Competitive Location Models - Location Science - page 368

14.4.8 UD1b, Plane, Nash Equilibria

Hurter and Lederer ( 1985 ) appear to have been among the few investigators to look
at the subgame-perfect Nash equilibrium on the plane. Their contribution includes
different cost functions for the firms and transportation costs that are proportional to
Euclidean distances. Firm are supposed to locate in a given convex set. The authors
show that there are no peripheral equilibrium locations. They also demonstrate
that the locations that minimize the social costs for serving the entire market are
a proper subset of equilibrium locations. Similarly, Tabuchi ( 1994 ) locates two
firms in the two-dimensional space and uses quadratic transportation costs. The
paper determines that for any convex set, there are no interior locational Nash
equilibria. The author then determines that in a rectangle, Nash equilibrium has
the facilities locate on opposite sides of the rectangle at their respective midpoints.
If the rectangle is very long, the Nash equilibrium is unique.
This is not the same as d'Aspremont's et al. ( 1979 ) result, as while this result
shows maximum differentiation in one direction, it has minimum differentiation
in the other. Lederer and Hurter ( 1986 ) consider customers located in a subset of
the two-dimensional plane with some typically nonuniform demand distribution
and firms facing different production and transportation costs. Firms use spatial
price discrimination and customer purchase goods from the cheapest source (a
number of tie-breaking rules is specified). The resulting “location, then price” game
has an equilibrium, and it is shown that identical firms (i.e., those with different
production and transportation costs) do not co-locate. The analysis is then extended
to nonidentical forms that locate on a disk, and again, there is no co-location.

14.4.9 UD1b, Networks, Nash Equilibria

Lederer and Thisse ( 1990 ) examine a competitive network location model, in
which firms determine their respective locations and chosen technologies in stage
1, and the prices in stage 2. The authors use spatial price discrimination. In
the usual backward recursion, the paper proves that for all first stage location
and technology choices, the second stage pricing game has an equilibrium. The
socially optimal location and technology choices of the first stage are also a Nash
equilibrium. However, locational Nash equilibria may exist that are not socially
optimal. An important feature is that if the transport cost function is concave, then
the equilibrium locations will satisfy the node property. Labbé and Hakimi ( 1991 )
also use delivered pricing and, in addition, a linear price-quantity relation. The two-
stage game locates facilities in stage 1, and determined quantities in stage 2. It turns
out that for any fixed pair of locations, the quantity game has an equilibrium. If it is
required that it is always profitable to supply any market of the graph with a positive
quantity of goods, then a location equilibrium exists at the nodes of the graph. If this
condition is not satisfied, then either a locational Nash equilibrium does not exist,
or it exists on the edges of the graph.