An Optimal Stopping Problem with Two Decision Makers

Abstract

We investigate in this paper an optimal stopping problem where two decision makers are involved in the selection of a single offer. Suppose that n offers are examined one at a time by both decision makers. At each stage, a decision must be taken: accept the current offer and stop the selection process, or discard it and examine the next offer. We assume that no recall of previously examined offers is allowed. A conflict arises when one decision maker decides to accept a currently inspected offer and the second decides to discard it. In such conflicting situations, a decision should be taken by defining a stopping rule for the group. We propose to stop the process if either decision maker accepts an offer. We develop the dynamic programming approach for this problem and state the optimal strategy for a fixed utility. Then, we propose an experimental investigation of the problem.

Dynamic programming and optimal control for vector-valued functions: A state-of-the-art review

“…In [30] a multiple objective optimal stopping problem is solved. The authors proposed to stop the selection process of the Bilateral Optimal Selection Problem (BOSP) if either DM decides to stop.…”

“…where Ev p (i, 1, k p ) is the maximum expected utility of DM p by following the optimal strategy when the ith offer, with relative ranks k p and k p is examined, Ev p c (i) is the expected utility of DM p when postponing the decision to the next stage and Ev p s (i, 1, k p ) is the expected utility of DM p when stopping at stage i and selecting the current offer with relative ranks k p and k p . In [30] the authors developed the game formulation of the problem through the backward recursive equations of each DM. Then, they studied the problem for a specific case of the proposed utility and they showed how each DM should foresee his opponent's individual decision.…”