Abstract
We generalize the classic dynamic single-item search model to a setting with multiple items and vector offers for subsets of items. We first show a computationally feasible way to solve the dynamic optimization problem, and then prove structural results. Although assignment is not generally monotonically increasing in offer value, we show that, in a special case “additive” model, monotonicity holds if costs are submodular. We examine how the thresholds for assignment change with the remaining items, and whether there are gains to grouping searches. Finally, we consider a stopping rule version of the problem with no subsets for sale, showing the optimal policy is myopic.
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Notes
Precisely, f is a submodular set function if, for sets S and T, \(f(S \cup T) + f(S \cap T) \le f(S) + f(T)\), and f is supermodular if \(f(S \cup T) + f(S \cap T) \ge f(S) + f(T)\).
Bruss and Ferguson (1997) also do not provide our solution to the optimality equation or computational approach. In addition, we extend their results about the stopping rule variation of the additive model. While Bruss and Ferguson (1997) show that a one-stage lookahead policy is optimal for the stopping rule variation of the additive model, we show that it is also optimal in the single applicant model, which is not a simple extension as the intuition is entirely different.
Specifically, the positions (items) should be interpreted here as workers and the applicants as sequentially arriving jobs, with the values of the value vector \(X_i\) being the value of assigning the job with vector \((X_1, \ldots ,X_n)\) to worker i.
Specifically, DLR supposed that there are n workers, with worker i having a specified value \(p_i\). Jobs appear sequentially, with each job having a value X that is chosen, independently from job to job, according to a known distribution function. If a worker with value \(p_i\) is assigned to a job with value x then a return \(p_i x\) is earned and the problem was to assign workers to jobs so as to maximize the expected sum of values. (DLR assumed that each job had to be assigned to some worker so the problem ended when n jobs had appeared.) Thus the model of DLR is a special case of our model in which the relative value of the applicants is fixed across periods, i.e., random value vectors are of the special form \((p_1X, p_2X, \ldots , p_n X)\) with X having a specified distribution.
Note that the additive model does allow the firm to take into account any complementarities between applicants and existing employees; these complementarities would be reflected in each applicant’s match value.
Note that even the more general model does not allow for complementarities across applicants who apply in different periods; this would require that the value of the value vector change as a result of past hiring decisions which would fundamentally change the nature of the classic search model.
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We appreciate support from the National Science Foundation (Ross through CMMI-1662442) and from the endowment in memory of B.F. Haley and E.S. Shaw (Dizon-Ross)
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Dizon-Ross, R., Ross, S.M. Dynamic search models with multiple items. Ann Oper Res 288, 223–245 (2020). https://doi.org/10.1007/s10479-019-03472-z
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DOI: https://doi.org/10.1007/s10479-019-03472-z