# Robot Swarms Decide under Perception Errors in Best-of-N Problems
## 1 Introduction  
Collective decision making is a wide-spread phenomena across both natural [1, 2, 3] and artificial systems [4, 5, 6]. In artificial systems, it has been intensively studied in the context of robot swarms, where a group of robots needs to establish an agreement while performing a particular task. For robot swarms, a large number of collective decision-making problems are defined in the form of consensus [7, 8], in which robots must agree on one of the available options. For a finite and discrete set of options, the problem is formulated as a discrete consensus problem. Nevertheless, when the quality of the options varies, the problem becomes a best-of-n problem [9, 10]. In best-of-n problems, robots are supposed to explore their environment looking for the available options, and measuring their quality values in a phase known as the exploration phase. The exploration phase is typically followed by an exploitation phase in which robots interact with their local neighborhoods to exchange their measurements and/or opinions and attempt to reach an agreement by following a particular voting approach. Many works have focused on consensus achievement in best-of-N decisions. In [11], the authors contributed a literature review on design methodologies in collective decision-making problems, in which they discussed best-of-N decisions in terms of option quality and cost. In [12], the authors investigate the impact of the population size and the number of options on the consensus speed in best-of-N decisions. In [13], the authors consider Byzantine robots in best-of-N decision problem. Robot can show arbitrarily faulty behavior or malicious behavior. The authors of [14] investigate the relative time invested by swarm individuals in option discovery and in signaling behaviors when generalizing the best-of-N problems to consider \(N>2\) options. Finally, the work presented in [15] paves the way for exploiting Bayesian algorithms to design collective decision making in robot swarms. The authors design a Bayesian algorithm to help robots judge the spatial distribution of a particular feature in a 2D environment. They show, by means of the accuracy-speed trade-off, that robots with fewer observations can help improve decision accuracy by reducing spatial correlations. In general, Bayesian algorithmsfor decision making in robot swarms can be a promising approach to deal with faulty measurements. Despite the large number of works focusing on consensus achievement in collective decision-making problems, very few assume individual perception flaws--i.e., errors while measuring the quality of the discovered options. As perception is the foundation of cognition since it provides the necessary input for the decision-making processes [16; 17; 18], perceptional errors may result in wrong decisions. In our study, robots' perceptional errors occur in two forms: (i) perceiving a higher quality value or (ii) perceiving a lower quality value of a discovered option. We assume a robot swarm that consists of three populations: accurate robots, overestimating robots, and underestimating robots. The overestimating robots are robots those for which their sensors generate erroneous measurements by increasing the quality value of the encountered option. On the contrary, underestimating robots are robots for which their sensors generate erroneous measurements by decreasing the quality value of the encountered option. Such inaccurate measurements can result from noisy sensor inputs, robot's malfunctioning logic, or intended intrusions [19; 20]. We consider a site selection problem as our best-of-n problem. In a site selection problem, robots need to select the best site (option) out of n sites (in our case n = 2). Sites are implemented in a 2D physical environment, in which robots explore and come back to a shared area referred to as the nest in order to communicate their findings and attempt to reach a consensus on the best site. The two sites (A and B) are assumed to have numeric qualities \(\gamma(l)\), \(l\in\{A,B\}\).