An Approach to Land Use Equilibrium by Potential
Game and Its Application to Land Oligopoly for
Parking and Its Dynamics
INU021005

Aoi Watanabe
awatanabe@bin.t.u-tokyo.ac.jp
Eiji Hato
hato@bin.t.u-tokyo.ac.jp

Abstract
It is essential to understand the formation mechanism of the current distribution of parking spaces
to implement proactive parking allocation policies. However, few studies have considered the land use
aspect of parking lots. In this study, to elucidate the generation mechanism of the parking lot distribution
in the city center, we integrate parking into an urban spatial equilibrium model. The uniqueness of ourmodel is that it explicitly considers agglomeration economies on a walking scale and does not assume
an explicit CBD. Thus, it enables us to deal with the externality that the location of the parking lot
breaks up the agglomeration of surrounding shops in the city center. Furthermore, we identify a globallystable solution by proving that we can interpret this equilibrium problem as a potential game. Although
the transformation to a potential game is generally impossible when there are multiple types of agents,
the unique nature of parking lots that we must come back to where we parked makes this possible. As
a result, depending on the nature of facilities other than parking lots located in a city, we found thatit may not be easy to voluntarily agglomerate parking lots. Also, the results suggest that we can onlychange the distribution of parking spaces by policies that can manipulate the choice of parking spaces
by visitors, such as toll tie-ups, street maintenance, and charging. Therefore, the relaxation of minimum
parking requirements cannot aﬀect the distribution characteristics of parking spaces.

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INTRODUCTION
The static traffic assignment problem has undergone a certain level of development with the proposal of Beckman’s optimization method in response to the Wardrop principle (1). On the other
hand, if we review the traffic network analysis from a game-theoretic point of view, we can reformulate it in the framework of non-cooperative games, and it has been shown that a potential
game (2) can describe it. Based on the hypothesis that a single potential function can explain any
player’s gain, the convergence of the solution is guaranteed by better response dynamics (3). It
is also known that the stability of the better response dynamics in the direction of time evolution
can be discussed in the same way for the dynamic traffic assignment problem (4). However, research on the analysis of solutions on transportation networks is limited to analyzing traffic flows
and day-to-day network equilibrium. No research applies such an analysis to urban land use, especially parking lot selection and location, where decisions are made concerning transportation
dynamics.
On the other hand, in the land-use model, since Anas (5), there has been an accumulation of analyses of location patterns resulting from interactions between different players, whichhave been interpreted in terms of game theory. In this paper, we propose an analysis method oflocation patterns using a potential game, focusing on the relationship between the location of parking lots and travel behavior, which manifests the interaction between transportation and land use,
based on the methodology of stability analysis of solutions based on game-theoretic approaches intransportation and land use research.
In the field of transportation, there is an accumulation of studies dealing with the interaction between transportation and land use (5, 6). These studies consider the interaction between thelocation of residential and commercial uses and the destinations, means, and routes of trips. Similarly, the location of a transportation node such as a station affects both traffic and land use (7).
However, few studies discuss the effects of parking lots, which are transportation nodes between
driving and walking, from both transportation and land use perspectives.
Most of the previous studies on parking lots focused only on the transportation aspect ofparking lots. In other words, most of the studies have focused on how to allocate the demand thatis concentrated in urban areas appropriately. For example, (8–10) study price-based demand management, (11–14) study reservation systems for parking lots, and (15–17) study trading systems
for parking permits. However, the distribution of parking lots is fixed in all of these studies. In
other words, the impact of the policies on the distribution of parking lots as land use has not been
considered.
In recent years, with the development of AVs and EVs, research has been conducted to optimize the placement of parking lots (18, 19). However, such research only deals with the methodology to find the optimal location and does not discuss whether it is possible to change parking lots
in real cities spontaneously. It has been pointed out that the area of parking lots in urban centers isvast (20), and it is essential to understand the location theory of parking lots as land use (21).
There are only three theoretical studies that consider the land use aspect of parking lots,
integrate parking lots into an urban spatial model, and discuss the changes in their location (22–24). All of them assume that people commute to the CBD by car in a monocentric city and discuss
how the distribution of residential areas and parking areas changes with the distance to the CBD
at a relatively wide scale due to locational equilibrium. These studies are excellent in that they
explicitly treat parking as a form of land use and incorporate the interaction between the locationof housing and parking and people’s choice of parking, route, and congestion. However, on a

Watanabe and Hato2

small walking scale in the city center, parking as land use has the externality of reducing the effectof increased profits from the agglomeration of shops (25, 26). These studies do not take theseexternalities into account.
This study deals with the location of parking lots in walking-scale areas in urban centers.
It explicitly deals with the interaction between the transportation aspect, in which visitors selectparking lots according to the distance to the destination, and the land use aspect, in which the location of a parking lot cancels out the increase in profits from the agglomeration of shops in thesurrounding area. Compared to previous location equilibrium models that incorporated parking
lots (22–24), this study deals with a smaller scale without assuming a definite CBD. We explicitly
incorporate the fact that parking lots have externalities that disrupt the agglomeration of surrounding shops using the location model for multiple agents considering agglomeration economies.
As a result, this study suggests that, depending on the nature of facilities other than parking
lots located in a city, it may be difficult to voluntarily agglomerate parking lots. In recent years, therelaxation of minimum parking requirements has been implemented to agglomerate parking lots.
The results suggest that such a policy cannot affect the distribution of parking spaces and that we
can only change the distribution of parking spaces by policies that can manipulate the choice ofparking spaces by visitors, such as toll tie-ups, street maintenance, and charging.
In addition, this study has a contribution to the stability analysis of the solution. This
study describes the location point selection behavior of two types of location agents, shops andparking lots, and analyzes the location distribution realized as static equilibrium. We can express
the equilibrium as a solution to a variational inequality, but since this is a location equilibrium
problem with multiple dependent agents, multiple local equilibrium solutions exist. This study
shows that we can interpret this equilibrium problem as a potential game by explicitly treating theconstraint that the parking lot imposes on visitors, which is that they must return to the place where
they parked their car. It enables us to concretely identify a globally stable solution among multiple
equilibria using stability analysis.
The equilibrium solution and its stability analysis have been studied extensively in traffic
assignment. In the traffic assignment problem, we describe the user equilibrium state as a variational inequality problem for the route traffic or link traffic (27, 28), which is mathematicallysimilar to the problem in this study. Since there are multiple equilibrium states under general conditions, many researchers have conducted stability analysis by constructing Lyapunov functions
for various day-to-day dynamics to determine which equilibrium state is plausible (29–33).
In general user equilibrium problems, there is no unified way to construct the Lyapunov
function. However, if the Jacobi matrix of the link cost function is symmetric, stability can be easily
discussed by interpreting the equilibrium problem as a potential game (2, 3). In potential games, thepotential function is a Lyapunov function of various dynamics (3). Therefore, in a user equilibrium
problem without inter-link interference (1) or a relatively simple location equilibrium problem witha single locating entity (34), we can identify a globally stable solution as an optimization problem
by attributing it to the potential game.
The symmetry of the cost function, which is an existing condition of the potential function,is generally not satisfied when there are multiple types of agents, as in the case of the problem
treated in this study. According to (31, 35), even in the case where there are multiple types oflocational agents, it is possible to transform the equilibrium problem into a potential game when
the profit of each agent does not explicitly depend on the location of other types of agents but
interacts only through endogenous variables as in (25, 36). In this study, however, the profit of theWatanabe and Hato

3

parking lot owner explicitly depends on the location of the shop owner. One of the contributionsof this study is that we show that even under such circumstances, we can interpret the original
equilibrium problem as a potential game by taking advantage of the characteristic of parking lotsthat visitors must return to the place where they parked their cars. Moreover, it allows us to seek aglobally stable solution specifically.
We organized the remainder of this paper as follows. In Section 2, we formulate the parking
lot location problem. Section 3 interprets the proposed parking lot location model as a potential
game, and we present an equilibrium solution analysis method to identify the globally stable state
by stochastic stability analysis. In section 4, we show the bifurcation of the globally stable stateby numerical calculations and show the implications of the parameters. In section 5, we present an
example of temporal change of parking lots and confirm the applicability of this model. Finally,
section 6 presents the conclusions of this paper and future issues.
MODELING
This section will formulate a parking location model on a walking scale using a spatial economic
model that considers agglomeration economies like (25, 26, 36). The difference from the previous
multi-agent spatial economic model is that the profit of the parking agents directly depends on thelocation of other kinds of agents.
Problem settings
We assume a circular city consisting of K location points, as shown in Figure 1(a), and let the setof location points be K ≡ {0, · · · , K − 1}. When we take the limit of the number of locations, thediscrete circular city becomes a continuous city, as in Figure 1(b). In the analysis of the solution,
we use the continuous circular city. The city’s total area is S, and we fix the area of each locationas S ≡ S/K. Although the circular city is seemingly unrealistic, it is superior in that it facilitates
analytical treatment and allows us to see the results arising from pure interaction only in the modelby ensuring the equivalence of each location. For this reason, the circular city is a typical settingin spatial economics (37–40).
We assume that there are only two types of location agents, shops and parking lots, inthe city. The set of shop agents is S ≡ {S1 , · · · , SM }, and the set of parking lot agents is P ≡{P1 , · · · , PN }. In other words, there are M of shop agents and N of parking lot agents. We express
the total number of location agents as N0 ≡ M + N. We denote the number of shops locatedat location i ∈ K as mi and the number of parking there as ni . m ≡ [mi ] and n ≡ [ni ] are vectors
representing the number of shops and parking lots at each location i ∈ K , respectively, and indicate
the distribution of shops and parking lots in the city. Each shop and each parking lot selects its
location in a way that maximizes its profit. Shops and parking lots in the city earn revenue fromvisitors to the city. People who visit the city from the outside of this circular city by car park at aparking lot in the city, walk to a shop in the city, purchase goods, and then return to the parking
lot where they parked. The term "shops" here is strictly a generic term for destinations for visitors,
including workplaces. We use the term "shops" for simplicity. Also, in some cases, shops provide
their parking lots. We can interpret such a situation in the framework of our model, as we will
describe in Chapter 5.
Figure 2 shows how land use and transportation interact. As for the land use aspect, shop
agents and parking agents choose their locations based on profit maximization. In this way, we candetermine the distribution of parking lots and shops in the city, as described in section 2.4. Various

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0

0

1

x

2
…
(a) circular city (the example of K=12)(b) continuous circular city

FIGURE 1: The settings of the city. We assume a circular city, consisting of K equivalent location
points as shown on the left side. When we take the limit of the number of locations, the discrete
circular city becomes a continuous city, as shown on the right side.
possible land-use patterns can be realized as equilibrium. On the other hand, as a transportation
aspect, visitors to the city choose a parking lot based on the distance to the shop, which determines
the demand for parking lots and shops at each location, as described in sections 2.2 and 2.3.
The behavior of parking lot agent
Visitors to the city by car, on their way to a shop, park their car in a parking lot somewhere in
the city. After purchasing at the shop, they return to the parking lot where they parked their car.
Visitors tend to choose parking lots that are close to their destinations. Therefore, the more shops
in the vicinity, the higher the revenue of the parking lot. Assuming that visitors choose a parking
lot based on the distance to the destination and the revenue per visitor is the same, we can drive
the parking profit as equation (1) by calculating the choice probability of each parking lot. Please
note that the parking profit does not depend on the parking distribution. This is because the model
already incorporates the disutility of having many parking lots around by the absence of shops
around. When the distribution of shops is m , and the parking Pn ∈ P is located at point i ∈ K , wem) as
can write the profit ΠPi n (mPnPm) = ∑ Di j · m j − sP · Ri .
Πi (m
(1)
j∈K
P
s is the land area (constant) required as a parking lot, and Ri is the land rent at the location i. Also,
DPij is a distance attenuation effect, which can be written using the parking-shop transportation cost
parameter σ > 0 as DPij = exp(−σ li j ). Since li j denotes the shortest distance between the location

points i and j, the closer the location is to many shops, the more the parking revenue increases.
This describes that visitors choose a parking lot based on the distance to their destination.
The larger the parking-shop transportation cost parameter σ is, the less profitable the parking lot will be if there are no more nearby shops. Conversely, when σ is small, the parking lot canbe profitable even if the shops are far apart because visitors can choose to park in a parking lotthat is far from their destination. The parking lot agent Pn ∈ P selects location i ∈ K with theWatanabe and Hato

5
: parking zone

Land Useshop/parking location

visitor's parking choice: mixed zone

・・・・・

agent's location choiceTransportation: shop zone

agent's proﬁt
car

car
park
walk

park
walk

FIGURE 2: Interactions between land use and transportation. The locators of shops and parking
lots choose their location points based on profit maximization, which determines parking lot distribution and shop distribution in the city. On the other hand, visitors to the city choose parking
lots based on the distance to the shops, which determines the demand for parking lots and shops.
maximum profit.
At first glance, the model does not deal with the capacity of parking lots, but it does deal
with it indirectly. In our model, the parking demand at each location point determines whether
the location point will be a parking lot or a shop. Since the capacity corresponds to the area,
determining the ratio of parking spaces to the area constraint of each location point corresponds toallocating the parking capacity to balance the parking demand of visitors to the city.
The behavior of shop agent
Because of the constraint of having to come back to the parking lot where we parked our car, there
is a customer exchange between the shop and the parking lot. Therefore, by formulating the profitof the parking lot as in equation (1), the direct profit by visitors from the parking lot among the
shop’s profits when the shop agent Sm ∈ S is located at location i ∈ K is derived asrS(2)∑ DPij · n j .
rP j∈K
Here, rP and rS are respectively the revenue per parking space user and the revenue per shop user
for a simple round trip between the shop and the parking lot. They are both constants.
Equation (2) is the revenue from the simple back and forth between one parking lot andone shop among the shop revenues. In fact, for shops, revenue increases due to the wandering
behavior of neighboring shops as they agglomerate (25, 26, 36). So, we assume that the increase inWatanabe and Hato

6

shops’ revenue is due to the agglomeration effect of shops. In other words, assuming that equation(2) describes only the parking lot selection behavior of visitors to the destination, we normalizerS = rP = 1 in equation (2) and assume that the shop agglomeration effect causes the incremental
shop revenue per user. We can describe the incremental revenue of the shop Sm ∈ S at location
point i ∈ K associated with shop agglomeration following Fujita and Ogawa (36) as(3)∑ DSij · m j .
j∈KHere, DSij is the distance attenuation effect and can be written using the inter-shop transportation
cost parameter τ > 0 as DSij = exp(−τli j ). In other words, the closer the location is to a large
number of shops, the higher the shop revenue will increase. The larger the inter-shop transportation
cost parameter τ, the greater the effect of shop agglomeration, and the more shops prefer more
substantial agglomeration.
From the above, the profit of a shop Sm ∈ S choosing location i ∈ K is as follows, accounting for the normalization of rS = rP = 1 described above:m, n ) = ∑ DPij · n j + ∑ DSij · m j − sS · Ri −C.
ΠSi m (m
(4)
j∈K
j∈K
S
Here, s is the land area (constant) required as a shop. Since shop maintenance costs are higher

than parking lot maintenance costs, we denote the increment of shop maintenance costs comparedto parking lot maintenance costs by C. The shop agent Sm ∈ S selects the location i ∈ K with the
maximum profit.
The difference between the proposed model and existing multi-agent spatial economic
models (25, 36) is that the profit of each agent explicitly depends on the location of another typem) depends on the shop distribution m , and theof agent. For example, the parking lot profit ΠPi n (mSmm, n ) also depends on the parking lot distribution n . In the existing models, theshop profit Πi (magents interact only through endogenous variables such as land rent. Also, the difference between
the proposed model and existing spatial equilibrium models incorporating parking (22–24) is thatthe proposed model does not assume CBD and explicitly considers agglomeration economies on awalking scale. We explicitly deal with the externality that the location of the parking lot breaks up
the agglomeration of shops.
Equilibrium conditions
Here, we show the equilibrium conditions resulting from the site selection based on the behavior
rules above. With the endogenously determined equilibrium profits of the shop and parking as ΠS∗P∗
andcan write the spatial equilibrium conditions as( Π , weSmS∗m, n ) if mi > 0Π = Πi (m∀i ∈ K , ∀Sm ∈ S (5)m, n ) if mi = 0
ΠS∗ ≥ ΠSi m (m(m) if ni > 0
ΠP∗ = ΠiPn (m∀i ∈ K , ∀Pn ∈ P. (6)Pm) if ni = 0
ΠP∗ ≥ Πi n (mAs a market equilibrium condition, if there is positive land rent, the area of supply anddemand will coincide in the land market. From this relation, the equilibrium land rent Ri is determined
endogenously:(
S = sS · mi + sP · ni if Ri > 0∀i ∈ K . (7)S ≥ sS · mi + sP · ni if Ri = 0

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Also,
we can write the conservation of the number of agents as ∑ mi = M
i∈K ∑ ni = N.

(8)

i∈K

We can obtain the parking distribution n as an equilibrium solution by solving the equilibrium problem of variational inequality type, denoted by equations (5)(6)(7)(8). However, thisvariational inequality type problem has multiple equilibria. Therefore, as described in the next
section, we replace the equilibrium problem with an equivalent optimization problem, making it
possible to identify a globally stable solution among the local equilibrium solutions using stochastic stability analysis.
ANALYSIS OF EQUILIBRIUM STATESThis section describes a method for identifying a globally stable solution to the equilibrium problem defined in the previous section.
Interpretation of the model as a potential gameWe can interpret The equilibrium problem formulated in section 2.4 as a population game in which
N0 ≡ M + N players choose a location under each profit maximization. We show that we caninterpret this population game as a potential game with the potential function (9). If we write down
the KKT condition, we can confirm that it is consistent with the equilibrium conditions in 2.4.
For simplicity, we assume that the difference between the maintenance cost of the shop and theparking lot C = 0. Even if C > 0, subsequent transformations to the equivalent potential function
maximization problem are still possible, and the results shown below have robustness for the value
of C.
max

m ≥00,nn≥00

subject tom, n ) =Z(m1∑ ∑ miDSij m j + ∑ ∑ niDPij m j
2 i∈K
j∈K
i∈K j∈K

sS · mi + sP · ni ≤ S(9)∀i ∈ K∑ mi = Mi∈K∑ ni = N.
i∈KWhen we can interpret the population game as the potential game, the set of Nash equilibria in the population game is consistent with the set of local maximum of potential function(41). In addition, under various day-to-day dynamics, such as stochastic limit in Logit dynamics
and full foresight dynamics, the global maximization point of the potential function coincides withthe globally stable solution (31, 35, 41). In other words, instead of solving the original equilibrium problem, we can get the globally stable solution from multiple equilibria by considering theequivalent potential function maximization problem.
The local optimum solutions include the global maximization point. In a circular city, we
can enumerate the candidates for equilibrium solutions by symmetry. Therefore, by concretely calculating and comparing the values of the potential function for each candidate equilibrium solution,a globally stable solution can be obtained.
Here, for the potential function to exist, the Jacobi matrix of the profit function must beWatanabe and Hato

8

symmetric. In general, the transformation to a potential game is not possible when profits explicitly
depend on other kinds of agents. The critical point is that this transformation is made possible by
the parking lot property that visitors must return to the place where they parked their car.
Calculation of the potential function
First, for calculating the potential function, we take the limit of the number of locations andperform a continuous approximation for cities that have been discrete before. The discrete city
becomes a continuous circular city K˜ ≡ [−S/2, S/2], as Figure 1(b) shows, and the locationx ∈ [−S/2, S/2] becomes a continuous value. In this case, since the total area is S, the area oflocation x ∈ K˜ is S = 1.
The shop distribution m and the parking distribution n in the city become a continuousfunction with m(x) and n(x), respectively, and the shop profit ΠSi m and the parking profit ΠPi n
become a continuous function with ΠSm (x) and ΠPn (x), respectively. Also, the land rent Ri becomesa continuous function with R(x). Then, the parking profit expressed in equation (1) and the shop
profit expressedin equation (4) areZ
ΠPn (x) =DP (x, y)m(y)dy − sP · R(x)∀x ∈ K˜ , ∀Pn ∈ P
ZK

˜

Sm

Π (x) =K˜P

D (x, y)n(y)dy +ZK˜DS (x, y)m(y)dy − sS · R(x) −C∀x ∈ K˜ , ∀Sm ∈ S .

Now, the distance decay effects can be written respectively as DP (x, y) = exp(−σ lxy ) and DS (x, y) =exp(−τlxy ). As before, σ is the parking-shop transportation cost parameter, and τ is the inter-shop
transportation cost parameter.
From the above, with continuous approximation, the potential function Z for C=0 can be
rewritten as
ZZ
ZZ
1
S
D (x, y)m(x)m(y)dxdy +DP (x, y)n(x)m(y)dxdy.
(10)
Z(m(x), n(x)) =2 K˜ ×K˜
K˜ ×K˜If we integrate the constraint equation of the land area over the whole space, we get the equationsS · M + sP · N ≤ S. Now, we can assume that there is no vacant land in a city in an equilibrium
state, where neither shops nor parking lots are located. This is because shops prefer to clusterwith other shops, and parking lots prefer to cluster with shops, so such vacant spaces are filled
in a state of equilibrium. Therefore, we can rewrite the constraint expression for the land area assS · M + sP · N = S.
Next, we consider the candidates of equilibrium solutions. Given the pattern of equilibrium
solutions in (36), the candidate equilibrium solutions for a circular city can be broadly divided
only into the following three categories, based on the symmetry of the location points, as shown in
Figure 3. There are three types of patterns: complete mixed pattern, J pole distribution patterns,
and incompletely integrated patterns. The complete mixed pattern, such as Figure 3(a), is a statein which parking lots and shops are mixed over the whole area. On the other hand, the J pole
distribution pattern, such as Figure 3(b), consists of J zones with only shops and J zones with only
parking lots. That is, Figure 3(b) is an example of J = 3. Also, non-trivial equilibrium solutions,such as Figure 3(c), in which a mixed area of shops and parking lots coexist with parking zones
and shop zones, appear in the transition phase between the complete mixed pattern and the J pole
distribution pattern. This study focuses on whether the parking lots agglomerate or disperse. Since
it is sufficient to know the bifurcation of the solution between the complete mixed pattern and theJ pole distribution pattern, we do not compute it explicitly in this study. From now on, we refer toWatanabe and Hato9

the complete mixed pattern as the J = 0 pole distribution pattern.

mixed zone

parking zone

parking zone
mixed zone

shop zone(a) Complete mixed pattern (J=0)(b) J pole distribution pattern (J=3)

shop zone(c) incompletely integrated pattern

FIGURE 3: The candidates of equilibrium solutions. There are three types of patterns: complete
mixed pattern, J pole distribution patterns, and incompletely integrated patterns by the symmetryof the location points.
The above equilibrium solutions for the complete mixed pattern J = 0 and the J pole distribution pattern are derived analytically by computing the multiple integrals (10). Here, we normalize it to sS = sP = 1 without losing generality. The potential function Z0 (m(x), n(x)) for a complete
mixed pattern is 




S
2MN
S
M2
1 − exp −τ ·+1 − exp −σ ·
.
(11)
Z0 =
Sτ
2
Sσ
2
On the other hand, the potential function
for a J pole distribution pattern is( ZJ (m(x), n(x)) )MβJ sinh [τbJ ]2N
δ̂J sinh [σ ĉJ ]
ZJ =
αJ −
+
γ̂J −
(12)
τ
τbJ
σ
σ ĉJ
where,MNbJ ≡ , ĉJ ≡
2J
2J
(
1
(J : odd)
αJ ≡J/2
1 − ζJ(J : even)(exp[τbJ ] + 2 sinh[−τbJ ] · λJ(J : odd)
βJ ≡J/2
exp[τbJ ] − ζJ exp[−τbJ ] + 2 sinh[−τbJ ] · λJ (J : even)  
d(J−1)/2eSjζJ ≡ exp −τ, λ J ≡ ∑ ζJ
J
j=0
(
J/2
−ηJ
(J : odd)γ̂J ≡0(J : even)
( J/2
−1/2
−ηJ exp[σ bJ ] + 2ηJsinh[−σ bJ ] · µ̂J (J : odd)
δ̂J ≡
−1/2
2ηJ
sinh[−σ bJ ] · µ̂J(J : even)

Watanabe and Hato

10


 
bJ/2c
S
j
ηJ ≡ exp −σ, µ̂J ≡ ∑ ηJ .
J
j=1
Please note that J → ∞ and J = 0 are equivalent states following immediately from the definition.
It is also be confirmed analytically by the fact that the expression of the potential function in thelimit of J → ∞ is consistent with the expression of J = 0.
NUMERICAL STUDIESThis section computes the potential function values for the candidate equilibrium solutions shownin the previous section. We identify the globally stable state for a set of parameters (τ, σ ) byselecting the state where the potential function is maximized among the candidate equilibrium
solutions.
Bifurcation of globally stable state
Figure 4 shows the bifurcation of the globally stable solution. It shows six different calculations
for two S = N0 = M + N values, each with three different parking ratios. To see the whole pictureof the branch, we change the range of σ . Naturally, the scale of the parameter varies with themagnitudes of M and N. What is essential is that there is no change in the qualitative nature of
the results mentioned below regardless of the magnitude of M and N. In this numerical study, we
calculate J = 0 − 10 pole distributions as a candidate equilibrium solution. We confirm that it issufficient to compute up to the J = 10 pole distribution for finding a globally stable solution bylooking at the change in the potential function for the change in J.
Now, we can see that the nature of the bifurcation of the globally stable solution depends
on the magnitude of the inter-shop transportation cost parameter τ, as shown in Figure 4. In otherwords, in the state where τ is large, the agglomeration and dispersion of parking spaces change
discontinuously in response to the variation of the parking-shop transportation cost parameter σ .
On the other hand, in the state where τ is small, the complete mixed pattern J = 0 is retained as a
globally stable solution if σ is greater than a threshold value, even if the parking-shop transportation cost parameter σ changes. Based on these results, Figure 5 shows the regional separation of
parking and shop zones in a circular city when σ is changed.
We confirmed that the qualitative nature of the bifurcation of solutions in our numericalresults is robust to changes in the parking ratio within a realistic range (Figure 4), to differences in
the cost of maintaining shops and parking lots C, and to whether or not the number of shops andparking lots is fixed.
First, we mention the case where there is a difference between the cost of maintaining ashop and the cost of maintaining a parking lot. This is the case where C > 0. When C > 0, we
m, n ) = 12 ∑i∈K ∑ j∈K mi DSij m j +can rewrite the potential function written in equation (9) as Z(m∑i∈K ∑ j∈K ni DPij m j −CM. It differs from Equation (9) only in the addition of the third term. This
term is a constant term when we fix the number of parking agents. Therefore, the value of C does
not affect the globally stable solution at all.
Next, we consider the case where both the parking lot owner and the shop owner can freelychange the type of business. In other words, we do not fix the number of parking lots N and shops
M but only fix the total number of parking lots and shops N0 = M + N. In this case, we can easily
derive that the corresponding potential game only changes the constraint on the conservation of thenumber of subjects (8) as Σi∈K (mi + ni ) = N0 . We have confirmed by numerical calculation that

Watanabe and Hato

11(a) M+N=100
Complete mixed
pattern (J=0)

0.15increasing J
sigma

sigma

0.15

0.20

0.10

J

0.20

Complete mixed
pattern (J=0)

0.15increasing J
sigma

0.20

0.10

10

Complete mixed
pattern (J=0)

98increasing J

7
6

0.10

5
4
3

0.05

0.05

0.05

2
1
0

0.1 0.2

0.3 0.4 0.5
tau

0.6

0.1 0.2

0.7 0.8(a-1) M : N = 9 : 1

0.3 0.4 0.5
tau

0.6

0.7 0.8

0.1 0.2(a-2) M : N = 7 : 3

0.3 0.4 0.5
tau

0.6

0.7 0.8(a-3) M : N = 5 : 5(b) M+N=20

0.2

increasing J

0.4

0.3

sigma

0.3

0.1

Complete mixed pattern
J=0

0.4

sigma

sigma

0.4

Complete mixed pattern
J=0

J

0.5

0.5

0.5

0.2

increasing J

0.1

10

Complete mixed pattern
J=0

9
8
7

0.3

6
5

0.2

4
3

increasing J

0.1

2
1
0

0.1

0.2

0.3

0.4 0.5tau

0.6(b-1) M : N = 9 : 1

0.7

0.8

0.1

0.2

0.3

0.4
tau

0.5

0.6

(b-2) M : N = 7 : 3

0.7

0.8

0.1

0.2

0.3

0.4
tau

0.5

0.6

0.7

0.8

(b-3) M : N = 5 : 5

FIGURE 4: Bifurcation of a globally stable state. There is no change in the qualitative nature of
the results regardless of the magnitude of M and N. In the state where τ is large, the agglomerationand dispersion of parking spaces change discontinuously in response to the variation of the parkingshop transportation cost parameter σ . On the other hand, in the state where τ is small, the complete
mixed pattern J = 0 is retained as a globally stable solution if σ is greater than a threshold value,even if the parking-shop transportation cost parameter σ changes.
the bifurcation property of the globally stable solution does not change even when the constraint ischanged.
Interpretation of the parameters
The two parameters τ and σ lead to a bifurcation of the globally stable solution in our model. Theinter-shop transportation cost parameter τ is a parameter that indicates the strength of the shop
agglomeration effect, and the larger τ is, the more shops prefer a more substantial agglomeration
among themselves. As mentioned in section 2.1, "shops" here are a generic term for destinations
for visitors to the city. Therefore, the inter-shops transportation cost parameter τ indicates the nature of the buildings distributed within the city, i.e., whether the city is downtown or a residential
area. Specifically, τ is large if the facilities located except the parking lot have a strong agglomeration effect, such as shops. On the other hand, τ is small if they have a weak agglomeration effect,
such as offices or residences.

Watanabe and Hato

12largeτ

・・・・・

J=1

J=2

J=3

J=4

J=5

J=6

J=0

・・・・・・・・・・・・・・・・・・・・・・・・・

smallτ

J=1

decreasing σ

J=2

J=0
: parking zone

J=0
: shop zone

: mixed zone

increasing σ

FIGURE 5: The illustration of changes in the city when σ is changed. The bifurcation of the
globally stable solution depends on the magnitude of the inter-shop transportation cost parameterτ.
On the other hand, the parking-shop transportation cost parameter σ indicates how muchthe parking lot prefers proximity to the shop. The larger the σ is, the less the parking lot will
generate revenue unless it is close to the shop. This character is easy to understand from thevisitor’s choice of parking space from the point of view. Parameter σ expresses how far the visitor
can tolerate the distance to the destination or the disutility of walking between the parking lot andthe shop. Therefore, when there is an attractive destination or when street space is well maintained,
the walking disutility is small, and therefore σ is small.
DISCUSSION (CASE STUDIES)In this section, we present an example of temporal change of parking lots in the central city ofMatsuyama, Japan, and confirm the applicability of this model based on the above parameter interpretations. Figure 6 shows the temporal change of parking distribution in the vicinity of Okaido
Street, the street with a large concentration of shops, and in the periphery, residential area. Ourmodel deals with the walking scale and the location points are spatially uniform (the parameter
takes only one value within a region). Therefore, we have extracted the areas of the walking scalethat can be considered spatially uniform. One is a downtown area, and the other is a residential
area.
Figure 7 shows a projection of the actual data onto the circumference to make it easier tocompare with the model’s results that uses a circular city. We made this figure as follows. First,
we draw the largest circle in the center of each figure of Figure 6 that does not extend beyond
the region of interest. Second, we divide the circle into 36 parts according to their diameters.
Then, we calculate the parking area SiP for each region i. Finally, for each region i, we calculate
SiP /(Σi Sip /36) as an indicator of the concentration of parking spaces. If the parking lots are evenly
distributed in the target area, this value will be 1.0 for all regions. If the value is greater than 1.0 in
one place, it means that there is a higher than average concentration of parking spaces in that area.
This figure allows for a more intuitive comparison with the results of the model using circular city.
Figure 8 shows the Hoover Index of parking distribution changes, an indicator of the degree

Watanabe and Hato

13(a) the vicinity of Okaido Street, the downtown areaN

: parking50m

N: parkingN: parking: parking

50m

50m

50m

N

Okaido Street

Okaido Street

Okaido Street

1990

2000

2010

Okaido Street

2019(b) the periphery, the residential areaN

: parkingN: parking

50m

50m

1990

N

: parking

50m

2000

N

: parking

50m

2010

2019

FIGURE 6: The temporal transition of the distribution of parking lots in the vicinity of Okaido
Street and that in the periphery. It is made based on a residential map. The vicinity of Okaido
Street is the downtown area, and the periphery is the residential area.
of agglomeration. We define the Hoover Index H as H ≡ 12 ∑i |ai − bi |. When the district is divided
into blocks, ai is the ratio of the parking area of block i to the total parking area of the entire
district. bi is the ratio of the size of block i to the size of the whole district. When the parking
distribution is entirely evenly distributed, i.e., in a completely mixed pattern with J = 0, H = 0.
On the other hand, it follows immediately from the definition that H = 1 when the parking spaces
are concentrated in a single pole (J = 1).
By comparing them, we can observe the following two characteristics. First, the distribution of parking spaces is more dispersed in the periphery, where there are more houses and offices
than in the vicinity of Okaido Street, downtown area. Also, the changes in dispersion and agglomeration are moderate. Secondly, in the vicinity of Okaido Street, the distribution of parking spaces
is more concentrated. The shift in diffusion and accumulation is also drastic: until about 2010,
parking lots moved toward dispersal, but since 2010, they have turned into aggregation.
In the periphery, many residences and offices are located other than parking lots, corre-Watanabe and Hato

14(a) the vicinity of Okaido Street, the downtown area

0.0

2.0

4.0
0.0

1990

2.0

4.0
0.0

2000

2.0

4.0
0.0

2010

2.0

4.0

2019

(b) the periphery, the residential area

0.0

1990

2.0

4.0
0.0

2000

2.0

4.0
0.0

2010

2.0

4.0
0.0

2.0

4.0

2019

FIGURE 7: The comparison between the temporal transition of parking distribution in the vicinity
of Okaido Street and that in the periphery. To facilitate comparison with the model’s results withthe circular city, we project the parking lots in the domain onto the circle. We express the parking
lot area percentage for each region on the circle with the mean value of 1.0.
sponding to a low τ state. In the condition where τ is small, the distribution of parking spaces isdispersed independent of σ , as shown in Figure 4. On the other hand, in the vicinity of Okaido
Street, a high proportion of shops other than parking lots has been maintained for a long time, corresponding to a state with a large τ. In the state where τ is large, the distribution of parking spaces
varies in dispersion and accumulation with the size of σ , as shown in Figure 4. In the vicinity
of Okaido Street, the aging of the shopping area decreased visitors, and the disutility of walking
on the street increased till 2010. In response to this situation, since 2013, the local government
has promoted large-scale renovation of the shopping area, creating more performing space, andcooperation policy between parking lots and shops. As a result, the walk-in disutility σ started todecrease, and the distribution of parking lots turned to agglomeration. Thus, for areas with large τ,efforts to reduce the travel disutility of visitors lead to a decrease in the parking-shop transportation
cost parameter σ , which affects the distribution of parking spaces. Thus, we can see that the actual
data are consistent with the results of this study.
These results provide some suggestions for real-world parking policies. By improving
streets and performing spaces, partnerships between shops and parking facilities, and charging forparking, we can shift visitors’ parking choices away from their destinations. As a result, in the
downtown area where τ is large, the distribution of parking lots can be spontaneously agglomerated by reducing σ . On the other hand, in the district where houses and offices are spread out, i.e., τ

15

0.7

Watanabe and Hato
the vicinity of Okaido Street, the downtown area
the periphery, the residential area

hoover index

0.6

dispersion

0.2

0.3

0.4

0.5

agglomeration

1990

2000

year

2010

2020

FIGURE 8: The changes in the Hoover Index of parking distribution, an indicator of the degreeof agglomeration. The distribution of parking spaces is more dispersed in the periphery. Also, thechange in dispersion and agglomeration is also drastic: until about 2010, parking lots were moving
toward dispersal, but since 2010, they have turned into agglomeration.

is small, it isn’t easy to cause the spontaneous aggregation of parking lots. In recent years, the minimum parking requirement has been relaxed to agglomerate the parking lots. However, this study
shows that these measures do not directly lead to the aggregation of parking lots. This is because
the relaxation of minimum parking requirements corresponds to a change in the ratio of parking
spaces. Still, the qualitative branching of the parking distribution is robust to the percentage ofparking spaces.
In this study, we did not specify the type of parking lot. However, the behavioral principle that visitors choose a parking lot depending on the distance to their destination is universal
regardless of the parking lot type. Thus, even if we consider curbside parking as a form of parking,the nature of the parking distribution in this paper is consistent. In addition, in real cities, shops
may provide their parking lots. We can also interpret such a situation within the framework of
our model. For example, a large commercial facility may have its parking lot nearby. This phenomenon also occurs in our model because the presence of a large store increases the demand forparking near the store, and a parking lot is created nearby. In some cases, popular shops attract
many customers regardless of whether there are other shops in the vicinity or not have their parking
lots. This situation corresponds to a condition in which the parameter τ is small, in the sense thatthe effect of increasing profits by agglomerating shops is low.
COUNCLUSION
In this study, we formulated a walking-scale parking location equilibrium model in the context ofthe multi-agent spatial economy model considering the economy of agglomeration. The parking lot
location is determined simultaneously as the location of shops considering the interaction between
land use and transportation. There has been no parking location model that does not assume
CBD and considers the economics of shop agglomeration at the walking scale. In this study, weWatanabe and Hato

16

focused on the symmetry of the traffic flow between the shop and the parking lot by considering
the particular constraint of the parking lot, which is the need to come back to the place where weparked. As a result, we show that we can interpret this parking location equilibrium problem as apotential game even though profits are explicitly dependent on the location of other agents. Thus,
we can analyze the globally stable solution by concretely calculating potential function values forcandidate equilibrium solutions.
The results of such analysis show a definite order to the occurrence of seemingly disorderedparking spaces in the city. We characterize parking location in urban areas by the inter-shop transportation cost parameter τ and the parking-shop transportation cost parameter σ . τ indicates thenature of the buildings distributed within the city, and σ expresses how far the visitor can tolerate
the distance to the destination or the disutility of walking between the parking lot and the shop.
Numerical calculations show that the globally stable solution is bifurcated. When τ is large, theagglomeration and dispersion of parking spaces change discontinuously in response to the variation of the parking-shop transportation cost parameter σ . On the other hand, when τ is small,the complete mixed pattern is retained as a globally stable solution if σ is greater than a thresholdvalue, even if the parking-shop transportation cost parameter σ changes. The fundamental natureof these results is robust to changes in the parking ratio within a realistic range, differences in thecost of maintaining shops and parking lots, and whether or not the number of shops and parking
lots are fixed. We qualitatively confirmed the applicability of this model using actual data.
These results also have implications for actual parking policies. For example, it is possible
to reduce the disutility of visitors by maintaining the streets, creating more performing space,
and the partnership between shops and parking lots. In a downtown area where τ is large, we
can spontaneously agglomerate the distribution of parking spaces by decreasing the parking-shop
transportation cost parameter σ . On the other hand, it is challenging to spontaneously cause theagglomeration of parking lots in areas with sprawling houses and offices, small τ areas. Also, theresults suggest that the relaxation of minimum parking requirements cannot affect the distribution
characteristics of parking spaces because we confirmed the robustness of branching properties tochanges in the parking ratio within a realistic range. We can only change the distribution of parking
spaces by policies that can manipulate the choice of parking spaces by visitors.
We should consider the following issues for the future. In this study, the generation of parking lot distribution is described as a problem to determine parking lots and shops in the context ofthe spatial economy model. However, the generation of the actual parking distribution occurs due
to each landowner’s use choice. Then, we should consider a method of faithfully describing eachlandowner’s land-use choice behavior using a behavioral model. Also, the model’s interpretability will increase if we loosen the assumption of static equilibrium and use dynamic equilibrium.
This is because the actual distribution of parking spaces may contain a provisional use of parking
spaces. Besides, based on the mechanism for generating the parking distribution, it is a futurechallenge to consider the mechanism design for manipulating the parking distribution.

Watanabe and Hato

17

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