Constrained Sequential Resource Allocation and Guessing Games

In this paper, we consider a constrained sequential resource allocation problem where an individual needs to accomplish a task by repeatedly guessing/investing a sufficient level of effort/input. If the investment falls short of a minimum required level that is unknown to the individual, she fails; with each unsuccessful attempt, the individual then increases the input and tries again until she succeeds. The objective is to complete the task with as little resources/cost as possible subject to a delay constraint. The optimal strategy lies in the proper balance between 1) selecting a level (far) below the minimum required and therefore having to try again, thus wasting resources, and 2) selecting a level (far) above the minimum required, and therefore, overshooting and wasting resources. A number of motivating applications arising from communication networks are provided. Assuming that the individual has no knowledge on the distribution of the minimum effort required to complete the task, we adopt a worst-case cost measure and a worst-case delay measure to formulate the above constrained optimization problem. We derive a class of optimal strategies, shown to be randomized, and obtain their performance as a function of the constraint.