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calculate the game equilibria, we resorted to an
algorithm based on the best responses of each player
to the other one's strategy, proposed by Godinho and
Dias (2010).
The algorithm was implemented in C
programming language, using LP Solve routines for
solving the linear programming problems (source:
http://lpsolve.sourceforge.net). For each instance, we
applied the algorithm twice for the game in which
player 1 has preferential rights. The first time we
chose a null strategy for player 1 (opening no
locations) as the starting point; the second time, we
chose a null strategy for player 2 as the starting
point. In fact, in a model without preferential rights,
the algorithm will often find solutions that are more
favorable to the player whose best response is
considered first (the algorithm will only find one
equilibrium, and the game may have several
equilibria, so the results may be somewhat biased by
the choice of the starting point, as shown in Godinho
and Dias, 2010).
However, in the problem here addressed, the
equilibrium solution that is found is usually
independent of the starting point of the algorithm;
moreover, when different starting points lead to
different equilibria, the differences in the players
payoffs in the two equilibria are small.
Test set 1 was used as a reference, the parameters
of the remaining test sets being defined as changes
over the parameters of this test set. For test set 1, we
defined a network with 100 nodes (that is, 100
possible locations for the customers), with both
players being able to open facilities at 48 of these
locations. The budget for each player was set to
1000, and the average cost of opening a facility was
set to 350.
Test sets 2-4 were designed to allow us to
analyze the impact of simultaneously changing the
number of potential locations for both players’
facilities. The number of potential locations for the
players’ facilities was set to 36, 24 and 12 in test sets
2, 3 and 4, respectively, and the other parameters’
values were identical to the ones used in test set 1.
The results obtained with test sets 1-4 are
summarized in Table 1. As expected, the average
payoffs of both players increase as the number of
potential facility locations increase, but this increase
takes place at a decreasing rate. This behavior occurs
both when there are preferential rights and when
they do not exist, and it is consistent with the results
of Godinho and Dias (2010). Both the benefit that
player 1 gets from having preferential rights and the
loss player 2 incurs when player 1 has such rights,
seem fairly stable in absolute terms. Since payoffs
increase with the number of potential locations, this
means that the relative gain of player 1 and the
relative loss of player 2 become less significant as
the number of potential locations increase.
This makes sense because an increase in the
number of potential locations leaves player 2 with
more places in which he can avoid player 1, and
provides player 1 with more interesting locations, so
he has a relatively smaller incentive to try to choose
the same locations as player 2.
Test sets 5-7 allow us to analyze the
consequences of changing the potential locations
available to just one of the players. Player 1 has 48
potential locations, and the number of potential
locations for player 2’s facilities is 48, 36, 24 and 12
in test sets 1, 5, 6 and 7, respectively. This is done
by randomly choosing a subset of G and
considering
2i
f
+∞
, for all facilities i in this
subset. The other parameters’ values were identical
to the ones used in test set 1. The results are
summarized in Table 2. As the number of locations
available to player 2 increases, player 2’s payoff
increases and player 1’s payoff tends to decrease.
The relative loss of player 2 from the preferential
rights of player 1 is fairly stable. In the case of
player 1, both the absolute and the relative gain
increase with the number of potential locations for
player 2. This means that, as player 2 gets more
Table 1: Summary of the results obtained with test sets 1-4.
Test
set
Potential
locations
Average return (with
preferential rights)
Average return (without
preferential rights)
Player 1 benefit from
preferential rights
Player 2 loss from player 1 rights
1
with
π
2
with
π
/
1
w out
π
/
2
w out
π
Absolute
/
11
with w out
ππ
−
Relative
/
11
/1
with w out
ππ
Absolute
/
22
w out with
ππ
−
Relative
/
22
1/
with w out
ππ
−
1 48 1427.8 952.8 1197.6 1196.9 230.2 19.2% 244.1 20.4%
2 36 1416.9 932.7 1179.2 1180.6 237.7 20.2% 247.9 21.0%
3 24 1310.8 813.4 1084.2 1060.7 226.6 20.9% 247.4 23.3%
4 12 1089.9 633.1 840.3 866.9 249.6 29.7% 233.7 27.0%
1
with
π
,
2
with
π
: average payoffs for player 1 and player 2, respectively, when player 1 has preferential rights;
/
1
w out
π
,
/
2
w out
π
: average payoffs for player 1 and player 2, respectively, when there are no preferential rights.
A TWO-PLAYER MODEL FOR THE SIMULTANEOUS LOCATION OF FRANCHISING SERVICES WITH
PREFERENTIAL RIGHTS
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