A bargaining approach for resolving the tradeoff between beneficial and harmful drug responses

In medical treatments, a fundamental dilemma often arises: on the one hand, an increase in a drug's dose could lead to a stronger, therapeutic treatment response, but on the other hand this could also lead to increased toxicity risks. In this paper, we propose to solve this dilemma using a Nash bargaining approach. To do so, we reformulate the tradeoff problem in an equivalent form as a dilemma between a drug's beneficial response and the drug's safety, where the dilemma then becomes a two-objective problem with safety and response as the objectives. Using a general receptor response model, we show that the set of all feasible outcomes associated with a drug's treatment is characterized as a convex set, which allows the dilemma to be solved as a bargaining problem between the two objectives. The Nash bargaining solution (NBS) is found in closed form, and interesting properties associated with the solution are presented. In particular, it is shown that one can interpret the NBS as corresponding to a treatment giving the maximal expected safe treatment response associated with the drug. Moreover, the condition when the NBS coincides with the maxmin solution for the two objectives is derived. Two approaches to control the NBS solution are then presented: first, by means of assigning bargaining powers to objectives; second, by means of assigning an upper-bound on the drug's dosage. The obtained NBS are compared with the dosage returning maximal drug response efficiency.