Introduction

Abstract

Decision making problems in decentralized organizations are often modeled as Stackelberg games, and they are formulated as two-level mathematical programming problems. In the Stackelberg game model, there are two players; the first player chooses a strategy at the start, and thereafter the second player with knowledge of the first player’s strategy determines a strategy of the second player self (Simaan and Cruz, 1973a). It is assumed that each of the two players completely knows the objective functions and the constraints of them. In the context of twolevel programming, the decision maker at the upper level first specifies a strategy, and then the decision maker at the lower level specifies a strategy so as to optimize the objective with full knowledge of the action of the decision maker at the upper level. For brevity, we abbreviate the decision makers at the upper level and at the lower level as the upper level DM and the lower level DM, respectively. Assuming that the lower level DM behaves rationally, that is, optimally responds to the decision of the upper level DM, the upper level DM also specifies the strategy so as to optimize the objective of self. Although a situation described as the above is called a Stackelberg equilibrium in the field of game theory or economics, in this book dealing with mathematical programming, we will refer to it as a Stackelberg solution.

Even if the objective functions of both decision makers and the common constraint functions are linear, such a two-level mathematical programming problem, i.e., a two-level linear programming problem is a non-convex programming problem with a special structure, and it is shown to be NP-hard (Jeroslow, 1985; Bard, 1991). In general, Stackelberg solutions do not satisfy Pareto optimality because of their noncooperative nature.