A TWO-PLAYER MODEL FOR THE SIMULTANEOUS

LOCATION OF FRANCHISING SERVICES

WITH PREFERENTIAL RIGHTS

Pedro Godinho

and Joana Dias

Faculdade de Economia and GEMF, Universidade de Coimbra, Av. Dias da Silva, 165, 3004-512 Coimbra, Portugal

Faculdade de Economia and Inesc-Coimbra, Universidade de Coimbra, Av. Dias da Silva, 165,

3004-512 Coimbra, Portugal

Keywords: Location problem, Competition, Simultaneous decisions, Game theory, Nash equilibrium.

Abstract: We consider the discrete location problems faced by two decision-makers, franchisees, that will have to

simultaneously decide where to locate their own services (unsure about the decisions of one another). All

services compete among themselves. At most one service can be located at each potential location. We

consider that one of the decision-makers has preferential rights meaning that if both decision-makers are

interested in the same location, only to this decision-maker will be given the permission to open the service.

We present a mathematical formulation and some conclusions based on computational results.

1 INTRODUCTION

Competitive location problems consider the situation

where it is not sufficient for a given decision maker

to consider only his own facilities when faced with a

location decision (throughout the paper we will refer

to facility and service interchangeably). Most of the

times, these facilities will compete with similar

facilities in the market, so that the customers’ share

that will be assigned to the decision-maker’s

facilities depends on his own choices as much as on

the competitors’ decisions. In this paper we work

with a competitive discrete location problem where

two decision makers (players) will have to decide

simultaneously where to locate their own facilities,

unsure about the decisions of one another.

Several authors have studied competitive

location problems (for a review see, for instance,

Plastria, 2001). Dobson and Karmarkar, 1987, study

a discrete competitive location problem in which

price and demand are fixed, and considering an

existing firm, a competitor, and clients that want to

minimize the distance traveled. Labbé and Hakimi,

1991, study the problem in which two firms have to

make decisions regarding the location of facilities

and also the quantities of a given commodity they

will make available. Vetta, 2002, also proposes a

location game where multiple decision makers

(service providers) start by deciding where to locate

their facilities and then define how much they

charge their customers. Hande et al, 2011, study a

sequential competitive location problem where the

follower can react to the decisions made by the

leader, adjusting the attractiveness of their own

services.

Among the competitive location linear

programming problems, most approaches either

consider that the firms already present in the market

will not be able to react to the decision-maker’s new

chosen locations or consider a Stackelberg problem,

where there is a follower that will react to a leader,

knowing what the leader has decided. These types of

problems differ significantly from the problem

tackled here. We consider a situation where a

franchiser intends to open new facilities in a given

area. There are two potential investors, and the

facilities to be open will compete among themselves.

They provide the same type of commodities to

consumers, at the same prices, and it is assumed that

customers patronize the closest available facility.

The franchiser defines the finite set of potential

locations for facilities, but he is not familiar with the

demand patterns of the area. So, he will define more

potential locations than he expects the investors to

choose, and leave the choices among them to the

investors, who are better acquainted with the area.

The franchiser payoff will be a percentage calculated

120

Godinho P. and Dias J..

A TWO-PLAYER MODEL FOR THE SIMULTANEOUS LOCATION OF FRANCHISING SERVICES WITH PREFERENTIAL RIGHTS.

DOI: 10.5220/0003717401200125

In Proceedings of the 1st International Conference on Operations Research and Enterprise Systems (ICORES-2012), pages 120-125

ISBN: 978-989-8425-97-3

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

over the total demand assigned to the new facilities.

Each investor is interested in maximizing the total

demand that is assigned to his own facilities. Each

investor is aware of the fixed costs incurred by

opening each and every facility, which can be

different for both investors. Each investor has a

budget constraint. They are also aware of the

demand associated with each customer. This demand

will not increase with distance, meaning that the

closer the assigned facility is of a given customer,

the greater the demand from the customer. At each

location at most one facility can be opened. If the

decision-makers were to decide sequentially, this

problem would be a sequential problem that could

be formulated as a bilevel linear programming

problem. But we consider that both decision-makers

will have to decide simultaneously. In this situation,

it will be necessary to clarify what happens if both

investors apply for the same location. As at most one

service can be opened, the franchiser will have to

decide what to do in this case. We consider that the

franchiser patronizes one investor, in detriment of

the other. For the sake of simplicity, consider that

the franchiser always chooses investor 1. This means

that if both apply for the same location, then the

franchiser will allow investor 1 to open the facility,

and investor 2 will not be able to do so. We can say

that investor 1 has preferential rights, which is

known by both decision-makers. This problem can

also be interpreted as a full information game

(because each player knows the payoffs and

strategies of the other), with a finite number of

players (the two decision-makers), and a finite

number of pure strategies (for each player, a pure

strategy can be defined as a particular combination

of locations, out of the set of potential new locations,

where the player chooses to open facilities). That is

why we will not distinguish between investor,

decision-maker and player, and will use these terms

interchangeably as having the same meaning.

We approach this simultaneous decision problem

from a mathematical programming point of view (a

preliminary mathematical formulation appeared in

the research report Dias and Godinho, 2011) and

from a game theory point of view. The game will

have at least one Nash equilibrium, possibly with

mixed strategies, that can be calculated

algorithmically. Some computational results are

presented and conclusions drawn.

2 PROBLEM FORMULATION

In this problem we are considering that both

decision-makers will decide simultaneously, without

knowing the decision made by the other. We will

also accommodate the existence of already opened

services. Let us consider that these services belong

to investor 1.

Consider the following definitions:

−

set of pre-existing facilities that belong to

investor 1;

−

set of potential locations for new facilities;

−

set of customers;

−

demand associated with customer j when he

is assigned to a facility located at i;

−

distance between customer j and location i;

−

fixed cost associated with investor p opening

a facility at location i (and such that

∀∈0,

)

−

percentage over the demand captured to be

paid to the franchiser by investor p;

−

maximum budget available to investor p.

We consider that demand will not increase with

distance. We will additionally assume that potential

locations at the same distance will capture the same

demand. This means that:

≤

⇒≥ ∀∈∪∀∈,, ,

ij kj ij kj

cc ddikFGjJ

(1)

⇒= ∀∈∪∀∈,, ,

ij kj ij kj

cc ddikFGjJ

(2)

Let us define the following decision variables:

1, if investor 1 either opens

a facility at or has a

pre-existing facility at

0, otherwise

iFG

⎧

⎪

∀∈ ∪

⎨

⎪

⎩

1, if investor 2 bids for

opening a facility at

0, otherwise

wiFG

⎧

⎪

∀∈ ∪

⎨

⎪

⎩

1, if investor 2 opens a facility at

0, otherwise

ziFG

⎧

∀∈ ∪

⎨

⎩

1, if client is assigned to

facility that belongs to

investor 1

0, otherwise

iFGjJ

⎧

⎪

∀∈ ∪ ∀∈

⎨

⎪

⎩

1, if client is assigned to

facility that belongs to

investor 2

0, otherwise

miFGjJ

⎧

⎪

∀∈ ∪ ∀∈

⎨

⎪

⎩

A TWO-PLAYER MODEL FOR THE SIMULTANEOUS LOCATION OF FRANCHISING SERVICES WITH

PREFERENTIAL RIGHTS

121

In most location problems, only binary variables

similar to

and

are needed. For this problem,

however, another set of variables,

, is essential to

allow the distinction between two different

situations: of bidding for and of being able to open a

facility. This distinction is not needed for investor 1:

he will open every facility that he bids for, because

he has preferential rights. But investor 2 can bid for

a location and still not be able to open a facility there

if investor 1 has also shown interest for the same

location.

In this problem, different sets of variables are

controlled by different stakeholders: investor 1

controls variables

∀∈,

yiG

; investor 2 controls

variables

∀∈,

wiG

; the franchiser controls

variables

∀∈,

ziG

(according to a predefined rule

known by both decision makers); customers control

variables

∀∈ ∪ ∈,,

ij ij

miFGjJ

(also according

to known rules – in this case resorting to the

minimum distance criteria). Decisions made by the

franchiser and by customers are not controlled by

the two decision-makers, despite the fact that these

decisions play a crucial role in the determination of

each players’ payoff. An important point to make is

that despite not being under their control, decision

makers are both fully aware of how these decisions

are made. As a matter of fact, once variables

and

are fixed it is possible to immediately compute

the corresponding values for

and

Each investor will make his own decisions

conditioned only by his own constraints. A set of

connection constraints is then considered that will

determine the values of the remaining variables

according to the pre-established rules.

Let us now formulate the problem, following the

representation introduced in Godinho and Dias,

2010:

Decision-Maker 1

∈∪ ∈

−

∑∑

(1 )

ij ij

iFGjJ

ax d x

(3)

Subject to:

∈

≤

∑

11ii

(4)

=∀∈1,

yiF

(5)

Decision-Maker 2

(1 )

ij ij

iF GjJ

ax d m

∈∪ ∈

−

∑∑

(6)

Subject to:

∈

≤

∑

fw O

(7)

wiF=∀∈

(8)

Connection restrictions

≤∀∈∪1,

yz iFG

(9)

≤

∀∈ ∪ ∈,,

ij i

mziFGjJ

(10)

≤

∀∈ ∪ ∈,,

ij i

yiFGjJ

(11)

∈∪

=∀∈

∑

()1,

ij ij

iFG

mjJ

(12)

1,,,

ij ij k k ij

mx zyiGjJkT

≤− − ∀∈ ∈ ∈

(13)

≤

∀∈ ∪,

zwiFG

(14)

{

}

{}

∈∀∈∪

∈

∀∈ ∪ ∈

∈

∀∈ ∪ ∈

,, 0,1,

0, 1 , ,

0,1 , ,

iii

zwy i F G

iFGjJ

miFGjJ

(15)

Regarding decision-maker 1, he will maximize

his payoff subject to the restriction that he has to

afford to open all the facilities he bids for (constraint

(4)). Constraint (5) guarantees that the existing

facilities will stay open. A similar objective function

is considered by decision-maker 2, and similar

constraints: a budget constraint (7) and a constraint

that does not allow him to bid for already opened

services (8). Constraint (9) guarantees that at most

one service is opened at each location. Customers

can only be assigned to opened facilities (constraints

(10) and(11)), and have to be assigned to exactly one

facility (12). Each customer is assigned to the closest

opened facility (13). We are not considering the

situation such that a customer is equally distant from

two or more opened facilities. This possibility can

easily be considered, assuming that the demand of a

customer is equally split by two or more opened

facilities (see Godinho and Dias, 2010).

Constraints (14) state that investor 2 can only

open facilities he has bid for.

Each solution to this problem is composed of a

set of

variable values, which we will denote as

vector

, and a set of

variables’ values, which

we will denote as vector

. Interpreting this

problem as a game,

is a strategy for player 1, and

is a strategy for player 2. An admissible solution

is a Nash equilibrium solution. In the case of a Nash

equilibrium with pure strategies, this means that

(

)

is admissible if

is a best response to

and vice-versa.

3 COMPUTATIONAL RESULTS

There is no obvious procedure for solving the

two-player simultaneous decision problem presented

in the previous section. Therefore, in order to

ICORES 2012 - 1st International Conference on Operations Research and Enterprise Systems

122

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

+∞

, 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

with

w out

Absolute

with w out

ππ

−

Relative

with w out

ππ

−

Absolute

w out with

ππ

−

Relative

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%

with

: average payoffs for player 1 and player 2, respectively, when player 1 has preferential rights;

w out

: 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

123

Table 2: Summary of the results obtained with test sets 1 (repeated for easier reference) and 5-7.

Test

set

Potential

locations for

player 2

Average return (with

preferential rights)

Average return (without

preferential rights)

Player 1 benefit from

preferential rights

Player 2 loss from player 1 rights

with

w out

Absolute

with w out

ππ

−

Relative

with w out

ππ

−

Absolute

w out with

ππ

−

Relative

with w out

ππ

−

1 48 1427.8 952.8 1197.6 1196.9 230.2 19.2% 244.1 20.4%

5 36 1419.1 916.8 1243.7 1152.0 175.4 14.1% 235.2 20.4%

6 24 1519.5 882.5 1365.7 1092.5 153.7 11.3% 210.0 19.2%

7 12 1543.6 742.3 1421.8 932.4 121.8 8.6% 190.1 20.4%

with

: average payoffs for player 1 and player 2, respectively, when player 1 has preferential rights;

w out

: average payoffs for player 1 and player 2, respectively, when there are no preferential rights.

Table 3: Summary of the results obtained with test sets 1 (repeated for easier reference) and 8-10.

Test

set

Player 2’s

budget

Average return (with

preferential rights)

Average return (without

preferential rights)

Player 1 benefit from

preferential rights

Player 2 loss from player 1 rights

with

w out

Absolute

with w out

ππ

−

Relative

with w out

ππ

−

Absolute

w out with

ππ

−

Relative

with w out

ππ

−

1 1000 1427.8 952.8 1197.6 1196.9 230.2 19.2% 244.1 20.4%

8 750 1567.4 763.8 1340.8 1000.9 226.6 16.9% 237.1 23.7%

9 500 1576.7 505.6 1425.2 675.4 151.4 10.6% 169.8 25.1%

10 250 1678.4 244.4 1576.4 348.0 102.0 6.5% 103.7 29.8%

with

: average payoffs for player 1 and player 2, respectively, when player 1 has preferential rights;

w out

: average payoffs for player 1 and player 2, respectively, when there are no preferential rights.

Table 4: Summary of the results obtained with test sets 1 (repeated for easier reference) and 11-13.

Test

set

Average

fixed facility

cost

Average return (with

preferential rights)

Average return (without

preferential rights)

Player 1 benefit from

preferential rights

Player 2 loss from player 1 rights

with

w out

Absolute

with w out

ππ

−

Relative

with w out

ππ

−

Absolute

w out with

ππ

−

Relative

with w out

ππ

−

11 175 1961.2 1150.1 1590.0 1580.0 371.2 23.3% 429.9 27.2%

12 262.5 1709.6 1089.9 1408.5 1429.0 301.1 21.4% 339.1 23.7%

1 350 1427.8 952.8 1197.6 1196.9 230.2 19.2% 244.1 20.4%

13 525 1109.6 832.4 965.0 989.0 144.6 15.0% 156.7 15.8%

with

: average payoffs for player 1 and player 2, respectively, when player 1 has preferential rights;

w out

: average payoffs for player 1 and player 2, respectively, when there are no preferential rights.

potential locations, it becomes more important to

player 1 to get preferential rights, in order to secure

exclusive benefits from the most interesting

locations.

Test sets 8-10, considered along with test set 1,

allow us to analyze the consequences of changing

the budget of a player, while keeping the other

player’s budget constant. We defined that player 1’s

budget is 1000, and set player 2’s budget to 1000,

750, 500 and 250 in test sets 1, 8, 9 and 10,

respectively, with all other parameters’ values held

constant. The results are summarized in Table 3.

As expected, player 2’s payoff increases when

his budget increases, and player 1’s payoff decreases

in that situation. The benefit from having

preferential rights becomes more significant for

player 1 as player 2’s budget increases. This means

that, as player 2 is able to build more facilities, it

becomes more important for player 1 to secure

exclusive benefits from the best locations. As for

player 2, the absolute loss from player 1’s rights

increases with his budget, but the relative payoff

reduction becomes less significant for higher

budgets.

Test sets 11-13, considered along with test set 1,

allow us to analyze what happens when the average

fixed cost of each facility changes and the players’

budgets are kept constant. We set the average cost of

each facility to 175, 262.5, 350 and 525 in test sets

11, 12, 1 and 13, respectively. The other parameters’

values were identical to the ones used in test set 1.

The results are summarized in Table 4. The payoffs

ICORES 2012 - 1st International Conference on Operations Research and Enterprise Systems

124

of both players decrease as the average cost of each

facility increases. When the average cost increases,

players are able to open less facilities, thus reducing

their payoffs. At the same time, the increase in

average facility cost reduces the absolute and

relative benefit player 1 gets from preferential rights,

and it also reduces the absolute and relative loss

incurred by player 2. In fact, with the increase in

average facility cost, and the consequent reduction

in the number of facilities, the level of competition

between players decreases, reducing the impact of

preferential rights.

4 CONCLUSIONS

We have introduced a simultaneous discrete location

problem with two decision-makers, in a franchising

environment, where one of the players has

preferential rights. This model has several

distinguishing features, namely the fact of

considering explicitly simultaneous decisions

instead of sequential decisions. We have formulated

the problem as a linear programming problem, and

have defined as admissible solutions those that are

Nash equilibrium solutions.

The computational results show us that if the

level of competition increases, then the importance

of having preferential rights also increases. The level

of competition is higher when there are fewer

potential locations for opening facilities, when fixed

opening costs decrease keeping the budget constant,

or when the budget sizes are similar.

The developed work raises other questions,

namely what happens if it is given to the player

without preferential rights the possibility of bidding

for more facilities than the ones he can afford. This

will be the subject of further research.

ACKNOWLEDGEMENTS

This research was partially supported by research

project PEst-C/EEI/UI0308/2011.

REFERENCES

Dias, Joana and P. Godinho. (2011). Some thoughts about

the simultaneous location of franchising services with

preferential rights. Inesc-Coimbra Research Report,

3-2011

Dobson, G. and U. S. Karmarkar. (1987). Competitive

Location on a Network. Operations Research, 35,

565-574.

Godinho, P. and J. Dias. (2010). A two-player competitive

discrete location model with simultaneous decisions.

European Journal of Operational Research, 207,

1419-1432

Hande, K. Aras and N. Altınel. (2011). Competitive

facility location problem with attractiveness

adjustment of the follower: A bilevel programming

model and its solution. European Journal of

Operational Research, 208, 206-220.

Labbé, M. and S. L. Hakimi. (1991). Market and

Locational Equilibrium for Two Competitors.

Operations Research, 39, 749-756.

Plastria, F. (2001). Static Competitive Facility Location:

An Overview of Optimization Approaches. European

Journal of Operational Research, 129, 461-470.

Vetta, A. (2002). Nash Equilibria in Competitive

Societies, with Applications to Facility Location,

Traffic Routing and Auctions. Proceedings of the 43rd

Annual IEEE Symposium on Foundations of Computer

Science (FOCS ‘02), IEEE Computer Society Press.

A TWO-PLAYER MODEL FOR THE SIMULTANEOUS LOCATION OF FRANCHISING SERVICES WITH

PREFERENTIAL RIGHTS

125