A Model of Competitive Signaling

Multiple candidates (senders) compete over an exogenous number of jobs. There are different tasks in which the candidates' unobservable ability determines their probability of success. We study a signaling game with multiple senders each choosing one task to perform, and one receiver who observes all task choices and performances (success or failure) and matches the senders to jobs. In order to analyze the effects of different levels of competition we consider two refinements of the concept of sequential equilibrium: (i) sequential equilibria that survive when varying the number of senders; (ii) sequential equilibria that are supported by out-of-the-equilibrium-path beliefs satisfying a monotonicity condition (implied by Banks and Sobel's divinity refinement). We show that the set of sequential equilibria includes simple pooling equilibria where all senders choose the same task, and these simple pooling equilibria are the only type of sequential equilibria that satisfies (i). The unique sequential equilibrium under both (i) and (ii) is a simple pooling equilibrium with every sender choosing the most informative task. If senders have a lower overall likelihood of success in more informative tasks, this unraveling towards conspicuousness is inefficient.