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Finally, let Q r c denote the amount of sales of retailer r =1, 2inconstituency
c and Q r = c∈C Qr c. Since each consumer buy a unit of the good,
we have that: Q 1 c + Q 2 c = n c and Q 1 + Q 2 = n.
In the next subsections we formalise the two stages of the decision problem:
3.1 First stage: location game
In this steps, we have the model has the following elements:
• AsetoftwoplayersR
• A set of pure strategies for each player, consisting of the set of feasible
potential opening constituencies C.
• Payoff functions that assign to a location pattern (c 1 ,c 2 )themarket
share obtained by the retailers. For each r = 1, 2,r = r, these
functions are given by
Q r 1
(c 1 ,c 2 )= n if c 2 1 = c 2
c∈C n
c θ∈Θ π c(θ)µ c (c r |θ,c r ) if c 1 = c 2
Note that if c 1 = c 2 all consumers choose the same location to buy the
good and, since both retailers are identical including prices, consumers choose
each establishment with equal probability obtaining each firm a share of
50% of the whole market. However, if firms choose different locations, then
aconsumeroftypeθ living in c will buy the good to the firm at r with
probability µ c (c r |θ,c r ) depending on the alternative locations to buy.
We can define the normalized market share of retailer r as
MS r (c 1 ,c 2 )= 1 n Qr (c 1 ,c 2 ) − 1 2
(1)
such that MS r (c 1 ,c 2 ) ∈ [− 1, 1 ] and represents the difference between retailer
2 2
r’s market share and a market share of 50%. We refer to MS r (c 1 ,c 2 )as
retailer r’s market share hereafter. Notice that, after normalization, we have
that MS 1 (c 1 ,c 2 )+MS 2 (c 1 ,c 2 ) = 0 and the location game can be considered
as a zero-sum game. Moreover, since retailers can guaranty 50% of the total
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