I study how organizations assign tasks to identify the best candidate to promote among a pool of workers. When only non-routine tasks are informative about a worker’s potential and non-routine tasks are scarce, the organization’s preferred promotion system is an index contest. Each worker is assigned a number that depends only on his own potential. The principal delegates the non-routine task to the worker whose current index is the highest and promotes the first worker whose type exceeds a threshold. Each worker’s threshold depends only on his own type. In this environment, task allocation and workers’ motivation interact through the organization’s promotion decisions. The organization designs the workers’ careers to both screen and develop talent. So competition is mediated by the allocation of tasks: who gets the opportunity to prove themselves is a determinant factor in promotions. Finally, features of the index contest can help understand the prevalence of fast-track promotion, the role of seniority, or when a group of workers is systemically advantaged.

We derive properties of value functions in mixed optimal control and stopping problems in which (i) the state variable may be multi-dimensional, (ii) the domain may be unbounded and irregular, and (iii) primitives may be time-dependent. We show that the value function is the unique Lp-solution of the Hamilton-Jacobi-Bellman equation and that it is twice parabolically differentiable a.e. and continuously differentiable in the state variable under general conditions that accommodate most economic applications. We show that the smooth-pasting property must hold everywhere with respect to all non-time variables and provide sufficient conditions under which smooth pasting must also hold with respect to time. Our results imply that numerical solutions obtained by standard methods converge uniformly to the value function.

(Supersedes and expands on Theorem 3 in Smoothness of Value functions in General Control-Stopping Diffusion Problems)

We establish the existence, uniqueness, and W1,2,p-regularity of solutions to fully nonlinear parabolic obstacle problems when the obstacle is the pointwise supremum of arbitrary functions in W1,2,p, and the operator is only assumed to be measurable in the state and time variables. The results hold for a large class of non-smooth obstacles, including all convex obstacles. Applied to stopping problems, they imply that the decision maker never stops at a convex kink of the stopping payoff. The proof relies on new W1,2,p-estimates for obstacle problems where the obstacle is the maximum of finitely many functions in W1,2,p.

We formulate a general optimal stopping problem that can represent a wide variety of non-stationary environments, e.g., where the decision maker’s patience, time pressure, and learning speed can change gradually and abruptly over time. We show that this problem has a well-defined solution under some mild regularity conditions. Furthermore, we characterize the shape of the stopping region in a large class of “monotone” environments. As a result, we obtain comparative statics on the timing and quality of decisions for many sequential sampling problems à la Wald. For example, we show that accuracy is increasing (decreasing) over time when (i) learning speed increases (decreases) in time, or (ii) the discount rate decreases (increases) over time (i.e., the decision maker values the future more over time), or (iii) time pressure decreases (increases) over time. Since our main comparative static results hold locally, we can also capture non-monotone relations between time and accuracy that consistently arise in both perceptual and cognitive testing.