Optimal stopping in the balls-and-bins problem

This paper considers a multistage balls-and-bins problem with optimal stopping connected with the job allocation model. There are $N$ steps. The player drops balls (tasks) randomly one at a time into available bins (servers). The game begins with only one empty bin. At each step, a new bin can appear with probability $p$. At step $n$ ($n=1,\ldots,N$), the player can choose to stop and receive the payoff or continue the process and move to the next step. If the player stops, then he/she gets $1$ for every bin with exactly one ball and loses $1/2$ for every bin with two or more balls. Empty bins do not count. At the last step, the player must stop the process. The player's aim is to find the stopping rule which maximizes the expected payoff. The optimal payoffs at each step are calculated. An approximate strategy depending on the number of steps is proposed. It is demonstrated that the payoff when using this strategy is close to the optimal payoff.