Competitive Location Models - Location Science - page 373

surplus and the profits of both firms. Starting with small values of the heterogeneity
factor , there is no equilibrium for mill pricing, and as increases, there are first
symmetric dispersed equilibria, and finally, for large values of , there is a unique
centrally agglomerated equilibrium. The case of uniform delivered demand just has
no equilibrium for small , and centrally agglomerated equilibria for larger values of
, and spatial discriminatory pricing has equilibria everywhere: outside the quartiles
for very small values of that move towards a central agglomeration for sufficiently
large values of .

14.4.19 UP1, Plane, Nash Equilibria and von Stackelberg

Solutions

Choi et al. ( 1990 ) frame their discussion in the context of product positioning.
Customers have a stochastic utility function that results in a logit model, and firms
maximize their profit. It is known that as long as the profit functions are pseudo-
concave, the game has a Nash equilibrium. The paper uses variational inequalities
to analyze computational aspects. The key contribution is a von Stackelberg game
with one leader and multiple followers. The solution of a von Stackelberg game
in continuous space cannot be a Nash equilibrium, as is often the case in discrete
spaces.

14.4.20 UP1, Network, Nash Equilibria

de Palma et al. ( 1989 ) investigate a very general model, in which n firms compete
with each other, and each locates n i facilities. Customers first choose a firm they
want to patronize, and then they patronize the closest facility of that firm. (Note the
similarity of this rule and Hakimi's “partially binary” choice rule.) The main result is
that if consumer tastes are “sufficiently heterogeneous,” then firm i will locate its n i
facilities at the n i -median. If a stronger condition on taste heterogeneity is satisfied,
then the resulting pattern—all firms locate their facilities at the n i -medians—is the
unique noncooperative Nash equilibrium. A special case is when all firms have the
same number of facilities to locate, in which case all firms will locate their facilities
at the same nodes, a case of minimum differentiation.

14.4.21 UP1, Network, von Stackelberg Solution

Benati ( 1999 ) discusses a maximum capture problem in the presence of heteroge-
neous customers. Given fixed demand, fixed and equal prices, as well as p leaders on