Knowledge that Transforms

This paper presents a new multiple leader-follower model that is a consistent extension of Stackelberg's leader-follower duopoly. The development contrasts with other existing extensions by demonstrating how the leader-firms can utilize the true reaction curve of the follower-firms; it also provides sufficient conditions for some useful convexity and differentiability properties of this function. For the proposed model, we conduct a static analysis and discuss the existence, uniqueness, and computation of an equilibrium solution, as well as study certain issues regarding the relative profits of leader and follower-firms. Since the Cournot oligopoly and the Stackelberg leader-follower models are special cases of this model, the analysis in this paper hopefully provides some further insights about these types of models.

Clark and Scarf (Clark, A., H. Scarf. 1960. Optimal policies for a multi-echelon inventory problem. Mgmt. Sci. 6 475–490.) characterize optimal policies in a two-echelon, two-location inventory model. We extend their result to the infinite-horizon case (for both discounted and average costs). The computations required are far easier than for the finite horizon problem. Further simplification is achieved for normal demands. We also consider the more interesting case of multiple locations at the lower echelon. We show that, under certain conditions, this problem can be closely approximated by a model with one such location. A rather simple computation thus yields both a near-optimal policy and a good approximation of the cost of the system.

We consider the Monte Carlo problem of generating points uniformly distributed within an arbitrary bounded (measurable) region. The class of Markovian methods considered generate points asymptotically uniformly distributed within the region. Computational experience suggests the methods are potentially superior to conventional rejection techniques for large dimensional regions.