A Stochastic Nash Equilibrium Problem for Medical Supply Competition

Abstract

In this paper, we study the competition of healthcare institutions for medical supplies in emergencies caused by natural disasters. In particular, we develop a two-stage procurement planning model in a random environment. We consider a pre-event policy, in which each healthcare institution seeks to minimize the purchasing cost of medical items and the transportation time from the first stage, and a recourse decision process to optimize the expected overall costs and the penalty for the prior plan, in response to each disaster scenario. Thus, each institution deals with a two-stage stochastic programming model that takes into account the unmet demand at the first stage, and the consequent penalty. Then, the institutions simultaneously solve their own stochastic optimization problems and reach a stable state governed by the stochastic Nash equilibrium concept. Moreover, we formulate the problem as a variational inequality; both the discrete and the general probability distribution cases are described. We also present an alternative formulation using infinite-dimensional duality tools. Finally, we discuss some numerical illustrations applying the progressive hedging algorithm.

On the study of multistage stochastic vector quasi-variational problems

“…However, in order to study the multistage stochastic vector quasi-variational problems introduced in the previous section, we point out that we cannot directly apply the nonlinear scalarization technique introduced in [12]. Indeed, in order to fit with our functional setting and with the ordering structure (6), we set (15) where ê : L G → R P is defined as…”

“…In particular, throughout the particular construction of ê in ( 16) and the structure of D(C), relation in (15) allows us to opportunely work in an expected values framework, that is, ξ perfectly fits with the time-uncertainty-information structure and the functional space in which we operate. As proved in Proposition 2.1 in [12], ξ is a well-defined function and its minimum is attained.…”

“…)) be a quasi-ordered space defined as in (6). Let ξ be defined as in (15). Then, x is a solution to (9) if and only if it holds…”

“…)) be a quasi-ordered space defined as in (6) and Assumptions A be satisfied. Let ξ be defined as in (15) and K ⊆ L G be nonempty, convex, and compact. Let Φ : K ⇒ L(L G , L P ) be upper semicontinuous with nonempty, convex, and compact values.…”

“…Indeed, to capture the dynamics that are essential to stochastic decision processes in response to an increasing level of information, Rockafellar and Wets [27] introduced a multistage stochastic variational problem. This formulation provides innovative and flexible tools to study real-life problems complicated by time, uncertainty, risk and allows us to capture the role of information in the recursive decision processes; see, e.g., [15,20,22,29]. The key concept of this new formulation turns out to be the nonanticipativity: some constraints have to be included in the formulation of the problem to take into account the partial information structure progressively revealed.…”