Competitive Location Models - Location Science - page 372

behavior is modeled by a function that relates a customer's attraction to a facility
to the sum of this customer's attractions to all facilities. This leads to a concave
fractional problem, which is solved by a branch-and-bound method and heuristic
concentration techniques.

14.4.17 UD2b, Network, von Stackelberg Solution

Aboolian et al. ( 2008 ) investigate a follower problem on a network with an
exponential attraction function. In order to capture a customer's demand, the
follower must be more attractive than the incumbent by a positive constant. The
variable production costs are the same everywhere, and the fixed location costs are
location-dependent. Co-location is not permitted. The model is loosely based on
work by Serra and ReVelle ( 1999 ). The node property does not hold. The authors
conjecture that there is a finite dominating set, but are unable to determine it in
this nonlinear integer program. Marianov et al. ( 2008 ) replace the distance by travel
time, and add waiting time as a competitive factor.
Consider now results relating to the probabilistic choice rules introduced in the
previous section. Most papers are written by economists, who are mainly interested
in the existence of Nash equilibria on a linear market.

14.4.18 UP1, Linear Market, Nash Equilibria

In all of these contributions, the parameter can be interpreted as the heterogeneity
of the customer tastes with respect to the product under consideration. de Palma
et al. ( 1987a ) use fixed and equal prices and unit transportation costs t (in a
linear cost function) in their triopoly model. Their main result is that for small
values of / t , there are no symmetric equilibria. As the value of / t increases,
there are symmetric dispersed equilibria, a further increase results in dispersed and
agglomerated equilibria, while for large values of / t , only agglomerated equilibria
exist. de Palma et al. ( 1985 ) consider the usual “first location, then price” game
with a linear transport cost function, and n facilities located on a linear market of
length L . The key result is that for large values of / tL , there is clustering of the
facilities at equilibrium, while small values of / tL lead to dispersion. Braid ( 1988 )
locates n firms on a line segment, on which the demand occurs at five even spaced
the facilities. de Palma et al. ( 1987b ) discuss a duopoly under delivered pricing
in their model with linear transportation costs with parameter t . Under sufficient
heterogeneity (i.e., > t /8), a centrally agglomerated location-price equilibrium
exists. The result generalizes to n firms.
Finally in this category, we find the contribution by Anderson et al. ( 1992 ), which
compares the three main pricing strategies in a duopoly setting. Transportation costs
are assumed to be linear, and social surplus is defined as the sum of customer