Download PDF - Ivie

supermarket placed in the same area where a consumer lives is T ,meanwhile
the driving time to commute to a different area is aT ,wherea>1.
Let p(c l |θ i ,c k ) denote the probability to buy in constituency c l of a consumer
of type θ i who lives in constituency c k . From the assumptions, it is
easy to check that:
1 k = l
p(c l |θ 1 ,c k ) =
0 k = l
a
k = l
p(c l |θ 2 ,c k ) =
a+1
1
k = l
a+1
Then, the expected number of consumers who will buy in region c k are
given by
π(c k )=n 1k +
a
a +1 n 2k + 1
a +1 n 2l
k =1, 2,l =1, 2,k = l.
Supermarkets’ problem to choose specific locations to open their new
premises, may be modelled then as zero-sum game. Total sales of a supermarket
depend then on its own location and on the location of its competitor.
Let us denote by π i (c l ,c k ) the total sale of company i =1, 2 when supermarket
1 is placed in region c l and supermarket 2 is placed in c k . We have
then,
π 1 (c l ,c k ) =
π 2 (c l ,c k ) =
1 n 2 1l + 1(n 2 21 + n 22 ) k = l
n 1l +
a n a+1 2l + 1 n a+1 2k k = l
1 n 2 1l + 1(n 2 21 + n 22 ) k = l
n 1k +
a n a+1 2k + 1 n a+1 2l k = l
The payoff can be written as a matrix as follows, where supermarket 1
chooses rows and supermarket 2 chooses columns:
c 1 c 2
1
c 1 n 2 11 + 1(n 2 21 + n 22 ), n 11 + a n a+1 21 + 1 n a+1 22,
1
n 2 11 + 1(n 2 21 + n 22 ) n 12 + a n a+1 22 + 1 n a+1 21
c 2 n 12 + a n a+1 22 + 1 n 1
a+1 21, n 2 12 + 1(n 2 21 + n 22 ),
n 11 +
a n a+1 21 + 1 n 1
a+1 22 n 2 12 + 1(n 2 21 + n 22 )
9