non-linear (XOR) problem, accuracy is maximized at an intermediate number of groups. ( C ) For the non-linear prediction problem with , within-group diversity (the entropy of individual opinions) falls while inter-group diversity (the entropy of group verdicts) rises, as the number of groups increases; both types of diversity are large only for an intermediate number of groups. Diversity is shown in the initial and in the stationary state. Parameters: , , , , , .
Figure 4: Deeper hierarchies solve more complex problems under global marginal feedback. ( A ) A three-layer population, illustrated with first-layer groups of size , second-layer groups of size , and a third-layer group of size . The population size is . Individual lies in first-layer group (opinion sum ), and lies in second-layer group (verdict sum ). Under global marginal feedback is rewarded only when their opinion is pivotal at both layers, that is when and . ( B ) On the XOR problem ( ), evolution under global marginal feedback (Eq. 6 ) produces the highest accuracy; the two forms of local marginal feedback, each retaining a single pivotality indicator improve accuracy but less so; and feedback alone performs no better than random chance. ( C ) Stationary accuracy against the first-layer group size , for three-layer populations with , and (fixed ). An intermediate group size is best, and most three-layer structures exceed the maximum accuracy of any two-layer structure in the same population (red dashed line). ( D ) Across problem complexities , the best three-layer population ( and ) predicts more accurately than the best two-layer population ( ). ( E ) Searching over all grouping structures, the maximum achievable accuracy grows with the number of layers: for two layers, for three, and for four ( ). Parameters: , ( B and C ), ( E ), , , , , , and ( B and D ).
Figure S1: Collective intelligence is robust to the way individuals form opinions. In the main text an individual forms the continuous opinion ; here we vary how individuals form opinions and re-measure the stationary accuracy of two-layer voting, across different groupings. Faint lines reproduce the accuracy of the baseline model from panel A for comparison. ( A ) The baseline model, (the same data as Fig. 3 B). ( B ) Individuals give a binary opinion from the sign of the observed factor, ; discarding the magnitude of the factor lowers the stationary accuracy. ( C ) Individuals give a thresholded binary opinion, , with strategy ; accuracy exceeds the baseline, especially for few groups. ( D ) Individuals give the sign of a quadratic function of the observed factor, . ( E ) A population mixing all four ways of forming opinions, where imitation copies a neighbor’s method together with its parameters. Across all cases, two-layer voting evolves collective intelligence, with peak accuracy at an intermediate number of groups. The qualitative results are driven by the population and reward schemes, not by the specific way individuals form opinions. Parameters: (two layers), , ( B ), , , .
Figure S2: Observing more factors raises the accuracy of collective prediction. Here each individual observes factors and forms a linear opinion from them, , with strategy . ( A to C ) Each individual observes one (the baseline model), two, and three factors, respectively. The more factors an individual observes, the more accurate the collective prediction, since individuals then process more information about the prediction problem. Parameters: (two layers), , , , .
Figure S3: Collective accuracy persists under unequal group sizes. For a fixed population size , group is given size proportional to , so that controls the inequality in group sizes and recovers equal groups. Two updating rules are shown: ( A ) an individual is drawn from the population and then a partner from ’s group; ( B ) a group is drawn and then two individuals within it. The two updating rules produce qualitatively similar stationary accuracies. When groups are few, greater inequality lowers accuracy; when groups are many, greater inequality raises it, because concentrating individuals into fewer, larger groups restores the within-group diversity that small groups lack. Parameters: , , , , , , .
Figure S4: Both layered voting and marginal feedback are required to solve non-linear decision problems. Figure 2 shows that single-layer voting evolved under feedback fails on non-linear prediction problems while two-layer voting evolved under marginal feedback succeeds. Here we test the two remaining combinations: single-layer voting under marginal feedback ( A and B ) and two-layer voting under feedback ( C and D ). ( A and C ) For a linear prediction problem, both combinations evolve accurate prediction. ( B and D ) In a non-linear prediction problem, both fail to exceed random chance. Layered voting and marginal feedback are therefore each necessary, and neither alone is sufficient, for non-linear prediction problems. Parameters: , , , , , , .
Figure S5: Marginal feedback rewards only pivotal opinions. ( A ) Under the feedback payoff scheme, , every individual is rewarded or penalized regardless of whether their opinion is pivotal. ( B ) The marginal feedback scheme, , adds the indicator , where the two terms cannot hold at once (except when ). The first term marks an individual whose removal would flip the group verdict (individual ); the second marks an individual a copy of whose opinion would flip it (individual ). Only such pivotal individuals are rewarded (or penalized). For illustration a single group is shown, with the sum of all opinions (i.e., ).
Figure S6: Either component of marginal feedback promotes collective intelligence. Marginal feedback decomposes into two terms: and . The first term represents rewards for individuals whose opinion currently determines the group verdict (i.e., the group verdict will change if is removed); and the second term represents rewards for individuals whose opinion has potential to correct a wrong verdict (or disrupt a currently correct verdict) if a second copy of this opinion were added into the group. Thus, the first term measures individuals’ present decisiveness and the second term measures their potential decisiveness. Either component alone raises accuracy to the same stationary level as the combination of the two; the combination converges faster. Parameters: , , , , .
Figure S7: Larger populations reach higher accuracy in non-linear prediction problems. Stationary accuracy against population size, for a non-linear prediction problem with factors. ( A ) A single-layer population cannot predict a non-linear prediction problem, so enlarging it does not help. ( B and C ) A two-layer population can be enlarged either by adding groups at fixed group size ( B ) or by enlarging groups at fixed group number ( C ); in both cases a larger population reaches higher stationary accuracy. Parameters: , , , , ( B ), ( C ), .
Figure S8: The optimal number of groups grows slowly with population size. ( A ) The number of groups that maximizes stationary accuracy increases with population size, and decreases with the number of factors at fixed population size. ( B ) The corresponding optimal group size grows only slowly with population size, or remains roughly constant. A growing population therefore improves its predictions mainly by forming new groups rather than by enlarging existing ones. Parameters: , , , , .
Figure S9: Global marginal feedback in multi-layer populations. ( A ) An -layer population: in layer , every entities form a higher-layer group whose verdict passes to layer , with . Writing for the layer- entity containing individual , global marginal feedback is , with pivotality indicators. ( B and C ) The probability that a random individual receives a nonzero payoff, given . In the two-layer case ( B ) it rises with the number of groups and is independent of the problem scale ; in the three-layer case ( C ) it is set mainly by the number of top-layer entities (i.e., ). Individuals in deeper hierarchies are rewarded less often, because of the extra indicator. ( D ) In two-layer populations, the stationary payoffs are small due to the pivotality indicator, especially for few groups. So stronger selection amplifies payoff differences and raises accuracy until saturation; with more groups, a lower selection intensity suffices. Main-text results are in the saturation regime. Parameters: , , ( C and D ), , , .
Figure S10: Collective accuracy is robust to the innovation rate. Stationary accuracy against the innovation rate , for two-layer populations with few groups ( ), an intermediate number ( ), and many groups ( ). Innovation adds strategic diversity within groups. When groups are few and large, within-group diversity is already high, so accuracy is nearly independent of the innovation rate, and a very high rate lowers it by adding noise. When groups are many and small, within-group diversity is low, so a higher innovation rate raises accuracy, until a very high rate again adds noise. Parameters: , , .
Figure S11: Two forms of marginal feedback reach the same accuracy. The marginal feedback of Eq. 5 and the global marginal feedback of Eq. 6 follow from requiring the Lyapunov function to decrease; we call them -induced. Requiring instead that decrease yields -induced forms, equal to the -induced forms times an extra indicator on the top-layer margin (see supplementary text section S4). ( A and B ) The two forms reach the same stationary accuracy in a two-layer ( A ) and a three-layer ( B ) population, so the extra top-layer indicator is redundant; the -induced form, with one fewer indicator, pays individuals more often and converges faster. Parameters: , ( A ), ( B ), ( A ), and ( B ), .
Figure S12: The approximation of the sign function and its derivative. ( A ) Here we use function to approximate the sign function. Parameter controls the characteristic length (bandwidth) of the transition region between and for . ( B ) For the function , we can compute its derivative. The derivative can be approximated by an indicator function , which equals if and equals otherwise.