Collective decision-making refers to the phenomenon whereby a collective of agents makes a choice in a way that, once made, it is no longer attributable to any of the individual agents. This phenomenon is widespread across natural and artificial systems and is studied in a number of different disciplines including psychology (Moscovici and Zavalloni, 1969; Hirokawa and Poole, 1996), biology (Camazine et al., 2001; Conradt and List, 2009; Couzin et al., 2011), and physics (Galam, 2008; Castellano et al., 2009). For example, social insects such as honeybees and ants are able to collectively choose and commit to a single suitable nest site using collective and distributed information processing (Franks et al., 2002). In a similar way, schools of fish, flocks of birds, and wild baboons are able to move coherently in a common direction using only local interactions with their neighbors (Okubo, 1986; Sumpter, 2010; Kao et al., 2014; Strandburg-Peshkin et al., 2015). A different situation arises in the context of other social insect colonies, where workers are able to collectively allocate themselves to a variety of tasks, such as foraging, brood care, and nest construction, and to change their allocation as a function of the colony needs (Pinter-Wollman et al., 2013; Gordon, 2016; Jandt and Gordon, 2016). The distinction between these two situations has been formalized in the context of swarm robotics by Brambilla et al. (2013) and organized in two categories: consensus achievement and task allocation (see Figure 1). The first category encompasses systems where agents aim at making a common decision on a certain matter (see Section 4 and Section 5), whereas the second category includes systems where agents allocate themselves to different tasks, with the objective to maximize the performance of the collective (Gerkey and Matarić, 2004; Liu et al., 2007; Correll, 2008; Berman et al., 2009). Understanding and designing both types of collective decision-making systems is pivotal for the development of robot swarms (Brambilla et al., 2013).

The field of swarm robotics aims at developing robotic systems that exhibit features similar to those that characterize natural self-organized systems (Brambilla et al., 2013; Dorigo et al., 2014). In particular, it aims at developing systems that are scalable to different swarm sizes (i.e., the number of robots), robust to a broad range of environmental conditions (e.g., same application but different environments), tolerant to failures of individual components (i.e., the robots), and offer flexible solutions to different goals (i.e., application scenarios). To obtain these features, swarm robotics systems are characterized by robots interacting only locally, without access to global information, and without a leader to coordinate the work activities. Similar to natural systems, swarm robotics systems achieve a desired collective behavior through self-organization.

Recent review articles have highlighted the intrinsic empirical nature of swarm robotics as one of the primary challenges of this field (Brambilla et al., 2013; Hamann et al., 2016). This fact is exacerbated by the lack of a formal engineering process that allows the designer to develop individual behaviors and interaction rules that generate a collective behavior with the desired characteristics. In our view, one important reason for this is the lack of agreement on the definition of what are the possible classes of problems for robot swarms and, consequently, we lack a formal understanding of each of these classes.

The goal of this article is to provide a contribution toward a formal understanding of swarm robotics problems. We focus on one specific class of problems, that is, on consensus achievement problems. This class of problems encompasses a wide set of application scenarios faced by robot swarms: whether the swarm needs to select the shortest path to traverse, the most suitable morphology to create, or the most favorable rendez-vous location, it first needs to address a consensus achievement problem (Christensen et al., 2007; Garnier et al., 2009; Montes de Oca et al., 2011). We further decompose this wide set of problems into two classes (cf. Figure 1), depending on the cardinality of the choices available to the swarm. When the possible choices of the swarm are finite and countable, we say that the consensus achievement problem is discrete. An example of a discrete problem is the selection of the shortest path connecting the entry of a maze to its exit (Szymanski et al., 2006). Alternatively, when the choices of the swarm are infinite and measurable, we say that the consensus achievement problem is continuous. For example, the selection of a common direction of motion by a swarm of agents flocking in a two- or three-dimensional space (Reynolds, 1987; Olfati-Saber et al., 2007) is a continuous problem.

In this article, we introduce the best-of-n problem, i.e., an abstraction capturing the structure and logic of discrete consensus achievement problems that need to be solved in several swarm robotics scenarios. First, we provide a taxonomy of possible variants of the best-of-n problem, irrespective of the specific application scenario and design solution. According to this taxonomy, we group together research studies in which the environment and the robot capabilities share common characteristics. In doing so, we identify which variants of the best-of-n problem have received less attention and thus require further research. Second, we provide a more in-depth review of the literature using an additional taxonomy that classifies research studies according to the design approach utilized to develop the collective decision-making strategy. This second classification of the literature allows us to discuss for each different design approach the domain of application and the level of portability of the resulting strategies.

Discrete consensus achievement problems similar to those faced by robot swarms have been studied in a number of different contexts. The community of artificial intelligence focused on decision-making approaches for cooperation in teams of agents and studied methods from the theory of decentralized partially observable Markov decision processes (Bernstein et al., 2002; Pynadath and Tambe, 2002). Discrete consensus achievement problems have been considered also in the context of the RoboCup soccer competition (Kitano et al., 1997). In this scenario, robots in a team are provided with a predefined set of plays and are required to agree on which play to execute. Different decision-making approaches have been developed to tackle this problem including centralized (Bowling et al., 2004) and decentralized (Kok and Vlassis, 2003; Kok et al., 2003) play-selection strategies. Other approaches to consensus achievement over discrete problems have been developed in the context of sensor fusion to perform distributed object classification (Kornienko et al., 2005a,b). These approaches, however, rely on sophisticated communication strategies and are suitable only for relatively small teams of agents. Finally, discrete consensus achievement problems are also studied by the community of statistical physics. Examples include models of collective motion in one-dimensional spaces (Czirók et al., 1999; Czirók and Vicsek, 2000; Yates et al., 2009) that describe the marching bands phenomenon of locust swarms (Buhl et al., 2006) as well as models of democratic voting and opinion dynamics (Galam, 2008; Castellano et al., 2009).

Continuous consensus achievement problems have been mainly studied in the context of collective motion, that is, flocking (Camazine et al., 2001). Flocking is the phenomenon whereby a collective of agents moves cohesively in a common direction. The selection of a shared direction of motion represents the consensus achievement problem. In swarm robotics, flocking has been studied in the context of both autonomous ground robots (Nembrini et al., 2002; Spears et al., 2004; Turgut et al., 2008; Ferrante et al., 2012, 2014) and unmanned aerial vehicles (Holland et al., 2005; Hauert et al., 2011) with a focus on developing control and communication strategies suitable for minimal and unreliable hardware. Apart from flocking, the swarm robotics community focused on spatial aggregation scenarios, where robots are required to aggregate in the same region of a continuous space (Trianni et al., 2003; Soysal and Şahin, 2007; Garnier et al., 2008; Gauci et al., 2014; Güzel and Kayakökü, 2017). Outside the swarm robotics community, the phenomenon of flocking is also studied within statistical physics (Szabó et al., 2006; Vicsek and Zafeiris, 2012) with the aim of defining a unifying theory of collective motion that equates several natural systems. A popular study is provided by the minimalist model of self-driven particles proposed by Vicsek et al. (1995). The community of control theory has intensively studied the problem of consensus achievement (Mesbahi and Egerstedt, 2010) with the objective of deriving optimal control strategies and proves their stability. In addition to flocking and tracking (Savkin and Teimoori, 2010; Cao and Ren, 2012), the consensus achievement problems studied in control theory include formation control (Ren et al., 2005), agreement on state variables (Hatano and Mesbahi, 2005), sensor fusion (Ren and Beard, 2008), as well as the selection of motion trajectories (Sartoretti et al., 2014). Continuous consensus achievement problems have been also studied in the context of wireless sensor networks with the aim of developing algorithms for distributed estimation of signals (Schizas et al., 2008a,b). More recently, continuous consensus achievement has been investigated using a network-theoretic perspective, which focuses on the signaling network emerging between interacting agents (Komareji and Bouffanais, 2013; Shang and Bouffanais, 2014).

The best-of-n problem requires a swarm of robots to make a collective decision over which option, out of n available options, offers the best alternative to satisfy the current needs of the swarm. We use the term options to abstract domain-specific concepts that are related to particular application scenarios (e.g., foraging patches, aggregation areas, traveling paths). We refer to the different options of the best-of-n problem using natural numbers, 1, …, n. Given a swarm of N robots, we say that the swarm has found a solution to a particular instance of the best-of-n problem as soon as it makes a collective decision for any option i ∈ {1, …, n}. A collective decision is represented by the establishment of a large majority M ≥ (1 − δ)N of robots that favor the same option i, where δ, 0 ≤ δ ≪ 0.5, represents a threshold set by the designer. The constraint δ ≪ 0.5 requires the opinions within the swarm to form a cohesive collective decision for a single option (i.e., the opinions are not spread over different options of the best-of-n problem). In the boundary case with δ = 0, we say that the swarm has reached a consensus decision, i.e., all robots of the swarm favor the same option i.

The best-of-n problem requires a swarm of robots to make a collective decision for the option i ∈ {1, …, n} that maximizes the resulting benefits for the collective and minimizes its costs. Each option i is characterized by a quality and by a cost that are function of one or more attributes of the target environment (Reid et al., 2015). For example, when searching for a new nest site, honeybees instinctively favor candidate sites with a certain volume, exposure, and height from the ground (Camazine et al., 1999); however, their search is limited to sites within a certain distance from the current nest location. In this example, the volume, exposure, and height from the ground of a candidate site represent the option qualities, while the distance from the current nest location to the candidate site location represents the option cost.

Let ρ_i be the opinion quality associated with each option i ∈ {1, …, n}. Without loss of generality, we consider the quality of each option i to be normalized in the interval (0;1]. Option i is a maximum quality option if ρ_i = 1. We use the term option quality as an abstraction to represent the quality of domain-specific attributes of primary concern for the objective of the swarm. These attributes are defined by the designer for the specific application scenario. Robots are programmed to actively measure and estimate their quality and to prefer options whose attributes have certain characteristics. For example, in a collective construction scenario, the focus of the swarm is often on the dimension of a candidate site for construction; differently, in a foraging scenario, the swarm usually focuses on the type, quality, or availability of food in a foraging patch. Once evaluated, the information carried by the option quality is used by the robots to directly influence or modulate the collective decision-making process in favor of the best option (Garnier et al., 2007a; Valentini et al., 2016b).

We define the option cost σ_i > 0 associated with each option i ∈ {1, …, n} as the cost in terms of the average time needed by a robot to obtain one sample of the quality ρ_i of option i. The option cost is a function of the characteristics of one or more attributes of the target environment. We will use the term option cost as an abstraction for the cost resulting from these domain-specific features. These attributes depend on the target scenario, and robots are not required to perform measurements to evaluate them. Instead, this cost biases the collective decision-making process indirectly: the bias is induced by the environment and is not under the control of individual robots. For example, when foraging, certain species of ants find the shortest traveling path between a pair of locations as a result of pheromone trails being reinforced more often on the shortest path (Goss et al., 1989). These ants do not measure the length of each path individually and do not lay more or less pheromone depending on the path they are on. However, the length of a path indirectly influences the amount of pheromone laid over the path by the ants. Note that other sources of cost such as the amount of energy consumed or the risk involved in exploring a certain option need to be considered as option cost only when they affect the time necessary to explore a certain option while otherwise they need to be considered during the estimation of the option quality.

We classify instances of the best-of-n problem in five different categories depending on how the option quality and the option cost are configured in the application scenario and perceived by the robots (cf. Figure 2). In general, the best-of-n problem is either symmetric or asymmetric with respect to both the option quality and option cost. If all options have the same quality (respectively, cost), we say that the best-of-n problem is symmetric with respect to the option quality (option cost). If at least two options of different quality (cost) exist, we say that the best-of-n problem has asymmetric option qualities (costs). When both option qualities and option costs are symmetric, the options of the best-of-n problem are equivalent to each other and the objective of the swarm is to make a collective decision for any of them. This problem is known in the literature as the symmetry-breaking problem (de Vries and Biesmeijer, 2002; Hamann et al., 2012). When the option qualities are symmetric but the option costs are not, the objective of the swarm is to make a collective decision for the option of minimum cost. In the opposite situation, i.e., asymmetric qualities but symmetric costs, the best option for the swarm corresponds to the option of maximum quality. Finally, when both option qualities and option costs are asymmetric, we further distinguish between two situations: in the first situation, the option qualities and the option costs are synergic and the best option has both maximum quality and minimum costs; in the second situation, they are antagonistic and the best option is characterized by a trade-off between quality and cost.

Finally, the option quality and the option cost can be either static or dynamic. This feature is particularly relevant to guide the choices of designers during the design of a collective decision-making strategy. When the option quality is static, designers favor collective decision-making strategies that results in consensus decisions (Parker and Zhang, 2009; Montes de Oca et al., 2011; Scheidler et al., 2016). Differently, when the option quality is dynamic, i.e., a function of time, designers favor strategies that result in a large majority of robots in the swarm favoring the same option without converging to consensus (Parker and Zhang, 2010; Arvin et al., 2014). In this case, the remaining minority of agents that are not aligned with the current collective decision keep exploring other options and possibly discover new ones, making the swarm adaptive to changes in the environment (Schmickl et al., 2009b). Additionally, a consensus decision corresponding to a large majority rather than unanimity allows swarm systems to swiftly react to perturbations as in the case of fish schools (Calovi et al., 2015).

The efforts of researchers in the last decade resulted in research contributions that span over a number of different design approaches. Brambilla et al. (2013), who surveyed the field of swarm robotics focusing on design methodologies, organized research studies in two categories, behavior-based and automatic design methods. In this section, we make use of a similar taxonomy to classify research studies according to the methodology used by designers to derive their collective decision-making strategies (see Figure 7). Differently from Brambilla et al. (2013), our focus is not on the design methodology but on the structure and functioning of the designed strategies.

We divide the design approaches used to address the best-of-n problem into two categories: bottom-up and top-down (Crespi et al., 2008). In a bottom-up approach, the designer develops the robot controller by hand, following a trial and error process where the robot controller is iteratively refined until the swarm behavior fulfills the requirements. Conversely, in a top-down approach, the controller for individual robots is derived directly from a high-level specification of the desired behavior of the swarm by means of automatic techniques, for example, as a result of an optimization process (Nolfi and Floreano, 2000; Bongard, 2013).

In a bottom-up approach (see Section 5.1), a typical design paradigm consists in defining different atomic behaviors that are combined together by the designer to obtain a probabilistic finite-state machine that represents the robot controller (Scheutz and Andronache, 2004). Each behavior used in the robot controller is implemented by a set of control rules that determine (i) how a robot works on a certain task and (ii) how it interacts with its neighbor robots and (iii) with the environment. We organize collective decision-making strategies designed by means of a bottom-up process in two categories (see Figure 7), according to how the control rules governing the interaction among robots have been defined. In the first category, that we call opinion-based approaches, robots have an explicit internal representation of their favored opinion, and the role of the designer is to define the control rules that determine how robots exchange opinions and how they change their own opinion. The main advantage of opinion-based approaches is that they result in strategies that are generic and can be applied to different application scenarios. In the second category, that we call ad hoc approaches, we consider research studies where the control rules governing the interaction between robots have been defined by the designer to address a specific task. As opposed to opinion-based approaches, control strategies belonging to this category are not explicitly designed to solve a consensus achievement problem; nonetheless, their execution by the robots of the swarm results in a collective decision. In this category, we consider research studies that focus on the problem of spatial aggregation and on the problem of navigation in unknown environments.

In a top-down approach (see Section 5.2), the robot controller is derived automatically from a high-level description of the desired swarm behavior. We organize research studies adopting a top-down approach in two categories: evolutionary robotics and automatic modular design (AutoMoDe). Evolutionary robotics (Nolfi and Floreano, 2000; Bongard, 2013) relies on evolutionary computation to obtain a neural network representing the robot controller. As a consequence, this design approach results in black-box controllers. In contrast, automatic modular design (Francesca et al., 2014) relies on optimization processes to combine behaviors chosen from a predefined set and obtain a robot controller that is represented by a probabilistic finite-state machine.

In this article, our aim was to improve our formal understanding of a given class of problems within swarm robotics. We divided collective decision-making problems in task allocation and consensus achievement, whereby the latter is further divided into discrete and continuous problems. We then focused on discrete consensus achievement. We formally defined the structure of the best-of-n problem and showed how this general framework covers a large number of specific application scenarios. We analyzed and surveyed the literature on discrete consensus achievement from two complementary points of view: the problem structure and the solution design.

In order to analyze the literature with a focus on the structure of the underlying cognitive problem, we first formalized the best-of-n problem. In the best-of-n problem, a swarm of robots is required to make a collective decision about which of a set of n available options offers the best alternative to satisfy its current needs. In the best-of-n problem, each option is characterized by an intrinsic quality and by a cost in terms of time necessary to evaluate that option. Depending on how quality and cost interact with each other, we distinguished between five different variants of the best-of-n problem and defined a problem-oriented taxonomy. Using this taxonomy, we surveyed the literature of swarm robotics and classified research studies according to the considered variant of the best-of-n problem.

As it emerged at the end of Section 4 and perhaps due to their simpler problem structure, the first three variants of the best-of-n problem have been the subject of a large portion of the literature. The first variant is the simplest form of best-of-n problem, whereby options have both equal quality and equal cost (i.e., symmetry-breaking problem), and the objective of the swarm is to make a decision for any of the available options. The second variant is characterized by options of equal quality but with different cost, and the objective of the swarm is to minimize the cost of the chosen option. We saw that, in this case, the environment has a key role in biasing the collective decision and no direct measurement by individual robots is required. In the third variant, options differ in their quality but have the same cost. A collective decision in favor of the best option requires individual robots of the swarm to measure (or sample) the quality of each option and to use this information to bias the collective decision-making process.

Less effort has been put in the study of the last two variants of the best-of-n problem. These two variants have asymmetries in both the option quality and the option cost and their interaction is either synergic or antagonistic. In the fourth variant, the interaction is synergic: options with higher quality have lower costs and the best option has both maximum quality and minimum cost. This is possibly the easiest type of best-of-n problem to solve from the perspective of the swarm because both the environment and the individual robots of the swarm bias the collective decision toward the best option. In the fifth variant, the interaction is antagonistic and the selection by the swarm of the option with highest quality is hindered by its cost. This variant of the best-of-n problem is the most challenging one. Probably because of its difficulty, it is the one that received the least attention from the swarm robotics community. For this reason, we encourage further research to tackle novel application scenarios within this variant of the best-of-n problem.

As discussed in Section 4.6, only a handful of research studies investigated application scenarios requiring the solution of a best-of-n problem with more than n = 2 options. While binary decision-making scenarios simplify the study and analysis of collective decision-making strategies, robot swarms will generally face best-of-n problems with a higher number of options. Moreover, some of the research results reviewed in this paper might not extend to the general case of n > 2 options. For this reason, we encourage further research to develop and study application scenarios characterized by more than 2 options.

In order to analyze the literature with a focus on the designed strategies, we divided research studies in two categories: bottom-up and top-down design approaches. We further organized each category in sub-categories. In the case of bottom-up design, we distinguished between opinion-based approaches and ad hoc control strategies (further organized in aggregation-based and navigation-based strategies). In the case of top-down design, we distinguished between evolutionary robotics and automatic modular design.

Aggregation-based strategies to collective decision-making have the advantage of functioning without the need of communication as they exploit implicit information transfer. However, aggregation as a means of communicating one’s own opinion provides a viable solution only when the options of the best-of-n problem are clearly separated in space from each other, which, as showed in Figure 5B, is not always the case. Similarly, navigation-based strategies can be applied only to scenarios in which the swarm is required to find the shortest-path connecting different locations. In contrast, automatic design approaches as evolutionary robotics and automatic modular design have the potential to be applicable to a larger set of consensus achievement scenarios. Evolutionary robotics, however, suffer from the reality gap between simulated and real robots. Moreover, it is difficult to derive predictive mathematical models for systems designed using artificial evolution. This latter limitation might also affect automatic modular design depending on the complexity of the resulting probabilistic finite-state machines. Opinion-based approaches offer a more general design methodology that can be applied and ported to different application scenarios. This higher level of generality, however, requires explicit information transfer and is obtained at the cost of allowing for robot-to-robot communication.

The work presented in this paper sets the basis for a principled understanding of discrete consensus agreement in robot swarms. The identified structure of the best-of-n problem provides designers with the means to understand, which design requirements characterize a certain application scenario, while the overview of the possible design approaches supports them in the selection of a design solution.

GV, EF, and MD equally contributed to the conception of the work, to the writing of the paper, and to its critical revision concerning intellectual content. GV performed most of the preliminary bibliographic research.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.