Collective decision with 100 Kilobots: speed versus accuracy in binary discrimination problems

Abstract

Achieving fast and accurate collective decisions with a large number of simple agents without relying on a central planning unit or on global communication is essential for developing complex collective behaviors. In this paper, we investigate the speed versus accuracy trade-off in collective decision-making in the context of a binary discrimination problem—i.e., how a swarm can collectively determine the best of two options. We describe a novel, fully distributed collective decision-making strategy that only requires agents with minimal capabilities and is faster than previous approaches. We evaluate our strategy experimentally, using a swarm of 100 Kilobots, and we study it theoretically, using both continuum and finite-size models. We find that the main factor affecting the speed versus accuracy trade-off of our strategy is the agents’ neighborhood size—i.e., the number of agents with whom the current opinion of each agent is shared. The proposed strategy and the associated theoretical framework can be used to design swarms that take collective decisions at a given level of speed and/or accuracy.

Appendix

We performed additional robot experiments to estimate the values of the average group size $\mathcal {G}$ in the two settings and that of the time $\sigma ^{-1}$ necessary to exploration a site. Each Kilobot records internally its series of exploration times and that of the number of neighbors at decision time. After an entire experiment, the acquired data are downloaded from the robots using a wired connection. We had to limit the number of experiments for data acquisition because it is very time-consuming. Specifically, we performed four runs: two runs to measure the actual average group size in the two settings ($\mathcal {G}_{\mathrm{max}}=5$ and $\mathcal {G}_{\mathrm{max}}=25$); and two runs to measure the average time required to complete the exploration of a site, again in the two settings.

Fig. 14

The figure shows the results form the statistical analysis performed on the additional robot experiments: a probability mass function of the group size $\mathcal {G}$ for the two parameter settings $\mathcal {G}_{\mathrm{max}}=5$ (green) and $\mathcal {G}_{\mathrm{max}}=25$ (purple), b probability density function of the time $\sigma ^{-1}$ necessary for a robot to explore a site (dotted line gives the average value of $\sigma ^{-1}$) (Color figure online)

Figure 14a reports the probability mass function of the actual neighborhood size $P(\mathcal {G})$ estimated from a single experimental run for each setting. When $\mathcal {G}_{\mathrm{max}}=25$ (purple histograms, 652 samples), the average group size estimated using this mass function was 8.57, while it was 4.4 for $\mathcal {G}_{\mathrm{max}}=5$ (green histograms, 682 samples). We have therefore a difference of almost a factor of two between the two averages. Concerning the exploration time, we have graphically shown in [48] that the probability density functions resulting from $\mathcal {G}_{\mathrm{max}}=5$ (504 samples) and $\mathcal {G}_{\mathrm{max}}=25$ (602 samples) were very similar (as one would expect). To further investigate this point, we performed a two-sample Kolmogorov–Smirnov test. The null hypothesis that the two samples come from a different distribution could not be rejected (p value $=0.5364$), which supports our original conclusion that data sets are consistent with each other. We therefore merged the two data sets to improve our estimate of the exploration time. Figure 14b shows the probability density function of the time $\sigma ^{-1}$ (where $\sigma$ is a rate, see Sect. 5) a robot spends to complete the exploration of a site. The average exploration time is 6.072 min (dotted line).