A model of competitive signaling with rich message spaces

We examine a signaling game with competition: multiple senders each choose one task to perform, and one receiver observes the senders’ tasks choices and their performance (success or failure) and matches them to a limited number of jobs. Importantly, the space of tasks is rich in the sense that there are arbitrarily close different tasks to any given task. Higher tasks are associated with a stronger screening power but a lower probability of success. Intuitively, choosing a slightly higher task than those chosen by others only slightly reduces the probability of success, while it could significantly (discretely) increase the likelihood of getting a job. This must be so under a trembling-hand refinement because of the richness of the task space: tasks that are very close to each other must give rise to similar beliefs in the receiver. Thus, pooling at the highest task is the only equilibrium outcome in many situations, such as when high tasks have a sufficiently strong screening power or when competition is sufficiently high.