Game Theoretic Models for Competition in Public Transit Services

Eddie Y. S. Chan

and Janny M. Y. Leung

Quintiq, Penang, Malaysia

Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong,

Shatin, New Territories, Hong Kong

Keywords:

Game Theory, Competition, Public Transit.

Abstract:

As metropolitan areas grow, the need to travel by the populace has increased the burden on the transport

systems, leading to increased trafﬁc congestion and environmental concerns. In this paper, we discuss some

game-theoretic models that can be used to investigate the competitive situation when several service providers

offer public transit services. The competition among the operators can be modelled by a class of games called

potential games, and we discuss mathematical programmes that can be used to ﬁnd the Nash equilibria for

these games. By examining the equilibrium solutions, we can investigate the relative merits and tradeoffs

for different structures of the transit networks, and the interplay between the services offered and the overall

ridership of the system.

1 INTRODUCTION

Metropolitan areas have accounted for the majority

of increases in population and economic growth in

recent decades. China’s phenomenal economic de-

velopment has been fuelled by growth in the major

cities, many of which has over 5 million in popula-

tion. Metropolitan areas account for over half of the

population, and a signiﬁcant majority of the GDP, of

the United States. As the geographical size and pop-

ulation of major metropolitan areas have increased,

much economic activity remain focussed in the cen-

tral business districts of the metropolises, thus the av-

erage travel distances for work have not decreased

as expected. The average commuting distance for

London is over 10 kilometres. The need to travel

by the populace has placed signiﬁcant burden on the

transport systems of metropolitan areas, leading to in-

creased trafﬁc congestion and attendant safety and en-

vironmental concerns.

Development of transport infrastructure and pub-

lic transit services have not kept apace with the

swell and sprawl of metropolitan areas, with seri-

ous congestion occurring in central business districts

and insufﬁcient coverage in peripheral areas. In

metropolises where public transit services are pro-

vided by private ﬁrms in a relatively free market, op-

erators tend to focus on high-proﬁt routes and outly-

ing smaller communities are under-served. In Hong

Kong, the already congested Central business district

is often jammed with half-empty double-decker buses

from all the bus operators, while bus services to satel-

lite communities in the New Territories are very in-

frequent and expensive.

In this paper, we discuss some game-theoretic

models that can be used to investigate the competitive

situation when several service providers offer public

transit services, and study the impact on the total set

of services offered to the public and the resultant level

of ridership of the system. The competition among

the operators can be modelled by a class of games

called potential games. We discuss mathematical pro-

grammes that can be used to ﬁnd the Nash equilibria

for these games. By examining the equilibria solu-

tions, we can examine the relative merits and tradeoffs

for different structures of the transit networks, and the

interplay between the services offered and the overall

ridership of the system. We hope that our modelling

and analysis may provide some insight on the types

and bundling of routes being offered by operators, and

the locations for transportation interchanges and hubs.

2 BACKGROUND

Whilst not explicitly acknowledged, concepts of

game theory have been pervasively used in trafﬁc

studies. As Fisk (1984) pointed out, the famous

133

Y. S. Chan E. and M. Y. Leung J..

Game Theoretic Models for Competition in Public Transit Services.

DOI: 10.5220/0005218201330139

In Proceedings of the International Conference on Operations Research and Enterprise Systems (ICORES-2015), pages 133-139

ISBN: 978-989-758-075-8

 2015 SCITEPRESS (Science and Technology Publications, Lda.)

Wardrop’s (1952) user-equilibrium principle is essen-

tially the condition for a Nash (1950) game-theoretic

equilibrium among road-users, since no driver can re-

duce his/her travel time by switching to a different

route choice. Wardrop’s principle has been a cor-

nerstone in road trafﬁc research for decades. For an

overview of trafﬁc equilibrium models, see Patriksson

and Labbe (2004).

Other researchers have developed speciﬁc game-

theoretic models for transport-related problems. Bell

(2000) investigates network reliability by studying a

zero-sum game between a cost-minimising driver and

a demon that sets the link costs. This game is a con-

cept game in the sense that the demon is not a real

player, and is used to explore the worst-case scenar-

ios faced by the driver. Other researchers have also

explored concept games among road users. James

(1998) studies a game among n road-users where

any player’s utility of using the road segment de-

creases when there are more users. Levinson (2005)

also studies congestion by investigating a game where

the players (drivers) choose their departure times.

Pedersen (2003) investigates road safety by a game

where players choose the behavioural level of driv-

ing aggression. All these games study the competi-

tion among road users. Holland and Prashker (2006)

give an excellent review of recent literature on non-

cooperative games in transport research.

Surprisingly, studies on the competitive situation

amongst public transit operators have received little

attention from transport researchers. Castell et al.

(2004) modelled a Stackelberg game between two au-

thorities (one determining ﬂow, and the other capaci-

ties) in a freight transport network, which is different

to a passenger transit network since the route choice

is not determined by the transportee (freight). Ac-

cording to Holland and Prashker (2006), the “small

number of such games is surprising, considering that

NCGT [non-cooperative game theory] seems a natu-

ral tool for analysing relations between authorities. ...

Trends such as tendering and privatisation, that have

a vital role on today’s transport agenda, also seem apt

to be modelled through games between authorities”.

Surprisingly, there has been very little research along

this line.

Some researchers have studied games between au-

thorities and travellers. Fisk (1984) investigates a

Stackelberg game between the authority that sets traf-

ﬁc signals and all travellers who then ﬁnds the user-

equilibrium solution. Chen and Ben-Akiva (1998)

investigates a similar game in a dynamic setting.

Reyniers (1992) studies a game between the railway

operators who sets the capacities for different fare

classes and the passengers who chooses which class

to use. Hollander et al. (2006) studies a game be-

tween the parking authority and travellers to explore

the incentives for public transport ridership. All these

games, however, only involve one operator/authority.

Only few researchers have investigated games with

several operators and passengers. Van Zuylen and

Taale (2004) studies a game with two authorities (one

for urban roads and one for ring roads) and the driv-

ing public. Fernandez and Marcotte (1992) and Fer-

nandez et al. (1993) investigated trafﬁc equilibrium

models involving car users, bus users and bus opera-

tors and presented algorithms for ﬁnding the equilib-

rium solutions; their models are general and consid-

ered trafﬁc congestion effects. The focus of our paper

is on the strategic competition among public transit

operators.

3 MODELS

We have made some preliminary investigation into

the strategic gaming situation among competing pub-

lic transit service providers. In our ﬁrst-cut model,

we assume that all the operators have the same cost

and price structure, and that the total ridership be-

tween each origin-destination pair is equally divided

among all the operators that service that particular

route. In this setting, a player of the game is the ser-

vice provider, and its strategy is the set of routes that

it chooses to offer service. Each player tries to max-

imize its total proﬁt, and a Nash equilibrium occurs

when no player can improve its proﬁt by unilaterally

changing the set of routes it services.

We can show that this game can be modelled as

what is known as a potential game (ﬁrst introduced

by Rosenthal, 1973) where the equilibrium can be

computed by solving an auxiliary mathematical pro-

gramme with a potential function as a surrogate ob-

jective. The solvability for the Nash equilibrium al-

lows us to make some comparisons between the com-

petitive equilibrium and a centralised monopolistic

proﬁt-maximizing operator and draw some insights.

3.1 Basic Competitive Model

We consider a game among n players (service

providers) and m possible routes (origin-destination

pairs). Player i’s strategy consists of a subset of the

routes S

⊆ M = {1, 2,··· , m}. For each route j, let k

be the number of players who choose to offer service

on the route, i.e., k

= |{i : j ∈ S

}|. Note that k

endogenously determined. Let a

denote the revenue

from the total ridership on route j, assumed exoge-

nously determined, and let δ

be the cost of operat-

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134

ing the route. Each player (service provider) that of-

fers service on this route earns an identical payoff of

) which depends on the total number of players

) who choose to serve that route:

) =

(

− δ

, k > 0,

0, k = 0.

Each player i selects a strategy to maximise his to-

tal proﬁt π

,· ·· , S

) =

∑

j∈S

). A pure

Nash (1950) equilibrium is a set of strategies

∗

,· ·· , S

∗

} such that each player cannot unilat-

erally improve his total proﬁt, that is,

∗

,· ·· , S

∗

i−1

∗

i+1

,· ·· , S

∗

) ≥

∗

,· ·· , S

∗

i−1

∗

i+1

,· ·· , S

∗

),∀ S

Following Rosenthal (1973), we can show that

the Nash equilibrium for this game can be obtained

by solving the following auxiliary mathematical pro-

gramme:

(G0) : Max

∑

j=1

∑

y=1

(y)

s.t.

∑

i=1

= k

, ∀ j = 1,...,m, (1)

∈ {0,1}, ∀i = 1,. .. ,n, j = 1,. .. ,m, (2)

where x

is 1 if player i offers service on route j. We

note that this is not a straightforward binary linear

programme, since the k

values are also variables, so

the objective is not a linear function.

Proof. We will show, by contradiction, that the solu-

tion of (G0) yields a Nash equilibrium. Let {x

∗

}

be an optimal solution to (G0) such that the associ-

ated strategy combination is not a Nash equilibrium.

Then for some player l, there is a strategy

such that

∑

j∈

∗

+ 1) >

∑

j∈S

∗

)

where S

∗

is the strategy used by l as indicated by the

values of x

∗ j

. Consider the new values { ˆx

} asso-

ciated with player l changing to the pure-strategy

(with the rest of the players not changing their strate-

gies). The objective value of (G0) for this solution

is:

∑

j=1

∑

y=1

(y)

∑

j=1

∗

∑

y=1

(y) +

∑

j∈

∗

+ 1) −

∑

j∈S

∗

)

{z }

∑

j=1

∗

∑

y=1

(y)

which contradict with the optimality of {x

∗ j

∗

We note that the result also holds when there is a

limit to the number of routes offered by the operators.

Equivalently, a solution of (G0) can be found by

solving the following integer programme (G1). Let

(

1, if k

= k,

0, otherwise.

We also deﬁne

(k) =

(

∑

z=1

(z), for k > 0,

0, if k = 0.

(G1) : Max

∑

j=1

∑

k=0

(k)y

s.t.

∑

i=1

∑

k=0

, ∀ j = 1,· ·· , m, (3)

∑

k=0

= 1, ∀ j = 1,· ·· , m, (4)

∈ {0,1}, ∀ i = 1,...,n, j = 1, ·· · ,m. (5)

Note that

(k) = p

(1) + p

(2) + ··· + p

(k)

= (a

− δ

) + (a

/2 − δ

) + ··· + (a

/k − δ

)

Therefore, P

(k) is at a maximum at

∗

= argmax{h : p

(h) > 0}.

Observation 1:

Operators will continue to ”enter the market” for a

route until it is no longer proﬁtable.

Therefore, as long as the marginal proﬁt is non-

negative, additional operators will offer service on a

route. This leads to better service for the customers,

but reduces the proﬁt for each operator. We compare

this to a monopolistic setting below.

3.2 Basic Proﬁt-Maximising Model

Consider the case when there is only one operator

which wants to maximize the total proﬁt. The opti-

mal choice of routes to offer service is given by the

solution to:

(G2) : Max

∑

j=1

∑

k=0

(k)y

GameTheoreticModelsforCompetitioninPublicTransitServices

135

such that

∑

i=1

∑

k=0

, ∀ j = 1,· ·· , m, (6)

∑

k=0

= 1, ∀ j = 1,· ·· , m, (7)

∈ {0,1}, ∀ i = 1,. .. ,n, j = 1,...,m, (8)

∈ {0,1}, ∀ k = 0, .. ., n, j = 1,. ..,m.(9)

Observation 2:

To maximize total proﬁt, the optimal solution is to

have at most one operator per route.

Proof. Since for k 6= 0,

(k) = k(a

/k − δ

) = a

− kδ

the objective is maximised when k

= 1 as long as

> δ

We note that while the monopolistic proﬁt-

maximising solution is to assign only one operator per

route, to determine an equitable allocation of routes

to operators is not a simple task but requires solving a

non-linear optimisation problem.

Observation 3:

The competitive equilibrium solution has as many as

possible operators offering service on each route un-

til it is no longer proﬁtable to do so, whereas the

centrally-controlled (total proﬁt maximizing) solution

has at most one operator per route.

3.3 Model with Player - and Route -

Dependent Operating Cost

In reality, each operator might have different operat-

ing costs on different routes. Hence we extend our

model to a congestion game with player- and route-

speciﬁc payoff functions which admit a potential. In

contrast to Milchtaich’s work [13], our model is not

a singleton game; players can choose more than one

route in their strategy sets.

As with the basic model, the revenue is shared

among all operators of a route, that is, the revenue

to each player i serving route j is r

) = a

for

> 0, where k

is the number of players offering

service on route j. Because of the cost structure,

the payoff to player i offering service on route j is

p(i, j,k

) = a

− δ

i j

, which depends on player i,

route j and the total number of player choosing that

particular route k

. We next show that a Nash equi-

librium of this problem can be found by solving the

following auxiliary problem:

(GD1) : Max

∑

j=1

∑

y=1

(y) −

∑

j=1

∑

i=1

i j

s.t.

∑

i=1

= k

, ∀ j = 1,...,m, (10)

∈ {0,1}, ∀i = 1,. .. ,n, j = 1,. .. ,m, (11)

Proof. Let x

∗

be an optimal solution of (GD1), and

let (S

∗

,· ·· , S

∗

) be the corresponding strategies of

the players. If this is not an equilibrium solution, then

for some player l, there exist another strategy

such

that

∗

,· ·· , S

∗

l−1

∗

l+1

,· ·· , S

∗

) >

∗

,· ·· , S

∗

l−1

∗

l+1

,· ·· , S

∗

that is,

∑

j∈

∗

+ 1)

− δ

l j

−

∑

j∈S

∗

− δ

l j

> 0 (∗∗)

Let ( ˆx,

k) be the solution of (GD1) corresponding to

∗

,· ·· , S

∗

l−1

∗

l+1

,· ·· , S

∗

). Then

∑

j=1

∑

y=1

(y) −

∑

j=1

∑

i=1

i j

ˆx

∑

j=1

∗

∑

y=1

(y) −

∑

j=1

∑

i=1

i j

∗ j

∑

j∈

∗

+ 1)

− δ

l j

−

∑

j∈S

∗

)

− δ

l j

| {z }

> 0 by (**)

∑

j=1

∗

∑

y=1

(y) −

∑

j=1

∑

i=1

i j

∗ j

Hence, we have another solution to (GD1) with a bet-

ter objective value contradicting the optimality of x

∗

Similar to the basic model, (GD1) can be formu-

lated as an equivalent integer programme:

(GD2) : Max

∑

j=1

∑

k=0

(k)y

−

∑

j=1

∑

k=0

i j

s.t.

∑

i=1

∑

k=0

, ∀ j = 1,· ·· , m, (12)

∑

k=0

= 1, ∀ j = 1,· ·· , m, (13)

∈ {0,1}, ∀ i = 1,. .. ,n, j = 1,··· , m, (14)

∈ {0,1}, ∀ k = 0, 1,. .. ,n, j = 1,...,m,(15)

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136

where R

(k) =

∑

y=1

(k) = a

+ a

/2 + ··· + a

/k.

Observation 4:

As for the basic model, operators will continue to en-

ter the market for a route until it is no longer prof-

itable.

The number of operators for route j is

∗

= max{h : a

/h − δ

[h] j

> 0},

where [·] is a permutation of {1,2,· ·· , n} such that

[1] j

≤ δ

[2] j

≤ ·· · ≤ δ

[n] j

4 NETWORK DESIGN

Using this initial framework, we have also explored

the impact of the network structure on the proﬁt for

the service providers. We consider a service area with

t townships and compared the equilibrium solution

for a network structure where direct point-to-point

services are offered between every pair of townships,

to the equilibrium for a hub-and-spoke network where

every route between any two townships involves an

interchange via a central hub. The ﬁrst one is a com-

plete network with t nodes and

∑

t−1

u=1

= t(t − 1)/2

links, one for each origin-destination pair. In the sec-

ond network structure, the routes offered are between

a township and the central hub, and the total ridership

from each origin (to all destinations) is consolidated

into the ridership from the origin to the central hub.

That is, the second is a hub-and-spoke network with

t + 1 nodes and t links. Figure 1 illustrate these two

networks with t = 6. For each service provider, the

proﬁt from a route depends on the operating cost of

offering the service, the total revenue due to the rid-

ership and the number of competitors also servicing

that route.

Figure 1: Network structures.

4.1 Hub-and-Spoke vs Complete

Network

We consider a simple case where all origin-

destination pairs have the same demand (that is, a

a for all j) and all operators have the same cost for

all routes (that is, δ

i j

= δ for all i and j). We assume

there is no loss of ridership between i and j whether

the route is direct or via a central hub. Therefore, all

demand from a node i is aggregated to the demand

on the link from the node to the hub. This is illus-

trated in Figure 2. For fair comparison between the

Figure 2: Demand on links.

two networks, the revenue on a route between a hub

and an origin/destination is half that of a direct route

between an origin and a destination. For the simplistic

case where the ridership between every pair of town-

ships are the same and all ﬁxed operating costs are the

same, the overall proﬁts depends on the ratio of rider-

ship to route operating cost. For a complete network

on t nodes, the total proﬁt in a competitive equilib-

rium is:

= t(t − 1)/2(a − δ

) where k = ba/kc.

For the hub-and-spoke network, the total proﬁt in a

competitive equilibrium is:

= t((t − 1)a/2 − δh) where h = b(t − 1)a/2c.

Therefore, π

is greater than π

a(t − 1)

2δ

c ≤

t − 1

It ca be observed that if the ratio a/δ is integer-valued

and t is odd, we have π

= π

. When this is not the

case, rounding has an impact, but not always favour-

ing one network structure over the other, as illustrated

in Figure 3.

GameTheoreticModelsforCompetitioninPublicTransitServices

137

Figure 3: Rounding Effects.

5 SUMMARY AND FUTURE

EXTENSIONS

In this paper, we investigate the competition among

operators of public transit using the framework of

congestion games. The Nash equilibria can be found

by solving an auxiliary integer programme. Using

this framework, we can draw some insights regard-

ing competition vs. centralization, and the impact of

network structure on the proﬁt of the operators.

The assumption that the network structure and the

total service bundle being offered will not affect the

overall ridership is perhaps too restrictive and unre-

alistic. This assumption implies that all passenger

who would travel from town X to town Y with di-

rect service will still travel even if the trip involves

an interchange via a central hub. Clearly, the con-

venience level, travel time and possibly travel cost

will not be the same for the two trips. Also, appar-

ently with wider service coverage and greater num-

ber of operators servicing a particular route (higher

frequency), it will attract more ridership. A more re-

alistic model would allow for ridership to depend on

origin-destination pairs, on the network infrastructure

and also on the set of transit services available. This

may lead to a bi-level model where not only do op-

erators compete with each other but the passengers

preferences and patronage depends on the set of ser-

vices offered by the transit operators. The upper-level

model would represent the strategic game among the

service providers as they select the services to be of-

fered to maximise their individual proﬁt. The lower-

level is the game between the public and the operators

as a group, in that the public may be diverted to other

forms of transport (e.g. taxis, private vehicles) if the

availability and service quality (e.g. interchanges re-

quired, circuitous routes, travel time) of the basket of

services offered by the operators are too low. The two

levels of the bi-level problem are interlinked since the

choice of the public to utilise public transit or not

would affect the potential ridership of the system and

thus impact the potential proﬁt of the operators.

The potential game framework is a non-

cooperative game framework. Operators may con-

sider to cooperate when their resources are limited

(which is mostly true). One operator may offers ser-

vices on a part of the network that serve as feeder

links to the service provided by another operator,

and vice versa. Developing a cooperative game-

theoretic model may help us to compare and contrast

the equilibrium solutions of both setting, the coopera-

tive game and non-cooperative settings. In a coopera-

tive game setting, it would also be interesting to inves-

tigate what is the appropriate proﬁt-sharing scheme to

induce higher proﬁts or more comprehensive services

for the public.

By investigating these extensions, we may ob-

tain further insights into the relationship among the

network infrastructure, competitive situation between

operators and the impact on the type and level of ser-

vices offered to the public. These relationships could

further guide us in decision making on possible in-

frastructural investments and incentives to offer both

operators and the riding public, which is very helpful

for the government authorities and to ensure a public

transit system that well-serves the public and beneﬁts

the community in terms of costs, convenience, qual-

ity, environmental impact and other concerns being

designed.

ACKNOWLEDGEMENTS

This research is partially supported by a GRF

grant from the Research Grants Council of the

Hong Kong Special Administrative Region, China

(CUHK414313).

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