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concepts of pure and mixed strategies . A pure strategy prescribes a certain course of
action (i.e., a decision) for a decision maker, while a mixed strategy will provide a
schedule of decision, associated with probabilities that indicate with what likelihood
a decision maker should use this strategy. The work by Caplin and Nalebuff
( 1991 ) outlines conditions under which a pure-strategy price equilibrium exists in a
locational game, while Dasgupta and Maskin ( 1986 ), who deal with discontinuous
payoff functions, describe conditions for the existence of mixed strategies.
A full sequential strategy has one player, the so-called leader , locate first,
followed by all other players, the followers , which locate later. This asymmetric
situation has originally been described by the economist von Stackelberg ( 1943 ).
The leader, when choosing his locations, will have to guard against the followers. If
all players have the same objective and the same perception of the demand structure,
this means that the leader will use a strategy to maximize the minimal market
share or profit he will obtain. On the other hand, the followers will have a chance
to observe the action of the leader and then react accordingly, meaning that they
solve a conditional optimization problem, in which they maximize their own market
share or profit, given that the leader has already located. Notice that the problem of
the follower is much easier to solve mathematically, as it is a simple optimization
problem. The problem of the leader, however, is a bilevel optimization problem, as
it requires the solution of the follower's problem as an input parameter.
The last major descriptor of competitive location models concerns customer
behavior . As a matter of fact, this aspect is the main leitmotif of this paper. The
first major distinction between different classes of models is between demand
allocation models and customer choice models. As the name suggests, in allocation
models the firm decides which facility is allocated to a customer. A typical example
would be the delivery of furniture to customers, who will receive the goods from
whatever warehouse the firm decides to deliver from. (Note that, strictly speaking,
the purchase of, say, a sofa, typically involves a mix of allocation and choice models:
when customers drive to a store to purchase the sofa is a choice model, while
the actual delivery of the sofa is an allocation model.) In scenarios of customer
choice, on the other hand, customers choose which facility or firm they want to deal
with. Often, the two models are referred to shipping and shopping. This paper deals
exclusively with customer choice models.
The manner in which customers choose which facility they patronize, is the
main subject of this contribution. The next section will provide a framework for
this decision. At this point, suffice it to say that while many, or even most, papers
use the “patronize the closest facility” (or cheapest, in case prices are different and
mill pricing is assumed), other models have been suggested. For instance, some
models include a (single-dimensional) parameter that measures the attractiveness of
a facility in contrast to other, competing facilities. Furthermore, an important and
fairly recent strand of research uses probabilistic choice rules, according to which
customers at the same location do not all behave in the same way. Similarly, it is
able to capture the fact that a customer, even if he and all of the competing facilities
remain in the same positions, will not always patronize the same facility.
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