Geoscience Reference
In-Depth Information
14.4
Results for Different Behavioral Assumptions
This section is organized along the lines of customer behavior. Each part will
examine one customer choice rule, followed by results in the literature regarding
Nash equilibria, followed by von Stackelberg solutions. The review in this section
will be organized along the lines of the customer choice rules outlined in the
previous section.
14.4.1
UD1a, Linear Market, Nash Equilibria
Stevens (
1961
) appears to have been the first to use game theory to reestablish
Hotelling result of minimal differentiation for fixed and equal prices. Recognizing
the complexity of the problem described in Hotelling's (
1929
) paper, some contrib-
utors decided to simplify matters. Eaton and Lipsey (
1975
) used fixed and equal
prices. While this assumption appears somewhat contrived, it is usually justified
by legislated pricing for essential goods. With this assumption, customer choice
rule UD1a (the “closest” rule) is applied. Given this assumption, Hotelling's result
of minimal differentiation is reestablished, as by moving towards its opponent, a
firm gains customers in the competitive region and does not lose customers in
its hinterland. The authors also extend the analysis to more than two firms. In
particular, they determine that for more than five firms, multiple equilibria exist,
and the only case without equilibria is the instance with three facilities. In particular,
the two outside facilities will push inwards so as to gain additional market shares,
thus squeezing the market of the inside firm to zero. This firm will counteract by
“leapfrogging” to the outside, become an outside facility itself, and start moving
inwards. Teitz (
1968
) referred to this behavior as “dancing equilibria.” Shaked
(
1975
) investigates the usual Hotelling model with fixed and equal prices, but three
facilities that employ mixed strategies. It turns out that an equilibrium exists, in
which all facilities randomize their strategies in the central half of the market.
In a follow-up paper, Shaked (
1982
) investigates the Hotelling model with
three firms locating one facility each, with fixed and equal prices, allowing mixed
strategies. It turns out that all firms will chose locations in the central half of the
market with equal probability. Cancian et al. (
1995
) consider a Hotelling model
with directional constraints, i.e., customers can only walk in one direction towards
the firm they want to patronize. The authors determine that with random arrival
times of the customers and two or more facilities, no equilibrium exists.
14.4.2
UD1a, Linear Market, von Stackelberg Solution
The first author to introduce sequential (and final) location decisions into the
discussion appears to have been Hay (
1976
). However, it was the contribution of
