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Gifford Brickley
Pestilus is the head priest of Quna, a vast kingdom in the southeast. He serves the Qunan royal family, but secretly worships an evil deity, Rasal. His schemes manipulated the king into exiling the crown prince. In the arena, Pestilus unleashes swarms of insects to infest and cripple his enemies, releasing his innermost darkest desires.
In this paper we study a generalized case of best-of-n model, which considers three kind of agents: zealots, individuals who remain stubborn and do not change their opinion; informed agents, individuals that can change their opinion, are able to assess the quality of the different options; and uninformed agents, individuals that can change their opinion but are not able to assess the quality of the different opinions. We study the consensus in different regimes: we vary the quality of the options, the percentage of zealots and the percentage of informed versus uninformed agents. We also consider two decision mechanisms: the voter and majority rule. We study this problem using numerical simulations and mathematical models, and we validate our findings on physical kilobot experiments. We find that (1) if the number of zealots for the lowest quality option is not too high, the decision-making process is driven toward the highest quality option; (2) this effect can be improved increasing the number of informed agents that can counteract the effect of adverse zealots; (3) when the two options have very similar qualities, in order to keep high consensus to the best quality it is necessary to have higher proportions of informed agents.
Collective decision-making is a collective behavior where a group of agents (or swarm) makes a joint decision using only local perception and communication, without any centralized leadership (Valentini et al., 2017). The distinctive feature of any collective decision-making process is that once the decision is made, it is no longer attributable to any of the individual agents participating to the process. The mechanisms underlying this type of processes are widely studied in behavioral biology (Camazine et al. 2001), in statistical physics (Bialek et al., 2012; Vicsek et al., 1995; Cavagna et al., 2018), social sciences (Kok et al., 2016), and more recently in behavioral economics (Bose et al., 2017). Collective decision-making is also studied in artificial systems such as robotic swarms, in which relatively simple autonomous robots interact to generate collective responses through self-organization processes (Hamann, 2018). In swarm robotics, examples of contexts where collective decision-making is studied are the following: (1) aggregation behavior, where a swarm has to aggregate either on a site among those available in the environment (Firat et al., 2020), or in any location of environments that do not offer specific aggregation sites (Gauci et al., 2014); (2) collective motion, where the group has to choose, among a virtually infinite number of options, a direction of motion (Couzin et al., 2005); (3) collective perception, where the relative abundance of certain environmental features is assessed by local measurements and communication among the agents (Valentini et al., 2016a).
In this paper, we contribute to develop a principled understanding of how the variability in individual quality awareness in the best-of-n problem with \(n=2\) changes the decision-making dynamics. We study a group of simulated agents required to collectively choose one of two options which differ in their quality. In brief, each agent in the swarm goes through two sequential phases that periodically repeat: the exploration phase and the dissemination phase. During the exploration phase, the agents explore the environment and evaluate the options quality. During the dissemination phase, the agents interact with each other, an interaction that consists of two steps. First, each agent advertises (i.e. communicates via local broadcast) the individually selected option. We call an option opinion when it is the currently selected option by a focal agent. Then, the focal agent may change its current opinion under the influence of the other agents according to the rules of the specific decision mechanism (or voting system). The above phases are executed by all agents in an asynchronous manner. This type of scenario has already been studied in the literature (refer to Prasetyo et al., 2019, for a recent model). Differently from previous work, in this study, groups are made of agents that differ with respect to either their capability to directly evaluate the quality of each option, or their flexibility in changing option through interactions with group mates during the dissemination phases. The original contribution of this research resides in an in-depth analysis of the decision-making dynamics developed by systematically varying the model parameters, and carried out with a large methodological toolkit made of simulation models, mathematical models, and physical robots experiments. Consistently with the literature (Valentini et al., 2017), we consider the two most commonly studied decision mechanisms: the voter model and the majority rule.
The paper is organized as follows. The methodologies are described in Sect. 3 with details of the mathematical model illustrated in Sect. 3.1. The results are reported in Sect. 4. The methodology used and the results obtained with physical robots are illustrated in Sect. 5. Finally, in Sect. 6 we draw our conclusions and we illustrate a research agenda for the future.
The two central concepts explored in this article are those of informed agency and zealotry. In this section, we will review the related work, across different fields, performed around these two concepts.
There are additional recent studies that analyzed the role of informed robots in an interdisciplinary context. Mann (2020) studies how the differences in information and differences in preferences among the agents affect the use and efficacy of social information, analyzing the collective behavior generated by rational agents with differing preferences. Another very recent paper (van Veen et al., 2020) studies the impact of overload of information on the accuracy and precision in collective decision-making. Berekmri and Zafeiris (2020) focus their attention to the role of the topology of interactions among agents in a collective decision-making process, finding that a fully connected topology promotes consensus, while a hierarchical structure favor accuracy, more than consensus.
In the context of physics, Colaiori and Castellano (2016) introduced zealots in a model of pairwise social influence for opinion dynamics, showing a rich phase diagram of the possible dynamics in presence of a small percentage of zealots. In the context of Internet social networks, Hunter and Zaman (2018) studied the best placement of zealots that maximizes the impact on the consensus dynamics of the population, showing that a small number of zealots can significantly influence the overall opinion dynamics and influence the consensus of the population over disputed issues, such as Brexit. Mistry et al. (2015), using the naming game as a decision mechanism, showed that even a very small minority can drive the opinion of a large population, if committed agents are more active than the others. However, this effect can be hindered if nodes with the same opinion are more connected with each other than with nodes with different opinion, producing a polarization inside the network.
Xie et al. (2011) showed the presence of a tipping point at which a minority of zealots is able to swing the initial majority opinion in a network. The study described by Masuda (2015) focused on zealots with the voter model to perform peer-to-peer opinion influence; however, differently from our work, zealots were nodes of a complex network. Galam and Jacobs (2007), introducing zealots in a majority model, showed that the system has spontaneous symmetry breaking when zealots numbers are symmetrical for the two options, while consensus toward one option emerged even with minimal unbalance in the number of zealots. In these studies, options did not have an intrinsic quality.
In a biologically inspired model, Couzin et al. (2011) show that strongly opinionated minorities (like groups of zealots) can drive the consensus of other individuals, but uninformed individuals spontaneously inhibit this process returning the consensus to the majority, favoring in this sense a democratic consensus. We found this work very inspiring and also found an interesting parallel between our and their results which we will explore.
Compared to the latest works in swarms (Canciani et al., 2019; Matre et al., 2020; Prasetyo et al., 2019; Primiero et al., 2018), to the best of our knowledge, in this paper we study for the first time the interplay between different option quality, zealot quantity and proportion of informed agents, by extending the preliminary studies in Prasetyo et al. (2020) and De Masi et al. (2020), in which either all agents or none of the agents were able to measure the quality of their opinion and disseminate differentially based on that. In particular, we introduce here the explicit distinction between informed and uninformed agents, and study for the first time the case in which these two types of agents co-exist in the swarm at the same time.
We focus on a classic best-of-n with \(n=2\) scenario, in which a population of agents is required to collectively choose one between two foraging sites: site A and B. As mentioned above, the distinctive features of this scenario is the heterogeneity of the population, with zealots, informed, and uninformed agents. The behavior of all agents is determined by the same finite state machine made of four possible states: two exploration states referred to as \(E_\mathrmA\) and \(E_\mathrmB\), and two dissemination states referred to as \(D_\mathrmA\) and \(D_\mathrmB\) (see Fig. 1). The agents behave asynchronously with respect to each other, meaning that at a given time agents may be in any of the above states. This asynchronicity is ensured by having stochastic switching times between states, as explained below. Thus, even if multiple agents start from the same state, they soon break this synchronicity because they will switch states at different times. When in any of the two exploration states, an agent moves randomly within a square arena for a time that is randomly extracted from an exponential distribution. Agents in state \(E_\mathrmA\) are those holding opinion A, while agents in \(E_\mathrmB\) are holding opinion B. In our minimalist simulation scenario, during exploration none of the agents can change opinion. At the end of the exploration state, every agent enters into the dissemination state. Zealots and informed agents disseminate their currently held opinion for a time randomly extracted from another exponential distribution, where the time parameter depends on the quality of site A, for those agents in state \(D_\mathrmA\), or on the quality of site B, for those agents in state \(D_\mathrmB\). Contrarily to zealots and informed agents, uninformed agents disseminate their opinion for a time that is exponentially distributed with a parameter that is fixed to 1. Thus the agent disseminates always proportionally to a default quality value of 1 that represents the lack of information on the quality. At the end of the dissemination state, informed and uninformed agents can change their mind based on the logic of a voting system. In this research work, we compare the dynamics generated by two different voting systems: the majority and the voter model. When the majority is in place, an agent samples the opinion of \(G-1\) randomly chosen neighbors, where G is the group size in the majority model, including the focal agent. A single agent changes opinion when the majority of the sampled neighbors hold an opinion different from its opinion. In situations where the agent has fewer than \(G-1\) agents, it skips the application of the decision rule and does not change its opinion. In this way, we are sure that the effect of the parameter G is well captured and studied. When the voter is in place, an agent samples the opinion of a randomly chosen neighbor. It changes opinion when the sampled neighbor holds an opinion different from its opinion. Contrary to informed and uniformed agents, zealots never undergo this opinion changing process. Thus, zealots never change their opinion.
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