Game Theory for the Evolution of Collective Intelligence

A swarm of bacteria exhibits social behavior by following a communication protocol called quorum sensing to coordinate gene expression and forage for food [2]. The animal kingdom is social, too, as birds and mammals have fine-tuned their collective behavior over millennia to tackle predation, gain vital resources, and gather information in dynamic environments [1].

In humans, collaboration within a group of individuals (as opposed to efforts made by a single individual) can be better suited to successfully tackle complex problems. This type of group behavior is known as collective intelligence. “The best real-world examples of collective intelligence are probably the stock markets and, more recently, prediction betting markets,” Joshua Plotkin of the University of Pennsylvania said. 

Engineers have extensively studied the phenomenon of collective intelligence and designed algorithms to allow a group of independent agents (e.g., servers) to combine limited information (e.g., distributed computation nodes) in an intelligent manner for successful collective outcomes. However, critical open questions remain. For instance, in the absence of any central planner, how do living beings in the natural world get together to tackle their problems?

To address this question, Plotkin and his colleagues Guocheng Wang, Qi Su, and Long Wang employ a game theoretic approach to demonstrate the evolution of collective intelligence in their new study [3].

To develop a mathematical model, the researchers consider the environmental outcome \(Y\) as a function of \(m\) random factors \(\{X_1, X_2, \textrm{...}X_m\}.\) Here, \(x_i\) represents the value of factor \(X_i\) at any given time over discrete time steps. Each random factor is sampled independently from a Gaussian distribution with mean zero and variance \({\sigma}_i^2\) (see Figure 1a). 

The researchers assume a linear relationship between the factors and the environmental outcome:

\[Y={\alpha}_0+{\alpha}_1X_1+ {\cdot}{\cdot}{\cdot}+{\alpha}_mX_m\tag1\]

The constant linear coefficients \(\alpha_i(i{\in}M)\) denote the weight of each factor \(X_i\)’s contribution to the outcome. These weights are unknown to the individuals.

“It would be helpful to think about this as, what is the temperature [outcome] tomorrow?” Plotkin said to provide an example. “There are so many random factors that will affect the temperature.”

The authors consider a population of \(n\) individuals who are tasked with predicting the environmental outcome based on observations of the factors, which serve as the determinants. Predicting the outcome by linearly combining various factors may, at first blush, seem straightforward. “Here’s what makes it hard,” Plotkin said. “Each individual in the population can only look at one factor.” 

Based on their observation of a single factor and personal belief about the correlation between the observed factor and the outcome, each individual makes a prediction, quantified by the following framework:

\[Y_A=c_Ax_{eA}.\tag2\]

This equation represents the personal prediction made by individual \(A,\) who has chosen to observe factor \(e_A\) with value \(x_{eA}.\) Here, \(c_A\) represents the individual’s belief. If \(c_A\) is positive, then \(A\) believes that the factor is positively correlated with the outcome \(Y.\) The absolute value of \(c_A\) reflects the strength that the individual believes this factor has for the outcome (see Figure 1b).

Aggregating all of the individual predictions produces a collective prediction for the group. Typically, this collective prediction is expressed as a function of individual predictions: \(\hat{Y}=g(Y_1+Y_2+{\cdot}{\cdot}{\cdot}+Y_n)\) (see Figure 1c). The authors use two different approaches to combine individual predictions into a group prediction: (i) average across the entire population or 
(ii) only average across clusters (i.e., average belief among individuals who observe the same factor). When averaging across the entire population, the collective prediction is denoted by

\[\hat{Y}=\frac{Y_1+Y_2+{\cdot}{\cdot}{\cdot}+Y_n}{n}\tag3\]

For each factor, the aggregate prediction can be expressed in terms of the predicted coefficients:

\[\hat{\alpha}_i=\frac{1}{n}\sum\nolimits_{A{\in}N_i}c_A.\tag4\]

Here, \(N_i\) denotes the group of individuals who observe the factor \(X_i.\) The collective prediction \(\hat{Y}\) can thus be expressed as \(\hat{Y}=\hat{\alpha}_0+\hat{\alpha}_1X_1+{\cdot}{\cdot}{\cdot}+\hat{\alpha}_mX_m.\)

In contrast, the clustering aggregation method is based on the framework that each individual only predicts the contribution of the factor that they have observed rather than predicting the final outcome. The predicted coefficient \(\hat{\alpha}_i\) is then calculated as the average belief of individuals who observe the same factor, instead of averaging over all the individuals:

\[\hat{\alpha}_i=\frac{1}{n_i}\sum\nolimits_{A{\in}N_i}c_A.\tag5\]

Here, \(n_i\) denotes the size of cluster \(i\) of individuals who observe the same factor \(X_i.\)

The collective prediction \(\hat{Y}\) is then determined by adding the predicted effects of all observed factors. Any unobserved factor (such that \(n_i=0\)) is excluded from the sum:

\[\hat{Y}=\sum\nolimits_{i=0}^m\frac{1}{n_i}\sum\nolimits_{A{\epsilon}N_i}Y_A.\tag6\]

After the individuals form their own predictions, they are rewarded with payoffs based on their performance. Individuals then imitate each other’s strategies, with a tendency to imitate those with greater payoffs — a process called social learning. In their paper, Plotkin and his colleagues explore varying payoff schemes that could foster collective intelligence via social learning.

They first considered an obvious payoff scheme, which rewards the “expert” who has made the most accurate prediction. This payoff structure is represented as 

\[{\pi}_A=-(Y_A-Y)^2,\tag7\]

such that individuals whose predictions \(Y_A\) are closer to the actual outcome \(Y\) will receive higher payoffs. However, such a payoff scheme presents a risk to the evolution of population-level behavior. Other individuals in the group will tend to imitate the expert and adopt the most lucrative strategy; while this approach may enhance collective accuracy in the initial stages of behavioral evolution, it can fall apart over time because little information from other unobserved factors (which may have an influence on the final outcome) are incorporated into the reward structure. “Since everyone will be copying one person’s strategy, that won’t work the next round,” Plotkin said. “It’s as if there’s not enough diversity of opinions in the population.”

Figure 2a displays the collapse in collective accuracy over time in the expert payoff scheme (purple curve). As shown in Figure 2b, simulations reveal that personal beliefs are focused on a limited number of factors, leading to poor collective accuracy.

To compensate for the shortcomings of the “expert” reward scheme, the researchers introduced a modified version called the “niche expert.” Under this structure, individuals imitate others who are good predictors, particularly when these predictors focus on “very rare” factors that impact the final outcome. “What this does is that it preserves diversity of the population,” Plotkin said. “People copy each other’s strategies, but they never end up all looking at the exact same factor because again, you don’t copy people who are looking at a common factor.”

The “niche expert” payoff structure is formulated as

\[{\pi}_A=-{\rho}_{eA}(Y_A-Y)^2,\tag8\]

where \({\rho}_{eA}\) denotes the proportion of the population that observes the same factor as individual \(A.\) Under the niche payoff structure with a diversity of observed factors, simulations demonstrated that personal beliefs fall close to the real coefficients (see Figure 2c). Assuming the clustering aggregation rule in (5) and using Monte Carlo simulations, Figure 2a shows that the niche expert scheme (blue curve) can drive the population to produce more accurate long-term predictions than the expert scheme. In fact, the niche expert scheme will indeed eventually ensure collective accuracy.

Both the expert and niche expert reward structures are based on the idea of incentivizing personal accuracy. However, would these reward structures be efficient even under diverse problem settings? For example, consider that rather than predicting the temperature, the problem has now been changed to predict how sunny it will be. While this modified problem may still depend on the same factors that were used to predict the temperature, the researchers mathematically demonstrate that the niche expert payoff structure may not be robust enough to tackle diverse problems.

The introduction of a perturbation or “environmental shock” leads to a drop in prediction accuracy for the niche expert scheme (represented by the blue curve in Figure 3a). A jolt to the system leads to a resampling of the coefficients to ensure that the population finds a solution to the new prediction task. However, the niche expert model does not immediately recover its high accuracy level because it does not allow for a diversity of beliefs after equilibrium is reached for the first time, as well as exhibits a reduction of belief variance within clusters (see Figure 3b). This is clearly not ideal. “In a changing environment where you have to keep adapting to new decisions, maintaining some exploration and variation is crucial,” Simon Levin, a Princeton University professor who was not associated with this study, said.

To strengthen their model’s robustness to various problem settings, the study authors propose another reward structure called the “feedback” payoff scheme. “By feedback, we mean how much did that person’s contribution move the collective prediction to the true value,” Plotkin said.

Under the feedback scheme, individuals whose contributions improve the overall accuracy of the collective prediction \(\hat{Y}\) are rewarded even if their individual predictions are not necessarily accurate. In this scheme, “You have to know three things — what was their individual prediction, what was the collective prediction, and what was the truth?” Plotkin said. 

The orange curve in Figure 3a demonstrates that when the model encounters a shock, the feedback scheme is able to rapidly adapt back to an almost perfect collective accuracy. “[This structure] rewards people who effectively get the wrong answer, but in the right direction,” Levin said. “That is people who, by choosing answers that are different than the average of the group, pull it back towards the middle and maintain the diversity in the system. I think it’s very clever.”

Under the feedback payoff scheme, the equilibrium state supports a diversity of beliefs that allows the model to bounce back quickly and converge to collective accuracy. The error of collective prediction is now modeled as 

\[\pi_A=Y_A(Y-\hat{Y}),\tag9\]

for the feedback payoff structure. If the collective prediction \(\hat{Y}\) is lower than the real outcome \(Y,\) then individuals with predictions larger than \(\hat{Y}\) will receive a higher payoff. This scheme rewards “reformers” in the group, whose predictions help bring the population average closer to the true outcome. The researchers used a Lyapunov function to prove that the feedback reward scheme is guaranteed to produce collective accuracy. These results demonstrate that identifying and rewarding good team players, rather than experts, enhances collective intelligence. Other advantages to the feedback payoff scheme include better performance when the factors are correlated (see Figure 3c). 

The current model framework can provide insights into the emergence of collective intelligence in real-world settings. For instance, it may be natural to consider a feedback payoff scheme in the stock market, where a trader’s prediction may move the market price towards the true outcome even if their individual predictions may not be accurate. In the future, the model could be extended to study collective intelligence when the relationship between the observable factors and the outcome is nonlinear, or when individuals intentionally report false value of factors to gain higher payoffs.

References
[1] Couzin, I.D. (2025). Collective intelligence in animals and robots. Nat. Commun., 16, 9574.
[2] Miguel, C.M.T.S., Santos, C.A., Lima, E.M.F., Rahme, L.G., & Pinto, U.M. (2026). Quorum sensing in bacteria: From mechanism to applications in foods. Curr. Opin. Food Sci., 69, 101394.
[3] Wang, G., Su, Q., Wang, L., & Plotkin, J.B. (2025). Individual incentives that promote collective intelligence. Proc. Natl. Acad. Sci., 122(51), e2516535122.