Supply chain coordination in spatial games

The notion that economies consist of the systems of cities originates from a theory by Christaller and Beckmann, based on a Löchean framework of retailers having an endogenously determined market area. In 1968 it was completed by Tinbergen, who has suggested production approach. This original theory explains the location pattern of the market and manufacturing centers selling goods to the local populations within given market areas. The theory was based on the idea of the urban area being a trading center. Our paper, influenced by the ideas, presents the quantitative method of building up the model of spatial hierarchy as the result of spatial games. The paper by Girlich: ‘On the Metric Transportation Problems and Their Solution’ gives us proper foundations for our present research. In this paper customer traveling problem is described, which helps us to define a market area. A mathematical description of market areas defines the border between two areas under the assumption of shortages of goods in at least one market area. The study is influenced by Grubbström's work on MRP optimization. In our paper the problem is formulated as a non-constant sum game, which is the case of the total market area being constant and the conditions for lower central places, described for different positions of this central place in an urban hierarchy, in the case of two-level production. Differential games are suggested when the total market area changes over the time.