Studying speed-accuracy trade-offs in best-of-n collective decision-making through heterogeneous mean-field modeling

TL;DR

This work models speed-accuracy trade-offs in best-of-2 collective decisions by introducing a generalized voting rule parameterized by pooling error $\alpha$, which encodes cognitive load, and by analyzing dynamics on networks using heterogeneous mean-field theory. The model unifies the weighted voter and weighted local majority-rule processes and reveals a three-regime bifurcation in well-mixed populations, with an intermediate cognitive load often maximizing accuracy but increasing decision time. Extending to networks, the study shows that topology (via excess-degree distributions) alters stability and that sparser, more heterogeneous networks can enhance collective accuracy at the expense of speed. Scale-free networks show pronounced accuracy gains with higher $\gamma$, while 2m-regular rings expose limitations of HMF but still illustrate that reduced connectivity can improve performance, offering design insights for robotics and understanding of biological collectives.

Abstract

To succeed in their objectives, groups of individuals must be able to make quick and accurate collective decisions on the best option among a set of alternatives with different qualities. Group-living animals aim to do that all the time. Plants and fungi are thought to do so too. Swarms of autonomous robots can also be programmed to make best-of-n decisions for solving tasks collaboratively. Ultimately, humans critically need it and so many times they should be better at it. Thanks to their mathematical tractability, simple models like the voter model and the local majority rule model have proven useful to describe the dynamics of such collective decision-making processes. To reach a consensus, individuals change their opinion by interacting with neighbors in their social network. At least among animals and robots, options with a better quality are exchanged more often and therefore spread faster than lower-quality options, leading to the collective selection of the best option. With our work, we study the impact of individuals making errors in pooling others' opinions caused, for example, by the need to reduce the cognitive load. Our analysis is grounded on the introduction of a model that generalizes the two existing models (local majority rule and voter model), showing a speed-accuracy trade-off regulated by the cognitive effort of individuals. We also investigate the impact of the interaction network topology on the collective dynamics. To do so, we extend our model and, by using the heterogeneous mean-field approach, we show the presence of another speed-accuracy trade-off regulated by network connectivity. An interesting result is that reduced network connectivity corresponds to an increase in collective decision accuracy.

Figures (18)

• Figure 1: The probability $P_\alpha(x)$ given by Eq. \ref{['eq:Px']} for representative values of the pooling error $\alpha$, which is a parameter inversely proportional to agents' cognitive load. For $\alpha=0$, our model corresponds to the (weighted) local majority rule model Krapivsky2003Valentini2015aamas, for $\alpha=1$ our model corresponds to the (weighted) voter model Clifford1973Valentini2014, for $0<\alpha<1$ our model interpolates between the two, and for $\alpha>1$ the agents change their opinion with little attention to others' opinion.
• Figure 2: Bifurcation diagrams showing the equilibria of the mean-field model of Eq. \ref{['eq:odea2']} as a function of $\alpha \in [0,2]$ for $Q_A=1$ and $Q_B=0.9$. Green dots show stable equilibria, red dots show unstable equilibria, and blue dots show the average asymptotic values of $\langle n_A\rangle/N$ obtained with stochastic numerical simulations of a population of $N=500$ agents interacting on a complete graph and initialized at $n_A(0)=100$. The match between mean-field model predictions and simulations is good.
• Figure 3: Stability diagrams and convergence time. In panel (a) we report the stability diagram of the mean-field model \ref{['eq:odea2']} as a function of the quality ratio $Q$ and the pooling error $\alpha$ (inversely proportional to the cognitive cost). The parameter space is divided into three regions: in the red region the population makes accurate collective decisions for any initial condition; in the blue region the population remains locked at indecision with agents' opinions fluctuating between the two options; in the green region, a consensus for either alternative is possible depending on the initial conditions, therefore mistakes are possible. The three regions are separated by the curves $\alpha=Q$ (white curve) and $\alpha=1/Q$ (black curve). Panels (b-c) show the results from simulations of $N=500$ agents interacting on a fully connected (all-to-all) network. For each couple $(Q,\alpha)$, we perform $100$ independent simulations with random initial configurations (i.e. the initial number of agents with opinion $A$ is uniformly drawn in $[0,N]$ at each run). In (b), the RGB color of each pixel is computed by assigning to the three values R, G, and B a value equal to the proportion of simulations that terminated at $n_A=500$, $n_A=0$, and $0<n_A<500$, respectively. In (c), the colormap shows the average number of timesteps needed to reach a consensus, i.e., $n_A=500$ or $n_B=500$, as a function of $Q$ and $\alpha$. The white region in the top-right corner indicates the absence of data as no simulations reached a consensus for either option.
• Figure 4: Schematic representation of the probabilities involved in the heterogeneous mean-field computations. The focal node $i$ has $k$ neighbors (degree $k$), denoted by $i_1,\dots,i_r,\dots,i_k$. Each neighbor, e.g., node $i_r$, has an excess degree $j_r$ with probability $q_{j_r}$ and therefore degree $j_r+1$. With probability $a_{j_r+1}$, she has opinion $A$, and therefore with probability $1-a_{j_r+1}$, she has opinion $B$.
• Figure 5: Stability diagrams (panels a-c), decision outcome from multiagent simulations (panels d-f) and convergence time (panels g-i) for collective decision-making on scale-free networks with $k_{min}=2$ as a function of the pooling error $\alpha$ and quality ratio $Q$. We present the results for three values of the exponent $\gamma$ regulating network connectivity: top row $\gamma=2.2$, central row $\gamma=2.6$, bottom row $\gamma=3.1$. (a-c) Left column panels show the convergence diagram of the mean-field model \ref{['eq:dakdt3b']}. The parameter space is divided into the same three regions of Fig. \ref{['fig:stability01complete']}a using the same color code. (d-i) Central and right column panels show the results of simulations (100 independent runs for each $(Q,\alpha)$ configuration) of $N=500$ agents interacting on a scale-free network with random initial configurations (i.e., $n_A(t=0) \sim \mathcal{U}(0,N)$) for $50\,000$ time steps. (d-f) Central column panels show the outcome of the collective decision-making process using the same RGB color code as Fig. \ref{['fig:stability01complete']}b. (g-i) Right column panels show the average number of timesteps needed to reach a consensus, i.e., $n_A=500$ or $n_B=500$. The top-right white region indicates the absence of data, as the system never reaches a consensus.