In group-living organisms, consensual decision of site selection results from the interplay between individual responses to site characteristics and to group-members. Individuals independently gather personal information by exploring their environment. Through social interaction, the presence of others provides public information that could be used by individuals and modulates the individual probability of joining/leaving a site. The way that individual's information processing and the network of interactions influence the dynamics of public information (depending on population size) that in turn affect discrimination in site quality is a central question. Using binary choice between sheltering sites of different quality, we demonstrate that cockroaches in group dramatically outperform the problem-solving ability of single individual. Such use of public information allows animals to discriminate between alternatives whereas isolated individuals are ineffective (i.e. the personal discrimination efficiency is weak). Our theoretical results, obtained from a mathematical model based on behavioral rules derived from experiments, highlight that the collective discrimination emerges from competing amplification processes relying on the modulation of the individual sheltering time without shelters comparison and communication modulation. Finally, we well demonstrated here the adaptive value of such decision algorithm. Without any behavioral change, the system is able to shift to a more effective strategy when alternatives are present: the modification of the spatio-temporal distributions of individuals leading to the collective selection of the best resource. This collective discrimination implying such parsimonious and widespread mechanism must be shared by many group living-species.